author | nipkow |
Sun, 15 Feb 2009 16:25:16 +0100 | |
changeset 29923 | 24f56736c56f |
parent 29920 | b95f5b8b93dd |
child 29925 | 17d1e32ef867 |
permissions | -rw-r--r-- |
12396 | 1 |
(* Title: HOL/Finite_Set.thy |
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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel |
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with contributions by Jeremy Avigad |
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*) |
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header {* Finite sets *} |
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theory Finite_Set |
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imports Nat Product_Type Power |
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begin |
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|
15392 | 12 |
subsection {* Definition and basic properties *} |
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inductive finite :: "'a set => bool" |
22262 | 15 |
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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22 |
proof - |
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28823 | 23 |
from assms have "A \<noteq> UNIV" by blast |
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thus ?thesis by blast |
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qed |
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lemma finite_induct [case_names empty insert, induct set: finite]: |
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"finite F ==> |
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P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F" |
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-- {* Discharging @{text "x \<notin> F"} entails extra work. *} |
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proof - |
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assume "P {}" and |
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insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)" |
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assume "finite F" |
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thus "P F" |
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proof induct |
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show "P {}" by fact |
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fix x F assume F: "finite F" and P: "P F" |
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show "P (insert x F)" |
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proof cases |
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assume "x \<in> F" |
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hence "insert x F = F" by (rule insert_absorb) |
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with P show ?thesis by (simp only:) |
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next |
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assume "x \<notin> F" |
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from F this P show ?thesis by (rule insert) |
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qed |
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qed |
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qed |
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lemma finite_ne_induct[case_names singleton insert, consumes 2]: |
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assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow> |
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\<lbrakk> \<And>x. P{x}; |
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\<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk> |
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\<Longrightarrow> P F" |
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using fin |
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proof induct |
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case empty thus ?case by simp |
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next |
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case (insert x F) |
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show ?case |
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proof cases |
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23389 | 63 |
assume "F = {}" |
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thus ?thesis using `P {x}` by simp |
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next |
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assume "F \<noteq> {}" |
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thus ?thesis using insert by blast |
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qed |
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qed |
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lemma finite_subset_induct [consumes 2, case_names empty insert]: |
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assumes "finite F" and "F \<subseteq> A" |
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and empty: "P {}" |
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and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)" |
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shows "P F" |
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proof - |
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from `finite F` and `F \<subseteq> A` |
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show ?thesis |
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proof induct |
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show "P {}" by fact |
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next |
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fix x F |
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assume "finite F" and "x \<notin> F" and |
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P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A" |
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show "P (insert x F)" |
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proof (rule insert) |
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from i show "x \<in> A" by blast |
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from i have "F \<subseteq> A" by blast |
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with P show "P F" . |
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show "finite F" by fact |
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show "x \<notin> F" by fact |
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qed |
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qed |
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qed |
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text{* A finite choice principle. Does not need the SOME choice operator. *} |
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lemma finite_set_choice: |
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"finite A \<Longrightarrow> ALL x:A. (EX y. P x y) \<Longrightarrow> EX f. ALL x:A. P x (f x)" |
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proof (induct set: finite) |
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case empty thus ?case by simp |
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101 |
next |
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case (insert a A) |
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then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto |
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show ?case (is "EX f. ?P f") |
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proof |
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show "?P(%x. if x = a then b else f x)" using f ab by auto |
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qed |
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108 |
qed |
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text{* Finite sets are the images of initial segments of natural numbers: *} |
112 |
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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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15392 | 116 |
using fin |
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proof induct |
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case empty |
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15510 | 119 |
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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121 |
qed |
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15392 | 122 |
next |
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case (insert a A) |
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23389 | 124 |
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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15392 | 130 |
thus ?case by blast |
131 |
qed |
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lemma nat_seg_image_imp_finite: |
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"!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A" |
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proof (induct n) |
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case 0 thus ?case by simp |
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137 |
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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146 |
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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150 |
qed |
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151 |
qed |
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152 |
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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) |
