author  haftmann 
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permissions  rwrr 
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(* Title: HOL/Library/Set_Algebras.thy 
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Author: Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM 

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*) 
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header {* Algebraic operations on sets *} 
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38622  7 
theory Set_Algebras 
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imports Main 
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begin 
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text {* 
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This library lifts operations like addition and multiplication to 
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sets. It was designed to support asymptotic calculations. See the 
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comments at the top of theory @{text BigO}. 

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*} 
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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" where 
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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" where 
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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 
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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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definition 
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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) where 
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"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) where 
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"a *o B = {c. \<exists>b\<in>B. c = a * b}" 

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

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instance set :: (semigroup_add) semigroup_add 
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by default (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 default (force simp add: set_plus_def add.commute) 
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instance set :: (monoid_add) monoid_add 
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by default (simp_all add: set_plus_def) 
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instance set :: (comm_monoid_add) comm_monoid_add 
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by default (simp_all add: set_plus_def) 
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instance set :: (semigroup_mult) semigroup_mult 
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by default (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 default (force simp add: set_times_def mult.commute) 
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instance set :: (monoid_mult) monoid_mult 
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by default (simp_all add: set_times_def) 
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instance set :: (comm_monoid_mult) comm_monoid_mult 
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by default (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: 
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"((a::'a::comm_monoid_add) +o C) + (b +o D) = (a + b) +o (C + D)" 

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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::'a::semigroup_add) +o (b +o C) = (a + b) +o C" 
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by (auto simp add: elt_set_plus_def add.assoc) 
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lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = a +o (B + C)" 
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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::'a::comm_monoid_add) +o D) = a +o (C + D)" 
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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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theorems 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::'a::plus set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F" 
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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::'a::comm_monoid_add) \<in> C \<Longrightarrow> a +o D \<subseteq> D + C" 
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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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168 

56899  169 
lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C" 
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170 
apply (frule set_plus_mono4) 
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171 
apply auto 
19736  172 
done 
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173 

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174 
lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C" 
19736  175 
by (auto simp add: elt_set_plus_def) 
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176 

56899  177 
lemma set_zero_plus2: "(0::'a::comm_monoid_add) \<in> A \<Longrightarrow> B \<subseteq> A + B" 
44142  178 
apply (auto simp add: set_plus_def) 
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179 
apply (rule_tac x = 0 in bexI) 
19736  180 
apply (rule_tac x = x in bexI) 
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181 
apply (auto simp add: ac_simps) 
19736  182 
done 
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183 

56899  184 
lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C \<Longrightarrow> (a  b) \<in> C" 
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185 
by (auto simp add: elt_set_plus_def ac_simps) 
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186 

56899  187 
lemma set_minus_imp_plus: "(a::'a::ab_group_add)  b : C \<Longrightarrow> a \<in> b +o C" 
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188 
apply (auto simp add: elt_set_plus_def ac_simps) 
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189 
apply (subgoal_tac "a = (a +  b) + b") 
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190 
apply (rule bexI, assumption) 
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191 
apply (auto simp add: ac_simps) 
19736  192 
done 
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193 

56899  194 
lemma set_minus_plus: "(a::'a::ab_group_add)  b \<in> C \<longleftrightarrow> a \<in> b +o C" 
195 
by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus) 

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196 

56899  197 
lemma set_times_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D" 
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198 
by (auto simp add: set_times_def) 
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199 

53596  200 
lemma set_times_elim: 
201 
assumes "x \<in> A * B" 

202 
obtains a b where "x = a * b" and "a \<in> A" and "b \<in> B" 

203 
using assms unfolding set_times_def by fast 

204 

56899  205 
lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C" 
19736  206 
by (auto simp add: elt_set_times_def) 
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207 

56899  208 
lemma set_times_rearrange: 
209 
"((a::'a::comm_monoid_mult) *o C) * (b *o D) = (a * b) *o (C * D)" 

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210 
apply (auto simp add: elt_set_times_def set_times_def) 
19736  211 
apply (rule_tac x = "ba * bb" in exI) 
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212 
apply (auto simp add: ac_simps) 
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213 
apply (rule_tac x = "aa * a" in exI) 
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214 
apply (auto simp add: ac_simps) 
19736  215 
done 
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216 

56899  217 
lemma set_times_rearrange2: 
218 
"(a::'a::semigroup_mult) *o (b *o C) = (a * b) *o C" 

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219 
by (auto simp add: elt_set_times_def mult.assoc) 
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220 

56899  221 
lemma set_times_rearrange3: 
222 
"((a::'a::semigroup_mult) *o B) * C = a *o (B * C)" 

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223 
apply (auto simp add: elt_set_times_def set_times_def) 
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224 
apply (blast intro: ac_simps) 
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225 
apply (rule_tac x = "a * aa" in exI) 
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226 
apply (rule conjI) 
19736  227 
apply (rule_tac x = "aa" in bexI) 
228 
apply auto 

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229 
apply (rule_tac x = "ba" in bexI) 
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230 
apply (auto simp add: ac_simps) 
19736  231 
done 
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232 

56899  233 
theorem set_times_rearrange4: 
234 
"C * ((a::'a::comm_monoid_mult) *o D) = a *o (C * D)" 

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235 
apply (auto simp add: elt_set_times_def set_times_def ac_simps) 
19736  236 
apply (rule_tac x = "aa * ba" in exI) 
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237 
apply (auto simp add: ac_simps) 
19736  238 
done 
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239 

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240 
theorems set_times_rearranges = set_times_rearrange set_times_rearrange2 
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241 
set_times_rearrange3 set_times_rearrange4 
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242 

56899  243 
lemma set_times_mono [intro]: "C \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D" 
19736  244 
by (auto simp add: elt_set_times_def) 
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245 

56899  246 
lemma set_times_mono2 [intro]: "(C::'a::times set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F" 
26814
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247 
by (auto simp add: set_times_def) 
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248 

56899  249 
lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D" 
26814
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250 
by (auto simp add: elt_set_times_def set_times_def) 
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251 

56899  252 
lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C \<Longrightarrow> a *o D \<subseteq> D * C" 
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253 
by (auto simp add: elt_set_times_def set_times_def ac_simps) 
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254 

56899  255 
lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D" 
256 
apply (subgoal_tac "a *o B \<subseteq> a *o D") 

19736  257 
apply (erule order_trans) 
258 
apply (erule set_times_mono3) 

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259 
apply (erule set_times_mono) 
19736  260 
done 
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261 

56899  262 
lemma set_times_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a *o C \<Longrightarrow> x \<in> a *o D" 
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263 
apply (frule set_times_mono) 
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264 
apply auto 
19736  265 
done 
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avigad
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266 

56899  267 
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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268 
apply (frule set_times_mono2) 
19736  269 
prefer 2 
270 
apply force 

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271 
apply assumption 
19736  272 
done 
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273 

56899  274 
lemma set_times_mono3_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> C * D" 
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275 
apply (frule set_times_mono3) 
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276 
apply auto 
19736  277 
done 
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278 

56899  279 
lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C" 
16908
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diff
changeset

280 
apply (frule set_times_mono4) 
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avigad
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changeset

281 
apply auto 
19736  282 
done 
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avigad
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diff
changeset

283 

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diff
changeset

284 
lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C" 
19736  285 
by (auto simp add: elt_set_times_def) 
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avigad
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286 

56899  287 
lemma set_times_plus_distrib: 
288 
"(a::'a::semiring) *o (b +o C) = (a * b) +o (a *o C)" 

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

289 
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs) 
16908
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avigad
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diff
changeset

290 

56899  291 
lemma set_times_plus_distrib2: 
292 
"(a::'a::semiring) *o (B + C) = (a *o B) + (a *o C)" 

26814
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diff
changeset

293 
apply (auto simp add: set_plus_def elt_set_times_def ring_distribs) 
19736  294 
apply blast 
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
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diff
changeset

295 
apply (rule_tac x = "b + bb" in exI) 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
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21404
diff
changeset

296 
apply (auto simp add: ring_distribs) 
19736  297 
done 
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
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diff
changeset

298 

56899  299 
lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D \<subseteq> a *o D + C * D" 
44142  300 
apply (auto simp add: 
26814
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diff
changeset

