author | desharna |
Mon, 13 Jun 2022 20:02:00 +0200 | |
changeset 75560 | aeb797356de0 |
parent 75450 | f16d83de3e4a |
child 77003 | ab905b5bb206 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Set_Algebras.thy |
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Author: Jeremy Avigad |
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Author: Kevin Donnelly |
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Author: Florian Haftmann, TUM |
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*) |
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section \<open>Algebraic operations on sets\<close> |
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theory Set_Algebras |
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imports Main |
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begin |
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text \<open> |
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This library lifts operations like addition and multiplication to sets. It |
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was designed to support asymptotic calculations. See the comments at the top |
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of \<^file>\<open>BigO.thy\<close>. |
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\<close> |
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instantiation set :: (plus) plus |
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begin |
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definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" |
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where set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}" |
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instance .. |
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end |
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instantiation set :: (times) times |
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begin |
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definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" |
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where set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}" |
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instance .. |
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end |
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instantiation set :: (zero) zero |
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begin |
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definition set_zero[simp]: "(0::'a::zero set) = {0}" |
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instance .. |
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end |
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instantiation set :: (one) one |
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begin |
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definition set_one[simp]: "(1::'a::one set) = {1}" |
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instance .. |
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end |
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definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "+o" 70) |
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where "a +o B = {c. \<exists>b\<in>B. c = a + b}" |
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definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "*o" 80) |
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where "a *o B = {c. \<exists>b\<in>B. c = a * b}" |
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abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infix "=o" 50) |
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where "x =o A \<equiv> x \<in> A" |
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instance set :: (semigroup_add) semigroup_add |
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by standard (force simp add: set_plus_def add.assoc) |
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instance set :: (ab_semigroup_add) ab_semigroup_add |
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by standard (force simp add: set_plus_def add.commute) |
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instance set :: (monoid_add) monoid_add |
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by standard (simp_all add: set_plus_def) |
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instance set :: (comm_monoid_add) comm_monoid_add |
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by standard (simp_all add: set_plus_def) |
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instance set :: (semigroup_mult) semigroup_mult |
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by standard (force simp add: set_times_def mult.assoc) |
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instance set :: (ab_semigroup_mult) ab_semigroup_mult |
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by standard (force simp add: set_times_def mult.commute) |
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instance set :: (monoid_mult) monoid_mult |
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by standard (simp_all add: set_times_def) |
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instance set :: (comm_monoid_mult) comm_monoid_mult |
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by standard (simp_all add: set_times_def) |
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lemma sumset_empty [simp]: "A + {} = {}" "{} + A = {}" |
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by (auto simp: set_plus_def) |
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lemma Un_set_plus: "(A \<union> B) + C = (A+C) \<union> (B+C)" and set_plus_Un: "C + (A \<union> B) = (C+A) \<union> (C+B)" |
