author | nipkow |
Thu, 31 Aug 2017 09:50:11 +0200 | |
changeset 66566 | a14bbbaa628d |
parent 63167 | 0909deb8059b |
child 67091 | 1393c2340eec |
permissions | -rw-r--r-- |
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(* Title: HOL/Algebra/Ring.thy |
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Author: Clemens Ballarin, started 9 December 1996 |
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Copyright: Clemens Ballarin |
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*) |
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theory Ring |
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imports FiniteProduct |
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begin |
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section \<open>The Algebraic Hierarchy of Rings\<close> |
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subsection \<open>Abelian Groups\<close> |
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record 'a ring = "'a monoid" + |
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zero :: 'a ("\<zero>\<index>") |
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add :: "['a, 'a] => 'a" (infixl "\<oplus>\<index>" 65) |
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text \<open>Derived operations.\<close> |
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definition |
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a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80) |
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where "a_inv R = m_inv \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>" |
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definition |
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a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65) |
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where "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> y)" |
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locale abelian_monoid = |
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fixes G (structure) |
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assumes a_comm_monoid: |
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"comm_monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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definition |
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finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where |
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"finsum G = finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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syntax |
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"_finsum" :: "index => idt => 'a set => 'b => 'b" |
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("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10) |
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translations |
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"\<Oplus>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finsum G (%i. b) A" |
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\<comment> \<open>Beware of argument permutation!\<close> |
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locale abelian_group = abelian_monoid + |
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assumes a_comm_group: |
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"comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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subsection \<open>Basic Properties\<close> |
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lemma abelian_monoidI: |
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fixes R (structure) |
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assumes a_closed: |
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"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R" |
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and zero_closed: "\<zero> \<in> carrier R" |
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and a_assoc: |
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"!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==> |
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(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
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and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x" |
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and a_comm: |
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"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x" |
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shows "abelian_monoid R" |
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by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms) |
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lemma abelian_groupI: |
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fixes R (structure) |
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assumes a_closed: |
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"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R" |
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and zero_closed: "zero R \<in> carrier R" |
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and a_assoc: |
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"!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==> |
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(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)" |
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and a_comm: |
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"!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x" |
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and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x" |
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and l_inv_ex: "!!x. x \<in> carrier R ==> EX y : carrier R. y \<oplus> x = \<zero>" |
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shows "abelian_group R" |
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by (auto intro!: abelian_group.intro abelian_monoidI |
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abelian_group_axioms.intro comm_monoidI comm_groupI |
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intro: assms) |
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lemma (in abelian_monoid) a_monoid: |
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"monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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by (rule comm_monoid.axioms, rule a_comm_monoid) |
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lemma (in abelian_group) a_group: |
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"group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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by (simp add: group_def a_monoid) |
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(simp add: comm_group.axioms group.axioms a_comm_group) |
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lemmas monoid_record_simps = partial_object.simps monoid.simps |
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text \<open>Transfer facts from multiplicative structures via interpretation.\<close> |
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sublocale abelian_monoid < |
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add: monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G" |
