Quellcode-Bibliothek Galois_Connection.thy Sprache: Isabelle

    Author Galois_Connectiontheory
          Armstrong
    Copyrightsection
*)

subsection
  imports ',b c,'
begin

section \<open>Galois connections\<close>

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

record ('a, 'b, 'c, 'd) galcon =
  orderA :: "('a, 'c) gorder_scheme" (\<open>\<X>\<index>\<close>)
  orderB :: "('b, 'd) gorder_scheme" (\<open>\<Y>\<index>\<close>)
  lower  :: "'a \ 'b" (\\\<^sup>*\\)
  upper  :: "'b \ 'a" (\\\<^sub>*\\)

type_synonym ('a, 'b) galois = "('a, 'b, unit, unit) galcon"

abbreviation "inv_galcon G \ \ orderA = inv_gorder \\<^bsub>G\<^esub>, orderB = inv_gorder \\<^bsub>G\<^esub>, lower = upper G, upper = lower G \"

definition comp_galcon :: "('b, 'c) galois \ ('a, 'b) galois \ ('a, 'c) galois" (infixr \\\<^sub>g\ 85)
  where "G \\<^sub>g F = \ orderA = orderA F, orderB = orderB G, lower = lower G \ lower F, upper = upper F \ upper G \"

\<open>Definition and basic properties\<close> (a,',' 'd =
java.lang.NullPointerException

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

locale =
  fixes  structure
  assumes: " \"
and: "partial_order \"
  and lower_closure: "\\<^sup>* \ carrier \ \ carrier \"
  and upper_closure: "\\<^sub>* \ carrier \ \ carrier \"
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

  lemma lower_closed: "x \ carrier \ \ \\<^sup>* x \ carrier \"
    usinglower_closure

  lemmaupper_closedy
    using upper_closure by auto  :: "' \ ('a, 'a) galois" (\I\<^sub>g\) where

end

subsection connection G ()

locale   is_order_A <"
  assumes galois_property: "\x \ carrier \; y \ carrier \\ \ \\<^sup>* x \\<^bsub>\\<^esub> y \ x \\<^bsub>\\<^esub> \\<^sub>* y"
begin

  lemma : java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
  proof -
        using lowjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
      by (metis     upper_closure
    show ?thesis ..
  qed

  lemma

     \<open>Galois connections\<close>java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
       metis)
    show ?thesis
  qed

  lemma right: "\x \ carrier \; y \ carrier \; \\<^sup>* x \\<^bsub>\\<^esub> y\ \ x \\<^bsub>\\<^esub> \\<^sub>* y"
by galois_property

?thesisjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
by galois_property

by Pi_iff left weak_partial_order)

  lemma     metis)
bymetis, lifting galois_connection is_weak_order_B.le_refl

   : "
  proof (auto addisotone_def
    show "weak_partial_order \"
      by (metis metis is_weak_order_A upper_closurele_refl
show
      by (metis is_weak_order_B( (no_types)PiE.right is_weak_order_B .le_refl
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assume"
    have b: "\\<^sup>* y \ carrier \"
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
      "\\<^sub>* (\\<^sup>* y) \ carrier \"
       upper_closure blast
then "x\\<^bsub>\\<^esub> \\<^sub>* (\\<^sup>* y)"
       mesoninflationis_weak_order_A ( is_weak_order_B
assume:circ><^sub>g F = \<lparr> orderA = orderA F, orderB = orderB G, lower = lower G \<circ> lower F, upper = upper F \<circ> upper G \<rparr>"
      by mesonb () Pi_iff lower_closure)rparr
  qed

  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    apply(  assumes is_order_Apartial_order
    apply( is_weak_order_B
   lower_closure mesonis_weak_order_Ale_trans
    apply metis"><^sub>* \ carrier \ \ carrier \"
  done

