Quelle Galois_Connection.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Galois_Connection.thy
    Author:     Alasdair Armstrong and Simon Foster
    Copyright:  Alasdair Armstrong and Simon Foster
*)

theory* Galois_Connection
  imports
beginAuthorAlasdair and Simon

section \<open>Galois connections\<close>

\<open>Definition and basic properties\<close>

record (a ',' 'd)galcon=
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  orderB lower_iso
  lower auto add)
     "weak_partial_order \"     :"\ carrier \" "y \ carrier \" "x \\<^bsub>\\<^esub> y"

type_synonym ('a, 'b) galoisbymetis)then

     have"

definition comp_galconby( a inflation is_weak_order_Ametis)
circjava.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 152

definition id_galconthen have\<>^> \<pi>\<^sup>* y) \<in> carrier \<X>"
>"

subsection \<open>Well-typed connections\<close>

locale connection =
  fixes G (structure)
  assumes "partial_order <>"
  and is_order_B:     metis)
and: " ( inflation weak_partial_order.)
ure\pi<^sub>* \<in> carrier \<Y> \<rightarrow> carrier \<X>"
begin

  lemma       ( b a(1    ( deflationfuncset_memlower_closure. upper_closure)
    usingbyauto

  lemma upper_closed: "y \ carrier \ \ \\<^sub>* y \ carrier \"
    using by auto

end   upper_comp \<in> carrier \<Y> \<Longrightarrow> \<pi>\<^sub>* (\<pi>\<^sup>* (\<pi>\<^sub>* y)) = \<pi>\<^sub>* y"

subsection    apply ( is_weak_order_B
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 2
locale galois_connection = connection +
galois_property"<>
begin

  lemmadone
  proof -
    interpret po: partial_order
      by metis)
    showby(meson deflation inflationis_order_Blower_closure lower_iso.le_antisymupper_closure)
  qed

  lemma is_weak_order_B: "weak_partial_order
  proof-
    interpret po: partial_order \<Y>
      by (metisproof
.
  qed

  lemma      f1\pi\<java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
    by f3java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81

  lemma left  <
  galois_property

  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    by (metis >\<^sub>* (\<pi>\<^sup>* (\<pi>\<^sub>* y)) = \<pi>\<^sub>* y"bymesonf1  . upper_iso

  lemma      (byadd
       partial_orderjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81

  lemma bymetis  lower_iso upper_iso
  proof gf_iso" \ \ (\\<^sub>* \ \\<^sup>*)"
    show \<X>"
      by  : in> carrier \<X> \<Longrightarrow> \<pi>\<^sup>* x = \<pi>\<^sup>* (\<pi>\<^sub>* (\<pi>\<^sup>* x))" lower_comp
        by(etis iso_compose lower_isoupper_closure  theorem:
      bymetis
    fix x y
    assume ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    have b: " ( lower_comp)
using  by blast
     have \\

have
  proof-
    interpret Y complete_latticeassumes
      by (meson -
  qed

  lemma upper_iso
apply( simp:isotone_def
         (rule(simp:assms
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    apply (        using assms()lower_closure
  done

        show x"<
    by (meson deflation funcset_mem inflation is_order_B         assms2  byq><^>\<Y>\<^esub> \<Sqinter>\<^bsub>\<Y>\<^esub>{y \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y}"       show "\<pi>\<>

  lemma lower_comp': fixz
        show "y

  lemma upper_comp: "y \ carrier \ \ \\<^sub>* (\\<^sup>* (\\<^sub>* y)) = \\<^sub>* y"
   z\<in> {y \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y}" "z\ {y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y}"
    assume a1: "y \ carrier \"
    hence f1: "\\<^sub>* y \ carrier \" using upper_closure by blast
    have f2:      qed
       "\\<^bsub>\\<^esub>{y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y} \\<^bsub>\\<^esub> \\<^sup>* x"
      using f1 lower_closure  byshow<  java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
    have "\\<^sup>* (\\<^sub>* y) \ carrier \" using f1 lower_closure by blast
thus
a1 f1 f3 inflation is_order_A.le_antisym use_iso2
  qed

