author ballarin Thu, 06 Jan 2011 21:06:17 +0100 changeset 41433 1b8ff770f02c parent 41432 3214c39777ab child 41434 710cdb9e0d17
Abelian group facts obtained from group facts via interpretation (sublocale).
 NEWS file | annotate | diff | comparison | revisions src/HOL/Algebra/FiniteProduct.thy file | annotate | diff | comparison | revisions src/HOL/Algebra/Ideal.thy file | annotate | diff | comparison | revisions src/HOL/Algebra/IntRing.thy file | annotate | diff | comparison | revisions src/HOL/Algebra/Ring.thy file | annotate | diff | comparison | revisions
```--- a/NEWS	Thu Jan 06 17:51:56 2011 +0100
+++ b/NEWS	Thu Jan 06 21:06:17 2011 +0100
@@ -498,6 +498,14 @@
INCOMPATIBILITY.

+*** HOL-Algebra ***
+
+* Theorems for additive ring operations (locale abelian_monoid and
+descendants) are generated by interpretation from their multiplicative
+counterparts.  This causes slight differences in the simp and clasets.
+INCOMPATIBILITY.
+
+
*** HOLCF ***

* The domain package now runs in definitional mode by default: The```
```--- a/src/HOL/Algebra/FiniteProduct.thy	Thu Jan 06 17:51:56 2011 +0100
+++ b/src/HOL/Algebra/FiniteProduct.thy	Thu Jan 06 21:06:17 2011 +0100
@@ -450,9 +450,9 @@

lemma finprod_cong:
"[| A = B; f \<in> B -> carrier G = True;
-      !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
+      !!i. i \<in> B =simp=> f i = g i |] ==> finprod G f A = finprod G g B"
(* This order of prems is slightly faster (3%) than the last two swapped. *)
-  by (rule finprod_cong') force+
+  by (rule finprod_cong') (auto simp add: simp_implies_def)

text {*Usually, if this rule causes a failed congruence proof error,
the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.```
```--- a/src/HOL/Algebra/Ideal.thy	Thu Jan 06 17:51:56 2011 +0100
+++ b/src/HOL/Algebra/Ideal.thy	Thu Jan 06 21:06:17 2011 +0100
@@ -155,12 +155,7 @@

lemma (in ring) oneideal:
shows "ideal (carrier R) R"
-apply (intro idealI  subgroup.intro)
-      apply (rule is_ring)
-     apply simp+
-  apply (fold a_inv_def, simp)
- apply simp+
-done
+  by (rule idealI) (auto intro: is_ring add.subgroupI)

lemma (in "domain") zeroprimeideal:
shows "primeideal {\<zero>} R"```
```--- a/src/HOL/Algebra/IntRing.thy	Thu Jan 06 17:51:56 2011 +0100
+++ b/src/HOL/Algebra/IntRing.thy	Thu Jan 06 21:06:17 2011 +0100
@@ -1,5 +1,6 @@
(*  Title:      HOL/Algebra/IntRing.thy
Author:     Stephan Hohe, TU Muenchen
+    Author:     Clemens Ballarin
*)

theory IntRing
@@ -20,17 +21,16 @@

subsection {* @{text "\<Z>"}: The Set of Integers as Algebraic Structure *}

-definition
+abbreviation
int_ring :: "int ring" ("\<Z>") where
-  "int_ring = \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"
+  "int_ring == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"

lemma int_Zcarr [intro!, simp]:
"k \<in> carrier \<Z>"
-  by (simp add: int_ring_def)
+  by simp

lemma int_is_cring:
"cring \<Z>"
-unfolding int_ring_def
apply (rule cringI)
apply (rule abelian_groupI, simp_all)
defer 1
@@ -62,30 +62,30 @@
and "pow \<Z> x n = x^n"
proof -
-- "Specification"
-  show "monoid \<Z>" proof qed (auto simp: int_ring_def)
+  show "monoid \<Z>" proof qed auto
then interpret int: monoid \<Z> .