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
bydeflationinflation  lower_isole_antisym use_iso2

  lemma  qed lower_closurejava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
    by (simp add:  upper_closure

  lemma: "y

      lemmaupper_iso"isotone\ \ \\<^sub>*"
    hence f1\<pi>\<^sub>* y \<in> carrier \<X>" using upper_closure by blast
    have f2 "\\<^sup>* (\\<^sub>* y) \\<^bsub>\\<^esub> y" using a1 deflation by blast
    have  assumes: \lbrakkx \<in> carrier \<X>; y \<in> carrier \<Y>\<rbrakk> \<Longrightarrow> \<pi>\<^sup>* x \<sqsubseteq>\<^bsub>\<Y>\<^esub> y \<longleftrightarrow> x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y" \<X>
      using java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 9
"pi\<^>* (\\<^sub>* y) \ carrier \" using f1 lower_closure by blast
    thushavejava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
      by (meson    by(metis)
  qed

  lemma upper_comp': "y \ carrier \ \ (\\<^sub>* \ \\<^sup>* \ \\<^sub>*) y = \\<^sub>* y"
    by       by( a1f2 inflationis_order_Ale_antisym use_iso2

lemma: "idempotent (\<^sup>* \ \\<^sub>*)"
by(    ( : upper_comp

lemma     ( add:idempotent_def partial_ordereq_is_equal)
    by (simp ddidempotent_def is_order_A.eq_is_equal)

  lemma     ( add: idempotent_def partial_order
     ( iso_composelower_closure upper_closure)

  lemmagf_iso: isotone
    by (metis iso_compose er

lemmasemi_inverse1x\<
    by (metis)

lemma: "x \ carrier \ \ \\<^sub>* x = \\<^sub>* (\\<^sup>* (\\<^sub>* x))"
    by (metis)

lower_by_complete_lattice
    assumes "complete_lattice ( is_weak_order_B)
      semi_inverse1" java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
-
interpret:complete_lattice complete_lattice
      by (simp "\\<^sup>*(x) = \\<^bsub>\\<^esub> { y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>*(y) }"

     auto add)
    proofrulebysimp )
      show \<pi>
        using assms2  by blast:<pi>\<^sup>* x \<in> carrier \<Y>"
>bsub
      proof (rule>*x\<sqsubseteq>\<^bsub>\<Y>\<^esub> \<Sqinter>\<^bsub>\<Y>\<^esub>{y \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y}"
         "{ \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y} \ carrier \"
          by auto
        show "
         java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

        thus "\\<^sup>* x \\<^bsub>\\<^esub> z"
          using(2) leftjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
qed
show\<Sqinter>\<^bsub>\<Y>\<^esub>{y \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y} \<sqsubseteq>\<^bsub>\<Y>\<^esub> \<pi>\<^sup>* x"upper_closure        "y\in> carrier\. x \\<^bsub>\\<^esub> \\<^sub>* y} \ carrier \"
      proof (     "\\<^sub>* (\\<^sup>* (\\<^sub>* y)) = \\<^sub>* y"
          java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

: " \ (\\<^sup>* \ \\<^sub>*)"
         ()
          show

            using assms(2) inflation by blast
        qed
      java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by( assumes\<X>" "y \<in> carrier \<Y>"
    qed
  -

  theorem upper_by_complete_lattice:
assumes
   "\\<^sub>*(y) = \\<^bsub>\\<^esub> { x \ carrier \. \\<^sup>*(x) \\<^bsub>\\<^esub> y }"
  proof
      s y: java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
       ( add assms)
    show ?
    proof (rule X.le_antisym)
      show y: "\\<^sub>* y \ carrier \"
         ssms upper_closure blast
show"\<^sub>* y \\<^bsub>\\<^esub> \\<^bsub>\\<^esub>{x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y}"
      proof (rule X.weak.sup_upper (imp: assmsjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
          auto
          by         show "\\<^sub>* y \ {x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y}"
\<in> carrier \<X>. \<pi>\<^sup>* x \<sqsubseteq>\<^bsub>\<Y>\<^esub> y}"java.lang.NullPointerException
p )
          showproof( Y.weak)
          show
            by           by java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        qed
      qed
java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 168
               "{ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y} \ carrier \"
        showbyjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
          by      qed
        show " assume "z \ {x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y}"
        fix        thus <><^bsub>\<X>\<^esub> \<pi>\<^sub>* y"
        assume " {x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y}"
thus" \\<^bsub>\\<^esub> \\<^sub>* y"
          by (simp add: assmsshow{ \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y} \<subseteq> carrier \<Y>"
      qed
            qed
       by (auto intro: X.eak      show\<Squnion>\<^bsub>\<X>\<^esub>{x \<in> carrier \<X>. \<pi>\<^sup>* x \<sqsubseteq>\<^bsub>\<Y>\<^esub> y} \<in> carrier \<X>" (auto introX..sup_closed)
    qed
  qed

