# Theory Set_Algebras

theory Set_Algebras
imports Main
```(*  Title:      HOL/Library/Set_Algebras.thy
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. See the comments at the top
of 🗏‹BigO.thy›.
›

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_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

instance set :: (comm_monoid_mult) comm_monoid_mult

lemma set_plus_intro [intro]: "a ∈ C ⟹ b ∈ D ⟹ a + b ∈ C + D"

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"

lemma set_plus_rearrange: "(a +o C) + (b +o D) = (a + b) +o (C + D)"
apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
apply (rule_tac x = "ba + bb" in exI)
apply (rule_tac x = "aa + a" in exI)
done

lemma set_plus_rearrange2: "a +o (b +o C) = (a + b) +o C"

lemma set_plus_rearrange3: "(a +o B) + C = a +o (B + C)"
apply (auto simp add: elt_set_plus_def set_plus_def)
apply (blast intro: ac_simps)
apply (rule_tac x = "a + aa" in exI)
apply (rule conjI)
apply (rule_tac x = "aa" in bexI)
apply auto
apply (rule_tac x = "ba" in bexI)
done

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

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"

lemma set_plus_mono2 [intro]: "C ⊆ D ⟹ E ⊆ F ⟹ C + E ⊆ D + F"
for C D E F :: "'a::plus set"

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"
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"
apply (subgoal_tac "a +o B ⊆ a +o D")
apply (erule order_trans)
apply (erule set_plus_mono3)
apply (erule set_plus_mono)
done

lemma set_plus_mono_b: "C ⊆ D ⟹ x ∈ a +o C ⟹ x ∈ a +o D"
apply (frule set_plus_mono)
apply auto
done

lemma set_plus_mono2_b: "C ⊆ D ⟹ E ⊆ F ⟹ x ∈ C + E ⟹ x ∈ D + F"
apply (frule set_plus_mono2)
prefer 2
apply force
apply assumption
done

lemma set_plus_mono3_b: "a ∈ C ⟹ x ∈ a +o D ⟹ x ∈ C + D"
apply (frule set_plus_mono3)
apply auto
done

lemma set_plus_mono4_b: "a ∈ C ⟹ x ∈ a +o D ⟹ x ∈ D + C"
apply (frule set_plus_mono4)
apply auto
done

lemma set_zero_plus [simp]: "0 +o C = C"

lemma set_zero_plus2: "0 ∈ A ⟹ B ⊆ A + B"
for A B :: "'a::comm_monoid_add set"
apply (rule_tac x = 0 in bexI)
apply (rule_tac x = x in bexI)
done

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

lemma set_minus_imp_plus: "a - b ∈ C ⟹ a ∈ b +o C"
apply (auto simp add: elt_set_plus_def ac_simps)
apply (subgoal_tac "a = (a + - b) + b")
apply (rule bexI)
apply assumption
done

lemma set_minus_plus: "a - b ∈ C ⟷ a ∈ b +o C"
apply (rule iffI)
apply (rule set_minus_imp_plus)
apply assumption
apply (rule set_plus_imp_minus)
apply assumption
done

lemma set_times_intro [intro]: "a ∈ C ⟹ b ∈ D ⟹ a * b ∈ C * D"

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"

lemma set_times_rearrange: "(a *o C) * (b *o D) = (a * b) *o (C * D)"
for a b :: "'a::comm_monoid_mult"
apply (auto simp add: elt_set_times_def set_times_def)
apply (rule_tac x = "ba * bb" in exI)
apply (rule_tac x = "aa * a" in exI)
done

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"
apply (auto simp add: elt_set_times_def set_times_def)
apply (blast intro: ac_simps)
apply (rule_tac x = "a * aa" in exI)
apply (rule conjI)
apply (rule_tac x = "aa" in bexI)
apply auto
apply (rule_tac x = "ba" in bexI)
done

theorem set_times_rearrange4: "C * (a *o D) = a *o (C * D)"
for a :: "'a::comm_monoid_mult"
apply (auto simp add: elt_set_times_def set_times_def ac_simps)
apply (rule_tac x = "aa * ba" in exI)
done

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"

lemma set_times_mono2 [intro]: "C ⊆ D ⟹ E ⊆ F ⟹ C * E ⊆ D * F"
for C D E F :: "'a::times set"

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"
apply (subgoal_tac "a *o B ⊆ a *o D")
apply (erule order_trans)
apply (erule set_times_mono3)
apply (erule set_times_mono)
done

lemma set_times_mono_b: "C ⊆ D ⟹ x ∈ a *o C ⟹ x ∈ a *o D"
apply (frule set_times_mono)
apply auto
done

lemma set_times_mono2_b: "C ⊆ D ⟹ E ⊆ F ⟹ x ∈ C * E ⟹ x ∈ D * F"
apply (frule set_times_mono2)
prefer 2
apply force
apply assumption
done

lemma set_times_mono3_b: "a ∈ C ⟹ x ∈ a *o D ⟹ x ∈ C * D"
apply (frule set_times_mono3)
apply auto
done

lemma set_times_mono4_b: "a ∈ C ⟹ x ∈ a *o D ⟹ x ∈ D * C"
for a x :: "'a::comm_monoid_mult"
apply (frule set_times_mono4)
apply auto
done

lemma set_one_times [simp]: "1 *o C = C"
for C :: "'a::comm_monoid_mult set"

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"
apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
apply blast
apply (rule_tac x = "b + bb" in exI)
done

lemma set_times_plus_distrib3: "(a +o C) * D ⊆ a *o D + C * D"
for a :: "'a::semiring"
apply (auto simp: elt_set_plus_def elt_set_times_def set_times_def set_plus_def ring_distribs)
apply auto
done

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"

lemma set_neg_intro2: "a ∈ C ⟹ - a ∈ (- 1) *o C"
for a :: "'a::ring_1"

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)"

lemma finite_set_times: "finite s ⟹ finite t ⟹ finite (s * t)"

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

end
```