author | huffman |
Thu, 04 Dec 2008 16:28:09 -0800 | |
changeset 28988 | 13d6f120992b |
parent 27368 | 9f90ac19e32b |
child 29233 | ce6d35a0bed6 |
permissions | -rw-r--r-- |
16932 | 1 |
(* Title: HOL/Library/SetsAndFunctions.thy |
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ID: $Id$ |
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Author: Jeremy Avigad and Kevin Donnelly |
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*) |
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header {* Operations on sets and functions *} |
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theory SetsAndFunctions |
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imports Plain |
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begin |
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text {* |
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This library lifts operations like addition and muliplication to sets and |
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functions of appropriate types. It was designed to support asymptotic |
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calculations. See the comments at the top of theory @{text BigO}. |
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*} |
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subsection {* Basic definitions *} |
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definition |
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set_plus :: "('a::plus) set => 'a set => 'a set" (infixl "\<oplus>" 65) where |
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"A \<oplus> B == {c. EX a:A. EX b:B. c = a + b}" |
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instantiation "fun" :: (type, plus) plus |
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begin |
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definition |
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func_plus: "f + g == (%x. f x + g x)" |
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||
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instance .. |
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end |
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definition |
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set_times :: "('a::times) set => 'a set => 'a set" (infixl "\<otimes>" 70) where |
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"A \<otimes> B == {c. EX a:A. EX b:B. c = a * b}" |
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instantiation "fun" :: (type, times) times |
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begin |
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definition |
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func_times: "f * g == (%x. f x * g x)" |
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instance .. |
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end |
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instantiation "fun" :: (type, zero) zero |
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begin |
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definition |
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func_zero: "0::(('a::type) => ('b::zero)) == %x. 0" |
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instance .. |
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end |
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instantiation "fun" :: (type, one) one |
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begin |
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definition |
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func_one: "1::(('a::type) => ('b::one)) == %x. 1" |
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instance .. |
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end |
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definition |
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elt_set_plus :: "'a::plus => 'a set => 'a set" (infixl "+o" 70) where |
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"a +o B = {c. EX b:B. c = a + b}" |
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definition |
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elt_set_times :: "'a::times => 'a set => 'a set" (infixl "*o" 80) where |
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"a *o B = {c. EX b:B. c = a * b}" |
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abbreviation (input) |
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elt_set_eq :: "'a => 'a set => bool" (infix "=o" 50) where |
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"x =o A == x : A" |
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80 |
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instance "fun" :: (type,semigroup_add)semigroup_add |
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by default (auto simp add: func_plus add_assoc) |
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instance "fun" :: (type,comm_monoid_add)comm_monoid_add |
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by default (auto simp add: func_zero func_plus add_ac) |
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instance "fun" :: (type,ab_group_add)ab_group_add |
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apply default |
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apply (simp add: fun_Compl_def func_plus func_zero) |
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apply (simp add: fun_Compl_def func_plus fun_diff_def diff_minus) |
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done |
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instance "fun" :: (type,semigroup_mult)semigroup_mult |
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apply default |
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apply (auto simp add: func_times mult_assoc) |
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done |
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instance "fun" :: (type,comm_monoid_mult)comm_monoid_mult |
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apply default |
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apply (auto simp add: func_one func_times mult_ac) |
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done |
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102 |
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instance "fun" :: (type,comm_ring_1)comm_ring_1 |
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apply default |
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apply (auto simp add: func_plus func_times fun_Compl_def fun_diff_def ext |
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func_one func_zero ring_simps) |
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apply (drule fun_cong) |
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apply simp |
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done |
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110 |
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interpretation set_semigroup_add: semigroup_add ["op \<oplus> :: ('a::semigroup_add) set => 'a set => 'a set"] |
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apply default |
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apply (unfold set_plus_def) |
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apply (force simp add: add_assoc) |
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done |
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116 |
