Theory Set_Algebras

(*  Title:      HOL/Library/Set_Algebras.thy
    Author:     Jeremy Avigad
    Author:     Kevin Donnelly
    Author:     Florian Haftmann, TUM
*)

section ‹Algebraic operations on sets›

theory Set_Algebras
  imports Main
begin

text ‹
  This library lifts operations like addition and multiplication to sets. It
  was designed to support asymptotic calculations for the now-obsolete BigO theory,
  but has other uses.
›

instantiation set :: (plus) plus
begin

definition plus_set :: "'a::plus set ⇒ 'a set ⇒ 'a set"
  where set_plus_def: "A + B = {c. ∃a∈A. ∃b∈B. c = a + b}"

instance ..

end

instantiation set :: (times) times
begin

definition times_set :: "'a::times set ⇒ 'a set ⇒ 'a set"
  where set_times_def: "A * B = {c. ∃a∈A. ∃b∈B. c = a * b}"

instance ..

end

instantiation set :: (zero) zero
begin

definition set_zero[simp]: "(0::'a::zero set) = {0}"

instance ..

end

instantiation set :: (one) one
begin

definition set_one[simp]: "(1::'a::one set) = {1}"

instance ..

end

definition elt_set_plus :: "'a::plus ⇒ 'a set ⇒ 'a set"  (infixl ‹+o› 70)
  where "a +o B = {c. ∃b∈B. c = a + b}"

definition elt_set_times :: "'a::times ⇒ 'a set ⇒ 'a set"  (infixl ‹*o› 80)
  where "a *o B = {c. ∃b∈B. c = a * b}"

abbreviation (input) elt_set_eq :: "'a ⇒ 'a set ⇒ bool"  (infix ‹=o› 50)
  where "x =o A ≡ x ∈ A"

instance set :: (semigroup_add) semigroup_add
  by standard (force simp add: set_plus_def add.assoc)

instance set :: (ab_semigroup_add) ab_semigroup_add
  by standard (force simp add: set_plus_def add.commute)

instance set :: (monoid_add) monoid_add
  by standard (simp_all add: set_plus_def)

instance set :: (comm_monoid_add) comm_monoid_add
  by standard (simp_all add: set_plus_def)

instance set :: (semigroup_mult) semigroup_mult
  by standard (force simp add: set_times_def mult.assoc)

instance set :: (ab_semigroup_mult) ab_semigroup_mult
  by standard (force simp add: set_times_def mult.commute)

instance set :: (monoid_mult) monoid_mult
  by standard (simp_all add: set_times_def)

instance set :: (comm_monoid_mult) comm_monoid_mult
  by standard (simp_all add: set_times_def)

lemma sumset_empty [simp]: "A + {} = {}" "{} + A = {}"
  by (auto simp: set_plus_def)

lemma Un_set_plus: "(A ∪ B) + C = (A+C) ∪ (B+C)" and set_plus_Un: "C + (A ∪ B) = (C+A) ∪ (C+B)"
  by (auto simp: set_plus_def)

lemma 
  fixes A :: "'a::comm_monoid_add set"
  shows insert_set_plus: "(insert a A) + B = (A+B) ∪ (((+)a) ` B)" and set_plus_insert: "B + (insert a A) = (B+A) ∪ (((+)a) ` B)"
  using add.commute by (auto simp: set_plus_def)

lemma set_add_0 [simp]:
  fixes A :: "'a::comm_monoid_add set"
  shows "{0} + A = A"
  by (metis comm_monoid_add_class.add_0 set_zero)

lemma set_add_0_right [simp]:
  fixes A :: "'a::comm_monoid_add set"
  shows "A + {0} = A"
  by (metis add.comm_neutral set_zero)

lemma card_plus_sing:
  fixes A :: "'a::ab_group_add set"
  shows "card (A + {a}) = card A"
proof (rule bij_betw_same_card)
  show "bij_betw ((+) (-a)) (A + {a}) A"
    by (fastforce simp: set_plus_def bij_betw_def image_iff)
qed

lemma set_plus_intro [intro]: "a ∈ C ⟹ b ∈ D ⟹ a + b ∈ C + D"
  by (auto simp add: set_plus_def)

lemma set_plus_elim:
  assumes "x ∈ A + B"
  obtains a b where "x = a + b" and "a ∈ A" and "b ∈ B"
  using assms unfolding set_plus_def by fast

lemma set_plus_intro2 [intro]: "b ∈ C ⟹ a + b ∈ a +o C"
  by (auto simp add: elt_set_plus_def)

lemma set_plus_rearrange: "(a +o C) + (b +o D) = (a + b) +o (C + D)"
  for a b :: "'a::comm_monoid_add"
    by (auto simp: elt_set_plus_def set_plus_def; metis group_cancel.add1 group_cancel.add2)

lemma set_plus_rearrange2: "a +o (b +o C) = (a + b) +o C"
  for a b :: "'a::semigroup_add"
  by (auto simp add: elt_set_plus_def add.assoc)

