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38622  1 
(* 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 
30738  8 
imports Main 
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begin 
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text {* 
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This library lifts operations like addition and muliplication 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 
25594  56 

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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 : C ==> b : D ==> a + b : C + D" 
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by (auto simp add: set_plus_def) 
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lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C" 
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by (auto simp add: elt_set_plus_def) 
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lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) + 
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(b +o D) = (a + b) +o (C + D)" 
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apply (auto simp add: elt_set_plus_def set_plus_def add_ac) 
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apply (rule_tac x = "ba + bb" in exI) 
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apply (auto simp add: add_ac) 
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apply (rule_tac x = "aa + a" in exI) 
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apply (auto simp add: add_ac) 
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done 
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lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = 
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(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 = 
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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: add_ac) 
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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: add_ac) 
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done 

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theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = 
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a +o (C + D)" 
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apply (auto simp add: elt_set_plus_def set_plus_def add_ac) 
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apply (rule_tac x = "aa + ba" in exI) 
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apply (auto simp add: add_ac) 

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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 <= D ==> a +o C <= 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) <= D ==> E <= F ==> 
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C + E <= D + F" 
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by (auto simp add: set_plus_def) 
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lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= 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) : C ==> 
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a +o D <= D + C" 
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by (auto simp add: elt_set_plus_def set_plus_def add_ac) 
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lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D" 
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146 
apply (subgoal_tac "a +o B <= a +o D") 
19736  147 
apply (erule order_trans) 
148 
apply (erule set_plus_mono3) 

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149 
apply (erule set_plus_mono) 
19736  150 
done 
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151 

19736  152 
lemma set_plus_mono_b: "C <= D ==> x : a +o C 
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153 
==> x : a +o D" 
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154 
apply (frule set_plus_mono) 
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155 
apply auto 
19736  156 
done 
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157 

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158 
lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==> 
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159 
x : D + F" 
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160 
apply (frule set_plus_mono2) 
19736  161 
prefer 2 
162 
apply force 

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163 
apply assumption 
19736  164 
done 
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165 

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166 
lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D" 
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167 
apply (frule set_plus_mono3) 
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168 
apply auto 
19736  169 
done 
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170 

19736  171 
lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==> 
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x : a +o D ==> x : D + C" 
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173 
apply (frule set_plus_mono4) 
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174 
apply auto 
19736  175 
done 
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176 

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177 
lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C" 
19736  178 
by (auto simp add: elt_set_plus_def) 
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179 

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180 
lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B" 
44142  181 
apply (auto simp add: set_plus_def) 
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182 
apply (rule_tac x = 0 in bexI) 
19736  183 
apply (rule_tac x = x in bexI) 
184 
apply (auto simp add: add_ac) 

185 
done 

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186 

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187 
lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a  b) : C" 
19736  188 
by (auto simp add: elt_set_plus_def add_ac diff_minus) 
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189 

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190 
lemma set_minus_imp_plus: "(a::'a::ab_group_add)  b : C ==> a : b +o C" 
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191 
apply (auto simp add: elt_set_plus_def add_ac diff_minus) 
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192 
apply (subgoal_tac "a = (a +  b) + b") 
19736  193 
apply (rule bexI, assumption, assumption) 
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194 
apply (auto simp add: add_ac) 
19736  195 
done 
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196 

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197 
lemma set_minus_plus: "((a::'a::ab_group_add)  b : C) = (a : b +o C)" 
19736  198 
by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus, 
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199 
assumption) 
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200 

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201 
lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D" 
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202 
by (auto simp add: set_times_def) 
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203 

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204 
lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C" 
19736  205 
by (auto simp add: elt_set_times_def) 
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206 

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207 
lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) * 
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208 
(b *o D) = (a * b) *o (C * D)" 
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209 
apply (auto simp add: elt_set_times_def set_times_def) 
19736  210 
apply (rule_tac x = "ba * bb" in exI) 
211 
apply (auto simp add: mult_ac) 

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212 
apply (rule_tac x = "aa * a" in exI) 
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213 
apply (auto simp add: mult_ac) 
19736  214 
done 
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215 

19736  216 
lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) = 
217 
(a * b) *o C" 

218 
by (auto simp add: elt_set_times_def mult_assoc) 

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219 

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220 
lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C = 
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221 
a *o (B * C)" 
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222 
apply (auto simp add: elt_set_times_def set_times_def) 
19736  223 
apply (blast intro: mult_ac) 
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224 
apply (rule_tac x = "a * aa" in exI) 
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225 
apply (rule conjI) 
19736  226 
apply (rule_tac x = "aa" in bexI) 
227 
apply auto 

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228 
apply (rule_tac x = "ba" in bexI) 
19736  229 
apply (auto simp add: mult_ac) 
230 
done 