15392 | 156 |
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lemma finite_Collect_less_nat[iff]: "finite{n::nat. n<k}" |
158 |
by(fastsimp simp: finite_conv_nat_seg_image) |
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26441 | 160 |
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subsubsection{* Finiteness and set theoretic constructions *} |
162 |
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" |
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by (induct set: finite) simp_all |
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A" |
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-- {* Every subset of a finite set is finite. *} |
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proof - |
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assume "finite B" |
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thus "!!A. A \<subseteq> B ==> finite A" |
|
171 |
proof induct |
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case empty |
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thus ?case by simp |
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174 |
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" |
178 |
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 . |
185 |
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) |
|
189 |
qed |
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190 |
qed |
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191 |
qed |
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192 |
||
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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) |
195 |
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lemma finite_Collect_disjI[simp]: |
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"finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})" |
198 |
by(simp add:Collect_disj_eq) |
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|
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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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29901 | 202 |
by (blast intro: finite_subset) |
203 |
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lemma finite_Collect_conjI [simp, intro]: |
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"finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}" |
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-- {* The converse obviously fails. *} |
|
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by(simp add:Collect_conj_eq) |
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12396 | 208 |
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29920 | 209 |
lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}" |
210 |
by(simp add: le_eq_less_or_eq) |
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211 |
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lemma finite_insert [simp]: "finite (insert a A) = finite A" |
213 |
apply (subst insert_is_Un) |
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apply (simp only: finite_Un, blast) |
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done |
216 |
||
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lemma finite_Union[simp, intro]: |
218 |
"\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)" |
|
219 |
by (induct rule:finite_induct) simp_all |
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220 |
||
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lemma finite_empty_induct: |
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assumes "finite A" |
223 |
and "P A" |
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224 |
and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" |
|
225 |
shows "P {}" |
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proof - |
227 |
have "P (A - A)" |
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228 |
proof - |
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23389 | 229 |
{ |
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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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} |
250 |
then show ?thesis by this (simp_all add: assms) |
|
12396 | 251 |
qed |
23389 | 252 |
then show ?thesis by simp |
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qed |
254 |
||
29901 | 255 |
lemma finite_Diff [simp]: "finite A ==> finite (A - B)" |
256 |
by (rule Diff_subset [THEN finite_subset]) |
|
257 |
||
258 |
lemma finite_Diff2 [simp]: |
|
259 |
assumes "finite B" shows "finite (A - B) = finite A" |
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260 |
proof - |
|
261 |
have "finite A \<longleftrightarrow> finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int) |
|
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also have "\<dots> \<longleftrightarrow> finite(A-B)" using `finite B` by(simp) |
|
263 |
finally show ?thesis .. |
|
264 |
qed |
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265 |
||
266 |
lemma finite_compl[simp]: |
|
267 |
"finite(A::'a set) \<Longrightarrow> finite(-A) = finite(UNIV::'a set)" |
|
268 |
by(simp add:Compl_eq_Diff_UNIV) |
|
12396 | 269 |
|
29916 | 270 |
lemma finite_Collect_not[simp]: |
29903 | 271 |
"finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)" |
272 |
by(simp add:Collect_neg_eq) |
|
273 |
||
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lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" |
275 |
apply (subst Diff_insert) |
|
276 |
apply (case_tac "a : A - B") |
|
277 |
apply (rule finite_insert [symmetric, THEN trans]) |
|
14208 | 278 |
apply (subst insert_Diff, simp_all) |
12396 | 279 |
done |
280 |
||
281 |
||
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text {* Image and Inverse Image over Finite Sets *} |
13825 | 283 |
|
284 |
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)" |
|
285 |
-- {* The image of a finite set is finite. *} |
|
22262 | 286 |
by (induct set: finite) simp_all |
13825 | 287 |
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lemma finite_surj: "finite A ==> B <= f ` A ==> finite B" |
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289 |
apply (frule finite_imageI) |
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apply (erule finite_subset, assumption) |
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291 |
done |
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292 |
|
13825 | 293 |
lemma finite_range_imageI: |
294 |
"finite (range g) ==> finite (range (%x. f (g x)))" |
|
27418 | 295 |
apply (drule finite_imageI, simp add: range_composition) |
13825 | 296 |
done |
297 |
||
12396 | 298 |
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" |
299 |
proof - |
|
300 |
have aux: "!!A. finite (A - {}) = finite A" by simp |
|
301 |
fix B :: "'a set" |
|
302 |
assume "finite B" |
|
303 |
thus "!!A. f`A = B ==> inj_on f A ==> finite A" |
|
304 |
apply induct |
|
305 |
apply simp |
|
306 |
apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") |
|
307 |
apply clarify |
|
308 |
apply (simp (no_asm_use) add: inj_on_def) |
|
14208 | 309 |
apply (blast dest!: aux [THEN iffD1], atomize) |
12396 | 310 |
apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) |
14208 | 311 |
apply (frule subsetD [OF equalityD2 insertI1], clarify) |
12396 | 312 |
apply (rule_tac x = xa in bexI) |
313 |
apply (simp_all add: inj_on_image_set_diff) |
|
314 |
done |
|
315 |
qed (rule refl) |
|
316 |
||
317 |
||
13825 | 318 |
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}" |
319 |
-- {* The inverse image of a singleton under an injective function |
|
320 |
is included in a singleton. *} |
|
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321 |
apply (auto simp add: inj_on_def) |
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322 |
apply (blast intro: the_equality [symmetric]) |
13825 | 323 |
done |
324 |
||
325 |
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)" |
|
326 |
-- {* The inverse image of a finite set under an injective function |
|
327 |
is finite. *} |
|
22262 | 328 |
apply (induct set: finite) |
21575 | 329 |
apply simp_all |
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330 |
apply (subst vimage_insert) |
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|
331 |
apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton]) |
13825 | 332 |
done |
333 |
||
334 |
||
15392 | 335 |
text {* The finite UNION of finite sets *} |
12396 | 336 |
|
337 |
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" |
|
22262 | 338 |
by (induct set: finite) simp_all |
12396 | 339 |
|
340 |
text {* |
|
341 |
Strengthen RHS to |
|
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|
342 |
@{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}? |
12396 | 343 |
|
344 |
We'd need to prove |
|
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|
345 |
@{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"} |
12396 | 346 |
by induction. *} |
347 |
||
29918 | 348 |
lemma finite_UN [simp]: |
349 |
"finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" |
|
350 |
by (blast intro: finite_UN_I finite_subset) |
|
12396 | 351 |
|
29920 | 352 |
lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow> |
353 |
finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})" |
|
354 |
apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})") |
|
355 |
apply auto |
|
356 |
done |
|
357 |
||
358 |
lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \<Longrightarrow> |
|
359 |
finite{x. EX y. P y & Q x y} = (ALL y. P y \<longrightarrow> finite{x. Q x y})" |
|
360 |
apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})") |
|
361 |
apply auto |
|
362 |
done |
|
363 |
||
364 |
||
17022 | 365 |
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)" |