-- "Carrier"
-  show "carrier \<Z> = UNIV" by (simp add: int_ring_def)
+  show "carrier \<Z> = UNIV" by simp

-- "Operations"
-  { fix x y show "mult \<Z> x y = x * y" by (simp add: int_ring_def) }
+  { fix x y show "mult \<Z> x y = x * y" by simp }
note mult = this
-  show one: "one \<Z> = 1" by (simp add: int_ring_def)
-  show "pow \<Z> x n = x^n" by (induct n) (simp, simp add: int_ring_def)+
+  show one: "one \<Z> = 1" by simp
+  show "pow \<Z> x n = x^n" by (induct n) simp_all
qed

interpretation int: comm_monoid \<Z>
where "finprod \<Z> f A = (if finite A then setprod f A else undefined)"
proof -
-- "Specification"
-  show "comm_monoid \<Z>" proof qed (auto simp: int_ring_def)
+  show "comm_monoid \<Z>" proof qed auto
then interpret int: comm_monoid \<Z> .

-- "Operations"
-  { fix x y have "mult \<Z> x y = x * y" by (simp add: int_ring_def) }
+  { fix x y have "mult \<Z> x y = x * y" by simp }
note mult = this
-  have one: "one \<Z> = 1" by (simp add: int_ring_def)
+  have one: "one \<Z> = 1" by simp
show "finprod \<Z> f A = (if finite A then setprod f A else undefined)"
proof (cases "finite A")
case True then show ?thesis proof induct
@@ -99,18 +99,22 @@
qed

interpretation int: abelian_monoid \<Z>
-  where "zero \<Z> = 0"
-    and "add \<Z> x y = x + y"
-    and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
+  where int_carrier_eq: "carrier \<Z> = UNIV"
+    and int_zero_eq: "zero \<Z> = 0"
+    and int_add_eq: "add \<Z> x y = x + y"
+    and int_finsum_eq: "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
proof -
-- "Specification"
-  show "abelian_monoid \<Z>" proof qed (auto simp: int_ring_def)
+  show "abelian_monoid \<Z>" proof qed auto
then interpret int: abelian_monoid \<Z> .

+  -- "Carrier"
+  show "carrier \<Z> = UNIV" by simp
+
-- "Operations"
-  { fix x y show "add \<Z> x y = x + y" by (simp add: int_ring_def) }
+  { fix x y show "add \<Z> x y = x + y" by simp }
note add = this
-  show zero: "zero \<Z> = 0" by (simp add: int_ring_def)
+  show zero: "zero \<Z> = 0" by simp
show "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
proof (cases "finite A")
case True then show ?thesis proof induct
@@ -124,30 +128,46 @@
qed

interpretation int: abelian_group \<Z>
-  where "a_inv \<Z> x = - x"
-    and "a_minus \<Z> x y = x - y"
+  (* The equations from the interpretation of abelian_monoid need to be repeated.
+     Since the morphisms through which the abelian structures are interpreted are
+     not the identity, the equations of these interpretations are not inherited. *)
+  (* FIXME *)
+  where "carrier \<Z> = UNIV"
+    and "zero \<Z> = 0"
+    and "add \<Z> x y = x + y"
+    and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
+    and int_a_inv_eq: "a_inv \<Z> x = - x"
+    and int_a_minus_eq: "a_minus \<Z> x y = x - y"
proof -
-- "Specification"
show "abelian_group \<Z>"
proof (rule abelian_groupI)
show "!!x. x \<in> carrier \<Z> ==> EX y : carrier \<Z>. y \<oplus>\<^bsub>\<Z>\<^esub> x = \<zero>\<^bsub>\<Z>\<^esub>"
-      by (simp add: int_ring_def) arith
-  qed (auto simp: int_ring_def)
+      by simp arith
+  qed auto
then interpret int: abelian_group \<Z> .
-
-- "Operations"
-  { fix x y have "add \<Z> x y = x + y" by (simp add: int_ring_def) }
+  { fix x y have "add \<Z> x y = x + y" by simp }
note add = this
-  have zero: "zero \<Z> = 0" by (simp add: int_ring_def)
+  have zero: "zero \<Z> = 0" by simp
{ fix x
have "add \<Z> (-x) x = zero \<Z>" by (simp add: add zero)
then show "a_inv \<Z> x = - x" by (simp add: int.minus_equality) }
note a_inv = this
show "a_minus \<Z> x y = x - y" by (simp add: int.minus_eq add a_inv)
-qed
+qed (simp add: int_carrier_eq int_zero_eq int_add_eq int_finsum_eq)+

