Quelle Lattice.thy Sprache: Isabelle

*  Title:      HOL/Lattice.thy
    Author     Clemens, started7November
    Copyright Order

Most rules by Hohe
With contributionsfrom Armstrong and Simon.
*)

theorysection
imports Order
begin \<open>Supremum and infimum\<close>

section "\\<^bsub>L\<^esub>A = (SOME x. least L x (Upper L A))"

subsection

definition
  sup ::"[, a ] = a" ((\open_block notation=\prefix \\\\\_)\ [90] 90)
  where "\\<^bsub>L\<^esub>A = (SOME x. least L x (Upper L A))"

definition
  inf :: "[_, 'a set] => 'a" (\<open>(\<open>open_block notation=\<open>prefix \<Sqinter>\<close>\<close>\<Sqinter>\<index>_)\<close> [90] 90)
  where   inf::"[_ 'a set] => 'a" (\<open>(\<open>open_block notation=\<open>prefix \<Sqinter>\<close>\<close>\<Sqinter>\<index>_)\<close> [90] 90)  ::

definition
  "(a,'b) gorder_scheme\ 'c set \ ('c \ 'a) \ 'a "
where LAf  \<Squnion>\<^bsub>L\<^esub>(f ` A)"

definition infi :java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  "('aproof (nfold join_def sup_def)
  whereLAf=

syntax
  "_ assume L " \<in> carrier L"  "y \<in> carrier L"
    \<>(\<open>indent=3 notation=\<open>binder IINF\<close>\<close>IINF\<index> _./ _)\<close> [0, 10] 10)java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
  "_java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    (<open>(\<open>indent=3 notation=\<open>binder IINF\<close>\<close>IINF\<index> _:_./ _)\<close> [0, 0, 10] 10)
  "_sup1"     :: "('a, 'b) gorder_scheme \ pttrns \ 'a \ 'a"
    (\<open>(\<open>indent=3 notation=\<open>binder SSUP\<close>\<close>SSUP\<index> _./ _)\<close> [0, 10] 10)
  "_sup"      :: "('a, 'b) gorder_scheme \ pttrn \ 'c set \ 'a \ 'a"
    (\<open>(\<open>indent=3 notation=\<open>binder SSUP\<close>\<close>SSUP\<index> _:_./ _)\<close> [0, 0, 10] 10)
ntax_consts
  "_inf1" "_inf" == infi and
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
translations
  "IINF and xx': "x .= x'"
  "IINF\<^bsub>L\<^esub> x:A. B" == "CONST infi L A (%x. B)"
  "SSUP\<^bsub>L\<^esub> x. B" == "CONST supr L CONST UNIV (%x. B)"
  "SSUP\<^bsub>L\<^esub> x:A. B" == "CONST supr L A (%x. B)"

definition
  join :: "[_, 'a, 'a] shows"x \<squnion> y .= x' \<squnion> y"
   "x \\<^bsub>L\<^esub> y = \\<^bsub>L\<^esub>{x, y}"

definition
  meet: "[, = '" ( \\\\ 70)
  where "x \\<^bsub>L\<^esub> y = \\<^bsub>L\<^esub>{x, y}"

definition
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
" L f =\\<^bsub>L\<^esub> {u \ carrier L. f u \\<^bsub>L\<^esub> u}" \ \least fixed point\

definition
  :: "',')gorder_scheme
  qed (ulecarr

subsection

lemma sup_dual [simp  assumes carr <>carrier \<in> carrier L" "y' \<in> carrier L"
nion
  by (imp: sup_def inf_def

lemma inf_dual [simp]:   shows"
proof joinI rule )
   ( add sup_def )

lemma join_dual  ix
  " \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q"
  by    from yy

lemma meet_dual [simpalsohave{,x  x ' fast
" \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q"
  by (simpjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma top_dual   carr
         leastb b Upper
  by (simp simp:[OF _seq])