366 |
by (simp add: Plus_def) |
|
367 |
||
15392 | 368 |
text {* Sigma of finite sets *} |
12396 | 369 |
|
370 |
lemma finite_SigmaI [simp]: |
|
371 |
"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" |
|
372 |
by (unfold Sigma_def) (blast intro!: finite_UN_I) |
|
373 |
||
15402 | 374 |
lemma finite_cartesian_product: "[| finite A; finite B |] ==> |
375 |
finite (A <*> B)" |
|
376 |
by (rule finite_SigmaI) |
|
377 |
||
12396 | 378 |
lemma finite_Prod_UNIV: |
379 |
"finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" |
|
380 |
apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") |
|
381 |
apply (erule ssubst) |
|
14208 | 382 |
apply (erule finite_SigmaI, auto) |
12396 | 383 |
done |
384 |
||
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
385 |
lemma finite_cartesian_productD1: |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
386 |
"[| finite (A <*> B); B \<noteq> {} |] ==> finite A" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
387 |
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
|
388 |
apply (drule_tac x=n in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
389 |
apply (drule_tac x="fst o f" in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
390 |
apply (auto simp add: o_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
391 |
prefer 2 apply (force dest!: equalityD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
392 |
apply (drule equalityD1) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
393 |
apply (rename_tac y x) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
394 |
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
|
395 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
396 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
397 |
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
|
398 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
399 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
400 |
lemma finite_cartesian_productD2: |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
401 |
"[| finite (A <*> B); A \<noteq> {} |] ==> finite B" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
402 |
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
|
403 |
apply (drule_tac x=n in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
404 |
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
|
405 |
apply (auto simp add: o_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
406 |
prefer 2 apply (force dest!: equalityD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
407 |
apply (drule equalityD1) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
408 |
apply (rename_tac x y) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
409 |
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
|
410 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
411 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
412 |
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
|
413 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
414 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
415 |
|
15392 | 416 |
text {* The powerset of a finite set *} |
12396 | 417 |
|
418 |
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" |
|
419 |
proof |
|
420 |
assume "finite (Pow A)" |
|
421 |
with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast |
|
422 |
thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp |
|
423 |
next |
|
424 |
assume "finite A" |
|
425 |
thus "finite (Pow A)" |
|
426 |
by induct (simp_all add: finite_UnI finite_imageI Pow_insert) |
|
427 |
qed |
|
428 |
||
29916 | 429 |
lemma finite_Collect_subsets[simp,intro]: "finite A \<Longrightarrow> finite{B. B \<subseteq> A}" |
430 |
by(simp add: Pow_def[symmetric]) |
|
15392 | 431 |
|
29918 | 432 |
|
15392 | 433 |
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A" |
434 |
by(blast intro: finite_subset[OF subset_Pow_Union]) |
|
435 |
||
436 |
||
26441 | 437 |
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
|
438 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
439 |
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*} |
29797 | 440 |
class finite = |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
441 |
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
|
442 |
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
|
443 |
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
|
444 |
|
27430 | 445 |
context finite |
446 |
begin |
|
447 |
||
448 |
lemma finite [simp]: "finite (A \<Colon> 'a set)" |
|
26441 | 449 |
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
|
450 |
|
27430 | 451 |
end |
452 |
||
26146 | 453 |
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
|
454 |
"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
|
455 |
|
26146 | 456 |
instance unit :: finite |
457 |
by default (simp add: UNIV_unit) |
|
458 |
||
459 |
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
|
460 |
"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
|
461 |
|
26146 | 462 |
instance bool :: finite |
463 |
by default (simp add: UNIV_bool) |
|
464 |
||
465 |
instance * :: (finite, finite) finite |
|
466 |
by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite) |
|
467 |
||
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
|
468 |
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
|
469 |
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
|
470 |
|
26146 | 471 |
instance "fun" :: (finite, finite) finite |
472 |
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
|
473 |
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
|
474 |
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
|
475 |
let ?graph = "%f::'a => 'b. {(x, y). y = f x}" |
26792 | 476 |
have "range ?graph \<subseteq> Pow UNIV" by simp |
477 |
moreover have "finite (Pow (UNIV :: ('a * 'b) set))" |
|
478 |
by (simp only: finite_Pow_iff finite) |
|
479 |
ultimately show "finite (range ?graph)" |
|
480 |
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
|
481 |
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
|
482 |
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
|
483 |
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
|
484 |
|
27981 | 485 |
instance "+" :: (finite, finite) finite |
486 |
by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite) |
|
487 |
||
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
|
488 |
|
15392 | 489 |
subsection {* A fold functional for finite sets *} |
490 |
||
491 |
text {* The intended behaviour is |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
492 |
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
493 |
if @{text f} is ``left-commutative'': |
15392 | 494 |
*} |
495 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
496 |
locale fun_left_comm = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
497 |
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
498 |
assumes fun_left_comm: "f x (f y z) = f y (f x z)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
499 |
begin |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
500 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
501 |
text{* On a functional level it looks much nicer: *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
502 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
503 |
lemma fun_comp_comm: "f x \<circ> f y = f y \<circ> f x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
504 |
by (simp add: fun_left_comm expand_fun_eq) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
505 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
506 |
end |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
507 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
508 |
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
509 |
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
510 |
emptyI [intro]: "fold_graph f z {} z" | |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
511 |
insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
512 |
\<Longrightarrow> fold_graph f z (insert x A) (f x y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
513 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
514 |
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
515 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
516 |
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
517 |
[code del]: "fold f z A = (THE y. fold_graph f z A y)" |
15392 | 518 |
|
15498 | 519 |
text{*A tempting alternative for the definiens is |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
520 |
@{term "if finite A then THE y. fold_graph f z A y else e"}. |
15498 | 521 |
It allows the removal of finiteness assumptions from the theorems |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
522 |
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}. |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
523 |
The proofs become ugly. It is not worth the effort. (???) *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
524 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
525 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
526 |
lemma Diff1_fold_graph: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
527 |
"fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
528 |
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
529 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
530 |
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
531 |
by (induct set: fold_graph) auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
532 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
533 |
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
534 |
by (induct set: finite) auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
535 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