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

theorem set_plus_rearrange4: "C + (a +o D) = a +o (C + D)"
  for a :: "'a::comm_monoid_add"
  by (metis add.commute set_plus_rearrange3)

lemmas set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
  set_plus_rearrange3 set_plus_rearrange4

lemma set_plus_mono [intro!]: "C ⊆ D ⟹ a +o C ⊆ a +o D"
  by (auto simp add: elt_set_plus_def)

lemma set_plus_mono2 [intro]: "C ⊆ D ⟹ E ⊆ F ⟹ C + E ⊆ D + F"
  for C D E F :: "'a::plus set"
  by (auto simp add: set_plus_def)

lemma set_plus_mono3 [intro]: "a ∈ C ⟹ a +o D ⊆ C + D"
  by (auto simp add: elt_set_plus_def set_plus_def)

lemma set_plus_mono4 [intro]: "a ∈ C ⟹ a +o D ⊆ D + C"
  for a :: "'a::comm_monoid_add"
  by (auto simp add: elt_set_plus_def set_plus_def ac_simps)

lemma set_plus_mono5: "a ∈ C ⟹ B ⊆ D ⟹ a +o B ⊆ C + D"
  using order_subst2 by blast

lemma set_plus_mono_b: "C ⊆ D ⟹ x ∈ a +o C ⟹ x ∈ a +o D"
  using set_plus_mono by blast

lemma set_zero_plus [simp]: "0 +o C = C"
  for C :: "'a::comm_monoid_add set"
  by (auto simp add: elt_set_plus_def)

lemma set_zero_plus2: "0 ∈ A ⟹ B ⊆ A + B"
  for A B :: "'a::comm_monoid_add set"
  using set_plus_intro by fastforce

lemma set_plus_imp_minus: "a ∈ b +o C ⟹ a - b ∈ C"
  for a b :: "'a::ab_group_add"
  by (auto simp add: elt_set_plus_def ac_simps)

lemma set_minus_imp_plus: "a - b ∈ C ⟹ a ∈ b +o C"
  for a b :: "'a::ab_group_add"
  by (metis add.commute diff_add_cancel set_plus_intro2)

lemma set_minus_plus: "a - b ∈ C ⟷ a ∈ b +o C"
  for a b :: "'a::ab_group_add"
  by (meson set_minus_imp_plus set_plus_imp_minus)

lemma set_times_intro [intro]: "a ∈ C ⟹ b ∈ D ⟹ a * b ∈ C * D"
  by (auto simp add: set_times_def)

lemma set_times_elim:
  assumes "x ∈ A * B"
  obtains a b where "x = a * b" and "a ∈ A" and "b ∈ B"
  using assms unfolding set_times_def by fast

lemma set_times_intro2 [intro!]: "b ∈ C ⟹ a * b ∈ a *o C"
  by (auto simp add: elt_set_times_def)

lemma set_times_rearrange: "(a *o C) * (b *o D) = (a * b) *o (C * D)"
  for a b :: "'a::comm_monoid_mult"
  by (auto simp add: elt_set_times_def set_times_def; metis mult.assoc mult.left_commute)

lemma set_times_rearrange2: "a *o (b *o C) = (a * b) *o C"
  for a b :: "'a::semigroup_mult"
  by (auto simp add: elt_set_times_def mult.assoc)

lemma set_times_rearrange3: "(a *o B) * C = a *o (B * C)"
  for a :: "'a::semigroup_mult"
  by (auto simp add: elt_set_times_def set_times_def; metis mult.assoc)

theorem set_times_rearrange4: "C * (a *o D) = a *o (C * D)"
  for a :: "'a::comm_monoid_mult"
  by (metis mult.commute set_times_rearrange3)

lemmas set_times_rearranges = set_times_rearrange set_times_rearrange2
  set_times_rearrange3 set_times_rearrange4

lemma set_times_mono [intro]: "C ⊆ D ⟹ a *o C ⊆ a *o D"
  by (auto simp add: elt_set_times_def)

lemma set_times_mono2 [intro]: "C ⊆ D ⟹ E ⊆ F ⟹ C * E ⊆ D * F"
  for C D E F :: "'a::times set"
  by (auto simp add: set_times_def)

lemma set_times_mono3 [intro]: "a ∈ C ⟹ a *o D ⊆ C * D"
  by (auto simp add: elt_set_times_def set_times_def)

lemma set_times_mono4 [intro]: "a ∈ C ⟹ a *o D ⊆ D * C"
  for a :: "'a::comm_monoid_mult"
  by (auto simp add: elt_set_times_def set_times_def ac_simps)