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and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G" |
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and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G" |
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by (rule a_monoid) auto |
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context abelian_monoid begin |
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lemmas a_closed = add.m_closed |
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lemmas zero_closed = add.one_closed |
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lemmas a_assoc = add.m_assoc |
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lemmas l_zero = add.l_one |
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lemmas r_zero = add.r_one |
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lemmas minus_unique = add.inv_unique |
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end |
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sublocale abelian_monoid < |
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add: comm_monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G" |
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and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G" |
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and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G" |
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and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G" |
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by (rule a_comm_monoid) (auto simp: finsum_def) |
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context abelian_monoid begin |
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lemmas a_comm = add.m_comm |
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lemmas a_lcomm = add.m_lcomm |
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lemmas a_ac = a_assoc a_comm a_lcomm |
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lemmas finsum_empty = add.finprod_empty |
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lemmas finsum_insert = add.finprod_insert |
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lemmas finsum_zero = add.finprod_one |
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lemmas finsum_closed = add.finprod_closed |
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lemmas finsum_Un_Int = add.finprod_Un_Int |
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lemmas finsum_Un_disjoint = add.finprod_Un_disjoint |
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lemmas finsum_addf = add.finprod_multf |
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lemmas finsum_cong' = add.finprod_cong' |
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lemmas finsum_0 = add.finprod_0 |
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lemmas finsum_Suc = add.finprod_Suc |
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lemmas finsum_Suc2 = add.finprod_Suc2 |
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lemmas finsum_add = add.finprod_mult |
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lemmas finsum_infinite = add.finprod_infinite |
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lemmas finsum_cong = add.finprod_cong |
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text \<open>Usually, if this rule causes a failed congruence proof error, |
63167 | 144 |
the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown. |
61382 | 145 |
Adding @{thm [source] Pi_def} to the simpset is often useful.\<close> |
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lemmas finsum_reindex = add.finprod_reindex |
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(* The following would be wrong. Needed is the equivalent of (^) for addition, |
27699 | 150 |
or indeed the canonical embedding from Nat into the monoid. |
151 |
||
27933 | 152 |
lemma finsum_const: |
27699 | 153 |
assumes fin [simp]: "finite A" |
154 |
and a [simp]: "a : carrier G" |
|
155 |
shows "finsum G (%x. a) A = a (^) card A" |
|
156 |
using fin apply induct |
|
157 |
apply force |
|
158 |
apply (subst finsum_insert) |
|
159 |
apply auto |
|
160 |
apply (force simp add: Pi_def) |
|
161 |
apply (subst m_comm) |
|
162 |
apply auto |
|
163 |
done |
|
164 |
*) |
|
165 |
||
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lemmas finsum_singleton = add.finprod_singleton |
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|
168 |
end |
|
169 |
||
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sublocale abelian_group < |
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add: group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G" |
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and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G" |
174 |
and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G" |
|
175 |
and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G" |
|
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by (rule a_group) (auto simp: m_inv_def a_inv_def) |
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|
55926 | 178 |
context abelian_group |
179 |
begin |
|
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|
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lemmas a_inv_closed = add.inv_closed |
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182 |
|
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lemma minus_closed [intro, simp]: |
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"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G" |
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by (simp add: a_minus_def) |
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186 |
|
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lemmas a_l_cancel = add.l_cancel |
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lemmas a_r_cancel = add.r_cancel |
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lemmas l_neg = add.l_inv [simp del] |
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lemmas r_neg = add.r_inv [simp del] |
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lemmas minus_zero = add.inv_one |
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lemmas minus_minus = add.inv_inv |
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lemmas a_inv_inj = add.inv_inj |
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lemmas minus_equality = add.inv_equality |
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195 |
|
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196 |
end |
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197 |
|
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198 |
sublocale abelian_group < |
61605 | 199 |
add: comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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200 |
rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G" |
55926 | 201 |
and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G" |
202 |
and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G" |
|
203 |
and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G" |
|
204 |
and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G" |
|
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205 |
by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def) |