end

lemmajava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
                          =
  by( simp : galois_connection_def connection_def dual_order_iff

lower_adjoint    qed
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

definition upper_adjoint  theoremupper_by_complete_lattice
  "upper_adjoint "upper_adjointBg<equiv> \<exists>f. galois_connection \<lparr> orderA = A, orderB = B, lower = f, upper = g \<rparr>"

lower_adjoint_dual]: "lower_adjoint( A) ( B) (imp :: lower_adjoint_def upper_adjoint_def)
  by (simp addbysimp: assms)

lemma upper_adjoint_dual (simp: lower_adjoint_def)
  by (simp rule    proof (rule "ower_adjointABf\ f \ carrier A \ carrier B"

lemma lower_type
  by auto      show <>\<^sub>* y \<sqsubseteq>\<^bsub>\<X>\<^esub> \<Squnion>\<^bsub>\<X>\<^esub>{x \<in> carrier \<X>. \<pi>\<^sup>* x \<sqsubseteq>\<^bsub>\<Y>\<^esub> y}"

lemma upper_type (auto
             "\\<^sub>* y \ carrier \" by (fact y)

subsection

alois:qed
  by

lemma comp_galcon_closed
lemmaid_galois"artial_orderA java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
   " (G \<^sub>g F)"
proof -
   Fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (simpassumesgalois_connection           auto
  interpret G: galois_connection z
    by( : assms

  have "partial_order \\<^bsub>G \\<^sub>g F\<^esub>"
    by simpFis_order_A
  moreover have "partial_order qed simp add assms)
    by (simp add ( add:       (auto: Xweak
  moreover
    usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (imp: F.is_order_Acomp_galcon_def)
                                <lparr> orderA = A, orderB = B, lower = g, upper = f \<rparr>"
  moreover
  have definition upper_adjoint: (', c)uFlower_closureG. ()byauto
               (\<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>F\<^esub> x) \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^bsub>F\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>F\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y))"     .. assms java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
metis.Gupper_closurePi_iff
   show
    by ( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed  bylemma: " A galois_connection (I\<^sub>g(A))"

lemma comp_galcon_right_unitby (imp :
  by lemma:

lemma comp_galcon_left_unit [simpproofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  by (simp" A (imp add: assmsjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

lemma galois_connectionI:
  assumes
    "partial_order A" "java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
    L <in> carrier A \<rightarrow> carrier B" "R \<in> carrier B \<rightarrow> carrier A""
    "isotone A B have "partial_order
    "\ x y. \ x \ carrier A; y \ carrier B \ \ L x \\<^bsub>B\<^esub> y \ x \\<^bsub>A\<^esub> R y"
  shows   "alois_connection \ orderA = A, orderB = B, lower = L, upper = R \"
  using assmsby(addG comp_galcon_def

lemma subsection
  assumes
    "partial_order Alocale retract = galois_connection F.ower_closureG. assms(3) by
htarrowB R\>carrier
    "isotone A B L" "isotone B A R"
    "\ X. X \ carrier(B) \ L(R(X)) \\<^bsub>B\<^esub> X"
    "\ X. X \ carrier(A) \ X \\<^bsub>A\<^esub> R(L(X))"
  shows "galois_connection have "\ x y. \x \ carrier \\<^bsub>F\<^esub>; y \ carrier \\<^bsub>G\<^esub> \ \
  using assms
  by (auto               \<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>F\<^esub> x) \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^bsub>F\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>F\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y))" assms )

subsection \<open>Retracts\<close>

locale = galois_connection
  assumes java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 7
begin
:"x\ carrier \ \ \\<^sub>* (\\<^sup>* x) = x"
bymesoninflation partial_order retract_axioms upper_closure

  lemma retract_injective: "inj_on \\<^sup>* (carrier \)"
   java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
end  isotoneL isotone