536 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
537 |
subsubsection{*From @{const fold_graph} to @{term fold}*} |
15392 | 538 |
|
15510 | 539 |
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})" |
19868 | 540 |
by (auto simp add: less_Suc_eq) |
15510 | 541 |
|
542 |
lemma insert_image_inj_on_eq: |
|
543 |
"[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; |
|
544 |
inj_on h {i. i < Suc m}|] |
|
545 |
==> A = h ` {i. i < m}" |
|
546 |
apply (auto simp add: image_less_Suc inj_on_def) |
|
547 |
apply (blast intro: less_trans) |
|
548 |
done |
|
549 |
||
550 |
lemma insert_inj_onE: |
|
551 |
assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" |
|
552 |
and inj_on: "inj_on h {i::nat. i<n}" |
|
553 |
shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n" |
|
554 |
proof (cases n) |
|
555 |
case 0 thus ?thesis using aA by auto |
|
556 |
next |
|
557 |
case (Suc m) |
|
23389 | 558 |
have nSuc: "n = Suc m" by fact |
15510 | 559 |
have mlessn: "m<n" by (simp add: nSuc) |
15532 | 560 |
from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE) |
27165 | 561 |
let ?hm = "Fun.swap k m h" |
15520 | 562 |
have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn |
563 |
by (simp add: inj_on_swap_iff inj_on) |
|
15510 | 564 |
show ?thesis |
15520 | 565 |
proof (intro exI conjI) |
566 |
show "inj_on ?hm {i. i < m}" using inj_hm |
|
15510 | 567 |
by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on) |
15520 | 568 |
show "m<n" by (rule mlessn) |
569 |
show "A = ?hm ` {i. i < m}" |
|
570 |
proof (rule insert_image_inj_on_eq) |
|
27165 | 571 |
show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp |
15520 | 572 |
show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) |
573 |
show "insert (?hm m) A = ?hm ` {i. i < Suc m}" |
|
574 |
using aA hkeq nSuc klessn |
|
575 |
by (auto simp add: swap_def image_less_Suc fun_upd_image |
|
576 |
less_Suc_eq inj_on_image_set_diff [OF inj_on]) |
|
15479 | 577 |
qed |
578 |
qed |
|
579 |
qed |
|
580 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
581 |
context fun_left_comm |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
582 |
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
|
583 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
584 |
lemma fold_graph_determ_aux: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
585 |
"A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
586 |
\<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x' |
15392 | 587 |
\<Longrightarrow> x' = x" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
588 |
proof (induct n arbitrary: A x x' h rule: less_induct) |
15510 | 589 |
case (less n) |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
590 |
have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
591 |
\<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
592 |
\<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
593 |
have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
594 |
and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+ |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
595 |
show ?case |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
596 |
proof (rule fold_graph.cases [OF Afoldx]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
597 |
assume "A = {}" and "x = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
598 |
with Afoldx' show "x' = x" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
599 |
next |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
600 |
fix B b u |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
601 |
assume AbB: "A = insert b B" and x: "x = f b u" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
602 |
and notinB: "b \<notin> B" and Bu: "fold_graph f z B u" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
603 |
show "x'=x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
604 |
proof (rule fold_graph.cases [OF Afoldx']) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
605 |
assume "A = {}" and "x' = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
606 |
with AbB show "x' = x" by blast |
15392 | 607 |
next |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
608 |
fix C c v |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
609 |
assume AcC: "A = insert c C" and x': "x' = f c v" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
610 |
and notinC: "c \<notin> C" and Cv: "fold_graph f z C v" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
611 |
from A AbB have Beq: "insert b B = h`{i. i<n}" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
612 |
from insert_inj_onE [OF Beq notinB injh] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
613 |
obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
614 |
and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
615 |
from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
616 |
from insert_inj_onE [OF Ceq notinC injh] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
617 |
obtain hC mC where inj_onC: "inj_on hC {i. i < mC}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
618 |
and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
619 |
show "x'=x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
620 |
proof cases |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
621 |
assume "b=c" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
622 |
then moreover have "B = C" using AbB AcC notinB notinC by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
623 |
ultimately show ?thesis using Bu Cv x x' IH [OF lessC Ceq inj_onC] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
624 |
by auto |
15392 | 625 |
next |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
626 |
assume diff: "b \<noteq> c" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
627 |
let ?D = "B - {c}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
628 |
have B: "B = insert c ?D" and C: "C = insert b ?D" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
629 |
using AbB AcC notinB notinC diff by(blast elim!:equalityE)+ |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
630 |
have "finite A" by(rule fold_graph_imp_finite [OF Afoldx]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
631 |
with AbB have "finite ?D" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
632 |
then obtain d where Dfoldd: "fold_graph f z ?D d" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
633 |
using finite_imp_fold_graph by iprover |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
634 |
moreover have cinB: "c \<in> B" using B by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
635 |
ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
636 |
hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
637 |
moreover have "f b d = v" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
638 |
proof (rule IH[OF lessC Ceq inj_onC Cv]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
639 |
show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp |
15392 | 640 |
qed |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
641 |
ultimately show ?thesis |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
642 |
using fun_left_comm [of c b] x x' by (auto simp add: o_def) |
15392 | 643 |
qed |
644 |
qed |
|
645 |
qed |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
646 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
647 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
648 |
lemma fold_graph_determ: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
649 |
"fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
650 |
apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
651 |
apply (blast intro: fold_graph_determ_aux [rule_format]) |
15392 | 652 |
done |
653 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
654 |
lemma fold_equality: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
655 |
"fold_graph f z A y \<Longrightarrow> fold f z A = y" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
656 |
by (unfold fold_def) (blast intro: fold_graph_determ) |
15392 | 657 |
|
658 |
text{* The base case for @{text fold}: *} |
|
659 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
660 |
lemma (in -) fold_empty [simp]: "fold f z {} = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
661 |
by (unfold fold_def) blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
662 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
663 |
text{* The various recursion equations for @{const fold}: *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
664 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
665 |
lemma fold_insert_aux: "x \<notin> A |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
666 |
\<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
667 |
(\<exists>y. fold_graph f z A y \<and> v = f x y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
668 |
apply auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
669 |
apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
670 |
apply (fastsimp dest: fold_graph_imp_finite) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
671 |
apply (blast intro: fold_graph_determ) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
672 |
done |
15392 | 673 |
|
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 |
lemma fold_insert [simp]: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
675 |
"finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
676 |
apply (simp add: fold_def fold_insert_aux) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
677 |
apply (rule the_equality) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
678 |
apply (auto intro: finite_imp_fold_graph |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
679 |
cong add: conj_cong simp add: fold_def[symmetric] fold_equality) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