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206 |
|
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207 |
lemmas (in abelian_group) minus_add = add.inv_mult |
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208 |
|
63167 | 209 |
text \<open>Derive an \<open>abelian_group\<close> from a \<open>comm_group\<close>\<close> |
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210 |
|
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211 |
lemma comm_group_abelian_groupI: |
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212 |
fixes G (structure) |
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213 |
assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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214 |
shows "abelian_group G" |
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215 |
proof - |
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216 |
interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" |
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217 |
by (rule cg) |
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218 |
show "abelian_group G" .. |
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219 |
qed |
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220 |
|
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|
61382 | 222 |
subsection \<open>Rings: Basic Definitions\<close> |
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223 |
|
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|
224 |
locale semiring = abelian_monoid R + monoid R for R (structure) + |
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225 |
assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
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==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z" |
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and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
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==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" |
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229 |
and l_null[simp]: "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>" |
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230 |
and r_null[simp]: "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>" |
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|
231 |
|
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232 |
locale ring = abelian_group R + monoid R for R (structure) + |
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|
233 |
assumes "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
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parents:
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changeset
|
234 |
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z" |
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|
235 |
and "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
43b1870b3e61
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|
236 |
==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" |
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237 |
|
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238 |
locale cring = ring + comm_monoid R |
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239 |
|
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240 |
locale "domain" = cring + |
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241 |
assumes one_not_zero [simp]: "\<one> ~= \<zero>" |
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242 |
and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==> |
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243 |
a = \<zero> | b = \<zero>" |
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244 |
|
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245 |
locale field = "domain" + |
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246 |
assumes field_Units: "Units R = carrier R - {\<zero>}" |
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247 |
|
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248 |
|
61382 | 249 |
subsection \<open>Rings\<close> |
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250 |
|
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251 |
lemma ringI: |
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252 |
fixes R (structure) |
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assumes abelian_group: "abelian_group R" |
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254 |
and monoid: "monoid R" |
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255 |
and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
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256 |
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z" |
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257 |
and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
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258 |
==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" |
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259 |
shows "ring R" |
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260 |
by (auto intro: ring.intro |
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|
261 |
abelian_group.axioms ring_axioms.intro assms) |
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262 |
|
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263 |
context ring begin |
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264 |
|
46721 | 265 |
lemma is_abelian_group: "abelian_group R" .. |
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266 |
|
46721 | 267 |
lemma is_monoid: "monoid R" |
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268 |
by (auto intro!: monoidI m_assoc) |
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269 |
|
46721 | 270 |
lemma is_ring: "ring R" |
26202 | 271 |
by (rule ring_axioms) |
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272 |
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273 |
end |
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274 |
|
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275 |
lemmas ring_record_simps = monoid_record_simps ring.simps |
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276 |
|
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277 |
lemma cringI: |
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278 |
fixes R (structure) |
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279 |
assumes abelian_group: "abelian_group R" |
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280 |
and comm_monoid: "comm_monoid R" |
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281 |
and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] |
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282 |
==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z" |
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283 |
shows "cring R" |
23350 | 284 |
proof (intro cring.intro ring.intro) |
285 |
show "ring_axioms R" |
|
63167 | 286 |
\<comment> \<open>Right-distributivity follows from left-distributivity and |
61382 | 287 |
commutativity.\<close> |
23350 | 288 |
proof (rule ring_axioms.intro) |
289 |
fix x y z |
|
290 |
assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R" |
|
291 |
note [simp] = comm_monoid.axioms [OF comm_monoid] |
|
292 |
abelian_group.axioms [OF abelian_group] |
|
293 |
abelian_monoid.a_closed |
|
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294 |