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interpretation set_semigroup_mult: semigroup_mult ["op \<otimes> :: ('a::semigroup_mult) set => 'a set => 'a set"] |
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apply default |
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apply (unfold set_times_def) |
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apply (force simp add: mult_assoc) |
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done |
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interpretation set_comm_monoid_add: comm_monoid_add ["{0}" "op \<oplus> :: ('a::comm_monoid_add) set => 'a set => 'a set"] |
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apply default |
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apply (unfold set_plus_def) |
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apply (force simp add: add_ac) |
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apply force |
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done |
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129 |
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interpretation set_comm_monoid_mult: comm_monoid_mult ["{1}" "op \<otimes> :: ('a::comm_monoid_mult) set => 'a set => 'a set"] |
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apply default |
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apply (unfold set_times_def) |
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apply (force simp add: mult_ac) |
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apply force |
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done |
136 |
||
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137 |
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subsection {* Basic properties *} |
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139 |
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lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D" |
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by (auto simp add: set_plus_def) |
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lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C" |
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by (auto simp add: elt_set_plus_def) |
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lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) \<oplus> |
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(b +o D) = (a + b) +o (C \<oplus> D)" |
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apply (auto simp add: elt_set_plus_def set_plus_def add_ac) |
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apply (rule_tac x = "ba + bb" in exI) |
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apply (auto simp add: add_ac) |
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apply (rule_tac x = "aa + a" in exI) |
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apply (auto simp add: add_ac) |
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done |
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154 |
|
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lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = |
156 |
(a + b) +o C" |
|
157 |
by (auto simp add: elt_set_plus_def add_assoc) |
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lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) \<oplus> C = |
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a +o (B \<oplus> C)" |
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apply (auto simp add: elt_set_plus_def set_plus_def) |
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apply (blast intro: add_ac) |
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apply (rule_tac x = "a + aa" in exI) |
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apply (rule conjI) |
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apply (rule_tac x = "aa" in bexI) |
166 |
apply auto |
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apply (rule_tac x = "ba" in bexI) |
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apply (auto simp add: add_ac) |
169 |
done |
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theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) = |
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a +o (C \<oplus> D)" |
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173 |
apply (auto intro!: subsetI simp add: elt_set_plus_def set_plus_def add_ac) |
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apply (rule_tac x = "aa + ba" in exI) |
175 |
apply (auto simp add: add_ac) |
|
176 |
done |
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theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2 |
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set_plus_rearrange3 set_plus_rearrange4 |
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avigad
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|
181 |
lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D" |
19736 | 182 |
by (auto simp add: elt_set_plus_def) |
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avigad
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diff
changeset
|
183 |
|
19736 | 184 |
lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==> |
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185 |
C \<oplus> E <= D \<oplus> F" |
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|
186 |
by (auto simp add: set_plus_def) |
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avigad
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changeset
|
187 |
|
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|
188 |
lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> D" |
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berghofe
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changeset
|
189 |
by (auto simp add: elt_set_plus_def set_plus_def) |
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avigad
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diff
changeset
|
190 |
|
19736 | 191 |
lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==> |
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|
192 |
a +o D <= D \<oplus> C" |
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changeset
|
193 |
by (auto simp add: elt_set_plus_def set_plus_def add_ac) |
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avigad
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diff
changeset
|
194 |
|
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changeset
|
195 |
lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C \<oplus> D" |
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avigad
parents:
diff
changeset
|
196 |
apply (subgoal_tac "a +o B <= a +o D") |
19736 | 197 |
apply (erule order_trans) |
198 |
apply (erule set_plus_mono3) |
|
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avigad
parents:
diff
changeset
|
199 |
apply (erule set_plus_mono) |
19736 | 200 |
done |
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avigad
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diff
changeset
|
201 |
|
19736 | 202 |
lemma set_plus_mono_b: "C <= D ==> x : a +o C |
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avigad
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|
203 |
==> x : a +o D" |
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avigad
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changeset
|
204 |
apply (frule set_plus_mono) |
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avigad
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diff
changeset
|
205 |
apply auto |
19736 | 206 |
done |
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avigad
parents:
diff
changeset
|
207 |
|
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changeset
|
208 |
lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C \<oplus> E ==> |
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|
209 |
x : D \<oplus> F" |
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avigad
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diff
changeset
|
210 |
apply (frule set_plus_mono2) |
19736 | 211 |
prefer 2 |
212 |
apply force |
|
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avigad
parents:
diff
changeset
|
213 |
apply assumption |
19736 | 214 |
done |
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avigad
parents:
diff
changeset
|
215 |
|
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changeset
|
216 |
lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C \<oplus> D" |
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avigad
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diff
changeset
|
217 |
apply (frule set_plus_mono3) |
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avigad
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diff
changeset
|
218 |
apply auto |
19736 | 219 |
done |
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avigad
parents:
diff
changeset
|
220 |
|
19736 | 221 |
lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==> |
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|
222 |
x : a +o D ==> x : D \<oplus> C" |
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avigad
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diff
changeset
|
223 |
apply (frule set_plus_mono4) |
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avigad
parents:
diff
changeset
|
224 |
apply auto |
19736 | 225 |
done |
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avigad
parents:
diff
changeset
|
226 |
|
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avigad
parents:
diff
changeset
|
227 |
lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C" |
19736 | 228 |
by (auto simp add: elt_set_plus_def) |
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avigad
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diff
changeset
|
229 |
|
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changeset
|
230 |
lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A \<oplus> B" |
b3e8d5ec721d
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changeset
|
231 |
apply (auto intro!: subsetI simp add: set_plus_def) |
16908
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avigad
parents:
diff
changeset
|
232 |
apply (rule_tac x = 0 in bexI) |
19736 | 233 |
apply (rule_tac x = x in bexI) |
234 |
apply (auto simp add: add_ac) |
|
235 |
done |
|
16908
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avigad
parents:
diff
changeset
|
236 |
|
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avigad
parents:
diff
changeset
|
237 |
lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C" |
19736 | 238 |
by (auto simp add: elt_set_plus_def add_ac diff_minus) |
16908
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avigad
parents:
diff
changeset
|
239 |
|
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
240 |
lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
241 |
apply (auto simp add: elt_set_plus_def add_ac diff_minus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
242 |
apply (subgoal_tac "a = (a + - b) + b") |
19736 | 243 |
apply (rule bexI, assumption, assumption) |
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
244 |
apply (auto simp add: add_ac) |
19736 | 245 |
done |
16908
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avigad
parents:
diff
changeset
|
246 |
|
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
247 |
lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)" |
19736 | 248 |
by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus, |
16908
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avigad
parents:
diff
changeset
|
249 |
assumption) |
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
250 |
|
26814
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berghofe
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changeset
|
251 |
lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C \<otimes> D" |
b3e8d5ec721d
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parents:
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changeset
|
252 |
by (auto simp add: set_times_def) |
16908
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avigad
parents:
diff
changeset
|
253 |
|
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
254 |
lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C" |
19736 | 255 |
by (auto simp add: elt_set_times_def) |
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
256 |
|
26814
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Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
257 |
lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) \<otimes> |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
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changeset
|
258 |
(b *o D) = (a * b) *o (C \<otimes> D)" |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
259 |
apply (auto simp add: elt_set_times_def set_times_def) |
19736 | 260 |
apply (rule_tac x = "ba * bb" in exI) |
261 |
apply (auto simp add: mult_ac) |
|
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
262 |
apply (rule_tac x = "aa * a" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
263 |
apply (auto simp add: mult_ac) |
19736 | 264 |
done |
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
265 |
|
19736 | 266 |
lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) = |
267 |
(a * b) *o C" |
|
268 |
by (auto simp add: elt_set_times_def mult_assoc) |
|
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
269 |
|
26814
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berghofe
parents:
25764
diff
changeset
|
270 |
lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C = |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
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changeset
|
271 |
a *o (B \<otimes> C)" |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
272 |
apply (auto simp add: elt_set_times_def set_times_def) |
19736 | 273 |
apply (blast intro: mult_ac) |
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
274 |
apply (rule_tac x = "a * aa" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
275 |
apply (rule conjI) |
19736 | 276 |
apply (rule_tac x = "aa" in bexI) |
277 |
apply auto |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
278 |
apply (rule_tac x = "ba" in bexI) |
19736 | 279 |
apply (auto simp add: mult_ac) |
280 |
done |
|
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
281 |
|
26814
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Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
282 |
theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) = |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
283 |
a *o (C \<otimes> D)" |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
284 |
apply (auto intro!: subsetI 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
|
285 |
mult_ac) |
19736 | 286 |
apply (rule_tac x = "aa * ba" in exI) |
287 |
apply (auto simp add: mult_ac) |
|
288 |
done |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
289 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
290 |
theorems set_times_rearranges = set_times_rearrange set_times_rearrange2 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
291 |
set_times_rearrange3 set_times_rearrange4 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