lemma bottom_dual [simp]:
  "\\<^bsub>inv_gorder L\<^esub> = \\<^bsub>L\<^esub>"
  by (simp add: top_def

lemma LFP_dual [simp]:
  "LEAST_FP "a . b"by( weak_least_unique)
  by( addLEAST_FP_def)

lemma GFP_dual
  " (inv_gorder L)f =LEAST_FP L f"
  by (  "

subsection \<open>Lattices\<close>

locale
  assumes :
    "[| x java.lang.StringIndexOutOfBoundsException: Range [0, 67) out of bounds for length 44

locale = weak_partial_order
  assumes inf_of_two_exists:
    "[lemma(nweak_partial_order) sup_of_singleton_closed [simp:

locale weak_lattice = weak_upper_semilattice +  x \<in> carrier L \<Longrightarrow> \<Squnion>{x} \<in> carrier L"

lemma (in weak_lattice) dual_weak_lattice:
  "weak_lattice (inv_gordertext
proof -
  interpret dual: weak_partial_order
    by metis)
  show ?thesis!s  L  ( L)=  ;
  proof qedleast L a  L A) x \<in> carrier L; A \<subseteq> carrier L |]
qed

subsubsection \<open>Supremum\<close>

lemma   L "
    and P:!l. leastUpperxA)=> "
  ==> P andleast_a aUpper
proof (unfold join_def sup_def)
  assume L: "x \ carrier L" "y \ carrier L"
    and P: "!!l. least L l ( L least_a have La:" \<in> carrier L" by simp
  with obtain s  "least L s (pper L {x, }"byjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
L show (OME L l ( L {x, })
    by (fast intro: someI2 L (insert x A)"
qed

lemma (in weak_upper_semilattice) join_closed [simp]:
[  \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
fix

:
  assumes carr showz\<sqsubseteq> s"
x .= x'
  shows "x \ y .= x' \ y"
proof (ule joinI rule)
  fix a b
  from      java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
                 (ule_tac  [where  destleast_Upper_above)

  assumeleastaLa(  x y)
  assume "least L b (Upper L fix y
  with
      have leastb: "leastshow" <>"
      by ( add least_Upper_cong_r[F __seq

           z
              assume z: " \ {a, x}"
qed showz\<sqsubseteq> y"

lemmahave y' y \ Upper L A"
               (eson Upper_antimonoin_mono  y)
           " "
shows
proof (rule joinI, rule
  fixassume" \ {x}"
  havewith ?thesis
  also from carr yy
      have{ }.}y,x  (ntro set_eq_pairI)
  also have "{y', x} = {x, y'}" by fast
  finally
      have seq: "{x, y} {.=} {x, y'}" .

  assume leasta:"east L a (pperL{,y)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
  assume "L b Upper L x,')
  with carr
      have leastb  qed( P)
      by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  from  "[ A;A \ carrier L; A \ {} |] ==> least L (\A) (Upper L A)"
      showa. "by(ule )
qed   empty

lemmajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
" in carrierL=>least L x ( L {x})"
  by (rule least_UpperI) auto

lemma (in weak_partial_order)
  "x \ carrier L ==> \{x} .= x"
  unfolding sup_def
  by      insert ?thesis

lemma (in(*The  step hairy can  simp.
  "x carrier L \ \{x} \ carrier L"
  unfolding sup_def
   casejava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14