lemma set_times_mono5: "a ∈ C ⟹ B ⊆ D ⟹ a *o B ⊆ C * D"
  by (meson dual_order.trans set_times_mono set_times_mono3)

lemma set_one_times [simp]: "1 *o C = C"
  for C :: "'a::comm_monoid_mult set"
  by (auto simp add: elt_set_times_def)

lemma set_times_plus_distrib: "a *o (b +o C) = (a * b) +o (a *o C)"
  for a b :: "'a::semiring"
  by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)

lemma set_times_plus_distrib2: "a *o (B + C) = (a *o B) + (a *o C)"
  for a :: "'a::semiring"
  by (auto simp: set_plus_def elt_set_times_def; metis distrib_left)

lemma set_times_plus_distrib3: "(a +o C) * D ⊆ a *o D + C * D"
  for a :: "'a::semiring"
  using distrib_right 
  by (fastforce simp add: elt_set_plus_def elt_set_times_def set_times_def set_plus_def)

lemmas set_times_plus_distribs =
  set_times_plus_distrib
  set_times_plus_distrib2

lemma set_neg_intro: "a ∈ (- 1) *o C ⟹ - a ∈ C"
  for a :: "'a::ring_1"
  by (auto simp add: elt_set_times_def)

lemma set_neg_intro2: "a ∈ C ⟹ - a ∈ (- 1) *o C"
  for a :: "'a::ring_1"
  by (auto simp add: elt_set_times_def)

lemma set_plus_image: "S + T = (λ(x, y). x + y) ` (S × T)"
  by (fastforce simp: set_plus_def image_iff)

lemma set_times_image: "S * T = (λ(x, y). x * y) ` (S × T)"
  by (fastforce simp: set_times_def image_iff)

lemma finite_set_plus: "finite s ⟹ finite t ⟹ finite (s + t)"
  by (simp add: set_plus_image)

lemma finite_set_times: "finite s ⟹ finite t ⟹ finite (s * t)"
  by (simp add: set_times_image)

lemma set_sum_alt:
  assumes fin: "finite I"
  shows "sum S I = {sum s I |s. ∀i∈I. s i ∈ S i}"
    (is "_ = ?sum I")
  using fin
proof induct
  case empty
  then show ?case by simp
next
  case (insert x F)
  have "sum S (insert x F) = S x + ?sum F"
    using insert.hyps by auto
  also have "… = {s x + sum s F |s. ∀ i∈insert x F. s i ∈ S i}"
    unfolding set_plus_def
  proof safe
    fix y s
    assume "y ∈ S x" "∀i∈F. s i ∈ S i"
    then show "∃s'. y + sum s F = s' x + sum s' F ∧ (∀i∈insert x F. s' i ∈ S i)"
      using insert.hyps
      by (intro exI[of _ "λi. if i ∈ F then s i else y"]) (auto simp add: set_plus_def)
  qed auto
  finally show ?case
    using insert.hyps by auto
qed

lemma sum_set_cond_linear:
  fixes f :: "'a::comm_monoid_add set ⇒ 'b::comm_monoid_add set"
  assumes [intro!]: "⋀A B. P A  ⟹ P B  ⟹ P (A + B)" "P {0}"
    and f: "⋀A B. P A  ⟹ P B ⟹ f (A + B) = f A + f B" "f {0} = {0}"
  assumes all: "⋀i. i ∈ I ⟹ P (S i)"
  shows "f (sum S I) = sum (f ∘ S) I"
proof (cases "finite I")
  case True
  from this all show ?thesis
  proof induct
    case empty
    then show ?case by (auto intro!: f)
  next
    case (insert x F)
    from ‹finite F› ‹⋀i. i ∈ insert x F ⟹ P (S i)› have "P (sum S F)"
      by induct auto
    with insert show ?case
      by (simp, subst f) auto
  qed
next
  case False
  then show ?thesis by (auto intro!: f)
qed

lemma sum_set_linear:
  fixes f :: "'a::comm_monoid_add set ⇒ 'b::comm_monoid_add set"
  assumes "⋀A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
  shows "f (sum S I) = sum (f ∘ S) I"
  using sum_set_cond_linear[of "λx. True" f I S] assms by auto

lemma set_times_Un_distrib:
  "A * (B ∪ C) = A * B ∪ A * C"
  "(A ∪ B) * C = A * C ∪ B * C"
  by (auto simp: set_times_def)

lemma set_times_UNION_distrib:
  "A * ⋃(M ` I) = (⋃i∈I. A * M i)"
  "⋃(M ` I) * A = (⋃i∈I. M i * A)"
  by (auto simp: set_times_def)

end