interpretation int: "domain" \<Z>
-  proof qed (auto simp: int_ring_def left_distrib right_distrib)
+  where "carrier \<Z> = UNIV"
+    and "zero \<Z> = 0"
+    and "add \<Z> x y = x + y"
+    and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
+    and "a_inv \<Z> x = - x"
+    and "a_minus \<Z> x y = x - y"
+proof -
+  show "domain \<Z>" by unfold_locales (auto simp: left_distrib right_distrib)
+qed (simp
+    add: int_carrier_eq int_zero_eq int_add_eq int_finsum_eq int_a_inv_eq int_a_minus_eq)+

text {* Removal of occurrences of @{term UNIV} in interpretation result
@@ -213,8 +233,8 @@

lemma int_Idl:
"Idl\<^bsub>\<Z>\<^esub> {a} = {x * a | x. True}"
-  apply (subst int.cgenideal_eq_genideal[symmetric]) apply (simp add: int_ring_def)
-  apply (simp add: cgenideal_def int_ring_def)
+  apply (subst int.cgenideal_eq_genideal[symmetric]) apply simp
+  apply (simp add: cgenideal_def)
done

lemma multiples_principalideal:
@@ -232,10 +252,8 @@
apply (rule int.genideal_ideal, simp)
apply (rule int_is_cring)
apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
- apply (simp add: int_ring_def)
apply clarsimp defer 1
apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
- apply (simp add: int_ring_def)
apply (elim exE)
proof -
fix a b x
@@ -336,7 +354,7 @@
shows "EX x. k = x * l + r"
proof -
from kIl[unfolded ZMod_def]
-      have "\<exists>xl\<in>Idl\<^bsub>\<Z>\<^esub> {l}. k = xl + r" by (simp add: a_r_coset_defs int_ring_def)
+      have "\<exists>xl\<in>Idl\<^bsub>\<Z>\<^esub> {l}. k = xl + r" by (simp add: a_r_coset_defs)
from this obtain xl
where xl: "xl \<in> Idl\<^bsub>\<Z>\<^esub> {l}"
and k: "k = xl + r"
@@ -382,7 +400,6 @@
unfolding ZMod_def
apply (rule a_repr_independence'[symmetric])
apply (simp add: int_Idl a_r_coset_defs)
-  apply (simp add: int_ring_def)
proof -
have "a = m * (a div m) + (a mod m)" by (simp add: zmod_zdiv_equality)
hence "a = (a div m) * m + (a mod m)" by simp
@@ -426,13 +443,13 @@
lemma ZFact_zero:
"carrier (ZFact 0) = (\<Union>a. {{a}})"
apply (insert int.genideal_zero)
-apply (simp add: ZFact_defs A_RCOSETS_defs r_coset_def int_ring_def ring_record_simps)
+apply (simp add: ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps)
done

