Theory Set_Algebras

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


header {* Algebraic operations on sets *}

theory Set_Algebras
imports Main
begin

text {*
This library lifts operations like addition and muliplication to
sets. It was designed to support asymptotic calculations. See the
comments at the top of theory @{text BigO}.
*}


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 default (force simp add: set_plus_def add.assoc)

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

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

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

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

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

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

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

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::'a::comm_monoid_add) +o C) +
(b +o D) = (a + b) +o (C + D)"

apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
apply (rule_tac x = "ba + bb" in exI)
apply (auto simp add: add_ac)
apply (rule_tac x = "aa + a" in exI)
apply (auto simp add: add_ac)
done

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

by (auto simp add: elt_set_plus_def add_assoc)

lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C =
a +o (B + C)"

apply (auto simp add: elt_set_plus_def set_plus_def)
apply (blast intro: add_ac)
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)
apply (auto simp add: add_ac)
done

theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) =
a +o (C + D)"

apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
apply (rule_tac x = "aa + ba" in exI)
apply (auto simp add: add_ac)
done

theorems 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::('a::plus) set) <= D ==> E <= F ==>
C + E <= D + F"

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::'a::comm_monoid_add) : C ==>
a +o D <= D + C"

by (auto simp add: elt_set_plus_def set_plus_def add_ac)

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::'a::comm_monoid_add) : C ==>
x : a +o D ==> x : D + C"

apply (frule set_plus_mono4)
apply auto
done

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

lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"
apply (auto simp add: set_plus_def)
apply (rule_tac x = 0 in bexI)
apply (rule_tac x = x in bexI)
apply (auto simp add: add_ac)
done

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

lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
apply (auto simp add: elt_set_plus_def add_ac diff_minus)
apply (subgoal_tac "a = (a + - b) + b")
apply (rule bexI, assumption, assumption)
apply (auto simp add: add_ac)
done

lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
assumption)

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::'a::comm_monoid_mult) *o C) *
(b *o D) = (a * b) *o (C * D)"

apply (auto simp add: elt_set_times_def set_times_def)
apply (rule_tac x = "ba * bb" in exI)
apply (auto simp add: mult_ac)
apply (rule_tac x = "aa * a" in exI)
apply (auto simp add: mult_ac)
done

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

by (auto simp add: elt_set_times_def mult_assoc)

lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C =
a *o (B * C)"

apply (auto simp add: elt_set_times_def set_times_def)
apply (blast intro: mult_ac)
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)
apply (auto simp add: mult_ac)
done

theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) =
a *o (C * D)"

apply (auto simp add: elt_set_times_def set_times_def
mult_ac)
apply (rule_tac x = "aa * ba" in exI)
apply (auto simp add: mult_ac)
done

theorems 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::('a::times) set) <= D ==> E <= F ==>
C * E <= D * F"

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::'a::comm_monoid_mult) : C ==>
a *o D <= D * C"

by (auto simp add: elt_set_times_def set_times_def mult_ac)

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::'a::comm_monoid_mult) : C ==>
x : a *o D ==> x : D * C"

apply (frule set_times_mono4)
apply auto
done

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

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

by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)

lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =
(a *o B) + (a *o C)"

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

lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=
a *o D + C * D"

apply (auto simp add:
elt_set_plus_def elt_set_times_def set_times_def
set_plus_def ring_distribs)
apply auto
done

theorems set_times_plus_distribs =
set_times_plus_distrib
set_times_plus_distrib2

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

by (auto simp add: elt_set_times_def)

lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
- a : (- 1) *o C"

by (auto simp add: elt_set_times_def)

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

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

lemma finite_set_plus:
assumes "finite s" and "finite t" shows "finite (s + t)"
using assms unfolding set_plus_image by simp

lemma finite_set_times:
assumes "finite s" and "finite t" shows "finite (s * t)"
using assms unfolding set_times_image by simp

lemma set_setsum_alt:
assumes fin: "finite I"
shows "setsum S I = {setsum s I |s. ∀i∈I. s i ∈ S i}"
(is "_ = ?setsum I")
using fin proof induct
case (insert x F)
have "setsum S (insert x F) = S x + ?setsum F"
using insert.hyps by auto
also have "...= {s x + setsum 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 + setsum s F = s' x + setsum 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 auto

lemma setsum_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 (setsum S I) = setsum (f o S) I"
proof cases
assume "finite I" from this all show ?thesis
proof induct
case (insert x F)
from `finite F` `!!i. i ∈ insert x F ==> P (S i)` have "P (setsum S F)"
by induct auto
with insert show ?case
by (simp, subst f) auto
qed (auto intro!: f)
qed (auto intro!: f)

lemma setsum_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 (setsum S I) = setsum (f o S) I"
using setsum_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 = UNION I (%i. A * M i)"
"UNION I M * A = UNION I (%i. M i * A)"
by (auto simp: set_times_def)

end