author  wenzelm 
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permissions  rwrr 
38622  1 
(* Title: HOL/Library/Set_Algebras.thy 
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Author: Jeremy Avigad 
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Author: Kevin Donnelly 

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Author: Florian Haftmann, TUM 

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*) 
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60500  7 
section \<open>Algebraic operations on sets\<close> 
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38622  9 
theory Set_Algebras 
63485  10 
imports Main 
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begin 
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60500  13 
text \<open> 
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This library lifts operations like addition and multiplication to sets. It 
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was designed to support asymptotic calculations. See the comments at the top 

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of \<^file>\<open>BigO.thy\<close>. 
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\<close> 
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instantiation set :: (plus) plus 
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begin 
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definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" 
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where set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}" 

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instance .. 
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end 
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instantiation set :: (times) times 
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begin 
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definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" 
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where set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}" 

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instance .. 
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end 
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instantiation set :: (zero) zero 
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begin 
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definition set_zero[simp]: "(0::'a::zero set) = {0}" 
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instance .. 
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end 
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instantiation set :: (one) one 
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begin 
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63473  51 
definition set_one[simp]: "(1::'a::one set) = {1}" 
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instance .. 
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end 
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definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "+o" 70) 
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where "a +o B = {c. \<exists>b\<in>B. c = a + b}" 

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definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "*o" 80) 
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where "a *o B = {c. \<exists>b\<in>B. c = a * b}" 

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63473  63 
abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infix "=o" 50) 
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where "x =o A \<equiv> x \<in> A" 

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instance set :: (semigroup_add) semigroup_add 
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by standard (force simp add: set_plus_def add.assoc) 
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instance set :: (ab_semigroup_add) ab_semigroup_add 
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by standard (force simp add: set_plus_def add.commute) 
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instance set :: (monoid_add) monoid_add 
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by standard (simp_all add: set_plus_def) 
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instance set :: (comm_monoid_add) comm_monoid_add 
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by standard (simp_all add: set_plus_def) 
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instance set :: (semigroup_mult) semigroup_mult 
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by standard (force simp add: set_times_def mult.assoc) 
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instance set :: (ab_semigroup_mult) ab_semigroup_mult 
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by standard (force simp add: set_times_def mult.commute) 
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instance set :: (monoid_mult) monoid_mult 
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by standard (simp_all add: set_times_def) 
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instance set :: (comm_monoid_mult) comm_monoid_mult 
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by standard (simp_all add: set_times_def) 
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lemma set_plus_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a + b \<in> C + D" 
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by (auto simp add: set_plus_def) 
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lemma set_plus_elim: 
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assumes "x \<in> A + B" 

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obtains a b where "x = a + b" and "a \<in> A" and "b \<in> B" 

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using assms unfolding set_plus_def by fast 

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lemma set_plus_intro2 [intro]: "b \<in> C \<Longrightarrow> a + b \<in> a +o C" 
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by (auto simp add: elt_set_plus_def) 
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lemma set_plus_rearrange: "(a +o C) + (b +o D) = (a + b) +o (C + D)" 
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for a b :: "'a::comm_monoid_add" 

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apply (auto simp add: elt_set_plus_def set_plus_def ac_simps) 
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apply (rule_tac x = "ba + bb" in exI) 
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apply (auto simp add: ac_simps) 
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apply (rule_tac x = "aa + a" in exI) 
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apply (auto simp add: ac_simps) 
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done 
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lemma set_plus_rearrange2: "a +o (b +o C) = (a + b) +o C" 
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for a b :: "'a::semigroup_add" 

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by (auto simp add: elt_set_plus_def add.assoc) 
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lemma set_plus_rearrange3: "(a +o B) + C = a +o (B + C)" 
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for a :: "'a::semigroup_add" 

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apply (auto simp add: elt_set_plus_def set_plus_def) 
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apply (blast intro: ac_simps) 
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apply (rule_tac x = "a + aa" in exI) 
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apply (rule conjI) 
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apply (rule_tac x = "aa" in bexI) 
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apply auto 

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apply (rule_tac x = "ba" in bexI) 
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apply (auto simp add: ac_simps) 
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done 
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theorem set_plus_rearrange4: "C + (a +o D) = a +o (C + D)" 
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for a :: "'a::comm_monoid_add" 

