src/HOL/Library/Set_Algebras.thy
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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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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 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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text {* Legacy syntax: *}
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abbreviation (input) set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<oplus>" 65) where
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  "A \<oplus> B \<equiv> A + B"
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abbreviation (input) set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<otimes>" 70) where
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  "A \<otimes> B \<equiv> A * B"
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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 : C ==> b : D ==> a + b : C \<oplus> 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) \<oplus>
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    (b +o D) = (a + b) +o (C \<oplus> 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) \<oplus> C =
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    a +o (B \<oplus> 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 \<oplus> ((a::'a::comm_monoid_add) +o D) =
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    a +o (C \<oplus> 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 \<oplus> E <= D \<oplus> 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 \<oplus> 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 \<oplus> 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 \<oplus> D"
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  apply (subgoal_tac "a +o B <= 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 <= D ==> x : a +o C
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    ==> x : 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 <= D ==> E <= F ==> x : C \<oplus> E ==>
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    x : D \<oplus> 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 : C ==> x : a +o D ==> x : C \<oplus> 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 ==>
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    x : a +o D ==> x : D \<oplus> 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) : A ==> B <= A \<oplus> 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: add_ac)
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  done
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lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
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  by (auto simp add: elt_set_plus_def add_ac diff_minus)
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lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
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  apply (auto simp add: elt_set_plus_def add_ac diff_minus)
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  apply (subgoal_tac "a = (a + - b) + b")
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   apply (rule bexI, assumption, assumption)
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  apply (auto simp add: add_ac)
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   203
  done
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lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
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  by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
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    assumption)
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lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C \<otimes> D"
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  by (auto simp add: set_times_def)
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lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
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  by (auto simp add: elt_set_times_def)
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lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) \<otimes>
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    (b *o D) = (a * b) *o (C \<otimes> 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: mult_ac)
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  apply (rule_tac x = "aa * a" in exI)
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  apply (auto simp add: mult_ac)
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   222
  done
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lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
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    (a * b) *o C"
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  by (auto simp add: elt_set_times_def mult_assoc)
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lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C =
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    a *o (B \<otimes> C)"
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  apply (auto simp add: elt_set_times_def set_times_def)
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   apply (blast intro: mult_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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   235
    apply auto
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  apply (rule_tac x = "ba" in bexI)
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   apply (auto simp add: mult_ac)
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   238
  done
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   239
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   240
theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
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    a *o (C \<otimes> D)"
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  apply (auto simp add: elt_set_times_def set_times_def
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    mult_ac)
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   apply (rule_tac x = "aa * ba" in exI)
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   245
   apply (auto simp add: mult_ac)
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   246
  done
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theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
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   249
  set_times_rearrange3 set_times_rearrange4
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lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
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  by (auto simp add: elt_set_times_def)
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lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
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    C \<otimes> E <= D \<otimes> F"
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   256
  by (auto simp add: set_times_def)
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diff changeset
   257
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   258
lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D"
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   259
  by (auto simp add: elt_set_times_def set_times_def)
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lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
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    a *o D <= D \<otimes> C"
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   263
  by (auto simp add: elt_set_times_def set_times_def mult_ac)
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diff changeset
   264
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   265
lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D"
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  apply (subgoal_tac "a *o B <= a *o D")
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   apply (erule order_trans)
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   268
   apply (erule set_times_mono3)
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  apply (erule set_times_mono)
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   270
  done
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lemma set_times_mono_b: "C <= D ==> x : a *o C
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    ==> x : a *o D"
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  apply (frule set_times_mono)
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  apply auto
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   276
  done
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lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==>
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diff changeset
   279
    x : D \<otimes> F"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   280
  apply (frule set_times_mono2)
19736
wenzelm
parents: 19656
diff changeset
   281
   prefer 2
wenzelm
parents: 19656
diff changeset
   282
   apply force
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   283
  apply assumption
19736
wenzelm
parents: 19656
diff changeset
   284
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   285
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   286
lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C \<otimes> D"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   287
  apply (frule set_times_mono3)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   288
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   289
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   290
19736
wenzelm
parents: 19656
diff changeset
   291
lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   292
    x : a *o D ==> x : D \<otimes> C"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   293
  apply (frule set_times_mono4)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   294
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   295
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   296
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   297
lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
19736
wenzelm
parents: 19656
diff changeset
   298
  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
   299
19736
wenzelm
parents: 19656
diff changeset
   300
lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
wenzelm
parents: 19656
diff changeset
   301
    (a * b) +o (a *o C)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 21404
diff changeset
   302
  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
   303
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   304
lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) =
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   305
    (a *o B) \<oplus> (a *o C)"
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   306
  apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
19736
wenzelm
parents: 19656
diff changeset
   307
   apply blast
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   308
  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
   309
  apply (auto simp add: ring_distribs)
19736
wenzelm
parents: 19656
diff changeset
   310
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   311
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   312
lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   313
    a *o D \<oplus> C \<otimes> D"
44142
8e27e0177518 avoid warnings about duplicate rules
huffman
parents: 40887
diff changeset
   314
  apply (auto simp add:
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   315
    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
   316
    set_plus_def ring_distribs)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   317
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   318
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   319
19380
b808efaa5828 tuned syntax/abbreviations;
wenzelm
parents: 17161
diff changeset
   320
theorems set_times_plus_distribs =
b808efaa5828 tuned syntax/abbreviations;
wenzelm
parents: 17161
diff changeset
   321
  set_times_plus_distrib
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   322
  set_times_plus_distrib2
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   323
19736
wenzelm
parents: 19656
diff changeset
   324
lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
wenzelm
parents: 19656
diff changeset
   325
    - a : C"
wenzelm
parents: 19656
diff changeset
   326
  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
   327
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   328
lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   329
    - a : (- 1) *o C"
19736
wenzelm
parents: 19656
diff changeset
   330
  by (auto simp add: elt_set_times_def)
wenzelm
parents: 19656
diff changeset
   331
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   332
lemma set_plus_image:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   333
  fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44142
diff changeset
   334
  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
   335
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
   336
text {* Legacy syntax: *}
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
   337
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
   338
abbreviation (input) setsum_set :: "('b \<Rightarrow> ('a::comm_monoid_add) set) \<Rightarrow> 'b set \<Rightarrow> 'a set" where
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
   339
   "setsum_set == setsum"
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
   340
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   341
lemma set_setsum_alt:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   342
  assumes fin: "finite I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   343
  shows "setsum_set S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   344
    (is "_ = ?setsum I")
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   345
using fin proof induct
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   346
  case (insert x F)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   347
  have "setsum_set S (insert x F) = S x \<oplus> ?setsum F"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   348
    using insert.hyps by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   349
  also have "...= {s x + setsum s F |s. \<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
   350
    unfolding set_plus_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   351
  proof safe
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   352
    fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   353
    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
   354
      using insert.hyps
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   355
      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
   356
  qed auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   357
  finally show ?case
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   358
    using insert.hyps by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   359
qed auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   360
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   361
lemma setsum_set_cond_linear:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   362
  fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   363
  assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A \<oplus> B)" "P {0}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   364
    and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   365
  assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   366
  shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   367
proof cases
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   368
  assume "finite I" from this all show ?thesis
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   369
  proof induct
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   370
    case (insert x F)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   371
    from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum_set S F)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   372
      by induct auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   373
    with insert show ?case
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   374
      by (simp, subst f) auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   375
  qed (auto intro!: f)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   376
qed (auto intro!: f)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   377
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   378
lemma setsum_set_linear:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   379
  fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   380
  assumes "\<And>A B. f(A) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   381
  shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   382
  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
   383
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   384
end