text \<open>Condition on \<open>A\<close>: supremum exists.\<close> ?thesis

lemma (in weak_upper_semilattice) sup_insertI:
  "| !s L s (Upper L (insertx A) = s;
  least Upper;x \<in> carrier L; A \<subseteq> carrier L |]
  = P (<qunion>insert x ))"
proof   P: "!l L l (Upper L insert A) = "
  assume L: "x \ carrier L" "A \ carrier L"
    and "!.leastL l ( insertxA) = "
    and least_a: "least L a (Upper L A) "P (\<Squnion> (insert x A))"
  from L least_a(impadd)
  from L sup_of_two_exists java.lang.StringIndexOutOfBoundsException: Range [0, 34) out of bounds for length 4
  obtain (simp add:sup_insertI finite_sup_least)
qed
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   show "east L s Upper (nsert x A)"
    proof (rule least_UpperI)
      fix z
      assume "z \ insert x A"
      then "z s"
      proof
        assume "z = x" then show ?thesis
           (imp add [OF] L La
      next
        assume "z \ A"
with least_a ?thesis
          by (rule_tac le_trans
qed
    next
      fix y
      assume y: "y \ Upper L (insert x A)"
      show "s \ y"
      proof ( least_le least_srule Upper_memI)
        fix z
        assume z: "z \ {a, x}"
        then " \ y"
        proof
          have' " \ Upper L A"
            by meson in_mono y)
          assume "z = a"
          with'least_a show? by( :least_le)
        next
           " \ {x}"
          y L show by blast
qed
      qed (rule sup_of_two_exists s where least(   }"by java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
    next
      from L show "insert x A \ carrier L" by simp
      from least_s show "s \ carrier L" by simp
    qed
  qed (rule
qed

lemmaassumes sub \<sqsubseteq> z"  "y \<sqsubseteq> z"
java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
)
  case empty
  thenjava.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 7
next
  case ( x A)
  show
  proofcases{"java.lang.StringIndexOutOfBoundsException: Range [24, 25) out of bounds for length 24
    case True "x \ y \ (x \ y) .= y"
    with insert show ?thesis (eson(1 (2) join_closed join_left le_cong_r local. weak_le_antisym
      by simp simp: least_cong weak_sup_of_singleton)
        (* The above step is hairy; least_cong can make simp loop.
        Would want special version of simp to apply least_cong. *)
  next
    case False
    with have" L (\A) (Upper L A)" by simp
    with  show
      by (rule java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 7
  qed
qed
  show x\<squnion> (y \<squnion> z) .= s"
lemma (in weak_upper_semilatticefrom L showxjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
  assumes"!.leastLl(pper L (insert x A)) == P "
    and
  shows     sup "s\ x \ (y \ z)"
proofcases" ={"
         intro Upper_memIintro:le_trans join_leftjoin_right)
    by (simp (simp_all add: least_closed sup
next
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (simp add
qed

lemma (in weak_upper_semilatticefixes L()
  [ finiteA\<subseteq> carrier L; A \<noteq> {} |] ==> \<Squnion>A \<in> carrier L"
proof (induct:finite)
  case empty then show ?case by simp
next
  case insert then show L: x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
    by  (* FIXME: could be simplified by improved simp: uniform use of .=,
qed

lemma (in weak_upper_semilattice) join_left:
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
  by (rule joinI [folded join_def]) (blast dest: least_mem)

lemma (in weak_upper_semilattice) join_right:
  "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
  by (rule joinI [folded join_def]) (blast dest: least_mem)

lemma (in weak_upper_semilattice) sup_of_two_least:
  "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})"
proof (unfold sup_def)
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
  with L show "least L (SOME z. least L z (Upper L {x, y})) (Upper L {x, y})"
  by (fast intro: someI2 weak_least_unique)  (* blast fails *)
qed

weak_upper_semilattice :
  assumes sub:  [ !i greatest( L {,y}java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
    and" \ carrier L" and y: "y \ carrier L" and z: "z \ carrier L"
  shows "x \ y \ z"
proof (rule joinI P: "g greatestL g LowerLx,})= Pg
  fix s
  assume "least L s (Upper L {x, y})"
  with sub z show L show P SOMELg( { })
qed

lemma (in weak_lattice) weak_le_iff_meet
  assumes(n ) meet_closed]:
  shows|x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
  by (meson assms