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apply (auto simp add: elt_set_plus_def set_plus_def ac_simps) 
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apply (rule_tac x = "aa + ba" in exI) 
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apply (auto simp add: ac_simps) 
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done 
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lemmas set_plus_rearranges = set_plus_rearrange set_plus_rearrange2 
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set_plus_rearrange3 set_plus_rearrange4 
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lemma set_plus_mono [intro!]: "C \<subseteq> D \<Longrightarrow> a +o C \<subseteq> a +o D" 
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by (auto simp add: elt_set_plus_def) 
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lemma set_plus_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F" 
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for C D E F :: "'a::plus set" 

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by (auto simp add: set_plus_def) 
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lemma set_plus_mono3 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> C + D" 
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by (auto simp add: elt_set_plus_def set_plus_def) 
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lemma set_plus_mono4 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> D + C" 
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for a :: "'a::comm_monoid_add" 

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by (auto simp add: elt_set_plus_def set_plus_def ac_simps) 
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lemma set_plus_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a +o B \<subseteq> C + D" 
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apply (subgoal_tac "a +o B \<subseteq> a +o D") 

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apply (erule order_trans) 
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apply (erule set_plus_mono3) 

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apply (erule set_plus_mono) 
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done 
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lemma set_plus_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a +o C \<Longrightarrow> x \<in> a +o D" 
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apply (frule set_plus_mono) 
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apply auto 
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done 
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lemma set_plus_mono2_b: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> x \<in> C + E \<Longrightarrow> x \<in> D + F" 
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apply (frule set_plus_mono2) 
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prefer 2 
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apply force 

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apply assumption 
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done 
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lemma set_plus_mono3_b: "a \<in> C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> C + D" 
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apply (frule set_plus_mono3) 
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apply auto 
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done 
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lemma set_plus_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C" 
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for a x :: "'a::comm_monoid_add" 

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apply (frule set_plus_mono4) 
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apply auto 
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done 
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lemma set_zero_plus [simp]: "0 +o C = C" 
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for C :: "'a::comm_monoid_add set" 

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by (auto simp add: elt_set_plus_def) 
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lemma set_zero_plus2: "0 \<in> A \<Longrightarrow> B \<subseteq> A + B" 
185 
for A B :: "'a::comm_monoid_add set" 

44142  186 
apply (auto simp add: set_plus_def) 
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187 
apply (rule_tac x = 0 in bexI) 
19736  188 
apply (rule_tac x = x in bexI) 
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189 
apply (auto simp add: ac_simps) 
19736  190 
done 
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191 

63473  192 
lemma set_plus_imp_minus: "a \<in> b +o C \<Longrightarrow> a  b \<in> C" 
193 
for a b :: "'a::ab_group_add" 

57514
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194 
by (auto simp add: elt_set_plus_def ac_simps) 
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diff
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195 

63473  196 
lemma set_minus_imp_plus: "a  b \<in> C \<Longrightarrow> a \<in> b +o C" 
197 
for a b :: "'a::ab_group_add" 

57514
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198 
apply (auto simp add: elt_set_plus_def ac_simps) 
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avigad
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diff
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199 
apply (subgoal_tac "a = (a +  b) + b") 
63473  200 
apply (rule bexI) 
201 
apply assumption 

202 
apply (auto simp add: ac_simps) 

19736  203 
done 
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204 

63473  205 
lemma set_minus_plus: "a  b \<in> C \<longleftrightarrow> a \<in> b +o C" 
206 
for a b :: "'a::ab_group_add" 

207 
apply (rule iffI) 

208 
apply (rule set_minus_imp_plus) 

209 
apply assumption 

210 
apply (rule set_plus_imp_minus) 

211 
apply assumption 

212 
done 

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diff
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213 

56899  214 
lemma set_times_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D" 
26814
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changeset

215 
by (auto simp add: set_times_def) 
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216 

53596  217 
lemma set_times_elim: 
218 
assumes "x \<in> A * B" 

219 
obtains a b where "x = a * b" and "a \<in> A" and "b \<in> B" 

220 
using assms unfolding set_times_def by fast 

221 

56899  222 
lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C" 
19736  223 
by (auto simp add: elt_set_times_def) 
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224 

63473  225 
lemma set_times_rearrange: "(a *o C) * (b *o D) = (a * b) *o (C * D)" 
226 
for a b :: "'a::comm_monoid_mult" 

26814
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227 
apply (auto simp add: elt_set_times_def set_times_def) 
19736  228 
apply (rule_tac x = "ba * bb" in exI) 
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229 
apply (auto simp add: ac_simps) 
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230 
apply (rule_tac x = "aa * a" in exI) 
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231 
apply (auto simp add: ac_simps) 
19736  232 
done 
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233 

