author wenzelm Wed Mar 05 21:51:30 2014 +0100 (2014-03-05) changeset 55926 3ef14caf5637 parent 55925 56165322c98b child 55927 30c41a8eca0e
more symbols;
 src/HOL/Algebra/AbelCoset.thy file | annotate | diff | revisions src/HOL/Algebra/Group.thy file | annotate | diff | revisions src/HOL/Algebra/IntRing.thy file | annotate | diff | revisions src/HOL/Algebra/Lattice.thy file | annotate | diff | revisions src/HOL/Algebra/Ring.thy file | annotate | diff | revisions src/HOL/Algebra/UnivPoly.thy file | annotate | diff | revisions
     1.1 --- a/src/HOL/Algebra/AbelCoset.thy	Wed Mar 05 20:07:43 2014 +0100
1.2 +++ b/src/HOL/Algebra/AbelCoset.thy	Wed Mar 05 21:51:30 2014 +0100
1.3 @@ -54,8 +54,8 @@
1.4  locale abelian_group_hom = G: abelian_group G + H: abelian_group H
1.5      for G (structure) and H (structure) +
1.6    fixes h
1.7 -  assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
1.8 -                                  (| carrier = carrier H, mult = add H, one = zero H |) h"
1.9 +  assumes a_group_hom: "group_hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
1.10 +                                  \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"
1.11
1.12  lemmas a_r_coset_defs =
1.13    a_r_coset_def r_coset_def
1.14 @@ -129,12 +129,12 @@
1.15      folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
1.16
1.17  lemma (in abelian_group) a_coset_join1:
1.18 -     "[| H +> x = H;  x \<in> carrier G;  subgroup H (|carrier = carrier G, mult = add G, one = zero G|) |] ==> x \<in> H"
1.19 +     "[| H +> x = H;  x \<in> carrier G;  subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> x \<in> H"
1.20  by (rule group.coset_join1 [OF a_group,
1.21      folded a_r_coset_def, simplified monoid_record_simps])
1.22
1.23  lemma (in abelian_group) a_solve_equation:
1.24 -    "\<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"
1.25 +    "\<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"
1.26  by (rule group.solve_equation [OF a_group,
1.27      folded a_r_coset_def, simplified monoid_record_simps])
1.28
1.29 @@ -535,8 +535,8 @@
1.30  lemma abelian_group_homI:
1.31    assumes "abelian_group G"
1.32    assumes "abelian_group H"
1.33 -  assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
1.34 -                                  (| carrier = carrier H, mult = add H, one = zero H |) h"
1.35 +  assumes a_group_hom: "group_hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
1.36 +                                  \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"
1.37    shows "abelian_group_hom G H h"
1.38  proof -
1.39    interpret G: abelian_group G by fact
1.40 @@ -636,7 +636,7 @@
1.41  theorem (in abelian_group_hom) A_FactGroup_iso:
1.42    "h  carrier G = carrier H
1.43     \<Longrightarrow> (\<lambda>X. the_elem (hX)) \<in> (G A_Mod (a_kernel G H h)) \<cong>
1.44 -          (| carrier = carrier H, mult = add H, one = zero H |)"
1.45 +          \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr>"
1.46  by (rule group_hom.FactGroup_iso[OF a_group_hom,
1.47      folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
1.48

     2.1 --- a/src/HOL/Algebra/Group.thy	Wed Mar 05 20:07:43 2014 +0100
2.2 +++ b/src/HOL/Algebra/Group.thy	Wed Mar 05 21:51:30 2014 +0100
2.3 @@ -721,7 +721,7 @@
2.4  text_raw {* \label{sec:subgroup-lattice} *}
2.5
2.6  theorem (in group) subgroups_partial_order:
2.7 -  "partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
2.8 +  "partial_order \<lparr>carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq>\<rparr>"
2.9    by default simp_all
2.10
2.11  lemma (in group) subgroup_self:
2.12 @@ -729,7 +729,7 @@
2.13    by (rule subgroupI) auto
2.14
2.15  lemma (in group) subgroup_imp_group:
2.16 -  "subgroup H G ==> group (G(| carrier := H |))"
2.17 +  "subgroup H G ==> group (G\<lparr>carrier := H\<rparr>)"
2.18    by (erule subgroup.subgroup_is_group) (rule group_axioms)
2.19
2.20  lemma (in group) is_monoid [intro, simp]:
2.21 @@ -737,7 +737,7 @@
2.22    by (auto intro: monoid.intro m_assoc)
2.23
2.24  lemma (in group) subgroup_inv_equality:
2.25 -  "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
2.26 +  "[| subgroup H G; x \<in> H |] ==> m_inv (G \<lparr>carrier := H\<rparr>) x = inv x"
2.27  apply (rule_tac inv_equality [THEN sym])
2.28    apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
2.29   apply (rule subsetD [OF subgroup.subset], assumption+)
2.30 @@ -766,7 +766,7 @@
2.31  qed
2.32
2.33  theorem (in group) subgroups_complete_lattice:
2.34 -  "complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"
2.35 +  "complete_lattice \<lparr>carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq>\<rparr>"
2.36      (is "complete_lattice ?L")
2.37  proof (rule partial_order.complete_lattice_criterion1)
2.38    show "partial_order ?L" by (rule subgroups_partial_order)
2.39 @@ -784,7 +784,7 @@
2.40      fix H
2.41      assume H: "H \<in> A"
2.42      with L have subgroupH: "subgroup H G" by auto
2.43 -    from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
2.44 +    from subgroupH have groupH: "group (G \<lparr>carrier := H\<rparr>)" (is "group ?H")
2.45        by (rule subgroup_imp_group)
2.46      from groupH have monoidH: "monoid ?H"
2.47        by (rule group.is_monoid)