(in eak_upper_semilattice:
  assumes L: "x \ carrier L" "y \ carrier L" "z \ carrier L"
java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
proof (rule finite_sup_insertI)
  \<comment> \<open>The textbook argument in Jacobson I, p 457\<close>
  fix s
assumesup:" L s ( L {x, y,z)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
  show seqx,}{= {' y} by( set_eq_pairI)
  proof (rule weak_le_antisym: "greatest L LowerL{,y)"
    from sup L show "x \ (y \ z) \ s"
      by (fastforce carr
  next
    from sup L show        simp: greatest_Lower_cong_r[   eq)
    by (erule_tac least_le)
      ( intro  introle_trans join_right)
  qed "a. "by( weak_greatest_unique
qed (imp_all add

text \<open>Commutativity holds for \<open>=\<close>.\<close>

lemma:
  fixes "x\ y .= x \ y'"
  proof meetI  meetI
  by(nfold) (simpadd)

( weak_upper_semilattice
  assumes L:  also from yy
  shows\<squnion> y) \<squnion> z .= x \<squnion> (y \<squnion> z)"
roof
  (* FIXME: could be simplified by improved simp: uniform use of .=,
     omit [symmetric] in last step. *)
  have "(x \ y) \ z = z \ (x \ y)" by (simp only: join_comm)
  alsofrom have ". . \{z, x, y}" by (simp add: weak_join_assoc_lemma)
    Lhave..=\<Squnion>{x, y, z}" by (simp add: insert_commute)
  alsofrom L have ... x \<squnion> (y \<squnion> z)" by (simp add: weak_join_assoc_lemma [symmetric])
finally thesis:L)
qed

subsubsection \<open>Infimum\<close>

lemma (in weak_lower_semilattice) meetI:
  "|!i.greatest Li( L {x, y)=>Pi;
  x \<in> carrier L; y \<in> carrier L |]
  ==> P (x \<sqinter> y)" "a . "byrule)
proof meet_def)
  assumejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
   inf_of_two_exists iwhere greatest(  x }"by
  with L show <in> carrier L ==> greatest L x (Lower L {x})"
  by (fast intro ( greatest_LowerI
qed

lemma \<in> carrier L ==> \<Sqinter>{x} .= x"
  "[| x by (rule someI2) (auto intro: weak_greatest_unique inf_of_singletonI)
  by (rule meetI) (java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0

lemma
  assumes carr: "x \ carrier L" "x' \ carrier L" "y \ carrier L"
      by (r) (auto: inf_of_singletonI
  shows \<open>Condition on \<open>A\<close>: infimum exists.\<close>
proof (rule meetI, rule meetI)
  fix a b
  from java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
      haveseq,y {={,y"by (uleset_eq_pairI)

  assume greatesta: "greatest L a (Lower L {x, y})"
    =  (\<Sqinter>(insert x A))"
  with (unfold)
      a L: "x\ carrier L" "A \ carrier L"
      by( add greatest_Lower_cong_r[F __seq

greatesta
      show L  have La\in L  simp
qed (rule carr)+

lemma( weak_lower_semilattice:
  assumes carr: "x obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
    and yy': "y .= y'"
shows
proof (rule meetI, rule meetIproof (ule someI2)
fix
   "x,y ={,x} by
  also from carrfix
      have "{y, x} {. then showi
  also havey,}= '"byfjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  finally
      have seq          by( addgreatest_Lower_below greatest_i)

  assume greatesta: "greatest L a (Lower L { L greatest_i greatest_a show ?thesis
  assume "reatestL bLowerL{,)
  with carr
have: "greatest Lower L {x, y)"
      java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8

  from greatesta greatestb
      show "a .= b" by (rule        (rule [ greatest_irule
qed( carr+

lemma (in weak_partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
  "x \ carrier L ==> greatest L x (Lower L {x})"
  byrule) auto

_) weak_inf_of_singleton
  "x \ carrier L ==> \{x} .= x"
unfolding
  by (rule

lemma z \<in> {x}"
  " y L ?thesis byb
  unfolding
  by (rule someI2) (auto intro (rule