lemma ZFact_one:
"carrier (ZFact 1) = {UNIV}"
-apply (simp only: ZFact_defs A_RCOSETS_defs r_coset_def int_ring_def ring_record_simps)
-apply (subst int.genideal_one[unfolded int_ring_def, simplified ring_record_simps])
+apply (simp only: ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps)
+apply (subst int.genideal_one)
apply (rule, rule, clarsimp)
apply (rule, rule, clarsimp)
apply (rule, clarsimp, arith)```
```--- a/src/HOL/Algebra/Ring.thy	Thu Jan 06 17:51:56 2011 +0100
+++ b/src/HOL/Algebra/Ring.thy	Thu Jan 06 21:06:17 2011 +0100
@@ -31,10 +31,23 @@
assumes a_comm_monoid:
"comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"

+definition
+  finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
+  "finsum G = finprod (| carrier = carrier G, mult = add G, one = zero G |)"

-text {*
-  The following definition is redundant but simple to use.
-*}
+syntax
+  "_finsum" :: "index => idt => 'a set => 'b => 'b"
+      ("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10)
+syntax (xsymbols)
+  "_finsum" :: "index => idt => 'a set => 'b => 'b"
+      ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
+syntax (HTML output)
+  "_finsum" :: "index => idt => 'a set => 'b => 'b"
+      ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
+translations
+  "\<Oplus>\<index>i:A. b" == "CONST finsum \<struct>\<index> (%i. b) A"
+  -- {* Beware of argument permutation! *}
+

locale abelian_group = abelian_monoid +
assumes a_comm_group:
@@ -85,248 +98,61 @@

lemmas monoid_record_simps = partial_object.simps monoid.simps

-lemma (in abelian_monoid) a_closed [intro, simp]:
-  "\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier G"
-  by (rule monoid.m_closed [OF a_monoid, simplified monoid_record_simps])
-
-lemma (in abelian_monoid) zero_closed [intro, simp]:
-  "\<zero> \<in> carrier G"
-  by (rule monoid.one_closed [OF a_monoid, simplified monoid_record_simps])
-
-lemma (in abelian_group) a_inv_closed [intro, simp]:
-  "x \<in> carrier G ==> \<ominus> x \<in> carrier G"
-  by (simp add: a_inv_def
-    group.inv_closed [OF a_group, simplified monoid_record_simps])
-
-lemma (in abelian_group) minus_closed [intro, simp]:
-  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
-  by (simp add: a_minus_def)
-
-lemma (in abelian_group) a_l_cancel [simp]:
-  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
-   (x \<oplus> y = x \<oplus> z) = (y = z)"
-  by (rule group.l_cancel [OF a_group, simplified monoid_record_simps])
-
-lemma (in abelian_group) a_r_cancel [simp]:
-  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
-   (y \<oplus> x = z \<oplus> x) = (y = z)"
-  by (rule group.r_cancel [OF a_group, simplified monoid_record_simps])
-
-lemma (in abelian_monoid) a_assoc:
-  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
-  (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
-  by (rule monoid.m_assoc [OF a_monoid, simplified monoid_record_simps])
-
-lemma (in abelian_monoid) l_zero [simp]:
-  "x \<in> carrier G ==> \<zero> \<oplus> x = x"
-  by (rule monoid.l_one [OF a_monoid, simplified monoid_record_simps])
-
-lemma (in abelian_group) l_neg:
-  "x \<in> carrier G ==> \<ominus> x \<oplus> x = \<zero>"
-  by (simp add: a_inv_def
-    group.l_inv [OF a_group, simplified monoid_record_simps])
-
-lemma (in abelian_monoid) a_comm:
-  "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
-  by (rule comm_monoid.m_comm [OF a_comm_monoid,
-    simplified monoid_record_simps])
-
-lemma (in abelian_monoid) a_lcomm:
-  "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
-   x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
-  by (rule comm_monoid.m_lcomm [OF a_comm_monoid,
-                                simplified monoid_record_simps])
-
-lemma (in abelian_monoid) r_zero [simp]:
-  "x \<in> carrier G ==> x \<oplus> \<zero> = x"
-  using monoid.r_one [OF a_monoid]
-  by simp
-
-lemma (in abelian_group) r_neg:
-  "x \<in> carrier G ==> x \<oplus> (\<ominus> x) = \<zero>"
-  using group.r_inv [OF a_group]
-  by (simp add: a_inv_def)
+text {* Transfer facts from multiplicative structures via interpretation. *}

-lemma (in abelian_group) minus_zero [simp]:
-  "\<ominus> \<zero> = \<zero>"
-  by (simp add: a_inv_def
-    group.inv_one [OF a_group, simplified monoid_record_simps])
-
-lemma (in abelian_group) minus_minus [simp]:
-  "x \<in> carrier G ==> \<ominus> (\<ominus> x) = x"
-  using group.inv_inv [OF a_group, simplified monoid_record_simps]
-  by (simp add: a_inv_def)
-
-lemma (in abelian_group) a_inv_inj:
-  "inj_on (a_inv G) (carrier G)"
-  using group.inv_inj [OF a_group, simplified monoid_record_simps]
-  by (simp add: a_inv_def)
-
-lemma (in abelian_group) minus_add:
-  "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
-  using comm_group.inv_mult [OF a_comm_group]
-  by (simp add: a_inv_def)
-
-lemma (in abelian_group) minus_equality:
-  "[| x \<in> carrier G; y \<in> carrier G; y \<oplus> x = \<zero> |] ==> \<ominus> x = y"
-  using group.inv_equality [OF a_group]
-  by (auto simp add: a_inv_def)
-
-lemma (in abelian_monoid) minus_unique:
-  "[| x \<in> carrier G; y \<in> carrier G; y' \<in> carrier G;
-      y \<oplus> x = \<zero>; x \<oplus> y' = \<zero> |] ==> y = y'"
-  using monoid.inv_unique [OF a_monoid]
-  by (simp add: a_inv_def)
-
-lemmas (in abelian_monoid) a_ac = a_assoc a_comm a_lcomm
-
-text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}
-lemma comm_group_abelian_groupI:
-  fixes G (structure)
-  assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
-  shows "abelian_group G"
-proof -
-  interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
-    by (rule cg)
-  show "abelian_group G" ..
-qed
-
-
-subsection {* Sums over Finite Sets *}
-
-text {*
-  This definition makes it easy to lift lemmas from @{term finprod}.
-*}
-
-definition
-  finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
-  "finsum G f A = finprod (| carrier = carrier G, mult = add G, one = zero G |) f A"
-
-syntax
-  "_finsum" :: "index => idt => 'a set => 'b => 'b"
-      ("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10)
-syntax (xsymbols)
-  "_finsum" :: "index => idt => 'a set => 'b => 'b"
-      ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
-syntax (HTML output)
-  "_finsum" :: "index => idt => 'a set => 'b => 'b"
-      ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
-translations
-  "\<Oplus>\<index>i:A. b" == "CONST finsum \<struct>\<index> (%i. b) A"
-  -- {* Beware of argument permutation! *}
+sublocale abelian_monoid <
+  add!: monoid "(| carrier = carrier G, mult = add G, one = zero G |)"
+  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
+    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
+    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
+  by (rule a_monoid) auto

context abelian_monoid begin

-(*
-  lemmas finsum_empty [simp] =
-    comm_monoid.finprod_empty [OF a_comm_monoid, simplified]
-  is dangeous, because attributes (like simplified) are applied upon opening
-  the locale, simplified refers to the simpset at that time!!!
+lemmas a_closed = add.m_closed
+lemmas zero_closed = add.one_closed
+lemmas a_assoc = add.m_assoc
+lemmas l_zero = add.l_one
+lemmas r_zero = add.r_one
+lemmas minus_unique = add.inv_unique

-  lemmas finsum_empty [simp] =
-    abelian_monoid.finprod_empty [OF a_abelian_monoid, folded finsum_def,
-      simplified monoid_record_simps]
-  makes the locale slow, because proofs are repeated for every
-  "lemma (in abelian_monoid)" command.
-  When lemma is used time in UnivPoly.thy from beginning to UP_cring goes down