680 |
done |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
681 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
682 |
lemma fold_fun_comm: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
683 |
"finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
684 |
proof (induct rule: finite_induct) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
685 |
case empty then show ?case by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
686 |
next |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
687 |
case (insert y A) then show ?case |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
688 |
by (simp add: fun_left_comm[of x]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
689 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
690 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
691 |
lemma fold_insert2: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
692 |
"finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
693 |
by (simp add: fold_insert fold_fun_comm) |
15392 | 694 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
695 |
lemma fold_rec: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
696 |
assumes "finite A" and "x \<in> A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
697 |
shows "fold f z A = f x (fold f z (A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
698 |
proof - |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
699 |
have A: "A = insert x (A - {x})" using `x \<in> A` by blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
700 |
then have "fold f z A = fold f z (insert x (A - {x}))" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
701 |
also have "\<dots> = f x (fold f z (A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
702 |
by (rule fold_insert) (simp add: `finite A`)+ |
15535 | 703 |
finally show ?thesis . |
704 |
qed |
|
705 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
706 |
lemma fold_insert_remove: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
707 |
assumes "finite A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
708 |
shows "fold f z (insert x A) = f x (fold f z (A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
709 |
proof - |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
710 |
from `finite A` have "finite (insert x A)" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
711 |
moreover have "x \<in> insert x A" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
712 |
ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
713 |
by (rule fold_rec) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
714 |
then show ?thesis by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
715 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
716 |
|
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
|
717 |
end |
15392 | 718 |
|
15480 | 719 |
text{* A simplified version for idempotent functions: *} |
720 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
721 |
locale fun_left_comm_idem = fun_left_comm + |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
722 |
assumes fun_left_idem: "f x (f x z) = f x z" |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
723 |
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
|
724 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
725 |
text{* The nice version: *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
726 |
lemma fun_comp_idem : "f x o f x = f x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
727 |
by (simp add: fun_left_idem expand_fun_eq) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
728 |
|
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
|
729 |
lemma fold_insert_idem: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
730 |
assumes fin: "finite A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
731 |
shows "fold f z (insert x A) = f x (fold f z A)" |
15480 | 732 |
proof cases |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
733 |
assume "x \<in> A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
734 |
then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
735 |
then show ?thesis using assms by (simp add:fun_left_idem) |
15480 | 736 |
next |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
737 |
assume "x \<notin> A" then show ?thesis using assms by simp |
15480 | 738 |
qed |
739 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
740 |
declare fold_insert[simp del] fold_insert_idem[simp] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
741 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
742 |
lemma fold_insert_idem2: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
743 |
"finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
744 |
by(simp add:fold_fun_comm) |
15484 | 745 |
|
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
|
746 |
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
|
747 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
748 |
subsubsection{* The derived combinator @{text fold_image} *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
749 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
750 |
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
751 |
where "fold_image f g = fold (%x y. f (g x) y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
752 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
753 |
lemma fold_image_empty[simp]: "fold_image f g z {} = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
754 |
by(simp add:fold_image_def) |
15392 | 755 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
756 |
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
|
757 |
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
|
758 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
759 |
lemma fold_image_insert[simp]: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
760 |
assumes "finite A" and "a \<notin> A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
761 |
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
762 |
proof - |
29223 | 763 |
interpret I: fun_left_comm "%x y. (g x) * y" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
764 |
by unfold_locales (simp add: mult_ac) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
765 |
show ?thesis using assms by(simp add:fold_image_def I.fold_insert) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
766 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
767 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
768 |
(* |
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
|
769 |
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
|
770 |
"finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)" |
22262 | 771 |
apply (induct set: finite) |
21575 | 772 |
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
|
773 |
apply (simp add: mult_left_commute [of x]) |
15392 | 774 |
done |
775 |
||
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
|
776 |
lemma fold_nest_Un_Int: |
15392 | 777 |
"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
|
778 |
==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)" |
22262 | 779 |
apply (induct set: finite) |
21575 | 780 |
apply simp |
15392 | 781 |
apply (simp add: fold_commute Int_insert_left insert_absorb) |
782 |
done |
|
783 |
||
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
|
784 |
lemma fold_nest_Un_disjoint: |
15392 | 785 |
"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
|
786 |
==> fold times g z (A Un B) = fold times g (fold times g z B) A" |
15392 | 787 |
by (simp add: fold_nest_Un_Int) |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
788 |
*) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
789 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
790 |
lemma fold_image_reindex: |
15487 | 791 |
assumes fin: "finite A" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
792 |
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A" |
15506 | 793 |
using fin apply induct |
15392 | 794 |
apply simp |
795 |
apply simp |
|
796 |
done |
|
797 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
798 |
(* |
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
|
799 |
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
|
800 |
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
|
801 |
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
|
802 |
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
|
803 |
*} |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour 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 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour 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 |
lemma fold_fusion: |
27611 | 806 |
assumes "ab_semigroup_mult g" |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
807 |
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
|
808 |
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
|
809 |
shows "h (fold g j w A) = fold times j (h w) A" |
27611 | 810 |
proof - |
29223 | 811 |
class_interpret ab_semigroup_mult [g] by fact |
27611 | 812 |
show ?thesis using fin hyp by (induct set: finite) simp_all |
813 |
qed |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
814 |
*) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
815 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
816 |
lemma fold_image_cong: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
817 |
"finite A \<Longrightarrow> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
818 |
(!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
819 |
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
820 |
apply simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
821 |
apply (erule finite_induct, simp) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
822 |