301 
elt_set_plus_def elt_set_times_def set_times_def 
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changeset

302 
set_plus_def ring_distribs) 
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diff
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303 
apply auto 
19736  304 
done 
16908
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avigad
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diff
changeset

305 

19380  306 
theorems set_times_plus_distribs = 
307 
set_times_plus_distrib 

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avigad
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diff
changeset

308 
set_times_plus_distrib2 
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avigad
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diff
changeset

309 

56899  310 
lemma set_neg_intro: "(a::'a::ring_1) \<in> ( 1) *o C \<Longrightarrow>  a \<in> C" 
19736  311 
by (auto simp add: elt_set_times_def) 
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avigad
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diff
changeset

312 

56899  313 
lemma set_neg_intro2: "(a::'a::ring_1) \<in> C \<Longrightarrow>  a \<in> ( 1) *o C" 
19736  314 
by (auto simp add: elt_set_times_def) 
315 

53596  316 
lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)" 
44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44142
diff
changeset

317 
unfolding set_plus_def by (fastforce simp: image_iff) 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

318 

53596  319 
lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)" 
320 
unfolding set_times_def by (fastforce simp: image_iff) 

321 

56899  322 
lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)" 
323 
unfolding set_plus_image by simp 

53596  324 

56899  325 
lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)" 
326 
unfolding set_times_image by simp 

53596  327 

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

328 
lemma set_setsum_alt: 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

329 
assumes fin: "finite I" 
47444
d21c95af2df7
removed "setsum_set", now subsumed by generic setsum
krauss
parents:
47443
diff
changeset

330 
shows "setsum S I = {setsum s I s. \<forall>i\<in>I. s i \<in> S i}" 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

331 
(is "_ = ?setsum I") 
56899  332 
using fin 
333 
proof induct 

334 
case empty 

335 
then show ?case by simp 

336 
next 

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

337 
case (insert x F) 
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47444
diff
changeset

338 
have "setsum S (insert x F) = S x + ?setsum F" 
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

339 
using insert.hyps by auto 
56899  340 
also have "\<dots> = {s x + setsum 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

341 
unfolding set_plus_def 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

342 
proof safe 
56899  343 
fix y s 
344 
assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i" 

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

345 
then show "\<exists>s'. y + setsum s F = s' x + setsum s' F \<and> (\<forall>i\<in>insert x F. s' i \<in> S i)" 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

346 
using insert.hyps 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

347 
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

348 
qed auto 
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset

349 
finally show ?case 
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using insert.hyps by auto 
56899  351 
qed 
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352 

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353 
lemma setsum_set_cond_linear: 
56899  354 
fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set" 
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355 
assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A + B)" "P {0}" 
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356 
and f: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}" 
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357 
assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)" 
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358 
shows "f (setsum S I) = setsum (f \<circ> S) I" 
56899  359 
proof (cases "finite I") 
360 
case True 

361 
from this all show ?thesis 

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362 
proof induct 
56899  363 
case empty 
364 
then show ?case by (auto intro!: f) 

365 
next 

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366 
case (insert x F) 
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367 
from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum S F)" 
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368 
by induct auto 
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369 
with insert show ?case 
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370 
by (simp, subst f) auto 
56899  371 
qed 
372 
next 

373 
case False 

374 
then show ?thesis by (auto intro!: f) 

375 
qed 

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376 

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377 
lemma setsum_set_linear: 
56899  378 
fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set" 
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379 
assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}" 
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380 
shows "f (setsum S I) = setsum (f \<circ> S) I" 
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381 
using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto 
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382 

47446  383 
lemma set_times_Un_distrib: 
384 
"A * (B \<union> C) = A * B \<union> A * C" 

385 
"(A \<union> B) * C = A * C \<union> B * C" 

56899  386 
by (auto simp: set_times_def) 
47446  387 

388 
lemma set_times_UNION_distrib: 

56899  389 
"A * UNION I M = (\<Union>i\<in>I. A * M i)" 
390 
"UNION I M * A = (\<Union>i\<in>I. M i * A)" 

391 
by (auto simp: set_times_def) 

47446  392 

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393 
end 