-  from 110 secs to 60 secs.
-*)
-
-lemma finsum_empty [simp]:
-  "finsum G f {} = \<zero>"
-  by (rule comm_monoid.finprod_empty [OF a_comm_monoid,
-    folded finsum_def, simplified monoid_record_simps])
+end

-lemma finsum_insert [simp]:
-  "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |]
-  ==> finsum G f (insert a F) = f a \<oplus> finsum G f F"
-  by (rule comm_monoid.finprod_insert [OF a_comm_monoid,
-    folded finsum_def, simplified monoid_record_simps])
-
-lemma finsum_zero [simp]:
-  "finite A ==> (\<Oplus>i\<in>A. \<zero>) = \<zero>"
-  by (rule comm_monoid.finprod_one [OF a_comm_monoid, folded finsum_def,
-    simplified monoid_record_simps])
+sublocale abelian_monoid <
+  add!: comm_monoid "(| carrier = carrier G, mult = add G, one = zero G |)"
+  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
+    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
+    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
+    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"
+  by (rule a_comm_monoid) (auto simp: finsum_def)

-lemma finsum_closed [simp]:
-  fixes A
-  assumes fin: "finite A" and f: "f \<in> A -> carrier G"
-  shows "finsum G f A \<in> carrier G"
-  apply (rule comm_monoid.finprod_closed [OF a_comm_monoid,
-    folded finsum_def, simplified monoid_record_simps])
-   apply (rule fin)
-  apply (rule f)
-  done
-
-lemma finsum_Un_Int:
-  "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
-     finsum G g (A Un B) \<oplus> finsum G g (A Int B) =
-     finsum G g A \<oplus> finsum G g B"
-  by (rule comm_monoid.finprod_Un_Int [OF a_comm_monoid,
-    folded finsum_def, simplified monoid_record_simps])
+context abelian_monoid begin

-lemma finsum_Un_disjoint:
-  "[| finite A; finite B; A Int B = {};
-      g \<in> A -> carrier G; g \<in> B -> carrier G |]
-   ==> finsum G g (A Un B) = finsum G g A \<oplus> finsum G g B"
-  by (rule comm_monoid.finprod_Un_disjoint [OF a_comm_monoid,
-    folded finsum_def, simplified monoid_record_simps])
-
-lemma finsum_addf:
-  "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
-   finsum G (%x. f x \<oplus> g x) A = (finsum G f A \<oplus> finsum G g A)"
-  by (rule comm_monoid.finprod_multf [OF a_comm_monoid,
-    folded finsum_def, simplified monoid_record_simps])
-
-lemma finsum_cong':
-  "[| A = B; g : B -> carrier G;
-      !!i. i : B ==> f i = g i |] ==> finsum G f A = finsum G g B"
-  by (rule comm_monoid.finprod_cong' [OF a_comm_monoid,
-    folded finsum_def, simplified monoid_record_simps]) auto
-
-lemma finsum_0 [simp]:
-  "f : {0::nat} -> carrier G ==> finsum G f {..0} = f 0"
-  by (rule comm_monoid.finprod_0 [OF a_comm_monoid, folded finsum_def,
-    simplified monoid_record_simps])
+lemmas a_comm = add.m_comm
+lemmas a_lcomm = add.m_lcomm
+lemmas a_ac = a_assoc a_comm a_lcomm

-lemma finsum_Suc [simp]:
-  "f : {..Suc n} -> carrier G ==>
-   finsum G f {..Suc n} = (f (Suc n) \<oplus> finsum G f {..n})"
-  by (rule comm_monoid.finprod_Suc [OF a_comm_monoid, folded finsum_def,
-    simplified monoid_record_simps])
-
-lemma finsum_Suc2:
-  "f : {..Suc n} -> carrier G ==>
-   finsum G f {..Suc n} = (finsum G (%i. f (Suc i)) {..n} \<oplus> f 0)"
-  by (rule comm_monoid.finprod_Suc2 [OF a_comm_monoid, folded finsum_def,
-    simplified monoid_record_simps])
+lemmas finsum_empty = add.finprod_empty
+lemmas finsum_insert = add.finprod_insert
+lemmas finsum_zero = add.finprod_one
+lemmas finsum_closed = add.finprod_closed
+lemmas finsum_Un_Int = add.finprod_Un_Int
+lemmas finsum_Un_disjoint = add.finprod_Un_disjoint
+lemmas finsum_addf = add.finprod_multf
+lemmas finsum_cong' = add.finprod_cong'
+lemmas finsum_0 = add.finprod_0
+lemmas finsum_Suc = add.finprod_Suc
+lemmas finsum_Suc2 = add.finprod_Suc2
