--- a/src/HOL/Groups.thy Thu Dec 26 22:47:49 2013 +0100
+++ b/src/HOL/Groups.thy Fri Dec 27 14:35:14 2013 +0100
@@ -90,10 +90,13 @@
locale comm_monoid = abel_semigroup +
fixes z :: 'a ("1")
assumes comm_neutral: "a * 1 = a"
+begin
-sublocale comm_monoid < monoid
+sublocale monoid
by default (simp_all add: commute comm_neutral)
+end
+
subsection {* Generic operations *}
@@ -148,19 +151,20 @@
class semigroup_add = plus +
assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)"
+begin
-sublocale semigroup_add < add!: semigroup plus
+sublocale add!: semigroup plus
by default (fact add_assoc)
+end
+
class ab_semigroup_add = semigroup_add +
assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
+begin
-sublocale ab_semigroup_add < add!: abel_semigroup plus
+sublocale add!: abel_semigroup plus
by default (fact add_commute)
-context ab_semigroup_add
-begin
-
lemmas add_left_commute [algebra_simps, field_simps] = add.left_commute
theorems add_ac = add_assoc add_commute add_left_commute
@@ -171,19 +175,20 @@
class semigroup_mult = times +
assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
+begin
-sublocale semigroup_mult < mult!: semigroup times
+sublocale mult!: semigroup times
by default (fact mult_assoc)
+end
+
class ab_semigroup_mult = semigroup_mult +
assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
+begin
-sublocale ab_semigroup_mult < mult!: abel_semigroup times
+sublocale mult!: abel_semigroup times
by default (fact mult_commute)
-context ab_semigroup_mult
-begin
-
lemmas mult_left_commute [algebra_simps, field_simps] = mult.left_commute
theorems mult_ac = mult_assoc mult_commute mult_left_commute
@@ -195,22 +200,28 @@
class monoid_add = zero + semigroup_add +
assumes add_0_left: "0 + a = a"
and add_0_right: "a + 0 = a"
+begin
-sublocale monoid_add < add!: monoid plus 0
+sublocale add!: monoid plus 0
by default (fact add_0_left add_0_right)+
+end
+
lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
-by (rule eq_commute)
+ by (fact eq_commute)
class comm_monoid_add = zero + ab_semigroup_add +
assumes add_0: "0 + a = a"
+begin
-sublocale comm_monoid_add < add!: comm_monoid plus 0
+sublocale add!: comm_monoid plus 0
by default (insert add_0, simp add: ac_simps)
-subclass (in comm_monoid_add) monoid_add
+subclass monoid_add
by default (fact add.left_neutral add.right_neutral)+
+end
+
class comm_monoid_diff = comm_monoid_add + minus +
assumes diff_zero [simp]: "a - 0 = a"
and zero_diff [simp]: "0 - a = 0"
@@ -265,22 +276,28 @@
class monoid_mult = one + semigroup_mult +
assumes mult_1_left: "1 * a = a"
and mult_1_right: "a * 1 = a"
+begin
-sublocale monoid_mult < mult!: monoid times 1
+sublocale mult!: monoid times 1
by default (fact mult_1_left mult_1_right)+
+end
+
lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
-by (rule eq_commute)
+ by (fact eq_commute)
class comm_monoid_mult = one + ab_semigroup_mult +
assumes mult_1: "1 * a = a"
+begin
-sublocale comm_monoid_mult < mult!: comm_monoid times 1
+sublocale mult!: comm_monoid times 1
by default (insert mult_1, simp add: ac_simps)
-subclass (in comm_monoid_mult) monoid_mult
+subclass monoid_mult
by default (fact mult.left_neutral mult.right_neutral)+
+end
+
class cancel_semigroup_add = semigroup_add +
assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"