63473  234 
lemma set_times_rearrange2: "a *o (b *o C) = (a * b) *o C" 
235 
for a b :: "'a::semigroup_mult" 

57512
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diff
changeset

236 
by (auto simp add: elt_set_times_def mult.assoc) 
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237 

63473  238 
lemma set_times_rearrange3: "(a *o B) * C = a *o (B * C)" 
239 
for a :: "'a::semigroup_mult" 

26814
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changeset

240 
apply (auto simp add: elt_set_times_def set_times_def) 
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241 
apply (blast intro: ac_simps) 
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changeset

242 
apply (rule_tac x = "a * aa" in exI) 
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avigad
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diff
changeset

243 
apply (rule conjI) 
19736  244 
apply (rule_tac x = "aa" in bexI) 
245 
apply auto 

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246 
apply (rule_tac x = "ba" in bexI) 
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247 
apply (auto simp add: ac_simps) 
19736  248 
done 
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249 

63473  250 
theorem set_times_rearrange4: "C * (a *o D) = a *o (C * D)" 
251 
for a :: "'a::comm_monoid_mult" 

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252 
apply (auto simp add: elt_set_times_def set_times_def ac_simps) 
19736  253 
apply (rule_tac x = "aa * ba" in exI) 
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254 
apply (auto simp add: ac_simps) 
19736  255 
done 
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256 

61337  257 
lemmas set_times_rearranges = set_times_rearrange set_times_rearrange2 
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258 
set_times_rearrange3 set_times_rearrange4 
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diff
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259 

56899  260 
lemma set_times_mono [intro]: "C \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D" 
19736  261 
by (auto simp add: elt_set_times_def) 
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262 

63473  263 
lemma set_times_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F" 
264 
for C D E F :: "'a::times set" 

26814
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diff
changeset

265 
by (auto simp add: set_times_def) 
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avigad
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diff
changeset

266 

56899  267 
lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D" 
26814
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changeset

268 
by (auto simp add: elt_set_times_def set_times_def) 
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269 

63473  270 
lemma set_times_mono4 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> D * C" 
271 
for a :: "'a::comm_monoid_mult" 

57514
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272 
by (auto simp add: elt_set_times_def set_times_def ac_simps) 
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avigad
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diff
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273 

56899  274 
lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D" 
275 
apply (subgoal_tac "a *o B \<subseteq> a *o D") 

19736  276 
apply (erule order_trans) 
277 
apply (erule set_times_mono3) 

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avigad
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diff
changeset

278 
apply (erule set_times_mono) 
19736  279 
done 
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avigad
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diff
changeset

280 

56899  281 
lemma set_times_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a *o C \<Longrightarrow> x \<in> a *o D" 
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avigad
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changeset

282 
apply (frule set_times_mono) 
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avigad
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diff
changeset

283 
apply auto 
19736  284 
done 
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avigad
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diff
changeset

285 

56899  286 
lemma set_times_mono2_b: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> x \<in> C * E \<Longrightarrow> x \<in> D * F" 
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avigad
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diff
changeset

287 
apply (frule set_times_mono2) 
19736  288 
prefer 2 
289 
apply force 

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avigad
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diff
changeset

290 
apply assumption 
19736  291 
done 
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avigad
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diff
changeset

292 

56899  293 
lemma set_times_mono3_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> C * D" 
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
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diff
changeset

294 
apply (frule set_times_mono3) 
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avigad
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diff
changeset

295 
apply auto 
19736  296 
done 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

297 

63473  298 
lemma set_times_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C" 
299 
for a x :: "'a::comm_monoid_mult" 

16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
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diff
changeset

300 
apply (frule set_times_mono4) 
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
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diff
changeset

301 
apply auto 
19736  302 
done 
16908
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avigad
parents:
diff
changeset

303 

63473  304 
lemma set_one_times [simp]: "1 *o C = C" 
305 
for C :: "'a::comm_monoid_mult set" 

19736  306 
by (auto simp add: elt_set_times_def) 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
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diff
changeset

307 

63473  308 
lemma set_times_plus_distrib: "a *o (b +o C) = (a * b) +o (a *o C)" 
309 
for a b :: "'a::semiring" 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
21404
diff
changeset

310 
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs) 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

311 

63473  312 
lemma set_times_plus_distrib2: "a *o (B + C) = (a *o B) + (a *o C)" 
313 
for a :: "'a::semiring" 

26814
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Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset

314 
apply (auto simp add: set_plus_def elt_set_times_def ring_distribs) 
19736  315 
apply blast 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