apply (simp add: subset_insert_iff, clarify) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
823 |
apply (subgoal_tac "finite C") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
824 |
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
825 |
apply (subgoal_tac "C = insert x (C - {x})") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
826 |
prefer 2 apply blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
827 |
apply (erule ssubst) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
828 |
apply (drule spec) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
829 |
apply (erule (1) notE impE) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
830 |
apply (simp add: Ball_def del: insert_Diff_single) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
831 |
done |
15392 | 832 |
|
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
|
833 |
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
|
834 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
835 |
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
|
836 |
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
|
837 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
838 |
lemma fold_image_Un_Int: |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
839 |
"finite A ==> finite B ==> |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
840 |
fold_image times g 1 A * fold_image times g 1 B = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
841 |
fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
842 |
by (induct set: finite) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
843 |
(auto simp add: mult_ac insert_absorb Int_insert_left) |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
844 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
845 |
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
|
846 |
"finite A ==> finite B ==> A Int B = {} ==> |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
847 |
fold_image times g 1 (A Un B) = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
848 |
fold_image times g 1 A * fold_image times g 1 B" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
849 |
by (simp add: fold_image_Un_Int) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
850 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
851 |
lemma fold_image_UN_disjoint: |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
852 |
"\<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
|
853 |
ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk> |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
854 |
\<Longrightarrow> fold_image times g 1 (UNION I A) = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
855 |
fold_image times (%i. fold_image times g 1 (A i)) 1 I" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
856 |
apply (induct set: finite, simp, atomize) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
857 |
apply (subgoal_tac "ALL i:F. x \<noteq> i") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
858 |
prefer 2 apply blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
859 |
apply (subgoal_tac "A x Int UNION F A = {}") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
860 |
prefer 2 apply blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
861 |
apply (simp add: fold_Un_disjoint) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
862 |
done |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
863 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
864 |
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
865 |
fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
866 |
fold_image times (split g) 1 (SIGMA x:A. B x)" |
15392 | 867 |
apply (subst Sigma_def) |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
868 |
apply (subst fold_image_UN_disjoint, assumption, simp) |
15392 | 869 |
apply blast |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
870 |
apply (erule fold_image_cong) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
871 |
apply (subst fold_image_UN_disjoint, simp, simp) |
15392 | 872 |
apply blast |
15506 | 873 |
apply simp |
15392 | 874 |
done |
875 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
876 |
lemma fold_image_distrib: "finite A \<Longrightarrow> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
877 |
fold_image times (%x. g x * h x) 1 A = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
878 |
fold_image times g 1 A * fold_image times h 1 A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
879 |
by (erule finite_induct) (simp_all add: mult_ac) |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
880 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
881 |
end |
22917 | 882 |
|
883 |
||
15402 | 884 |
subsection {* Generalized summation over a set *} |
885 |
||
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29223
diff
changeset
|
886 |
interpretation comm_monoid_add!: comm_monoid_mult "0::'a::comm_monoid_add" "op +" |
28823 | 887 |
proof qed (auto intro: add_assoc add_commute) |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
888 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
889 |
definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
890 |
where "setsum f A == if finite A then fold_image (op +) f 0 A else 0" |
15402 | 891 |
|
19535 | 892 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21249
diff
changeset
|
893 |
Setsum ("\<Sum>_" [1000] 999) where |
19535 | 894 |
"\<Sum>A == setsum (%x. x) A" |
895 |
||
15402 | 896 |
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is |
897 |
written @{text"\<Sum>x\<in>A. e"}. *} |
|
898 |
||
899 |
syntax |
|
17189 | 900 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) |
15402 | 901 |
syntax (xsymbols) |
17189 | 902 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 903 |
syntax (HTML output) |
17189 | 904 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 905 |
|
906 |
translations -- {* Beware of argument permutation! *} |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
907 |
"SUM i:A. b" == "CONST setsum (%i. b) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
908 |
"\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A" |
15402 | 909 |
|
910 |
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter |
|
911 |
@{text"\<Sum>x|P. e"}. *} |
|
912 |
||
913 |
syntax |
|
17189 | 914 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) |
15402 | 915 |
syntax (xsymbols) |
17189 | 916 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 917 |
syntax (HTML output) |
17189 | 918 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 919 |
|
920 |
translations |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
921 |
"SUM x|P. t" => "CONST setsum (%x. t) {x. P}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
922 |
"\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}" |
15402 | 923 |
|
924 |
print_translation {* |
|
925 |
let |
|
19535 | 926 |
fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = |
927 |
if x<>y then raise Match |
|
928 |
else let val x' = Syntax.mark_bound x |
|
929 |
val t' = subst_bound(x',t) |
|
930 |
val P' = subst_bound(x',P) |
|
931 |
in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end |
|
932 |
in [("setsum", setsum_tr')] end |
|
15402 | 933 |
*} |
934 |
||
19535 | 935 |
|
15402 | 936 |
lemma setsum_empty [simp]: "setsum f {} = 0" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
937 |
by (simp add: setsum_def) |
15402 | 938 |
|
939 |
lemma setsum_insert [simp]: |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
940 |
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
941 |
by (simp add: setsum_def) |
15402 | 942 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
943 |
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
944 |
by (simp add: setsum_def) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
945 |
|
15402 | 946 |
lemma setsum_reindex: |
947 |
"inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
948 |
by(auto simp add: setsum_def comm_monoid_add.fold_image_reindex dest!:finite_imageD) |
15402 | 949 |
|
950 |
lemma setsum_reindex_id: |
|
951 |
"inj_on f B ==> setsum f B = setsum id (f ` B)" |
|
952 |
by (auto simp add: setsum_reindex) |
|
953 |
||
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
954 |
lemma setsum_reindex_nonzero: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
955 |
assumes fS: "finite S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
956 |
and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
957 |
shows "setsum h (f ` S) = setsum (h o f) S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
958 |
using nz |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
959 |
proof(induct rule: finite_induct[OF fS]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
960 |
case 1 thus ?case by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
961 |
next |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
962 |
case (2 x F) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
963 |
{assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
964 |
then obtain y where y: "y \<in> F" "f x = f y" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
965 |
from "2.hyps" y have xy: "x \<noteq> y" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
966 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
967 |
from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
968 |
have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
969 |
also have "\<dots> = setsum (h o f) (insert x F)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
970 |
unfolding setsum_insert[OF `finite F` `x\<notin>F`] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
971 |
using h0 |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
972 |