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by (auto simp: set_plus_def) |
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lemma |
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fixes A :: "'a::comm_monoid_add set" |
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shows insert_set_plus: "(insert a A) + B = (A+B) \<union> (((+)a) ` B)" and set_plus_insert: "B + (insert a A) = (B+A) \<union> (((+)a) ` B)" |
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using add.commute by (auto simp: set_plus_def) |
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lemma set_add_0 [simp]: |
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shows "{0} + A = A" |
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by (metis comm_monoid_add_class.add_0 set_zero) |
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lemma set_add_0_right [simp]: |
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shows "A + {0} = A" |
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by (metis add.comm_neutral set_zero) |
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lemma card_plus_sing: |
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fixes A :: "'a::ab_group_add set" |
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shows "card (A + {a}) = card A" |
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proof (rule bij_betw_same_card) |
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show "bij_betw ((+) (-a)) (A + {a}) A" |
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by (fastforce simp: set_plus_def bij_betw_def image_iff) |
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qed |
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lemma set_plus_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a + b \<in> C + D" |
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by (auto simp add: set_plus_def) |
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lemma set_plus_elim: |
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assumes "x \<in> A + B" |
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obtains a b where "x = a + b" and "a \<in> A" and "b \<in> B" |
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using assms unfolding set_plus_def by fast |
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||
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lemma set_plus_intro2 [intro]: "b \<in> C \<Longrightarrow> a + b \<in> a +o C" |
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by (auto simp add: elt_set_plus_def) |
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lemma set_plus_rearrange: "(a +o C) + (b +o D) = (a + b) +o (C + D)" |
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for a b :: "'a::comm_monoid_add" |
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by (auto simp: elt_set_plus_def set_plus_def; metis group_cancel.add1 group_cancel.add2) |
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lemma set_plus_rearrange2: "a +o (b +o C) = (a + b) +o C" |
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for a b :: "'a::semigroup_add" |
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by (auto simp add: elt_set_plus_def add.assoc) |
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lemma set_plus_rearrange3: "(a +o B) + C = a +o (B + C)" |
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for a :: "'a::semigroup_add" |
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by (auto simp add: elt_set_plus_def set_plus_def; metis add.assoc) |
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theorem set_plus_rearrange4: "C + (a +o D) = a +o (C + D)" |
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for a :: "'a::comm_monoid_add" |
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by (metis add.commute set_plus_rearrange3) |
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lemmas set_plus_rearranges = set_plus_rearrange set_plus_rearrange2 |
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changeset
|
147 |
set_plus_rearrange3 set_plus_rearrange4 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
148 |
|
56899 | 149 |
lemma set_plus_mono [intro!]: "C \<subseteq> D \<Longrightarrow> a +o C \<subseteq> a +o D" |
19736 | 150 |
by (auto simp add: elt_set_plus_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
151 |
|
63473 | 152 |
lemma set_plus_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F" |
153 |
for C D E F :: "'a::plus set" |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
154 |
by (auto simp add: set_plus_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
155 |
|
56899 | 156 |
lemma set_plus_mono3 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> C + D" |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
157 |
by (auto simp add: elt_set_plus_def set_plus_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
158 |
|
63473 | 159 |
lemma set_plus_mono4 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> D + C" |
160 |
for a :: "'a::comm_monoid_add" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
161 |
by (auto simp add: elt_set_plus_def set_plus_def ac_simps) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
162 |
|
56899 | 163 |
lemma set_plus_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a +o B \<subseteq> C + D" |
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
164 |
using order_subst2 by blast |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
165 |
|
56899 | 166 |
lemma set_plus_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a +o C \<Longrightarrow> x \<in> a +o D" |
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
167 |