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231 

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232 
theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) = 
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233 
a *o (C * D)" 
44142  234 
apply (auto simp add: elt_set_times_def set_times_def 
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235 
mult_ac) 
19736  236 
apply (rule_tac x = "aa * ba" in exI) 
237 
apply (auto simp add: mult_ac) 

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 

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

19736  246 
lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==> 
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247 
C * E <= D * F" 
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248 
by (auto simp add: set_times_def) 
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249 

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250 
lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D" 
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251 
by (auto simp add: elt_set_times_def set_times_def) 
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252 

19736  253 
lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==> 
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254 
a *o D <= D * C" 
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255 
by (auto simp add: elt_set_times_def set_times_def mult_ac) 
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avigad
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diff
changeset

256 

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257 
lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D" 
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258 
apply (subgoal_tac "a *o B <= a *o D") 
19736  259 
apply (erule order_trans) 
260 
apply (erule set_times_mono3) 

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diff
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261 
apply (erule set_times_mono) 
19736  262 
done 
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263 

19736  264 
lemma set_times_mono_b: "C <= D ==> x : a *o C 
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265 
==> x : a *o D" 
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266 
apply (frule set_times_mono) 
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267 
apply auto 
19736  268 
done 
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avigad
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diff
changeset

269 

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270 
lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==> 
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271 
x : D * F" 
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272 
apply (frule set_times_mono2) 
19736  273 
prefer 2 
274 
apply force 

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275 
apply assumption 
19736  276 
done 
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changeset

277 

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278 
lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D" 
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279 
apply (frule set_times_mono3) 
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avigad
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280 
apply auto 
19736  281 
done 
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changeset

282 

19736  283 
lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==> 
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284 
x : a *o D ==> x : D * C" 
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changeset

285 
apply (frule set_times_mono4) 
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avigad
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diff
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286 
apply auto 
19736  287 
done 
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avigad
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diff
changeset

288 

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289 
lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C" 
19736  290 
by (auto simp add: elt_set_times_def) 
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diff
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291 

19736  292 
lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)= 
293 
(a * b) +o (a *o C)" 

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294 
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs) 
16908
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diff
changeset

295 

47445
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296 
lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) = 
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297 
(a *o B) + (a *o C)" 
26814
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298 
apply (auto simp add: set_plus_def elt_set_times_def ring_distribs) 
19736  299 
apply blast 
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avigad
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300 
apply (rule_tac x = "b + bb" in exI) 
23477
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changeset

301 
apply (auto simp add: ring_distribs) 
19736  302 
done 
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avigad
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diff
changeset

303 

47445
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304 
lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <= 
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305 
a *o D + C * D" 
44142  306 
apply (auto simp add: 
26814
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changeset

307 
elt_set_plus_def elt_set_times_def set_times_def 
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set_plus_def ring_distribs) 
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apply auto 
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done 
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theorems set_times_plus_distribs = 
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set_times_plus_distrib 

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set_times_plus_distrib2 
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lemma set_neg_intro: "(a::'a::ring_1) : ( 1) *o C ==> 
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 a : C" 

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by (auto simp add: elt_set_times_def) 

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lemma set_neg_intro2: "(a::'a::ring_1) : C ==> 
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 a : ( 1) *o C" 
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by (auto simp add: elt_set_times_def) 
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lemma set_plus_image: 
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fixes S T :: "'n::semigroup_add set" shows "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)" 
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unfolding set_plus_def by (fastforce simp: image_iff) 
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lemma set_setsum_alt: 
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assumes fin: "finite I" 
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shows "setsum S I = {setsum s I s. \<forall>i\<in>I. s i \<in> S i}" 
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(is "_ = ?setsum I") 
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using fin proof induct 
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case (insert x F) 
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have "setsum S (insert x F) = S x + ?setsum F" 
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using insert.hyps by auto 
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also have "...= {s x + setsum s F s. \<forall> i\<in>insert x F. s i \<in> S i}" 
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unfolding set_plus_def 
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proof safe 
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fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i" 
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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)" 
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using insert.hyps 
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by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def) 
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qed auto 
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finally show ?case 
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using insert.hyps by auto 
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qed auto 
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347 

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lemma setsum_set_cond_linear: 
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fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set" 
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assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A + B)" "P {0}" 
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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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assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)" 
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shows "f (setsum S I) = setsum (f \<circ> S) I" 
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proof cases 
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assume "finite I" from this all show ?thesis 
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proof induct 
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case (insert x F) 
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from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum S F)" 
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by induct auto 
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with insert show ?case 
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by (simp, subst f) auto 
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qed (auto intro!: f) 
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qed (auto intro!: f) 
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364 

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lemma setsum_set_linear: 
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366 
fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set" 
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assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}" 
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shows "f (setsum S I) = setsum (f \<circ> S) I" 
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using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto 
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370 

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371 
end 