apply simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
973 |
apply (rule "2.hyps"(3)) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
974 |
apply (rule_tac y="y" in "2.prems") |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
975 |
apply simp_all |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
976 |
done |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
977 |
finally have ?case .} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
978 |
moreover |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
979 |
{assume fxF: "f x \<notin> f ` F" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
980 |
have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
981 |
using fxF "2.hyps" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
982 |
also have "\<dots> = setsum (h o f) (insert x F)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
983 |
unfolding setsum_insert[OF `finite F` `x\<notin>F`] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
984 |
apply simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
985 |
apply (rule cong[OF refl[of "op + (h (f x))"]]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
986 |
apply (rule "2.hyps"(3)) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
987 |
apply (rule_tac y="y" in "2.prems") |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
988 |
apply simp_all |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
989 |
done |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
990 |
finally have ?case .} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
991 |
ultimately show ?case by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
992 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
993 |
|
15402 | 994 |
lemma setsum_cong: |
995 |
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
996 |
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_image_cong) |
15402 | 997 |
|
16733
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
998 |
lemma strong_setsum_cong[cong]: |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
999 |
"A = B ==> (!!x. x:B =simp=> f x = g x) |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
1000 |
==> setsum (%x. f x) A = setsum (%x. g x) B" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1001 |
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong) |
16632
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
1002 |
|
15554 | 1003 |
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"; |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1004 |
by (rule setsum_cong[OF refl], auto); |
15554 | 1005 |
|
15402 | 1006 |
lemma setsum_reindex_cong: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1007 |
"[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1008 |
==> setsum h B = setsum g A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1009 |
by (simp add: setsum_reindex cong: setsum_cong) |
15402 | 1010 |
|
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1011 |
|
15542 | 1012 |
lemma setsum_0[simp]: "setsum (%i. 0) A = 0" |
15402 | 1013 |
apply (clarsimp simp: setsum_def) |
15765 | 1014 |
apply (erule finite_induct, auto) |
15402 | 1015 |
done |
1016 |
||
15543 | 1017 |
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0" |
1018 |
by(simp add:setsum_cong) |
|
15402 | 1019 |
|
1020 |
lemma setsum_Un_Int: "finite A ==> finite B ==> |
|
1021 |
setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" |
|
1022 |
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1023 |
by(simp add: setsum_def comm_monoid_add.fold_image_Un_Int [symmetric]) |
15402 | 1024 |
|
1025 |
lemma setsum_Un_disjoint: "finite A ==> finite B |
|
1026 |
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" |
|
1027 |
by (subst setsum_Un_Int [symmetric], auto) |
|
1028 |
||
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1029 |
lemma setsum_mono_zero_left: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1030 |
assumes fT: "finite T" and ST: "S \<subseteq> T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1031 |
and z: "\<forall>i \<in> T - S. f i = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1032 |
shows "setsum f S = setsum f T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1033 |
proof- |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1034 |
have eq: "T = S \<union> (T - S)" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1035 |
have d: "S \<inter> (T - S) = {}" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1036 |
from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1037 |
show ?thesis |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1038 |
by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1039 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1040 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1041 |
lemma setsum_mono_zero_right: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1042 |
assumes fT: "finite T" and ST: "S \<subseteq> T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1043 |
and z: "\<forall>i \<in> T - S. f i = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1044 |
shows "setsum f T = setsum f S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1045 |
using setsum_mono_zero_left[OF fT ST z] by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1046 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1047 |
lemma setsum_mono_zero_cong_left: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1048 |
assumes fT: "finite T" and ST: "S \<subseteq> T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1049 |
and z: "\<forall>i \<in> T - S. g i = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1050 |
and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1051 |
shows "setsum f S = setsum g T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1052 |
proof- |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1053 |
have eq: "T = S \<union> (T - S)" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1054 |
have d: "S \<inter> (T - S) = {}" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1055 |
from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1056 |
show ?thesis |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1057 |
using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1058 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1059 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1060 |
lemma setsum_mono_zero_cong_right: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1061 |
assumes fT: "finite T" and ST: "S \<subseteq> T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1062 |
and z: "\<forall>i \<in> T - S. f i = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1063 |
and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1064 |
shows "setsum f T = setsum g S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1065 |
using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1066 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1067 |
lemma setsum_delta: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1068 |
assumes fS: "finite S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1069 |
shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1070 |
proof- |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1071 |
let ?f = "(\<lambda>k. if k=a then b k else 0)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1072 |
{assume a: "a \<notin> S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1073 |
hence "\<forall> k\<in> S. ?f k = 0" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1074 |
hence ?thesis using a by simp} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1075 |
moreover |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1076 |
{assume a: "a \<in> S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1077 |
let ?A = "S - {a}" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1078 |
let ?B = "{a}" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1079 |
have eq: "S = ?A \<union> ?B" using a by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1080 |
have dj: "?A \<inter> ?B = {}" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1081 |
from fS have fAB: "finite ?A" "finite ?B" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1082 |
have "setsum ?f S = setsum ?f ?A + setsum ?f ?B" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1083 |
using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1084 |
by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1085 |
then have ?thesis using a by simp} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1086 |
ultimately show ?thesis by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1087 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1088 |
lemma setsum_delta': |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1089 |
assumes fS: "finite S" shows |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1090 |
"setsum (\<lambda>k. if a = k then b k else 0) S = |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1091 |
(if a\<in> S then b a else 0)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1092 |
using setsum_delta[OF fS, of a b, symmetric] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1093 |
by (auto intro: setsum_cong) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1094 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1095 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1096 |
(*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
|
1097 |
the lhs need not be, since UNION I A could still be finite.*) |
15402 | 1098 |
lemma setsum_UN_disjoint: |
1099 |
"finite I ==> (ALL i:I. finite (A i)) ==> |
|
1100 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
|
1101 |
setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1102 |
by(simp add: setsum_def comm_monoid_add.fold_image_UN_disjoint cong: setsum_cong) |