316 
apply (rule_tac x = "b + bb" in exI) 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
21404
diff
changeset

317 
apply (auto simp add: ring_distribs) 
19736  318 
done 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

319 

63473  320 
lemma set_times_plus_distrib3: "(a +o C) * D \<subseteq> a *o D + C * D" 
321 
for a :: "'a::semiring" 

322 
apply (auto simp: elt_set_plus_def elt_set_times_def set_times_def set_plus_def ring_distribs) 

16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

323 
apply auto 
19736  324 
done 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

325 

61337  326 
lemmas set_times_plus_distribs = 
19380  327 
set_times_plus_distrib 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
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diff
changeset

328 
set_times_plus_distrib2 
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

329 

63473  330 
lemma set_neg_intro: "a \<in> ( 1) *o C \<Longrightarrow>  a \<in> C" 
331 
for a :: "'a::ring_1" 

19736  332 
by (auto simp add: elt_set_times_def) 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset

333 

63473  334 
lemma set_neg_intro2: "a \<in> C \<Longrightarrow>  a \<in> ( 1) *o C" 
335 
for a :: "'a::ring_1" 

19736  336 
by (auto simp add: elt_set_times_def) 
337 

53596  338 
lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)" 
63473  339 
by (fastforce simp: set_plus_def image_iff) 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

340 

53596  341 
lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)" 
63473  342 
by (fastforce simp: set_times_def image_iff) 
53596  343 

56899  344 
lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)" 
63473  345 
by (simp add: set_plus_image) 
53596  346 

56899  347 
lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)" 
63473  348 
by (simp add: set_times_image) 
53596  349 

64267  350 
lemma set_sum_alt: 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

351 
assumes fin: "finite I" 
64267  352 
shows "sum S I = {sum s I s. \<forall>i\<in>I. s i \<in> S i}" 
353 
(is "_ = ?sum I") 

56899  354 
using fin 
355 
proof induct 

356 
case empty 

357 
then show ?case by simp 

358 
next 

40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

359 
case (insert x F) 
64267  360 
have "sum S (insert x F) = S x + ?sum F" 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

361 
using insert.hyps by auto 
64267  362 
also have "\<dots> = {s x + sum s F s. \<forall> i\<in>insert x F. s i \<in> S i}" 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

363 
unfolding set_plus_def 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

364 
proof safe 
56899  365 
fix y s 
366 
assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i" 

64267  367 
then show "\<exists>s'. y + sum s F = s' x + sum s' F \<and> (\<forall>i\<in>insert x F. s' i \<in> S i)" 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

368 
using insert.hyps 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

369 
by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def) 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

370 
qed auto 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

371 
finally show ?case 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

372 
using insert.hyps by auto 
56899  373 
qed 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

374 

64267  375 
lemma sum_set_cond_linear: 
56899  376 
fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set" 
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47444
diff
changeset

377 
assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A + B)" "P {0}" 
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47444
diff
changeset

378 
and f: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}" 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

379 
assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)" 
64267  380 
shows "f (sum S I) = sum (f \<circ> S) I" 
56899  381 
proof (cases "finite I") 
382 
case True 

383 
from this all show ?thesis 

40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

384 
proof induct 
56899  385 
case empty 
386 
then show ?case by (auto intro!: f) 

387 
next 

40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

388 
case (insert x F) 
64267  389 
from \<open>finite F\<close> \<open>\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)\<close> have "P (sum S F)" 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

390 
by induct auto 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

391 
with insert show ?case 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
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392 
by (simp, subst f) auto 
56899  393 
qed 
394 
next 

395 
case False 

396 
then show ?thesis by (auto intro!: f) 

397 
qed 

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Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
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parents:
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398 

64267  399 
lemma sum_set_linear: 
56899  400 
fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set" 
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Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
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401 
assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}" 
64267  402 
shows "f (sum S I) = sum (f \<circ> S) I" 
403 
using sum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto 

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Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
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diff
changeset

404 

47446  405 
lemma set_times_Un_distrib: 
406 
"A * (B \<union> C) = A * B \<union> A * C" 

407 
"(A \<union> B) * C = A * C \<union> B * C" 

56899  408 
by (auto simp: set_times_def) 
47446  409 

410 
lemma set_times_UNION_distrib: 

56899  411 
"A * UNION I M = (\<Union>i\<in>I. A * M i)" 
412 
"UNION I M * A = (\<Union>i\<in>I. M i * A)" 

413 
by (auto simp: set_times_def) 

47446  414 

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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
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415 
end 