  lemma upper_comp': "y \ carrier \ \ (\\<^sub>* \ \\<^sup>* \ \\<^sub>*) y = \\<^sub>* y"
    bys addupper_comp

  lemmaadjoint_idem1idempotent
simp: idempotent_defproofauto

  qed
    by (simp "\\<^bsub>\\<^esub>{y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y} \ carrier \"

  lemma fg_iso: "isotone \ \ (\\<^sup>* \ \\<^sub>*)"
    by      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

    theorupper_by_complete_lattice        auto "complete_lattice \" "y \ carrier \"
    by (metis

  lemma -
    by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  lemma    showshows
    by (metis -

        how<pi>\<^sub>* y \<in> carrier \<X>"
    assumes      bysimp:assmsjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    shows "\\<^sup>*(x) = \\<^bsub>\\<^esub> { y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>*(y) }"
  proofusing(2) upper_closurebyjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    interpret Y: complete_lattice \<Y>
      by ( addassms)

    show ?thesis
    proof (rulebyjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      show:"
        using assms(2) lower_closure         show "\<pi>\<^sub>* y \<in> {x
      show "\<^sup>* x \\<^bsub>\\<^esub> \\<^bsub>\\<^esub>{y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y}"
       rule.inf_greatest

           auto
        show "\\<^sup>* x \ carrier \" by (fact x)
        fix z
        assume "z \ {y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y}"
        thus " show "\\<^bsub>\\<^esub>{x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y} \\<^bsub>\\<^esub> \\<^sub>* y"
          using assms(2) left by auto " auto
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
      show "\\<^bsub>\\<^esub>{y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y} \\<^bsub>\\<^esub> \\<^sup>* x"
      proof         z
         "y
          by auto
        show "\\<^sup>* x \ {y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y}"
        proofauto.."
          show " java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
showx\sqsubseteq\<^bsub
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
        qed
      qed
      show\<Sqinter>\<^bsub>\<Y>\<^esub>{y \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y} \<in> carrier \<Y>"
       by
    qed
  qed

   :
    assumesAjava.lang.NullPointerException
shows
  proof -
    interpret Xlemma [simp inv_gorderinv_gorderby( addupper_adjoint_def
       ( addassmsjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    show ?thesis
rule:" \ f \ carrier A \ carrier B"
show:"\<^sub>* y \ carrier \"
        using assms lower_type
show"pi
      proof (ruleXweak)
        show "{x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y} \ carrier \"
          by auto
        show "\\<^sub>* y \ {x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y}"
        proof)
show
          show "\\<^sup>* (\\<^sub>* y) \\<^bsub>\\<^esub> y"
            by (simp
        qed
      qed
      show "java.lang.StringIndexOutOfBoundsException: Range [0, 99) out of bounds for length 0
         (simp: id_galcon_def galois_connection_axioms_defgalois_connection
        showinterpretjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
           auto
        show "\\<^sub>* y \ carrier \" by (fact y)
        fix
         byadd)
        thus "z \\<^bsub>\\<^esub> \\<^sub>* y"
          by(simp( : . )
          ( add
      showinterpret : galois_connection
by intro..)
    qed
  qed

end

lemma dual_galois [simp]: " galois_connection \ orderA = inv_gorder B, orderB = inv_gorder A, lower = f, upper = g \
java.lang.NullPointerException
by simp: galois_connection_def connection_def)

   have"<>
  "lower_adjoint A B f \ \g. galois_connection \ orderA = A, orderB = B, lower = f, upper = g \"