using set_plus_mono by blast |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
168 |
|
63473 | 169 |
lemma set_zero_plus [simp]: "0 +o C = C" |
170 |
for C :: "'a::comm_monoid_add set" |
|
19736 | 171 |
by (auto simp add: elt_set_plus_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
172 |
|
63473 | 173 |
lemma set_zero_plus2: "0 \<in> A \<Longrightarrow> B \<subseteq> A + B" |
174 |
for A B :: "'a::comm_monoid_add set" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
175 |
using set_plus_intro by fastforce |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
176 |
|
63473 | 177 |
lemma set_plus_imp_minus: "a \<in> b +o C \<Longrightarrow> a - b \<in> C" |
178 |
for a b :: "'a::ab_group_add" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
179 |
by (auto simp add: elt_set_plus_def ac_simps) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
180 |
|
63473 | 181 |
lemma set_minus_imp_plus: "a - b \<in> C \<Longrightarrow> a \<in> b +o C" |
182 |
for a b :: "'a::ab_group_add" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
183 |
by (metis add.commute diff_add_cancel set_plus_intro2) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
184 |
|
63473 | 185 |
lemma set_minus_plus: "a - b \<in> C \<longleftrightarrow> a \<in> b +o C" |
186 |
for a b :: "'a::ab_group_add" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
187 |
by (meson set_minus_imp_plus set_plus_imp_minus) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
188 |
|
56899 | 189 |
lemma set_times_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D" |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
190 |
by (auto simp add: set_times_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
191 |
|
53596 | 192 |
lemma set_times_elim: |
193 |
assumes "x \<in> A * B" |
|
194 |
obtains a b where "x = a * b" and "a \<in> A" and "b \<in> B" |
|
195 |
using assms unfolding set_times_def by fast |
|
196 |
||
56899 | 197 |
lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C" |
19736 | 198 |
by (auto simp add: elt_set_times_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
199 |
|
63473 | 200 |
lemma set_times_rearrange: "(a *o C) * (b *o D) = (a * b) *o (C * D)" |
201 |
for a b :: "'a::comm_monoid_mult" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
202 |
by (auto simp add: elt_set_times_def set_times_def; metis mult.assoc mult.left_commute) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
203 |
|
63473 | 204 |
lemma set_times_rearrange2: "a *o (b *o C) = (a * b) *o C" |
205 |
for a b :: "'a::semigroup_mult" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
56899
diff
changeset
|
206 |
by (auto simp add: elt_set_times_def mult.assoc) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
207 |
|
63473 | 208 |
lemma set_times_rearrange3: "(a *o B) * C = a *o (B * C)" |
209 |
for a :: "'a::semigroup_mult" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
210 |
by (auto simp add: elt_set_times_def set_times_def; metis mult.assoc) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
211 |
|
63473 | 212 |
theorem set_times_rearrange4: "C * (a *o D) = a *o (C * D)" |
213 |
for a :: "'a::comm_monoid_mult" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
214 |
by (metis mult.commute set_times_rearrange3) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
215 |
|
61337 | 216 |
lemmas set_times_rearranges = set_times_rearrange set_times_rearrange2 |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
217 |
set_times_rearrange3 set_times_rearrange4 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
218 |
|
56899 | 219 |
lemma set_times_mono [intro]: "C \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D" |
19736 | 220 |
by (auto simp add: elt_set_times_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
221 |
|
63473 | 222 |
lemma set_times_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F" |
223 |
for C D E F :: "'a::times set" |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
224 |
by (auto simp add: set_times_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
225 |
|
56899 | 226 |
lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D" |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
227 |
by (auto simp add: elt_set_times_def set_times_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
228 |
|
63473 | 229 |
lemma set_times_mono4 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> D * C" |
230 |
for a :: "'a::comm_monoid_mult" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
231 |
by (auto simp add: elt_set_times_def set_times_def ac_simps) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
232 |
|
56899 | 233 |
lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D" |
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
234 |
by (meson dual_order.trans set_times_mono set_times_mono3) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
235 |
|
63473 | 236 |
lemma set_one_times [simp]: "1 *o C = C" |
237 |
for C :: "'a::comm_monoid_mult set" |
|
19736 | 238 |
by (auto simp add: elt_set_times_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
239 |
|
63473 | 240 |
lemma set_times_plus_distrib: "a *o (b +o C) = (a * b) +o (a *o C)" |
241 |
for a b :: "'a::semiring" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
21404
diff
changeset
|
242 |
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
243 |
|
63473 | 244 |
lemma set_times_plus_distrib2: "a *o (B + C) = (a *o B) + (a *o C)" |
245 |
for a :: "'a::semiring" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
246 |
by (auto simp: set_plus_def elt_set_times_def; metis distrib_left) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
247 |
|
63473 | 248 |
lemma set_times_plus_distrib3: "(a +o C) * D \<subseteq> a *o D + C * D" |