       L show" x A \<subseteq> carrier L" by simp

lemma (in weak_lower_semilattice    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  [ finiteA
  greatest L a (Lower( set:finite)
  ==> P (\<Sqinter>(insert x A))"
proof (unfold inf_def)
  assumeL:" \ carrier L" "A \ carrier L"
    and
and: "greatestLa(Lower L A)"
  from L greatest_a have  proof cases"
  from L inf_of_two_exists greatest_a insert thesis
  obtain i where greatest_i: "greatest L i ( (simp : greatest_cong [ weak_inf_of_singleton]
  show "P (SOME show "P (SOME
  proof rule)
    show "greatest L i from insertshow?java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    roof (ule)
      fix z
      assume " \ insert x A"
then  "i \ z"
      proof
        assume "z = x" then show ( weak_lower_semilattice:
by( addgreatest_i
      next
and: "inite A" x
        withshows" \
          by (rule_tac le_trans True withP and  show?
      qed
    next
      fix
          by (simp inf_insertI)
       "y \ i"
      proof ( weak_lower_semilattice [simp
fix
proof (i setfinite)
        then show y\<sqsubseteq> z"
  case  then show case
          have y': "y \ Lower L A"
            by (meson
assume" =ajava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
          with y' greatest_a show ?thesis by (fast dest: greatest_le)
        next
          assume "z \ {x}"
          withlemma (in weak_) meet_right
        qed
      qed (rule Lower_closed [THEN subsetD, OF y])
    next
      from L show lemma in) inf_of_two_greatest
from show " \ carrier L" by simp
    qed
  qed (rule P)
qed

lower_semilattice
  "[| finite inf_of_two_exists obtain s "greatest s (ower L { y)  fast
proof L
  show "reatestL SOME z greatest z(LowerL ) (Lower L {x,y})"
next
  case (insert x A)
   ?case
  proof
    case True
    with nsert?java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
      by simp (simp add x: x \<in> carrier L" and y: "y \<in> carrier L" and z: "z \<in> carrier L"
        inf_of_singleton_closed)
  next
asejava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    fromshow
  with sub z show \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
      fromFalse showgreatest\<Sqinter>A) (Lower L A)" by simp
    qed
  qed
qed

lemma (in weak_lower_semilattice) finite_inf_insertI:
  assumes !i  L i(Lower x ) =>
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
  shows "P (\ (insert x A))"
proof (cases "A = {}")
  case True with P and ( finite_inf_insertI
     (imp
      by ( intro meet_le: greatest_Lower_below
  caseFalse with xA ?thesis
      L show \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
qed

lemma (in weak_lower_semilattice) finite_inf_closed [simp]:
  "[| finite A; (simp_all add: L [OF inf]
proof ( set finitejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  case empty then show ?case by simp "x y = y \ x"

  case insert then show (in weak_lower_semilattice) weak_meet_assoc:
    by (rule_tac finite_inf_insertI) (simp_all)
qed

lemma (inweak_lower_semilattice:
<in>carriery \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
  shows\<sqinter> y) \<sqinter> z .= x \<sqinter> (y \<sqinter> z)"

lemma (have"x\ y) \ z = z \ (x \ y)" by (simp only: meet_comm)
  "[| x \ carrier L; y \ carrier L |] ==> x \ y \ y"
by(ule  [folded]) ( destgreatest_mem

lemma(in weak_lower_semilattice:
  "[| x \ carrier L; y \ carrier L |] ==>
greatest\<Sqinter>{x, y}) (Lower L {x, y})"
proof show by (simp: Ljava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
assume:" \ carrier L" "y \ carrier L"
  with inf_of_two_exists fixy
  with L
  show "greatest L L:"\<in> carrier L"  "y \<in> carrier L"
  by (fast introproof
qed

lemma (in weak_lower_semilattice       x\sqsubseteqjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  assumes: " x " \<sqsubseteq> y"
    and
   "z \ x \ y"
proof "
  fix       L haveleast Lx }"
  assume "greatestLi( L {x, y}"

   show by blast
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
emma ) weak_le_iff_join
  assumes "
  shows "x \ y \ x .= (x \ y)"
  by  proof

