src/HOL/Library/Set_Algebras.thy
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fewer aliases for toplevel theorem statements;
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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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section \<open>Algebraic operations on sets\<close>
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theory Set_Algebras
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imports Main
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begin
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text \<open>
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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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\<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" 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 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:
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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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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::'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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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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  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::'a::comm_monoid_add) +o C = C"
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  by (auto simp add: elt_set_plus_def)
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lemma set_zero_plus2: "(0::'a::comm_monoid_add) \<in> A \<Longrightarrow> B \<subseteq> A + B"
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  apply (auto simp add: set_plus_def)
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  apply (rule_tac x = 0 in bexI)
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   apply (rule_tac x = x in bexI)
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    apply (auto simp add: ac_simps)
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   182
  done
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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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lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C \<Longrightarrow> a \<in> b +o C"
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  apply (auto simp add: elt_set_plus_def ac_simps)
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  apply (subgoal_tac "a = (a + - b) + b")
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   apply (rule bexI, assumption)
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  apply (auto simp add: ac_simps)
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   192
  done
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lemma set_minus_plus: "(a::'a::ab_group_add) - b \<in> C \<longleftrightarrow> a \<in> b +o C"
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   195
  by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus)
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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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lemma set_times_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_times_def by fast
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   204
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lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C"
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  by (auto simp add: elt_set_times_def)
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lemma set_times_rearrange:
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  "((a::'a::comm_monoid_mult) *o C) * (b *o D) = (a * b) *o (C * D)"
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  apply (auto simp add: elt_set_times_def set_times_def)
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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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   213
  apply (rule_tac x = "aa * a" in exI)
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  apply (auto simp add: ac_simps)
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   215
  done
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lemma set_times_rearrange2:
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  "(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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lemma set_times_rearrange3:
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  "((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)
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   227
   apply (rule_tac x = "aa" in bexI)
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   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)
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   231
  done
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theorem set_times_rearrange4:
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  "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)
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   236
   apply (rule_tac x = "aa * ba" in exI)
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   237
   apply (auto simp add: ac_simps)
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   238
  done
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   239
61337
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   240
lemmas 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 \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D"
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   244
  by (auto simp add: elt_set_times_def)
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   245
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   246
lemma set_times_mono2 [intro]: "(C::'a::times set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
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   247
  by (auto simp add: set_times_def)
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   248
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   249
lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D"
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   250
  by (auto simp add: elt_set_times_def set_times_def)
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   251
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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
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lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D"
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   256
  apply (subgoal_tac "a *o B \<subseteq> a *o D")
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   257
   apply (erule order_trans)
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   258
   apply (erule set_times_mono3)
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  apply (erule set_times_mono)
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   260
  done
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   261
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lemma set_times_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a *o C \<Longrightarrow> x \<in> a *o D"
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  apply (frule set_times_mono)
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   264
  apply auto
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   265
  done
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   266
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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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  apply (frule set_times_mono2)
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   269
   prefer 2
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   270
   apply force
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  apply assumption
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  done
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   273
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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
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   277
  done
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diff changeset
   278
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   279
lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
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   280
  apply (frule set_times_mono4)
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diff changeset
   281
  apply auto
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parents: 19656
diff changeset
   282
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   283
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   284
lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
19736
wenzelm
parents: 19656
diff changeset
   285
  by (auto simp add: elt_set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   286
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   287
lemma set_times_plus_distrib:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   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
parents: 21404
diff changeset
   289
  by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   290
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   291
lemma set_times_plus_distrib2:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   292
  "(a::'a::semiring) *o (B + C) = (a *o B) + (a *o C)"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   293
  apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
19736
wenzelm
parents: 19656
diff changeset
   294
   apply blast
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   295
  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
   296
  apply (auto simp add: ring_distribs)
19736
wenzelm
parents: 19656
diff changeset
   297
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   298
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   299
lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D \<subseteq> a *o D + C * D"
44142
8e27e0177518 avoid warnings about duplicate rules
huffman
parents: 40887
diff changeset
   300
  apply (auto simp add:
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   301
    elt_set_plus_def elt_set_times_def set_times_def
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   302
    set_plus_def ring_distribs)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   303
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   304
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   305
61337
4645502c3c64 fewer aliases for toplevel theorem statements;
wenzelm
parents: 60679
diff changeset
   306
lemmas set_times_plus_distribs =
19380
b808efaa5828 tuned syntax/abbreviations;
wenzelm
parents: 17161
diff changeset
   307
  set_times_plus_distrib
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   308
  set_times_plus_distrib2
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   309
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   310
lemma set_neg_intro: "(a::'a::ring_1) \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
19736
wenzelm
parents: 19656
diff changeset
   311
  by (auto simp add: elt_set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   312
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   313
lemma set_neg_intro2: "(a::'a::ring_1) \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
19736
wenzelm
parents: 19656
diff changeset
   314
  by (auto simp add: elt_set_times_def)
wenzelm
parents: 19656
diff changeset
   315
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   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
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   319
lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   320
  unfolding set_times_def by (fastforce simp: image_iff)
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   321
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   322
lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   323
  unfolding set_plus_image by simp
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   324
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   325
lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   326
  unfolding set_times_image by simp
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   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
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   332
  using fin
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   333
proof induct
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   334
  case empty
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   335
  then show ?case by simp
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   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
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   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
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   343
    fix y s
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   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
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   350
    using insert.hyps by auto
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   351
qed
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   352
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   353
lemma setsum_set_cond_linear:
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   354
  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
   355
  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
   356
    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
   357
  assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
47444
d21c95af2df7 removed "setsum_set", now subsumed by generic setsum
krauss
parents: 47443
diff changeset
   358
  shows "f (setsum S I) = setsum (f \<circ> S) I"
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   359
proof (cases "finite I")
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   360
  case True
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   361
  from this all show ?thesis
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   362
  proof induct
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   363
    case empty
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   364
    then show ?case by (auto intro!: f)
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   365
  next
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   366
    case (insert x F)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   367
    from \<open>finite F\<close> \<open>\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)\<close> have "P (setsum S F)"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   368
      by induct auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   369
    with insert show ?case
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   370
      by (simp, subst f) auto
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   371
  qed
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   372
next
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   373
  case False
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   374
  then show ?thesis by (auto intro!: f)
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   375
qed
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   376
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   377
lemma setsum_set_linear:
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   378
  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
   379
  assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
47444
d21c95af2df7 removed "setsum_set", now subsumed by generic setsum
krauss
parents: 47443
diff changeset
   380
  shows "f (setsum S I) = setsum (f \<circ> S) I"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   381
  using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   382
47446
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   383
lemma set_times_Un_distrib:
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   384
  "A * (B \<union> C) = A * B \<union> A * C"
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   385
  "(A \<union> B) * C = A * C \<union> B * C"
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   386
  by (auto simp: set_times_def)
47446
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   387
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   388
lemma set_times_UNION_distrib:
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   389
  "A * UNION I M = (\<Union>i\<in>I. A * M i)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   390
  "UNION I M * A = (\<Union>i\<in>I. M i * A)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   391
  by (auto simp: set_times_def)
47446
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   392
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   393
end