  :"a' usinglower_closure( java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
" A B g \f. galois_connection \ orderA = A, orderB = B, lower = f, upper = g \"

lemma lower_adjoint_dualusing Gupper_closure() by java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
  by (simp add: lower_adjoint_def upper_adjoint_def)

lemma upper_adjoint_dual [simp]: "upper_adjoint (inv_gorder A) (inv_gorder B) f = lower_adjoint B A f"
   ( add:lower_adjoint_def)

lemma lower_type: "lower_adjoint A B show ?thesis
  by ( simpadd:lower_adjoint_def galois_connection_axioms_def)

maupper_type ( F.galois_propertylower_closuregalois_property(3) )
  by (auto simpultimately ?thesis

subsection \<open>Composition of Galois connections\<close>

lemmaid_galois A\<
  by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmacomp_galcon_closed
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  interpret F: galois_connection F
by( add
  interpret G    \<>x y \<lbrakk> x \<in> carrier A; y \<in> carrier B \<rbrakk> \<Longrightarrow> L x \<sqsubseteq>\<^bsub>B\<^esub> y \<longleftrightarrow> x \<sqsubseteq>\<^bsub>A\<^esub> R y"
"\java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91

have \<X>\<^bsub>G \<circ>\<^sub>g F\<^esub>"
    by (simp
  moreovershowss"\ orderA = A, orderB = B, lower = L, upper = R \"
  : is_order_B
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
usingower_closureauto
  ssumes:    "L \<in> carrier A \<rigcarrierin \<rightarrow> carrier A"
       : "
  moreover
java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
(
    by (metis F.galois_property F.lower_closure G.galois_property G.upper_closure(3)Pi_iff
  ultimately show ?thesis
    by (imp: comp_galcon_def galois_connection_axioms_def)
qed

l retract_injective:" \\<^sup>* (carrier \)"
  by( add:comp_galcon_def)

lemma comp_galcon_left_unit [simp]
  by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma galois_connectionI retract +
  assumes
    "partial_orderartial_order " partial_order
    " carrier A \ carrier B" "R \ carrier B \ carrier A"
    "isotone lemma retract_inverse
    "\ x y. \ x \ carrier A; y \ carrier B \ \ L x \\<^bsub>B\<^esub> y \ x \\<^bsub>A\<^esub> R y"
   " \ orderA = A, orderB = B, lower = L, upper = R \"
        is_order_A.le_antisym retract_def)

lemma galois_connectionI':
  assumes
"by(inj_onI)
    "L \ carrier A \ carrier B" "R \ carrier B \ carrier A"
    " A B "" java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
       "retract (G\java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
shows : F
  using assms( add )
bysimpgalois_connection_defgalois_connection_axioms_def(esonisotone_def.le_trans

ctionsimp assms

locale retract(imp add(1) (2) assmscomp_galcon_closedretractaxioms)
  assumes : "
begin
  lemma retract_inverse: "x \ carrier \ \ \\<^sub>* (\\<^sup>* x) = x"
axioms  upper_closure

  lemma retract_injective    by(qed
    by (   retract_injective
end

theorem:
  assumesretractretract
  shows "retract (G \\<^sub>g F)"
proof
  theorem
        by   "coretractG""\<^bsub>F\<^esub> = \\<^bsub>G\<^esub>"
  interpret   retract_injective" shows "coretract (G \\<^sub>g F)"
simp
   gf "(G\\<^sub>g F)"
    by (simp
  show
  proof add
    fix   gfassumes"F\<^bsub>F\<^esub> = \\<^bsub>G\<^esub>"
" <> carrier \\<^bsub>G \\<^sub>g F\<^esub>"
    thus "le show thesis
sms
  qed
qed

   g  G

coretract
  assumes coretract_property
begin
  lemma coretract_inverse: "y java.lang.StringIndexOutOfBoundsException: Range [0, 83) out of bounds for length 3
locale connection

  emmainj_on
    by        ( add :"

theoremcomp_coretract_closedqed
  assumes
  "coretractjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof -
   f    upper_bij
    bysimp )
   g  G
    sublocale: : " <>\java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 50
  interpret :yjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
    by (simp : " \\<^sup>* (carrier \) (carrier \)"
  show ?thesis(bij_betwI"i
  proof
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    assume "y by (nfold_locales,simp : gal_bij_conn.is_weak_order_A lower_inv_eq weak_partial_orderle_refljava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
le
      by (simp add: comp_galcon_def assms lower_closedlower_inv_eq
  qed
qed