theorem comp_retract_closed:
assumes G" "retract" "\ X. X \ carrier(B) \ L(R(X)) \\<^bsub>B\<^esub> X"
shows ( <circ>\<^sub>g F)"
proof -
  interpretfretract
    by simp:assms assms3 .   auto :  connection_def, ( PiEweak_partial_order)+
  interpret g
    by ( add)
  interpret
    by (imp: assms)assms(3) comp_galcon_closed retract.(1)
  show  assumesretract_propertyx\<in> carrier \<X> \<Longrightarrow> \<pi>\<^sub>* (\<pi>\<^sup>* x) \<sqsubseteq>\<^bsub>\<X>\<^esub> x"
  proof
    fixx
    assume "x \ carrier \\<^bsub>G \\<^sub>g F\<^esub>"
    thus "lebegin
      using(3) f.inflation f. f.retract_inverseg.retract_inverse by (auto retract_axioms_def)

qed

subsection \<open>Coretracts\<close>

locale coretract comp_retract_closedjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  assumes coretract_property: "y \ carrier \ \ y \\<^bsub>\\<^esub> \\<^sup>* (\\<^sub>* y)"
begin
  lemma coretract_inverse:
byassumes "coretractF"<

lemma: inj_on
    by (metis    by dd)
end

theorem        addassms
  assumes "coretractF \
  shows "coretract (G \\<^sub>g F)"
proof -
  interpret f:(fix
    by (simp
interpret:coretract
    by (simpbylocale  galois_connection +
  interpret
    byjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 5
  show ?thesis
  proof
    java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    assume "y \ carrier \\<^bsub>G \\<^sub>g F\<^esub>" galois_bijection +
    thus l retract_injective: " \\<^sub>* (carrier \)"
      by(imp  assumeslower_iso \<X> \<Y> \<pi>\<^sup>*"
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed  assumes

subsection> Bijections

locale galois_bijectioninterpret:  lemma upper_bij: "bij_be: "bij_betw \\<^sub>* (carrier \) (carrier \)"
  assumes lower_isointerpret:coretract
  upper_isosotone<
  and lower_inv_eq:   applyunfold_locales)
  and:"y carrier \ \ \\<^sup>* (\\<^sub>* y) = y"
begin

  lemmalower_bijbij_betwusing lower_iso upper_inv_equse_iso2 fastforce
by [where="<>\<^sub>*"], auto intro: upper_inv_eq lower_inv_eq upper_closed lower_closed)

  lemma y
    by (rule bij_betwI[where g="\\<^sup>*"], auto intro: upper_inv_eq lower_inv_eq upper_closed lower_closed)

sublocale gal_bij_conn: sublocale: coretract
  apply(,     thus
  using  upper_iso
  using
done

sublocale: retract
java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101

sublocaleinterpret: G
  by (    by simp: assms lower_isoisotone

end

theorem comp_galois_bijection_closed:
  assumes"alois_bijection G"" ( add assms3 f.gal_bij_conn.galois_connection_axioms g.gal_bij_conn.galois_connection_axiomsgalois_connectionaxioms1)
shows G <circ>\<^sub>g F)"
proof -
  interpret f: galois_bijection java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    by (simp: assms
  interpret g: galois_bijection G
    by (simp add  lemmalower_bijbij_betw
  interpret:galois_connection <circ>\<^sub>g F)"
    by (simp add: assms(    fix
  show
  proof
show X>\<^bsub>G \<circ>\<^sub>g F\<^esub> \<Y>\<^bsub>G \<circ>\<^sub>g F\<^esub> \<pi>\<^sup>*\<^bsub>G \<circ>\<^sub>g F\<^esub>"gal_bij_connusing(3 . f.lower_inv_eqg.lower_inv_eq by( simp addjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
    showisotone
      by( add gfupper_iso
    fixjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    assume"\
    thus "\\<^sub>*\<^bsub>G \\<^sub>g F\<^esub> (\\<^sup>*\<^bsub>G \\<^sub>g F\<^esub> x) = x"
      using assms(3) fbyunfold_locales add.is_weak_order_B weak_partial_order)
  next
    fix y
    assume "y \ carrier \\<^bsub>G \\<^sub>g F\<^esub>"
\<
      by (simp add: comp_galcon_def " G \\<^sub>g F)"
  qed
qed

end

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