+lemmas finsum_add = add.finprod_mult

-lemma finsum_add [simp]:
-  "[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
-     finsum G (%i. f i \<oplus> g i) {..n::nat} =
-     finsum G f {..n} \<oplus> finsum G g {..n}"
-  by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def,
-    simplified monoid_record_simps])
-
-lemma finsum_cong:
-  "[| A = B; f : B -> carrier G;
-      !!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B"
-  by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def,
-    simplified monoid_record_simps]) (auto simp add: simp_implies_def)
-
+lemmas finsum_cong = add.finprod_cong
text {*Usually, if this rule causes a failed congruence proof error,
the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
Adding @{thm [source] Pi_def} to the simpset is often useful. *}

-lemma finsum_reindex:
-  assumes fin: "finite A"
-    shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow>
-        inj_on h A ==> finsum G f (h ` A) = finsum G (%x. f (h x)) A"
-  using fin apply induct
-  apply (auto simp add: finsum_insert Pi_def)
-done
+lemmas finsum_reindex = add.finprod_reindex

-(* The following is wrong.  Needed is the equivalent of (^) for addition,
+(* The following would be wrong.  Needed is the equivalent of (^) for addition,
or indeed the canonical embedding from Nat into the monoid.

lemma finsum_const:
@@ -343,18 +169,60 @@
done
*)

-(* By Jesus Aransay. *)
-
-lemma finsum_singleton:
-  assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
-  shows "(\<Oplus>j\<in>A. if i = j then f j else \<zero>) = f i"
-  using i_in_A finsum_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<zero>)"]
-    fin_A f_Pi finsum_zero [of "A - {i}"]
-    finsum_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<zero>)" "(\<lambda>i. \<zero>)"]
-  unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
+lemmas finsum_singleton = add.finprod_singleton

end

+sublocale abelian_group <
+  add!: group "(| carrier = carrier G, mult = add G, one = zero G |)"
+  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
+    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
+    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
+    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"
+  by (rule a_group) (auto simp: m_inv_def a_inv_def)
+
+context abelian_group begin
+
+lemmas a_inv_closed = add.inv_closed
+
+lemma minus_closed [intro, simp]:
+  "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
+  by (simp add: a_minus_def)
+
+lemmas a_l_cancel = add.l_cancel
+lemmas a_r_cancel = add.r_cancel
+lemmas l_neg = add.l_inv [simp del]
+lemmas r_neg = add.r_inv [simp del]
+lemmas minus_zero = add.inv_one
+lemmas minus_minus = add.inv_inv
+lemmas a_inv_inj = add.inv_inj
+lemmas minus_equality = add.inv_equality
+
+end
+
+sublocale abelian_group <
+  add!: comm_group "(| carrier = carrier G, mult = add G, one = zero G |)"
+  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
+    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
+    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
+    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"
+    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"
+  by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def)
+
+lemmas (in abelian_group) minus_add = add.inv_mult
+
+text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}
+
+lemma comm_group_abelian_groupI:
+  fixes G (structure)
+  assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
+  shows "abelian_group G"
+proof -
+  interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
+    by (rule cg)
+  show "abelian_group G" ..
+qed
+

subsection {* Rings: Basic Definitions *}

@@ -389,18 +257,22 @@
by (auto intro: ring.intro
abelian_group.axioms ring_axioms.intro assms)

