(* 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 \<Rightarrow> 'a set \<Rightarrow> 'a set" where
set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
instance ..
end
instantiation set :: (times) times
begin
definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>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 \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "+o" 70) where
"a +o B = {c. \<exists>b\<in>B. c = a + b}"
definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "*o" 80) where
"a *o B = {c. \<exists>b\<in>B. c = a * b}"
abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infix "=o" 50) where
"x =o A \<equiv> x \<in> 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 \<in> A + B"
obtains a b where "x = a + b" and "a \<in> A" and "b \<in> 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)
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)
apply (subgoal_tac "a = (a + - b) + b")
apply (rule bexI, 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 \<in> A * B"
obtains a b where "x = a * b" and "a \<in> A" and "b \<in> 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 = (\<lambda>(x, y). x + y) ` (S \<times> T)"
unfolding set_plus_def by (fastforce simp: image_iff)
lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> 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. \<forall>i\<in>I. s i \<in> 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. \<forall> i\<in>insert x F. s i \<in> S i}"
unfolding set_plus_def
proof safe
fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
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)"
using insert.hyps
by (intro exI[of _ "\<lambda>i. if i \<in> 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 \<Rightarrow> ('b::comm_monoid_add) set"
assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A + B)" "P {0}"
and f: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"
assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
shows "f (setsum S I) = setsum (f \<circ> S) I"
proof cases
assume "finite I" from this all show ?thesis
proof induct
case (insert x F)
from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> 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 "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
shows "f (setsum S I) = setsum (f \<circ> S) I"
using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
lemma set_times_Un_distrib:
"A * (B \<union> C) = A * B \<union> A * C"
"(A \<union> B) * C = A * C \<union> 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