249 |
for a :: "'a::semiring" |
|
75450
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
250 |
using distrib_right |
f16d83de3e4a
Tidied up some super-messy proofs
paulson <lp15@cam.ac.uk>
parents:
69313
diff
changeset
|
251 |
by (fastforce simp add: elt_set_plus_def elt_set_times_def set_times_def set_plus_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
252 |
|
61337 | 253 |
lemmas set_times_plus_distribs = |
19380 | 254 |
set_times_plus_distrib |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
255 |
set_times_plus_distrib2 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
256 |
|
63473 | 257 |
lemma set_neg_intro: "a \<in> (- 1) *o C \<Longrightarrow> - a \<in> C" |
258 |
for a :: "'a::ring_1" |
|
19736 | 259 |
by (auto simp add: elt_set_times_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
260 |
|
63473 | 261 |
lemma set_neg_intro2: "a \<in> C \<Longrightarrow> - a \<in> (- 1) *o C" |
262 |
for a :: "'a::ring_1" |
|
19736 | 263 |
by (auto simp add: elt_set_times_def) |
264 |
||
53596 | 265 |
lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)" |
63473 | 266 |
by (fastforce simp: set_plus_def image_iff) |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
267 |
|
53596 | 268 |
lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)" |
63473 | 269 |
by (fastforce simp: set_times_def image_iff) |
53596 | 270 |
|
56899 | 271 |
lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)" |
63473 | 272 |
by (simp add: set_plus_image) |
53596 | 273 |
|
56899 | 274 |
lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)" |
63473 | 275 |
by (simp add: set_times_image) |
53596 | 276 |
|
64267 | 277 |
lemma set_sum_alt: |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
278 |
assumes fin: "finite I" |
64267 | 279 |
shows "sum S I = {sum s I |s. \<forall>i\<in>I. s i \<in> S i}" |
280 |
(is "_ = ?sum I") |
|
56899 | 281 |
using fin |
282 |
proof induct |
|
283 |
case empty |
|
284 |
then show ?case by simp |
|
285 |
next |
|
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
286 |
case (insert x F) |
64267 | 287 |
have "sum S (insert x F) = S x + ?sum F" |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
288 |
using insert.hyps by auto |
64267 | 289 |
also have "\<dots> = {s x + sum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}" |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
290 |
unfolding set_plus_def |
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
291 |
proof safe |
56899 | 292 |
fix y s |
293 |
assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i" |
|
64267 | 294 |
then show "\<exists>s'. y + sum s F = s' x + sum s' F \<and> (\<forall>i\<in>insert x F. s' i \<in> S i)" |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
295 |
using insert.hyps |
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
296 |
by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def) |
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
297 |
qed auto |
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
298 |
finally show ?case |
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
299 |
using insert.hyps by auto |
56899 | 300 |
qed |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
301 |
|
64267 | 302 |
lemma sum_set_cond_linear: |
56899 | 303 |
fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set" |
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47444
diff
changeset
|
304 |
assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A + B)" "P {0}" |
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47444
diff
changeset
|
305 |
and f: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}" |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
306 |
assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)" |
64267 | 307 |
shows "f (sum S I) = sum (f \<circ> S) I" |
56899 | 308 |
proof (cases "finite I") |
309 |
case True |
|
310 |
from this all show ?thesis |
|
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
311 |
proof induct |
56899 | 312 |
case empty |
313 |
then show ?case by (auto intro!: f) |
|
314 |
next |
|
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
315 |
case (insert x F) |
64267 | 316 |
from \<open>finite F\<close> \<open>\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)\<close> have "P (sum S F)" |
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
317 |
by induct auto |
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
318 |
with insert show ?case |
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
319 |
by (simp, subst f) auto |
56899 | 320 |
qed |
321 |
next |
|
322 |
case False |
|
323 |
then show ?thesis by (auto intro!: f) |
|
324 |
qed |
|
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
325 |
|
64267 | 326 |
lemma sum_set_linear: |
56899 | 327 |
fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set" |
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47444
diff
changeset
|
328 |
assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}" |
64267 | 329 |
shows "f (sum S I) = sum (f \<circ> S) I" |
330 |
using sum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto |
|
40887
ee8d0548c148
Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents:
39302
diff
changeset
|
331 |
|
47446 | 332 |
lemma set_times_Un_distrib: |
333 |
"A * (B \<union> C) = A * B \<union> A * C" |
|
334 |
"(A \<union> B) * C = A * C \<union> B * C" |
|
56899 | 335 |
by (auto simp: set_times_def) |
47446 | 336 |
|
337 |
lemma set_times_UNION_distrib: |
|
69313 | 338 |
"A * \<Union>(M ` I) = (\<Union>i\<in>I. A * M i)" |
339 |
"\<Union>(M ` I) * A = (\<Union>i\<in>I. M i * A)" |
|
56899 | 340 |
by (auto simp: set_times_def) |
47446 | 341 |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
342 |
end |