|
23350 | 295 |
from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z" |
296 |
by (simp add: comm_monoid.m_comm [OF comm_monoid.intro]) |
|
297 |
also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr) |
|
298 |
also from R have "... = z \<otimes> x \<oplus> z \<otimes> y" |
|
299 |
by (simp add: comm_monoid.m_comm [OF comm_monoid.intro]) |
|
300 |
finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" . |
|
301 |
qed (rule l_distr) |
|
302 |
qed (auto intro: cring.intro |
|
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27699
diff
changeset
|
303 |
abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
304 |
|
27699 | 305 |
(* |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
306 |
lemma (in cring) is_comm_monoid: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
307 |
"comm_monoid R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
308 |
by (auto intro!: comm_monoidI m_assoc m_comm) |
27699 | 309 |
*) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
310 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
311 |
lemma (in cring) is_cring: |
26202 | 312 |
"cring R" by (rule cring_axioms) |
23350 | 313 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
314 |
|
61382 | 315 |
subsubsection \<open>Normaliser for Rings\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
316 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
317 |
lemma (in abelian_group) r_neg2: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
318 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
319 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
320 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
321 |
then have "(x \<oplus> \<ominus> x) \<oplus> y = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
322 |
by (simp only: r_neg l_zero) |
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
323 |
with G show ?thesis |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
324 |
by (simp add: a_ac) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
325 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
326 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
327 |
lemma (in abelian_group) r_neg1: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
328 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
329 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
330 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
331 |
then have "(\<ominus> x \<oplus> x) \<oplus> y = y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
332 |
by (simp only: l_neg l_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
333 |
with G show ?thesis by (simp add: a_ac) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
334 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
335 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
336 |
context ring begin |
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
337 |
|
61382 | 338 |
text \<open> |
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
339 |
The following proofs are from Jacobson, Basic Algebra I, pp.~88--89. |
61382 | 340 |
\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
341 |
|
59851
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
342 |
sublocale semiring |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
343 |
proof - |
59851
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
344 |
note [simp] = ring_axioms[unfolded ring_def ring_axioms_def] |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
345 |
show "semiring R" |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
346 |
proof (unfold_locales) |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
347 |
fix x |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
348 |
assume R: "x \<in> carrier R" |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
349 |
then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x" |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
350 |
by (simp del: l_zero r_zero) |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
351 |
also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
352 |
finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" . |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
353 |
with R show "\<zero> \<otimes> x = \<zero>" by (simp del: r_zero) |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
354 |
from R have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)" |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
355 |
by (simp del: l_zero r_zero) |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
356 |
also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
357 |
finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" . |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
358 |
with R show "x \<otimes> \<zero> = \<zero>" by (simp del: r_zero) |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
359 |
qed auto |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
360 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
361 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
362 |
lemma l_minus: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
363 |
"[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
364 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
365 |
assume R: "x \<in> carrier R" "y \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
366 |
then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr) |
44677 | 367 |
also from R have "... = \<zero>" by (simp add: l_neg) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
368 |
finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
369 |
with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp |
21896
9a7949815a84
Experimenting with interpretations of "definition".
ballarin
parents:
20318
diff
changeset
|
370 |
with R show ?thesis by (simp add: a_assoc r_neg) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
371 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
372 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
373 |
lemma r_minus: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
374 |
"[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
375 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
376 |
assume R: "x \<in> carrier R" "y \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
377 |
then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr) |
44677 | 378 |
also from R have "... = \<zero>" by (simp add: l_neg) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
379 |
finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
380 |
with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
381 |
with R show ?thesis by (simp add: a_assoc r_neg ) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
382 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
383 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
384 |
end |
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
385 |
|
23957
54fab60ddc97
Interpretation of rings (as integers) maps defined operations to defined
ballarin
parents:
23350
diff
changeset
|
386 |
lemma (in abelian_group) minus_eq: |
54fab60ddc97
Interpretation of rings (as integers) maps defined operations to defined
ballarin
parents:
23350
diff
changeset
|
387 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
388 |
by (simp only: a_minus_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
389 |
|
61382 | 390 |
text \<open>Setup algebra method: |
