--- a/src/HOL/Algebra/AbelCoset.thy Wed Mar 05 20:07:43 2014 +0100
+++ b/src/HOL/Algebra/AbelCoset.thy Wed Mar 05 21:51:30 2014 +0100
@@ -54,8 +54,8 @@
locale abelian_group_hom = G: abelian_group G + H: abelian_group H
for G (structure) and H (structure) +
fixes h
- assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
- (| carrier = carrier H, mult = add H, one = zero H |) h"
+ assumes a_group_hom: "group_hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
+ \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"
lemmas a_r_coset_defs =
a_r_coset_def r_coset_def
@@ -129,12 +129,12 @@
folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
lemma (in abelian_group) a_coset_join1:
- "[| H +> x = H; x \<in> carrier G; subgroup H (|carrier = carrier G, mult = add G, one = zero G|) |] ==> x \<in> H"
+ "[| H +> x = H; x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> x \<in> H"
by (rule group.coset_join1 [OF a_group,
folded a_r_coset_def, simplified monoid_record_simps])
lemma (in abelian_group) a_solve_equation:
- "\<lbrakk>subgroup H (|carrier = carrier G, mult = add G, one = zero G|); x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<oplus> x"
+ "\<lbrakk>subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>; x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<oplus> x"
by (rule group.solve_equation [OF a_group,
folded a_r_coset_def, simplified monoid_record_simps])
@@ -535,8 +535,8 @@
lemma abelian_group_homI:
assumes "abelian_group G"
assumes "abelian_group H"
- assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
- (| carrier = carrier H, mult = add H, one = zero H |) h"
+ assumes a_group_hom: "group_hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
+ \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"
shows "abelian_group_hom G H h"
proof -
interpret G: abelian_group G by fact
@@ -636,7 +636,7 @@
theorem (in abelian_group_hom) A_FactGroup_iso:
"h ` carrier G = carrier H
\<Longrightarrow> (\<lambda>X. the_elem (h`X)) \<in> (G A_Mod (a_kernel G H h)) \<cong>
- (| carrier = carrier H, mult = add H, one = zero H |)"
+ \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr>"
by (rule group_hom.FactGroup_iso[OF a_group_hom,
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
--- a/src/HOL/Algebra/Group.thy Wed Mar 05 20:07:43 2014 +0100
+++ b/src/HOL/Algebra/Group.thy Wed Mar 05 21:51:30 2014 +0100
@@ -721,7 +721,7 @@
text_raw {* \label{sec:subgroup-lattice} *}
theorem (in group) subgroups_partial_order:
- "partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
+ "partial_order \<lparr>carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq>\<rparr>"
by default simp_all
lemma (in group) subgroup_self:
@@ -729,7 +729,7 @@
by (rule subgroupI) auto
lemma (in group) subgroup_imp_group:
- "subgroup H G ==> group (G(| carrier := H |))"
+ "subgroup H G ==> group (G\<lparr>carrier := H\<rparr>)"
by (erule subgroup.subgroup_is_group) (rule group_axioms)
lemma (in group) is_monoid [intro, simp]:
@@ -737,7 +737,7 @@
by (auto intro: monoid.intro m_assoc)
lemma (in group) subgroup_inv_equality:
- "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
+ "[| subgroup H G; x \<in> H |] ==> m_inv (G \<lparr>carrier := H\<rparr>) x = inv x"
apply (rule_tac inv_equality [THEN sym])
apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
apply (rule subsetD [OF subgroup.subset], assumption+)
@@ -766,7 +766,7 @@
qed
theorem (in group) subgroups_complete_lattice:
- "complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
+ "complete_lattice \<lparr>carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq>\<rparr>"
(is "complete_lattice ?L")
proof (rule partial_order.complete_lattice_criterion1)
show "partial_order ?L" by (rule subgroups_partial_order)
@@ -784,7 +784,7 @@
fix H
assume H: "H \<in> A"
with L have subgroupH: "subgroup H G" by auto
- from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
+ from subgroupH have groupH: "group (G \<lparr>carrier := H\<rparr>)" (is "group ?H")
by (rule subgroup_imp_group)
from groupH have monoidH: "monoid ?H"
by (rule group.is_monoid)
--- a/src/HOL/Algebra/IntRing.thy Wed Mar 05 20:07:43 2014 +0100
+++ b/src/HOL/Algebra/IntRing.thy Wed Mar 05 21:51:30 2014 +0100
@@ -23,7 +23,7 @@
abbreviation
int_ring :: "int ring" ("\<Z>") where
- "int_ring == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
+ "int_ring == \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"
lemma int_Zcarr [intro!, simp]:
"k \<in> carrier \<Z>"
@@ -183,27 +183,27 @@
by simp_all
interpretation int (* FIXME [unfolded UNIV] *) :
- partial_order "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
- where "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
- and "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
- and "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
+ partial_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
+ where "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
+ and "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
+ and "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
proof -
- show "partial_order (| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
+ show "partial_order \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
by default simp_all
- show "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
+ show "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
by simp
- show "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
+ show "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
by simp
- show "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
+ show "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
by (simp add: lless_def) auto
qed
interpretation int (* FIXME [unfolded UNIV] *) :
- lattice "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
- where "join (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = max x y"
- and "meet (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = min x y"