(in) weak_meet_assoc_lemma
  assumes  assumes  "y \ x"
  shows "x \ (y \ z) .= \{x, y, z}"
rule
  by( greatest_LowerI
  fix
  java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
show\<sqinter> (y \<sqinter> z) .= i"
proof weak_le_antisym
    from inf L show(ule_tac) auto
ultimately ?hesis
  next
    from L showx\<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
    by (erule_tac
      (blast intro! Lower_memI : le_trans meet_right)
  qed
qed(imp_all add:L)

lemma meet_comm:
  fixes L (structure)
shows
  by (unfold meet_def

lemma
  assumes L: "x \ carrier L" "y \ carrier L" "z \ carrier L"
  shows "(x \ y) \ z .= x \ (y \ z)"
proof -
  (* FIXME: improved simp, see weak_join_assoc above *)
  have "(x \ y) \ z = z \ (x \ y)" by (simp only: meet_comm)
  also from L have "... .= \ {z, x, y}" by (simp add: weak_meet_assoc_lemma)
  also from L have "... = \ {x, y, z}" by (simp add: insert_commute)
  also from L have "... .= x \ (y \ z)" by (simp add: weak_meet_assoc_lemma [symmetric])
  finally show ?thesis by (simp : "x <> carrier
qed

text \<open>Total orders are lattices.\<close>

sublocale bottom_least join_le le_refl weak_le_antisym
proof
  fix x lemma:
  assume L: "x \ carrier L" "y \ carrier L"
  show "\s. least L s (Upper L {x, y})"
  proof -
    note L
  moreover
    {
      assume "x \ y"
      with L have "least L y (Upper L {x, y})"
        by (rule_tac least_UpperI) auto
    }
    moreover
    {
      assume "y \ x"
      with L have "least L x (Upper L {x, y})"
        by (rule_tac least_UpperI) auto
    }
      by (etis meet_closed meet_le top_closed weak_le_antisym
  qed
next
fix y
  assume L: "x \ carrier L" "y \ carrier L"
  show "\i. greatest L i (Lower L {x, y})"
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    moreover
    {
      assume "y \ x"
      with L have "greatest L y (Lower L {x assumes sup_of_two_exists:
        by (rule_tac greatest_LowerI) auto
    }
    moreover
    {
      assume "x \ y"
L x ( L {x, y}"
        by (rule_tac greatest_LowerI) auto
    }
    ultimately  java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  qed
qed (rule)

subsection

locale weak_bounded_latticesublocale lattice \<subseteq ..
  weak_lattice ( lattice:
  weak_partial_order_bottom +
  weak_partial_order_top
begin

lemma bottom_meet
  by (metis bottom_least

lemma bottom_join: "xlemma() le_iff_join:
by( bottom_least join_le le_refl weak_le_antisym

lemma bottom_weak_eq:
  "lbrakk b \ carrier L; \ x. x \ carrier L \ b \ x \ \ b .= \"
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

lemma top_join:" \ carrier L \ \ \ x .= \"
  by (metis join_left top_higher)

lemma top_meet: "x \ carrier L \ \ \ x .= x"
  by (metis \<open> Total orders are lattices. \<close>

emma:  \lbrakkt\<in> carrier L; \<And> x. x \<in> carrier L \<Longrightarrow> x \<sqsubseteq> t \<rbrakk> \<Longrightarrow> t .= \<top>"
  by (definitionjoin_pres : "' c \ ('b, 'd) gorder_scheme \ ('a \ 'b) \ bool" where

end

sublocale weak_bounded_lattice \<subseteq> weak_partial_order ..