15402 | 1103 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1104 |
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
|
1105 |
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*} |
15402 | 1106 |
lemma setsum_Union_disjoint: |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1107 |
"[| (ALL A:C. finite A); |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1108 |
(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
|
1109 |
==> setsum f (Union C) = setsum (setsum f) C" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1110 |
apply (cases "finite C") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1111 |
prefer 2 apply (force dest: finite_UnionD simp add: setsum_def) |
15402 | 1112 |
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
|
1113 |
apply (unfold Union_def id_def, assumption+) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1114 |
done |
15402 | 1115 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1116 |
(*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
|
1117 |
the rhs need not be, since SIGMA A B could still be finite.*) |
15402 | 1118 |
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
17189 | 1119 |
(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1120 |
by(simp add:setsum_def comm_monoid_add.fold_image_Sigma split_def cong:setsum_cong) |
15402 | 1121 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1122 |
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
|
1123 |
lemma setsum_cartesian_product: |
17189 | 1124 |
"(\<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
|
1125 |
apply (cases "finite A") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1126 |
apply (cases "finite B") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1127 |
apply (simp add: setsum_Sigma) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1128 |
apply (cases "A={}", simp) |
15543 | 1129 |
apply (simp) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1130 |
apply (auto simp add: setsum_def |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1131 |
dest: finite_cartesian_productD1 finite_cartesian_productD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1132 |
done |
15402 | 1133 |
|
1134 |
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1135 |
by(simp add:setsum_def comm_monoid_add.fold_image_distrib) |
15402 | 1136 |
|
1137 |
||
1138 |
subsubsection {* Properties in more restricted classes of structures *} |
|
1139 |
||
1140 |
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1141 |
apply (case_tac "finite A") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1142 |
prefer 2 apply (simp add: setsum_def) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1143 |
apply (erule rev_mp) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1144 |
apply (erule finite_induct, auto) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1145 |
done |
15402 | 1146 |
|
1147 |
lemma setsum_eq_0_iff [simp]: |
|
1148 |
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1149 |
by (induct set: finite) auto |
15402 | 1150 |
|
1151 |
lemma setsum_Un_nat: "finite A ==> finite B ==> |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1152 |
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" |
15402 | 1153 |
-- {* For the natural numbers, we have subtraction. *} |
29667 | 1154 |
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) |
15402 | 1155 |
|
1156 |
lemma setsum_Un: "finite A ==> finite B ==> |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1157 |
(setsum f (A Un B) :: 'a :: ab_group_add) = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1158 |
setsum f A + setsum f B - setsum f (A Int B)" |
29667 | 1159 |
by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) |
15402 | 1160 |
|
1161 |
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) = |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1162 |
(if a:A then setsum f A - f a else setsum f A)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1163 |
apply (case_tac "finite A") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1164 |
prefer 2 apply (simp add: setsum_def) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1165 |
apply (erule finite_induct) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1166 |
apply (auto simp add: insert_Diff_if) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1167 |
apply (drule_tac a = a in mk_disjoint_insert, auto) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1168 |
done |
15402 | 1169 |
|
1170 |
lemma setsum_diff1: "finite A \<Longrightarrow> |
|
1171 |
(setsum f (A - {a}) :: ('a::ab_group_add)) = |
|
1172 |
(if a:A then setsum f A - f a else setsum f A)" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1173 |
by (erule finite_induct) (auto simp add: insert_Diff_if) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1174 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1175 |
lemma setsum_diff1'[rule_format]: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1176 |
"finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1177 |
apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1178 |
apply (auto simp add: insert_Diff_if add_ac) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1179 |
done |
15552
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
1180 |
|
15402 | 1181 |
(* By Jeremy Siek: *) |
1182 |
||
1183 |
lemma setsum_diff_nat: |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1184 |
assumes "finite B" and "B \<subseteq> A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1185 |
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1186 |
using assms |
19535 | 1187 |
proof induct |
15402 | 1188 |
show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp |
1189 |
next |
|
1190 |
fix F x assume finF: "finite F" and xnotinF: "x \<notin> F" |
|
1191 |
and xFinA: "insert x F \<subseteq> A" |
|
1192 |
and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F" |
|
1193 |
from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp |
|
1194 |
from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x" |
|
1195 |
by (simp add: setsum_diff1_nat) |
|
1196 |
from xFinA have "F \<subseteq> A" by simp |
|
1197 |
with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp |
|
1198 |
with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x" |
|
1199 |
by simp |
|
1200 |
from xnotinF have "A - insert x F = (A - F) - {x}" by auto |
|
1201 |
with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x" |
|
1202 |
by simp |
|
1203 |
from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp |
|
1204 |
with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" |
|
1205 |
by simp |
|
1206 |
thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp |
|
1207 |
qed |
|
1208 |
||
1209 |
lemma setsum_diff: |
|
1210 |
assumes le: "finite A" "B \<subseteq> A" |
|
1211 |
shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))" |
|
1212 |
proof - |
|
1213 |
from le have finiteB: "finite B" using finite_subset by auto |
|
1214 |
show ?thesis using finiteB le |
|
21575 | 1215 |
proof induct |
19535 | 1216 |
case empty |
1217 |
thus ?case by auto |
|
1218 |
next |
|
1219 |
case (insert x F) |
|
1220 |
thus ?case using le finiteB |
|
1221 |
by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb) |
|
15402 | 1222 |
qed |
19535 | 1223 |
qed |
15402 | 1224 |
|
1225 |
lemma setsum_mono: |
|
1226 |
assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))" |
|
1227 |
shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)" |
|
1228 |
proof (cases "finite K") |
|
1229 |
case True |
|
1230 |
thus ?thesis using le |
|
19535 | 1231 |
proof induct |
15402 | 1232 |
case empty |
1233 |
thus ?case by simp |
|
1234 |
next |
|
1235 |
case insert |
|
19535 | 1236 |
thus ?case using add_mono by fastsimp |
15402 | 1237 |
qed |
1238 |
next |
|
1239 |
case False |
|
1240 |
thus ?thesis |
|
1241 |
by (simp add: setsum_def) |
|
1242 |
qed |
|
1243 |
||
15554 | 1244 |
lemma setsum_strict_mono: |
19535 | 1245 |
fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}" |
1246 |
assumes "finite A" "A \<noteq> {}" |
|
1247 |
and "!!x. x:A \<Longrightarrow> f x < g x" |
|
1248 |
shows "setsum f A < setsum g A" |
|
1249 |
using prems |
|
15554 | 1250 |
proof (induct rule: finite_ne_induct) |
1251 |
case singleton thus ?case by simp |
|
1252 |
next |
|
1253 |
case insert thus ?case by (auto simp: add_strict_mono) |
|
1254 |
qed |
|
1255 |
||
15535 | 1256 |
lemma setsum_negf: |
19535 | 1257 |
"setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A" |
15535 | 1258 |
proof (cases "finite A") |
22262 | 1259 |
case True thus ?thesis by (induct set: finite) auto |
15535 | 1260 |
next |
1261 |
case False thus ?thesis by (simp add: setsum_def) |
|
1262 |
qed |
|
15402 | 1263 |
|
15535 | 1264 |
lemma setsum_subtractf: |
19535 | 1265 |
"setsum (%x. ((f x)::'a::ab_group_add) - g x) A = |
1266 |
setsum f A - setsum g A" |
|
15535 | 1267 |
proof (cases "finite A") |
1268 |
case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf) |
|
1269 |
next |
|
1270 |
case False thus ?thesis by (simp add: setsum_def) |
|
1271 |
qed |
|
15402 | 1272 |
|
15535 | 1273 |
lemma setsum_nonneg: |
19535 | 1274 |
assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x" |
1275 |
shows "0 \<le> setsum f A" |
|
15535 | 1276 |
proof (cases "finite A") |
1277 |
case True thus ?thesis using nn |
|
21575 | 1278 |
proof induct |
19535 | 1279 |
case empty then show ?case by simp |
1280 |
next |
|
1281 |
case (insert x F) |
|
1282 |
then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono) |
|
1283 |
with insert show ?case by simp |
|
1284 |
qed |
|
15535 | 1285 |
next |
1286 |
case False thus ?thesis by (simp add: setsum_def) |
|
1287 |
qed |
|
15402 | 1288 |
|
15535 |