|
|
|
|
|
Quellcode-Bibliothek
© Kompilation durch diese Firma
[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]
Datei:
Set_Algebras.thy
Sprache: Isabelle
|
|
(* Title: HOL/Library/Set_Algebras.thy
Author: Jeremy Avigad
Author: Kevin Donnelly
Author: Florian Haftmann, TUM
*)
section \<open>Algebraic operations on sets\<close>
theory Set_Algebras
imports Main
begin
text \<open>
This library lifts operations like addition and multiplication to sets. It
was designed to support asymptotic calculations. See the comments at the top
of \<^file>\<open>BigO.thy\<close>.
\<close>
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 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"
apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
apply (rule_tac x = "ba + bb" in exI)
apply (auto simp add: ac_simps)
apply (rule_tac x = "aa + a" in exI)
apply (auto simp add: ac_simps)
done
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"
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)
apply (auto simp add: ac_simps)
done
theorem set_plus_rearrange4: "C + (a +o D) = a +o (C + D)"
for a :: "'a::comm_monoid_add"
apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
apply (rule_tac x = "aa + ba" in exI)
apply (auto simp add: ac_simps)
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"
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"
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"
for a x :: "'a::comm_monoid_add"
apply (frule set_plus_mono4)
apply auto
done
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"
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: ac_simps)
done
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"
apply (auto simp add: elt_set_plus_def ac_simps)
apply (subgoal_tac "a = (a + - b) + b")
apply (rule bexI)
apply assumption
apply (auto simp add: ac_simps)
done
lemma set_minus_plus: "a - b \ C \ a \ b +o C"
for a b :: "'a::ab_group_add"
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"
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"
apply (auto simp add: elt_set_times_def set_times_def)
apply (rule_tac x = "ba * bb" in exI)
apply (auto simp add: ac_simps)
apply (rule_tac x = "aa * a" in exI)
apply (auto simp add: ac_simps)
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)
apply (auto simp add: ac_simps)
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)
apply (auto simp add: ac_simps)
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"
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"
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"
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"
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 +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"
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 = (\(x, y). x + y) ` (S \ T)"
by (fastforce simp: set_plus_def image_iff)
lemma set_times_image: "S * T = (\(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 _ "\i. 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 \<open>finite F\<close> \<open>\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)\<close> 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 "\x. 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
¤ Dauer der Verarbeitung: 0.20 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
|
|
|
|
|