+ lattice "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
+ where "join \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = max x y"
+ and "meet \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = min x y"
proof -
- let ?Z = "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
+ let ?Z = "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
show "lattice ?Z"
apply unfold_locales
apply (simp add: least_def Upper_def)
@@ -225,7 +225,7 @@
qed
interpretation int (* [unfolded UNIV] *) :
- total_order "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
+ total_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
by default clarsimp
--- a/src/HOL/Algebra/Lattice.thy Wed Mar 05 20:07:43 2014 +0100
+++ b/src/HOL/Algebra/Lattice.thy Wed Mar 05 21:51:30 2014 +0100
@@ -1278,7 +1278,7 @@
subsubsection {* The Powerset of a Set is a Complete Lattice *}
theorem powerset_is_complete_lattice:
- "complete_lattice (| carrier = Pow A, eq = op =, le = op \<subseteq> |)"
+ "complete_lattice \<lparr>carrier = Pow A, eq = op =, le = op \<subseteq>\<rparr>"
(is "complete_lattice ?L")
proof (rule partial_order.complete_latticeI)
show "partial_order ?L"
--- a/src/HOL/Algebra/Ring.thy Wed Mar 05 20:07:43 2014 +0100
+++ b/src/HOL/Algebra/Ring.thy Wed Mar 05 21:51:30 2014 +0100
@@ -19,7 +19,7 @@
definition
a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80)
- where "a_inv R = m_inv (| carrier = carrier R, mult = add R, one = zero R |)"
+ where "a_inv R = m_inv \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"
definition
a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)
@@ -28,11 +28,11 @@
locale abelian_monoid =
fixes G (structure)
assumes a_comm_monoid:
- "comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"
+ "comm_monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
definition
finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
- "finsum G = finprod (| carrier = carrier G, mult = add G, one = zero G |)"
+ "finsum G = finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
syntax
"_finsum" :: "index => idt => 'a set => 'b => 'b"
@@ -50,7 +50,7 @@
locale abelian_group = abelian_monoid +
assumes a_comm_group:
- "comm_group (| carrier = carrier G, mult = add G, one = zero G |)"
+ "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
subsection {* Basic Properties *}
@@ -87,11 +87,11 @@
intro: assms)
lemma (in abelian_monoid) a_monoid:
- "monoid (| carrier = carrier G, mult = add G, one = zero G |)"
+ "monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
by (rule comm_monoid.axioms, rule a_comm_monoid)
lemma (in abelian_group) a_group:
- "group (| carrier = carrier G, mult = add G, one = zero G |)"
+ "group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
by (simp add: group_def a_monoid)
(simp add: comm_group.axioms group.axioms a_comm_group)
@@ -100,10 +100,10 @@
text {* Transfer facts from multiplicative structures via interpretation. *}
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"
+ add!: monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
+ where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
+ and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
+ and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
by (rule a_monoid) auto
context abelian_monoid begin
@@ -118,11 +118,11 @@
end
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"
+ add!: comm_monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
+ where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
+ and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
+ and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
+ and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
by (rule a_comm_monoid) (auto simp: finsum_def)
context abelian_monoid begin
@@ -173,14 +173,15 @@
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"
+ add!: group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
+ where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
+ and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
+ and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
+ and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
by (rule a_group) (auto simp: m_inv_def a_inv_def)
-context abelian_group begin
+context abelian_group
+begin
lemmas a_inv_closed = add.inv_closed
@@ -200,12 +201,12 @@
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"
+ add!: comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
+ where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
+ and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
+ and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
+ and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
+ and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = 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
--- a/src/HOL/Algebra/UnivPoly.thy Wed Mar 05 20:07:43 2014 +0100
+++ b/src/HOL/Algebra/UnivPoly.thy Wed Mar 05 21:51:30 2014 +0100
@@ -57,7 +57,7 @@
where "up R = {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero>\<^bsub>R\<^esub> n f)}"
definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
- where "UP R = (|
+ where "UP R = \<lparr>
carrier = up R,
mult = (%p:up R. %q:up R. %n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),
one = (%i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),
@@ -65,7 +65,7 @@
add = (%p:up R. %q:up R. %i. p i \<oplus>\<^bsub>R\<^esub> q i),
smult = (%a:carrier R. %p:up R. %i. a \<otimes>\<^bsub>R\<^esub> p i),
monom = (%a:carrier R. %n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),
- coeff = (%p:up R. %n. p n) |)"
+ coeff = (%p:up R. %n. p n)\<rparr>"
text {*
Properties of the set of polynomials @{term up}.
@@ -1814,7 +1814,7 @@
definition
INTEG :: "int ring"
- where "INTEG = (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
+ where "INTEG = \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"
lemma INTEG_cring: "cring INTEG"
by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI