--- a/src/HOL/Library/Set_Algebras.thy Tue Jul 12 22:54:37 2016 +0200
+++ b/src/HOL/Library/Set_Algebras.thy Wed Jul 13 14:28:15 2016 +0200
@@ -1,5 +1,7 @@
(* Title: HOL/Library/Set_Algebras.thy
- Author: Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
+ Author: Jeremy Avigad
+ Author: Kevin Donnelly
+ Author: Florian Haftmann, TUM
*)
section \<open>Algebraic operations on sets\<close>
@@ -11,14 +13,14 @@
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 theory \<open>BigO\<close>.
+ comments at the top of @{file "BigO.thy"}.
\<close>
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}"
+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 ..
@@ -27,8 +29,8 @@
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}"
+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 ..
@@ -37,8 +39,7 @@
instantiation set :: (zero) zero
begin
-definition
- set_zero[simp]: "(0::'a::zero set) = {0}"
+definition set_zero[simp]: "(0::'a::zero set) = {0}"
instance ..
@@ -47,21 +48,20 @@
instantiation set :: (one) one
begin
-definition
- set_one[simp]: "(1::'a::one set) = {1}"
+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_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}"
+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"
+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 standard (force simp add: set_plus_def add.assoc)
@@ -98,19 +98,21 @@
lemma set_plus_intro2 [intro]: "b \<in> C \<Longrightarrow> a + b \<in> 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)"
+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 (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::'a::semigroup_add) +o (b +o C) = (a + b) +o C"
+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::'a::semigroup_add) +o B) + C = a +o (B + C)"
+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)
@@ -121,7 +123,8 @@
apply (auto simp add: ac_simps)
done
-theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = a +o (C + D)"
+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)
@@ -133,13 +136,15 @@
lemma set_plus_mono [intro!]: "C \<subseteq> D \<Longrightarrow> a +o C \<subseteq> a +o D"
by (auto simp add: elt_set_plus_def)
-lemma set_plus_mono2 [intro]: "(C::'a::plus set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
+lemma set_plus_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
+ for C D E F :: "'a::plus set"
by (auto simp add: set_plus_def)
lemma set_plus_mono3 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> C + D"
by (auto simp add: elt_set_plus_def set_plus_def)
-lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) \<in> C \<Longrightarrow> a +o D \<subseteq> D + C"
+lemma set_plus_mono4 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> 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 \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a +o B \<subseteq> C + D"
@@ -166,33 +171,45 @@
apply auto
done
-lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
+lemma set_plus_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
+ for a x :: "'a::comm_monoid_add"
apply (frule set_plus_mono4)
apply auto
done
-lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
+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::comm_monoid_add) \<in> A \<Longrightarrow> B \<subseteq> A + B"
+lemma set_zero_plus2: "0 \<in> A \<Longrightarrow> B \<subseteq> 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::'a::ab_group_add) : b +o C \<Longrightarrow> (a - b) \<in> C"
+lemma set_plus_imp_minus: "a \<in> b +o C \<Longrightarrow> a - b \<in> C"
+ for a b :: "'a::ab_group_add"
by (auto simp add: elt_set_plus_def ac_simps)
-lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C \<Longrightarrow> a \<in> b +o C"
+lemma set_minus_imp_plus: "a - b \<in> C \<Longrightarrow> a \<in> 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, assumption)
- apply (auto simp add: ac_simps)
+ apply (rule bexI)
+ apply assumption
+ apply (auto simp add: ac_simps)
done
-lemma set_minus_plus: "(a::'a::ab_group_add) - b \<in> C \<longleftrightarrow> a \<in> b +o C"
- by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus)
+lemma set_minus_plus: "a - b \<in> C \<longleftrightarrow> a \<in> 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 \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D"
by (auto simp add: set_times_def)
@@ -205,8 +222,8 @@
lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> 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)"
+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)
@@ -214,12 +231,12 @@
apply (auto simp add: ac_simps)
done
-lemma set_times_rearrange2:
- "(a::'a::semigroup_mult) *o (b *o C) = (a * b) *o C"
+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::'a::semigroup_mult) *o B) * C = a *o (B * C)"
+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)
@@ -230,8 +247,8 @@
apply (auto simp add: ac_simps)
done
-theorem set_times_rearrange4:
- "C * ((a::'a::comm_monoid_mult) *o D) = a *o (C * D)"
+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)
@@ -243,13 +260,15 @@
lemma set_times_mono [intro]: "C \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D"
by (auto simp add: elt_set_times_def)
-lemma set_times_mono2 [intro]: "(C::'a::times set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
+lemma set_times_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
+ for C D E F :: "'a::times set"
by (auto simp add: set_times_def)
lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D"
by (auto simp add: elt_set_times_def set_times_def)
-lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C \<Longrightarrow> a *o D \<subseteq> D * C"
+lemma set_times_mono4 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> 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 \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D"
@@ -276,30 +295,31 @@
apply auto
done
-lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
+lemma set_times_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
+ for a x :: "'a::comm_monoid_mult"
apply (frule set_times_mono4)
apply auto
done
-lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
+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::'a::semiring) *o (b +o C) = (a * b) +o (a *o C)"
+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::'a::semiring) *o (B + C) = (a *o B) + (a *o C)"
+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::'a::semiring) +o C) * D \<subseteq> 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)
+lemma set_times_plus_distrib3: "(a +o C) * D \<subseteq> 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
@@ -307,23 +327,25 @@
set_times_plus_distrib
set_times_plus_distrib2
-lemma set_neg_intro: "(a::'a::ring_1) \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
+lemma set_neg_intro: "a \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
+ for a :: "'a::ring_1"
by (auto simp add: elt_set_times_def)
-lemma set_neg_intro2: "(a::'a::ring_1) \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
+lemma set_neg_intro2: "a \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
+ for a :: "'a::ring_1"
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)
+ by (fastforce simp: set_plus_def 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)
+ by (fastforce simp: set_times_def image_iff)
lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)"
- unfolding set_plus_image by simp
+ by (simp add: set_plus_image)
lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)"
- unfolding set_times_image by simp
+ by (simp add: set_times_image)
lemma set_setsum_alt:
assumes fin: "finite I"