391 |
compute distributive normal form in locale contexts\<close> |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
392 |
|
48891 | 393 |
ML_file "ringsimp.ML" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
394 |
|
61382 | 395 |
attribute_setup algebra = \<open> |
58811 | 396 |
Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true) |
397 |
-- Scan.lift Args.name -- Scan.repeat Args.term |
|
398 |
>> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts)) |
|
61382 | 399 |
\<close> "theorems controlling algebra method" |
47701 | 400 |
|
61382 | 401 |
method_setup algebra = \<open> |
58811 | 402 |
Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac) |
61382 | 403 |
\<close> "normalisation of algebraic structure" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
404 |
|
59851
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
405 |
lemmas (in semiring) semiring_simprules |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
406 |
[algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
407 |
a_closed zero_closed m_closed one_closed |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
408 |
a_assoc l_zero a_comm m_assoc l_one l_distr r_zero |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
409 |
a_lcomm r_distr l_null r_null |
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
410 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
411 |
lemmas (in ring) ring_simprules |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
412 |
[algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
413 |
a_closed zero_closed a_inv_closed minus_closed m_closed one_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
414 |
a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
415 |
r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
416 |
a_lcomm r_distr l_null r_null l_minus r_minus |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
417 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
418 |
lemmas (in cring) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
419 |
[algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
420 |
_ |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
421 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
422 |
lemmas (in cring) cring_simprules |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
423 |
[algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] = |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
424 |
a_closed zero_closed a_inv_closed minus_closed m_closed one_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
425 |
a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
426 |
r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
427 |
a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
428 |
|
59851
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
429 |
lemma (in semiring) nat_pow_zero: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
430 |
"(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
431 |
by (induct n) simp_all |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
432 |
|
59851
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
433 |
context semiring begin |
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
434 |
|
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
435 |
lemma one_zeroD: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
436 |
assumes onezero: "\<one> = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
437 |
shows "carrier R = {\<zero>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
438 |
proof (rule, rule) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
439 |
fix x |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
440 |
assume xcarr: "x \<in> carrier R" |
47409 | 441 |
from xcarr have "x = x \<otimes> \<one>" by simp |
442 |
with onezero have "x = x \<otimes> \<zero>" by simp |
|
443 |
with xcarr have "x = \<zero>" by simp |
|
444 |
then show "x \<in> {\<zero>}" by fast |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
445 |
qed fast |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
446 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
447 |
lemma one_zeroI: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
448 |
assumes carrzero: "carrier R = {\<zero>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
449 |
shows "\<one> = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
450 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
451 |
from one_closed and carrzero |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
452 |
show "\<one> = \<zero>" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
453 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
454 |
|
46721 | 455 |
lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)" |
456 |
apply rule |
|
457 |
apply (erule one_zeroI) |
|
458 |
apply (erule one_zeroD) |
|
459 |
done |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
460 |
|
46721 | 461 |
lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)" |
27717
21bbd410ba04
Generalised polynomial lemmas from cring to ring.
ballarin
parents:
27714
diff
changeset
|
462 |
by (simp add: carrier_one_zero) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
463 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
464 |
end |
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
465 |
|
61382 | 466 |
text \<open>Two examples for use of method algebra\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
467 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
468 |
lemma |
27611 | 469 |
fixes R (structure) and S (structure) |
470 |
assumes "ring R" "cring S" |
|
471 |
assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S" |
|
472 |
shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c" |
|
473 |
proof - |
|
29237 | 474 |
interpret ring R by fact |
475 |
interpret cring S by fact |
|
27611 | 476 |
from RS show ?thesis by algebra |
477 |
qed |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
478 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
479 |
lemma |
27611 | 480 |
fixes R (structure) |
481 |
assumes "ring R" |
|
482 |
assumes R: "a \<in> carrier R" "b \<in> carrier R" |
|
483 |
shows "a \<ominus> (a \<ominus> b) = b" |
|
484 |
proof - |
|
29237 | 485 |
interpret ring R by fact |
27611 | 486 |
from R show ?thesis by algebra |
487 |
qed |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
488 |
|
35849 | 489 |
|
61382 | 490 |
subsubsection \<open>Sums over Finite Sets\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
491 |
|
59851
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
492 |
lemma (in semiring) finsum_ldistr: |
61384 | 493 |
"[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
494 |
finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A" |
22265 | 495 |
proof (induct set: finite) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
496 |
case empty then show ?case by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
497 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
498 |
case (insert x F) then show ?case by (simp add: Pi_def l_distr) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
499 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
500 |
|
59851
43b1870b3e61
added locale for semirings
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
58811
diff
changeset
|
501 |
lemma (in semiring) finsum_rdistr: |
61384 | 502 |
"[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
503 |
a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A" |
22265 | 504 |
proof (induct set: finite) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
505 |
case empty then show ?case by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
506 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
507 |
case (insert x F) then show ?case by (simp add: Pi_def r_distr) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
508 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
509 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
510 |
|
61382 | 511 |
subsection \<open>Integral Domains\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
512 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
513 |
context "domain" begin |
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
514 |
|
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
515 |
lemma zero_not_one [simp]: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
516 |
"\<zero> ~= \<one>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
517 |
by (rule not_sym) simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
518 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
519 |
lemma integral_iff: (* not by default a simp rule! *) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
520 |
"[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
521 |
proof |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
522 |
assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
523 |
then show "a = \<zero> | b = \<zero>" by (simp add: integral) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
524 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
525 |
assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
526 |
then show "a \<otimes> b = \<zero>" by auto |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
527 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
528 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
529 |
lemma m_lcancel: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
530 |
assumes prem: "a ~= \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
531 |
and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
532 |
shows "(a \<otimes> b = a \<otimes> c) = (b = c)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
533 |
proof |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
534 |
assume eq: "a \<otimes> b = a \<otimes> c" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
535 |
with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
536 |
with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
537 |
with prem and R have "b \<ominus> c = \<zero>" by auto |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
538 |
with R have "b = b \<ominus> (b \<ominus> c)" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
539 |
also from R have "b \<ominus> (b \<ominus> c) = c" by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
540 |
finally show "b = c" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
541 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
542 |
assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
543 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
544 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
545 |
lemma m_rcancel: |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
546 |
assumes prem: "a ~= \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
547 |
and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
548 |
shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
549 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
550 |
from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
551 |
with R show ?thesis by algebra |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
552 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
553 |
|
41433
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
554 |
end |
1b8ff770f02c
Abelian group facts obtained from group facts via interpretation (sublocale).
ballarin
parents:
35849
diff
changeset
|
555 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
556 |
|
61382 | 557 |
subsection \<open>Fields\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
558 |
|
61382 | 559 |
text \<open>Field would not need to be derived from domain, the properties |
560 |
for domain follow from the assumptions of field\<close> |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
561 |
lemma (in cring) cring_fieldI: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
562 |
assumes field_Units: "Units R = carrier R - {\<zero>}" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
563 |
shows "field R" |
28823 | 564 |
proof |
47409 | 565 |
from field_Units have "\<zero> \<notin> Units R" by fast |
566 |
moreover have "\<one> \<in> Units R" by fast |
|
567 |
ultimately show "\<one> \<noteq> \<zero>" by force |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
568 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
569 |
fix a b |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
570 |
assume acarr: "a \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
571 |
and bcarr: "b \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
572 |
and ab: "a \<otimes> b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
573 |
show "a = \<zero> \<or> b = \<zero>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
574 |
proof (cases "a = \<zero>", simp) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
575 |
assume "a \<noteq> \<zero>" |
47409 | 576 |
with field_Units and acarr have aUnit: "a \<in> Units R" by fast |
577 |
from bcarr have "b = \<one> \<otimes> b" by algebra |
|
578 |
also from aUnit acarr have "... = (inv a \<otimes> a) \<otimes> b" by simp |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
579 |
also from acarr bcarr aUnit[THEN Units_inv_closed] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
580 |
have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra |
47409 | 581 |
also from ab and acarr bcarr aUnit have "... = (inv a) \<otimes> \<zero>" by simp |
582 |
also from aUnit[THEN Units_inv_closed] have "... = \<zero>" by algebra |
|
583 |
finally have "b = \<zero>" . |
|
584 |
then show "a = \<zero> \<or> b = \<zero>" by simp |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
585 |
qed |
23350 | 586 |
qed (rule field_Units) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
587 |
|
61382 | 588 |
text \<open>Another variant to show that something is a field\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
589 |
lemma (in cring) cring_fieldI2: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
590 |
assumes notzero: "\<zero> \<noteq> \<one>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
591 |
and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
592 |
shows "field R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
593 |
apply (rule cring_fieldI, simp add: Units_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
594 |
apply (rule, clarsimp) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
595 |
apply (simp add: notzero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
596 |
proof (clarsimp) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
597 |
fix x |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
598 |
assume xcarr: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
599 |
and "x \<noteq> \<zero>" |
47409 | 600 |
then have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex) |
601 |
then obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>" by fast |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
602 |
from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm) |
47409 | 603 |
with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
604 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
605 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
606 |
|
61382 | 607 |
subsection \<open>Morphisms\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
608 |
|
35847 | 609 |
definition |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
610 |
ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set" |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
611 |
where "ring_hom R S = |
61384 | 612 |
{h. h \<in> carrier R \<rightarrow> carrier S & |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
613 |
(ALL x y. x \<in> carrier R & y \<in> carrier R --> |
35847 | 614 |
h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) & |
615 |
h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}" |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
616 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
617 |
lemma ring_hom_memI: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
618 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
619 |
assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
620 |
and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
621 |
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
622 |
and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
623 |
h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
624 |
and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
625 |
shows "h \<in> ring_hom R S" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27699
diff
changeset
|
626 |
by (auto simp add: ring_hom_def assms Pi_def) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
627 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
628 |
lemma ring_hom_closed: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
629 |
"[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
630 |
by (auto simp add: ring_hom_def funcset_mem) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
631 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
632 |
lemma ring_hom_mult: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
633 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
634 |
shows |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
635 |
"[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
636 |
h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
637 |
by (simp add: ring_hom_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
638 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
639 |
lemma ring_hom_add: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
640 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
641 |
shows |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
642 |
"[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==> |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
643 |
h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
644 |
by (simp add: ring_hom_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
645 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
646 |
lemma ring_hom_one: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
647 |
fixes R (structure) and S (structure) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
648 |
shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
649 |
by (simp add: ring_hom_def) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
650 |
|
61565
352c73a689da
Qualifiers in locale expressions default to mandatory regardless of the command.
ballarin
parents:
61384
diff
changeset
|
651 |
locale ring_hom_cring = R?: cring R + S?: cring S |
29237 | 652 |
for R (structure) and S (structure) + |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
653 |
fixes h |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
654 |
assumes homh [simp, intro]: "h \<in> ring_hom R S" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
655 |
notes hom_closed [simp, intro] = ring_hom_closed [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
656 |
and hom_mult [simp] = ring_hom_mult [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
657 |
and hom_add [simp] = ring_hom_add [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
658 |
and hom_one [simp] = ring_hom_one [OF homh] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
659 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
660 |
lemma (in ring_hom_cring) hom_zero [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
661 |
"h \<zero> = \<zero>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
662 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
663 |
have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
664 |
by (simp add: hom_add [symmetric] del: hom_add) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
665 |
then show ?thesis by (simp del: S.r_zero) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
666 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
667 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
668 |
lemma (in ring_hom_cring) hom_a_inv [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
669 |
"x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
670 |
proof - |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
671 |
assume R: "x \<in> carrier R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
672 |
then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
673 |
by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
674 |
with R show ?thesis by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
675 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
676 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
677 |
lemma (in ring_hom_cring) hom_finsum [simp]: |
61384 | 678 |
"f \<in> A \<rightarrow> carrier R ==> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
679 |
h (finsum R f A) = finsum S (h o f) A" |
60112
3eab4acaa035
finprod takes 1 in case of infinite sets => remove several "finite A" assumptions
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
59851
diff
changeset
|
680 |
by (induct A rule: infinite_finite_induct, auto simp: Pi_def) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
681 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
682 |
lemma (in ring_hom_cring) hom_finprod: |
61384 | 683 |
"f \<in> A \<rightarrow> carrier R ==> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
684 |
h (finprod R f A) = finprod S (h o f) A" |
60112
3eab4acaa035
finprod takes 1 in case of infinite sets => remove several "finite A" assumptions
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
59851
diff
changeset
|
685 |
by (induct A rule: infinite_finite_induct, auto simp: Pi_def) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
686 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
687 |
declare ring_hom_cring.hom_finprod [simp] |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
688 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
689 |
lemma id_ring_hom [simp]: |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
690 |
"id \<in> ring_hom R R" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
691 |
by (auto intro!: ring_hom_memI) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
692 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset
|
693 |
end |