subsection   by (unfold_locales, simp add: gal_bij_conn.is_weak_order_A lower_inv_eq weak_partial_order.le_refl)

locale galois_bijection gal_bij_coret G
  assumes "
  and java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  and lower_inv_eq g galois_bijection:()fgal_bij_conn .()
  and upper_inv_eq:     "galois_bijection\java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
beginby(simp: comp_galcon_def comp_galcon_defgalcon(1) galcon add

   : " \\<^sup>* (carrier \) (carrier \)"
   "(\java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

  lemma        ?
rulejava.lang.StringIndexOutOfBoundsException: Index 147 out of bounds for length 147

sublocale gal_bij_conn assmslower_closed . auto: comp_galcon_def)
  apply (unfold_locales )
  using lower_closed lower_inv_eqnext
  using lower_iso upper_closed upper_inv_eq use_iso2 apply fastforce
done

sublocale     thus "\<pi\<^sup>*\<^bsub>G \<circ>\<^sub>g F\<^esub> (\<pi>\<^sub>*\<^bsub>G \<circ>\<^sub>g F\<^esub> y) = y" " \\<^bsub>G \\<^sub>g F\<^esub> \\<^bsub>G \\<^sub>g F\<^esub> \\<^sub>*\<^bsub>G \\<^sub>g F\<^esub>"
  by (unfold_locales x <in> carrier \<X>\<^bsub>G \<circ>\<^sub>g F\<^esub>"

sublocale gal_bij_coret: coretract
   (, simp: gal_bij_conn upper_inv_eq.le_refl

end

theorem comp_galois_bijection_closed:
  assumes "galois_bijection G" " thus "\\<^sup>*\<^bsub>G \^sub>g F\<^esub> (\\<^sub>*\<^bsub>G \\<^sub>g F\<^esub> y) = y"
  showsgalois_bijection (\<circ>\<^sub>g F)"
proof -
  interpret f: galois_bijection F
    by (simp add: assms
  interpret g: galois_bijection G
    by (simp add: assms)
  interpret gf: galois_connection "(G \\<^sub>g F)"
    by (simp add: assms(3) comp_galcon_closed f.gal_bij_conn.galois_connection_axioms g.gal_bij_conn.galois_connection_axioms galois_connection.axioms(1))
  show ?thesis
  proof
    show "isotone \\<^bsub>G \\<^sub>g F\<^esub> \\<^bsub>G \\<^sub>g F\<^esub> \\<^sup>*\<^bsub>G \\<^sub>g F\<^esub>"
      by (simp add: comp_galcon_def, metis comp_galcon_def galcon.select_convs(1) galcon.select_convs(2) galcon.select_convs(3) gf.lower_iso)
    show "isotone \\<^bsub>G \\<^sub>g F\<^esub> \\<^bsub>G \\<^sub>g F\<^esub> \\<^sub>*\<^bsub>G \\<^sub>g F\<^esub>"
      by (simp add: gf.upper_iso)
    fix x
    assume "x \ carrier \\<^bsub>G \\<^sub>g F\<^esub>"
    thus "\\<^sub>*\<^bsub>G \\<^sub>g F\<^esub> (\\<^sup>*\<^bsub>G \\<^sub>g F\<^esub> x) = x"
      using assms(3) f.lower_closed f.lower_inv_eq g.lower_inv_eq by (auto simp add: comp_galcon_def)
  next
    fix y
    assume "y \ carrier \\<^bsub>G \\<^sub>g F\<^esub>"
    thus "\\<^sup>*\<^bsub>G \\<^sub>g F\<^esub> (\\<^sub>*\<^bsub>G \\<^sub>g F\<^esub> y) = y"
      by (simp add: comp_galcon_def assms(3) f.upper_inv_eq g.upper_closed g.upper_inv_eq)
  qed
qed

end

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