292 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
293 |
lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D" |
19736 | 294 |
by (auto simp add: elt_set_times_def) |
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
295 |
|
19736 | 296 |
lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==> |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
297 |
C \<otimes> E <= D \<otimes> F" |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
298 |
by (auto simp add: set_times_def) |
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
299 |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
300 |
lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D" |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
301 |
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
|
302 |
|
19736 | 303 |
lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==> |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
304 |
a *o D <= D \<otimes> C" |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
305 |
by (auto simp add: elt_set_times_def set_times_def mult_ac) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
306 |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
307 |
lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
308 |
apply (subgoal_tac "a *o B <= a *o D") |
19736 | 309 |
apply (erule order_trans) |
310 |
apply (erule set_times_mono3) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
311 |
apply (erule set_times_mono) |
19736 | 312 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
313 |
|
19736 | 314 |
lemma set_times_mono_b: "C <= D ==> x : a *o C |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
315 |
==> x : a *o D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
316 |
apply (frule set_times_mono) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
317 |
apply auto |
19736 | 318 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
319 |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
320 |
lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==> |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
321 |
x : D \<otimes> F" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
322 |
apply (frule set_times_mono2) |
19736 | 323 |
prefer 2 |
324 |
apply force |
|
16908
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avigad
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325 |
apply assumption |
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done |
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avigad
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diff
changeset
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327 |
|
26814
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berghofe
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328 |
lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C \<otimes> D" |
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329 |
apply (frule set_times_mono3) |
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330 |
apply auto |
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done |
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avigad
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changeset
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332 |
|
19736 | 333 |
lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==> |
26814
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berghofe
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changeset
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334 |
x : a *o D ==> x : D \<otimes> C" |
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avigad
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changeset
|
335 |
apply (frule set_times_mono4) |
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avigad
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336 |
apply auto |
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done |
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avigad
parents:
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changeset
|
338 |
|
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avigad
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changeset
|
339 |
lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C" |
19736 | 340 |
by (auto simp add: elt_set_times_def) |
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avigad
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changeset
|
341 |
|
19736 | 342 |
lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)= |
343 |
(a * b) +o (a *o C)" |
|
23477
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tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
21404
diff
changeset
|
344 |
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs) |
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avigad
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changeset
|
345 |
|
26814
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berghofe
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diff
changeset
|
346 |
lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) = |
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berghofe
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changeset
|
347 |
(a *o B) \<oplus> (a *o C)" |
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berghofe
parents:
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diff
changeset
|
348 |
apply (auto simp add: set_plus_def elt_set_times_def ring_distribs) |
19736 | 349 |
apply blast |
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avigad
parents:
diff
changeset
|
350 |
apply (rule_tac x = "b + bb" in exI) |
23477
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tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
21404
diff
changeset
|
351 |
apply (auto simp add: ring_distribs) |
19736 | 352 |
done |
16908
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avigad
parents:
diff
changeset
|
353 |
|
26814
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Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
354 |
lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <= |
b3e8d5ec721d
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berghofe
parents:
25764
diff
changeset
|
355 |
a *o D \<oplus> C \<otimes> D" |
19736 | 356 |
apply (auto intro!: subsetI simp add: |
26814
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Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
357 |
elt_set_plus_def elt_set_times_def set_times_def |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25764
diff
changeset
|
358 |
set_plus_def ring_distribs) |
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avigad
parents:
diff
changeset
|
359 |
apply auto |
19736 | 360 |
done |
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
361 |
|
19380 | 362 |
theorems set_times_plus_distribs = |
363 |
set_times_plus_distrib |
|
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
364 |
set_times_plus_distrib2 |
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
365 |
|
19736 | 366 |
lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==> |
367 |
- a : C" |
|
368 |
by (auto simp add: elt_set_times_def) |
|
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
369 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
370 |
lemma set_neg_intro2: "(a::'a::ring_1) : C ==> |
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
371 |
- a : (- 1) *o C" |
19736 | 372 |
by (auto simp add: elt_set_times_def) |
373 |
||
16908
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Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
374 |
end |