     3.1 --- a/src/HOL/Algebra/IntRing.thy	Wed Mar 05 20:07:43 2014 +0100
3.2 +++ b/src/HOL/Algebra/IntRing.thy	Wed Mar 05 21:51:30 2014 +0100
3.3 @@ -23,7 +23,7 @@
3.4
3.5  abbreviation
3.6    int_ring :: "int ring" ("\<Z>") where
3.7 -  "int_ring == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
3.8 +  "int_ring == \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"
3.9
3.10  lemma int_Zcarr [intro!, simp]:
3.11    "k \<in> carrier \<Z>"
3.12 @@ -183,27 +183,27 @@
3.13    by simp_all
3.14
3.15  interpretation int (* FIXME [unfolded UNIV] *) :
3.16 -  partial_order "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
3.17 -  where "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
3.18 -    and "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
3.19 -    and "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
3.20 +  partial_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
3.21 +  where "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
3.22 +    and "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
3.23 +    and "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
3.24  proof -
3.25 -  show "partial_order (| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
3.26 +  show "partial_order \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
3.27      by default simp_all
3.28 -  show "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
3.29 +  show "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
3.30      by simp
3.31 -  show "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
3.32 +  show "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
3.33      by simp
3.34 -  show "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
3.35 +  show "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
3.36      by (simp add: lless_def) auto
3.37  qed
3.38
3.39  interpretation int (* FIXME [unfolded UNIV] *) :
3.40 -  lattice "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
3.41 -  where "join (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = max x y"
3.42 -    and "meet (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = min x y"
3.43 +  lattice "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
3.44 +  where "join \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = max x y"
3.45 +    and "meet \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = min x y"
3.46  proof -
3.47 -  let ?Z = "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
3.48 +  let ?Z = "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
3.49    show "lattice ?Z"
3.50      apply unfold_locales
3.51      apply (simp add: least_def Upper_def)
3.52 @@ -225,7 +225,7 @@
3.53  qed
3.54
3.55  interpretation int (* [unfolded UNIV] *) :
3.56 -  total_order "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
3.57 +  total_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
3.58    by default clarsimp
3.59
3.60

     4.1 --- a/src/HOL/Algebra/Lattice.thy	Wed Mar 05 20:07:43 2014 +0100
4.2 +++ b/src/HOL/Algebra/Lattice.thy	Wed Mar 05 21:51:30 2014 +0100
4.3 @@ -1278,7 +1278,7 @@
4.4  subsubsection {* The Powerset of a Set is a Complete Lattice *}
4.5
4.6  theorem powerset_is_complete_lattice:
4.7 -  "complete_lattice (| carrier = Pow A, eq = op =, le = op \<subseteq> |)"
4.8 +  "complete_lattice \<lparr>carrier = Pow A, eq = op =, le = op \<subseteq>\<rparr>"
4.9    (is "complete_lattice ?L")
4.10  proof (rule partial_order.complete_latticeI)
4.11    show "partial_order ?L"