-lemma (in ring) is_abelian_group:
+context ring begin
+
+lemma is_abelian_group:
"abelian_group R"
..

-lemma (in ring) is_monoid:
+lemma is_monoid:
"monoid R"
by (auto intro!: monoidI m_assoc)

-lemma (in ring) is_ring:
+lemma is_ring:
"ring R"
by (rule ring_axioms)

+end
+
lemmas ring_record_simps = monoid_record_simps ring.simps

lemma cringI:
@@ -449,7 +321,7 @@
assume G: "x \<in> carrier G" "y \<in> carrier G"
then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
by (simp only: r_neg l_zero)
-  with G show ?thesis
+  with G show ?thesis
by (simp add: a_ac)
qed

@@ -462,11 +334,13 @@
with G show ?thesis by (simp add: a_ac)
qed

+context ring begin
+
text {*
-  The following proofs are from Jacobson, Basic Algebra I, pp.~88--89
+  The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
*}

-lemma (in ring) l_null [simp]:
+lemma l_null [simp]:
"x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
proof -
assume R: "x \<in> carrier R"
@@ -477,7 +351,7 @@
with R show ?thesis by (simp del: r_zero)
qed

-lemma (in ring) r_null [simp]:
+lemma r_null [simp]:
"x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
proof -
assume R: "x \<in> carrier R"
@@ -488,7 +362,7 @@
with R show ?thesis by (simp del: r_zero)
qed

-lemma (in ring) l_minus:
+lemma l_minus:
"[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
proof -
assume R: "x \<in> carrier R" "y \<in> carrier R"
@@ -499,7 +373,7 @@
with R show ?thesis by (simp add: a_assoc r_neg)
qed

-lemma (in ring) r_minus:
+lemma r_minus:
"[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
proof -
assume R: "x \<in> carrier R" "y \<in> carrier R"
@@ -510,6 +384,8 @@
with R show ?thesis by (simp add: a_assoc r_neg )
qed

+end
+
lemma (in abelian_group) minus_eq:
"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
by (simp only: a_minus_def)
@@ -539,12 +415,13 @@
r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus

-
lemma (in cring) nat_pow_zero:
"(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
by (induct n) simp_all

-lemma (in ring) one_zeroD:
+context ring begin
+
+lemma one_zeroD:
assumes onezero: "\<one> = \<zero>"
shows "carrier R = {\<zero>}"
proof (rule, rule)
@@ -559,7 +436,7 @@
thus "x \<in> {\<zero>}" by fast
qed fast

-lemma (in ring) one_zeroI:
+lemma one_zeroI:
assumes carrzero: "carrier R = {\<zero>}"
shows "\<one> = \<zero>"
proof -
@@ -567,14 +444,16 @@
show "\<one> = \<zero>" by simp
qed

-lemma (in ring) carrier_one_zero:
+lemma carrier_one_zero:
shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
by (rule, erule one_zeroI, erule one_zeroD)

-lemma (in ring) carrier_one_not_zero:
+lemma carrier_one_not_zero:
shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
by (simp add: carrier_one_zero)

+end
+
text {* Two examples for use of method algebra *}

lemma
@@ -623,11 +502,13 @@

subsection {* Integral Domains *}

-lemma (in "domain") zero_not_one [simp]:
+context "domain" begin
+
+lemma zero_not_one [simp]:
"\<zero> ~= \<one>"
by (rule not_sym) simp

-lemma (in "domain") integral_iff: (* not by default a simp rule! *)
+lemma integral_iff: (* not by default a simp rule! *)
"[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)"
proof
assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
@@ -637,7 +518,7 @@
then show "a \<otimes> b = \<zero>" by auto
qed

-lemma (in "domain") m_lcancel:
+lemma m_lcancel:
assumes prem: "a ~= \<zero>"
and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
@@ -653,7 +534,7 @@
assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
qed

-lemma (in "domain") m_rcancel:
+lemma m_rcancel:
assumes prem: "a ~= \<zero>"
and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
@@ -662,6 +543,8 @@
with R show ?thesis by algebra
qed

+end
+

subsection {* Fields *}
```