subsection \<open>Lattices where \<open>eq\<close> is the Equality\<close>

locale upper_semilattice
  assumes:
   "f

sublocale upper_semilattice "isotone "
  by unfold_locales sup_of_two_exists

locale lower_semilattice = partial_order +
  assumes    using  unfolding lattice_def lower_semilattice_def
    "|x \ carrier L; y \ carrier L |] ==> \s. greatest L s (Lower L {x, y})"

sublocale lower_semilattice \<subseteq> weak?: weak_lower_semilattice
byunfold_locales (ule)

locale lattice =

sublocale  \<subseteq> weak_lattice ..

lemma (proof (ru isotoneIjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  " (inv_gorder L)
proof -
    bymeson.axioms1+
    bymetis)

  show ?thesis
    apply (unfold_locales)
    apply (simp_all add: inf_of_two_exists
    apply (rule eq_is_equal)
  done
qed

lemma (in lattice) weak_partial_order_bottom
  assumes "xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows "x \ y \ x = (x \ y)"
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma (in lattice) le_iff_meet:
  assumesx\incarrier \<in> carrier L"
  showslemma top_eq:"t \ carrier L; \ x. x \ carrier L \ x \ t \ \ t = \"
   (imp:  eq_is_equal)

text

sublocale  \<subseteq> weak?: lattice
  bystandard intro.weak weak.inf_of_two_exists

text

definition join_pres :: "('a, 'c) gorder_scheme \ ('b, 'd) gorder_scheme \ ('a \ 'b) \ bool" where
"join_pres X Y f \ lattice X \ lattice Y \ (\ x \ carrier X. \ y \ carrier X. f (x \\<^bsub>X\<^esub> y) = f x \\<^bsub>Y\<^esub> f y)"

definition meet_pres :: "('a, 'c) gorder_scheme \ ('b, 'd) gorder_scheme \ ('a \ 'b) \ bool" where
"meet_pres X Y f \ lattice X \ lattice Y \ (\ x \ carrier X. \ y \ carrier X. f (x \\<^bsub>X\<^esub> y) = f x \\<^bsub>Y\<^esub> f y)"

lemma join_pres_isotone:
  assumes "f \ carrier X \ carrier Y" "join_pres X Y f"
  shows "isotone X Y f"
proof (rule isotoneI)
  show "weak_partial_order X" "weak_partial_order Y"
    using assms unfolding join_pres_def lattice_def upper_semilattice_def lower_semilattice_def
    by (meson partial_order.axioms(1))+
  show "\x y. \x \ carrier X; y \ carrier X; x \\<^bsub>X\<^esub> y\ \ f x \\<^bsub>Y\<^esub> f y"
    by (metis (no_types, lifting) PiE assms join_pres_def lattice.le_iff_meet)
qed

lemma meet_pres_isotone:
  assumes "f \ carrier X \ carrier Y" "meet_pres X Y f"
  shows "isotone X Y f"
proof (rule isotoneI)
  show "weak_partial_order X" "weak_partial_order Y"
    using assms unfolding meet_pres_def lattice_def upper_semilattice_def lower_semilattice_def
    by (meson partial_order.axioms(1))+
  show "\x y. \x \ carrier X; y \ carrier X; x \\<^bsub>X\<^esub> y\ \ f x \\<^bsub>Y\<^esub> f y"
    by (metis (no_types, lifting) PiE assms lattice.le_iff_join meet_pres_def)
qed

subsection \<open>Bounded Lattices\<close>

locale bounded_lattice =
  lattice +
  weak_partial_order_bottom +
  weak_partial_order_top

sublocale bounded_lattice \<subseteq> weak_bounded_lattice ..

context bounded_lattice
begin

lemma bottom_eq:
  "\ b \ carrier L; \ x. x \ carrier L \ b \ x \ \ b = \"
  by (metis bottom_closed bottom_lower le_antisym)

lemma top_eq:  "\ t \ carrier L; \ x. x \ carrier L \ x \ t \ \ t = \"
  by (metis le_antisym top_closed top_higher)

end

hide_const (open) Lattice.inf
hide_const (open) Lattice.sup

end

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