     5.1 --- a/src/HOL/Algebra/Ring.thy	Wed Mar 05 20:07:43 2014 +0100
5.2 +++ b/src/HOL/Algebra/Ring.thy	Wed Mar 05 21:51:30 2014 +0100
5.3 @@ -19,7 +19,7 @@
5.4
5.5  definition
5.6    a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _" [81] 80)
5.7 -  where "a_inv R = m_inv (| carrier = carrier R, mult = add R, one = zero R |)"
5.8 +  where "a_inv R = m_inv \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"
5.9
5.10  definition
5.11    a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)
5.12 @@ -28,11 +28,11 @@
5.13  locale abelian_monoid =
5.14    fixes G (structure)
5.15    assumes a_comm_monoid:
5.16 -     "comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"
5.17 +     "comm_monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.18
5.19  definition
5.20    finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
5.21 -  "finsum G = finprod (| carrier = carrier G, mult = add G, one = zero G |)"
5.22 +  "finsum G = finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.23
5.24  syntax
5.25    "_finsum" :: "index => idt => 'a set => 'b => 'b"
5.26 @@ -50,7 +50,7 @@
5.27
5.28  locale abelian_group = abelian_monoid +
5.29    assumes a_comm_group:
5.30 -     "comm_group (| carrier = carrier G, mult = add G, one = zero G |)"
5.31 +     "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.32
5.33
5.34  subsection {* Basic Properties *}
5.35 @@ -87,11 +87,11 @@
5.36      intro: assms)
5.37
5.38  lemma (in abelian_monoid) a_monoid:
5.39 -  "monoid (| carrier = carrier G, mult = add G, one = zero G |)"
5.40 +  "monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.41  by (rule comm_monoid.axioms, rule a_comm_monoid)
5.42
5.43  lemma (in abelian_group) a_group:
5.44 -  "group (| carrier = carrier G, mult = add G, one = zero G |)"
5.45 +  "group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.46    by (simp add: group_def a_monoid)
5.47      (simp add: comm_group.axioms group.axioms a_comm_group)
5.48
5.49 @@ -100,10 +100,10 @@
5.50  text {* Transfer facts from multiplicative structures via interpretation. *}
5.51
5.52  sublocale abelian_monoid <
5.53 -  add!: monoid "(| carrier = carrier G, mult = add G, one = zero G |)"
5.54 -  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
5.55 -    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
5.56 -    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
5.57 +  add!: monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.58 +  where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
5.59 +    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
5.60 +    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
5.61    by (rule a_monoid) auto
5.62
5.63  context abelian_monoid begin
5.64 @@ -118,11 +118,11 @@
5.65  end
5.66
5.67  sublocale abelian_monoid <
5.68 -  add!: comm_monoid "(| carrier = carrier G, mult = add G, one = zero G |)"
5.69 -  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
5.70 -    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
5.71 -    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
5.72 -    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"
5.73 +  add!: comm_monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.74 +  where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
5.75 +    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
5.76 +    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
5.77 +    and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
5.78    by (rule a_comm_monoid) (auto simp: finsum_def)
5.79
5.80  context abelian_monoid begin
5.81 @@ -173,14 +173,15 @@
5.82  end
5.83
5.84  sublocale abelian_group <
5.85 -  add!: group "(| carrier = carrier G, mult = add G, one = zero G |)"
5.86 -  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
5.87 -    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
5.88 -    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
5.89 -    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"
5.90 +  add!: group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.91 +  where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
5.92 +    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
5.93 +    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
5.94 +    and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
5.95    by (rule a_group) (auto simp: m_inv_def a_inv_def)
5.96
5.97 -context abelian_group begin
5.98 +context abelian_group
5.99 +begin
5.100
5.102
5.103 @@ -200,12 +201,12 @@
5.104  end
5.105
5.106  sublocale abelian_group <
5.107 -  add!: comm_group "(| carrier = carrier G, mult = add G, one = zero G |)"
5.108 -  where "carrier (| carrier = carrier G, mult = add G, one = zero G |) = carrier G"
5.109 -    and "mult (| carrier = carrier G, mult = add G, one = zero G |) = add G"
5.110 -    and "one (| carrier = carrier G, mult = add G, one = zero G |) = zero G"
5.111 -    and "m_inv (| carrier = carrier G, mult = add G, one = zero G |) = a_inv G"
5.112 -    and "finprod (| carrier = carrier G, mult = add G, one = zero G |) = finsum G"
5.113 +  add!: comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
5.114 +  where "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
5.115 +    and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
5.116 +    and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
5.117 +    and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
5.118 +    and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
5.119    by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def)
5.120

     6.1 --- a/src/HOL/Algebra/UnivPoly.thy	Wed Mar 05 20:07:43 2014 +0100
6.2 +++ b/src/HOL/Algebra/UnivPoly.thy	Wed Mar 05 21:51:30 2014 +0100
6.3 @@ -57,7 +57,7 @@
6.4    where "up R = {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero>\<^bsub>R\<^esub> n f)}"
6.5
6.6  definition UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
6.7 -  where "UP R = (|
6.8 +  where "UP R = \<lparr>
6.9     carrier = up R,
6.10     mult = (%p:up R. %q:up R. %n. \<Oplus>\<^bsub>R\<^esub>i \<in> {..n}. p i \<otimes>\<^bsub>R\<^esub> q (n-i)),
6.11     one = (%i. if i=0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>),
6.12 @@ -65,7 +65,7 @@
6.13     add = (%p:up R. %q:up R. %i. p i \<oplus>\<^bsub>R\<^esub> q i),
6.14     smult = (%a:carrier R. %p:up R. %i. a \<otimes>\<^bsub>R\<^esub> p i),
6.15     monom = (%a:carrier R. %n i. if i=n then a else \<zero>\<^bsub>R\<^esub>),
6.16 -   coeff = (%p:up R. %n. p n) |)"
6.17 +   coeff = (%p:up R. %n. p n)\<rparr>"
6.18
6.19  text {*
6.20    Properties of the set of polynomials @{term up}.
6.21 @@ -1814,7 +1814,7 @@
6.22
6.23  definition
6.24    INTEG :: "int ring"
6.25 -  where "INTEG = (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
6.26 +  where "INTEG = \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"
6.27
6.28  lemma INTEG_cring: "cring INTEG"
6.29    by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI