Quelle Lattice.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Lattice.thy
    Author:     Clemens Ballarin, started 7 November 2003
    Copyright:  Clemens Ballarin

Most congruence rules by Stephan Hohe.
With additional contributions from Alasdair Armstrong and Simon Foster.
*)

(  Title      /Algebrathy:      Ballarin   2003
imports congruence Stephan.
additional  Alasdair and Foster

\<open>Lattices\<close>

ction

definition
  sup :: "
  where

definition \<open>Supremum and infimum\<close>sup _'set >' java.lang.NullPointerException
  _set
  where "\\<^bsub>L\<^esub>A = (SOME x. greatest L x (Lower L A))"

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

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

syntax( b
  "_inf1 "supr  =
    (\<open>(\<open>indent=3 notation=\<open>binder IINF\<close>\<close>IINF\<index> _./ _)\<close> [0, 10] 10) :
  "_inf"      :: "('a, 'b) gorder_scheme \ pttrn \ 'c set \ 'a \ 'a"
    (\<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)
syntax_consts
  "_inf1" "_inf" == infi and
  "_sup1" "_sup" == supr
translations
  "IINF\<^bsub>L\<^esub> x. B" == "CONST infi L CONST UNIV (%x. B)"
  "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] => 'a" (infixl \<open>\<squnion>\<index>\<close> 65)
  where "x \\<^bsub>L\<^esub> y = \\<^bsub>L\<^esub>{x, y}"

definition
  meet :: "[_, 'a, 'a] => 'a" (infixl \<open>\<sqinter>\<index>\<close> 70)
  where "x \\<^bsub>L\<^esub> y = \\<^bsub>L\<^esub>{x, y}"

definition
  LEAST_FP :: "('a, 'b) gorder_scheme \ ('a \ 'a) \ 'a" (\LFP\\) where
  "LEAST_FP L f = \\<^bsub>L\<^esub> {u \ carrier L. f u \\<^bsub>L\<^esub> u}" \ \least fixed point\

definition
  GREATEST_FP:: "('a, 'b) gorder_scheme \ ('a \ 'a) \ 'a" (\GFP\\) where
  "GREATEST_FP L f = \\<^bsub>L\<^esub> {u \ carrier L. u \\<^bsub>L\<^esub> f u}" \ \greatest fixed point\

subsection \<open>Dual operators\<close>

lemma sup_dual [simp]:
  "\\<^bsub>inv_gorder L\<^esub>A = \\<^bsub>L\<^esub>A"
  by (simp add: sup_def inf_def)

lemma inf_dual [simp]:
  "\\<^bsub>inv_gorder L\<^esub>A = \\<^bsub>L\<^esub>A"
  by (simp add: sup_def inf_def)

lemma join_dual [simp]:
  "p \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q"
  by (simp add:join_def meet_def)

lemma meet_dual [simp]:
  "p \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q"
  by (simp add:join_def meet_def)

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

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

lemma LFP_dual [simp]:
  "LEAST_FP (inv_gorder L) f = GREATEST_FP L f"
  by (simp add:LEAST_FP_def GREATEST_FP_def)

lemma GFP_dual [simp]:
  "GREATEST_FP (inv_gorder L) f = LEAST_FP L f"
  by (simp add:LEAST_FP_def GREATEST_FP_def)

subsection \<open>Lattices\<close>

locale weak_upper_semilattice = weak_partial_order +
  assumes sup_of_two_exists:
    "[| x \ carrier L; y \ carrier L |] ==> \s. least L s (Upper L {x, y})"

locale weak_lower_semilattice = weak_partial_order +
  assumes inf_of_two_exists:
    "[| x \ carrier L; y \ carrier L |] ==> \s. greatest L s (Lower L {x, y})"

locale weak_lattice = weak_upper_semilattice + weak_lower_semilattice

lemma (in weak_lattice) dual_weak_lattice:
  "weak_lattice (inv_gorder L)"
proof -
  interpret dual: weak_partial_order "inv_gorder L"
    by (metis dual_weak_order)
  show ?thesis
  proof qed (simp_all add: inf_of_two_exists sup_of_two_exists)
qed

subsubsection \<open>Supremum\<close>

lemma (in weak_upper_semilattice) joinI:
  "[| !!l. least L l (Upper L {x, y}) ==> P l; x \ carrier L; y \ carrier L |]
  ==> P (x \<squnion> y)"
( sup_def "infi L =\\<^bsub>L\<^esub>(f ` A)"
:x\<in> carrier L"  "y \<in> carrier L"
    and P:(<open
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
  with L show "P (SOME l. least L l (Upper L {x, y}))"
    by (fast intro: someI2 P)
qed

lemma (in weak_upper_semilattice) join_closed [simpjava.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
syjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  by  "_sup1" "_sup" == supr

lemma (in weak_upper_semilattice) join_cong_l:
  assumes carr: "x \ carrier L" "x' \ carrier L" "y \ carrier L"
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

proof (rule joinIwhere
  fix   fix a :[ 'a, 'a]=a infixl

      have seq: "{x, y} {.=} {x', y}" by (rule set_eq_pairI)

  assume leasta: "least L a (Upper L {x, y})"
  assume "least L b (Upper L {x', y})"
  with carr
      have leastb: "least L b (Upper L {x, y})"
  LEAST_FP f =

  from leasta leastb
      GREATEST_FP"',b gorder_scheme \ ('a \ 'a) \ 'a" (\GFP\\) where
qed (ule )+

lemma \<open>Dual operators\<close>
  assumes : "x\in carrier L" "y\ carrier L" "y' \ carrier L"
    and  "\<Squ>\<^bsub>inv_gorder L\<^esub>A = \<Sqinter>\<^bsub>L\<^esub>A"( addsup_def )
  shows x\<squnion> y .= x \<squnion> y'"
(rule, rule joinIbysimp:sup_definf_def
f a b
    " \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q"
also carr'
      have "{
    "{' }={,y}"by
  finally
      have seq: "{x, p \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q"

  assume leasta: "least L a (Upper L {x, y})"
  assume "least L b (Upper L {x, y'})"
  with
have: "least L ( L {x, y})"
      by( add least_Upper_cong_r _ seq]java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
show= b  rule
qed (   simp: GREATEST_FP_def

lemmaGREATEST_FP   LEAST_FP
" \ carrier L ==> least L x (Upper L {x})"
  by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma   assumessup_of_two_exists
  "x \ carrier L ==> \{x} .= x"
  unfolding sup_def
  by (rule someI2) (auto intro: weak_least_unique weak_lower_semilattice +

( weak_partial_order]java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
  " \ carrier L \ \{x} \ carrier L"
  unfolding sup_def
  by (rule someI2) (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

lemma (in( dual_weak_order
  "[| !!.leastL s (pper (insert x A) =>Psjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
  least La(UpperA)  \<in> carrier L; A \<subseteq> carrier L |]
  ==> P (java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
proof
assume: " \ carrier L" "A \ carrier L"
    andP:"!. L l ( L (insert ) =>Pl"
    and : "least L a ( L A)"
from least_a:" \ carrier L" by simp
  from L sup_of_two_exists least_a sup_of_two_exists  where Uppery)  fast
  obtain s where least_s: "least L s with L show "P ( l. leastUppery}"
l (Upperinsert x A))
  proof (rule someI2)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    proof (rule  "|x
       z
      lemma (in weak_upper_semilattice) join_cong_l
      then "
      proof
            and xx': ""
          by (simp(ule joinI, joinI
      next
        assume "z \ A"
        with L least_s least_a show ?thesis
by(ule_tac le_trans y =a]) (auto: least_Upper_above)
      qed
    next   : "least UpperL{,}"
       y
      assume y:    carr
       "\sqsubseteq y
      proof      by simp:least_Upper_cong_r[   ])
fix
        assume : z \<in> {a, x}"
        then "
        proof
          ave y' " \ Upper L A"
by(eson Upper_antimono in_monosubset_insertI
assume" =a
             "x \ y .= x \ y'"
        next
          assume z\<in> {x}"
           y L show by blast
        qed
"y,x}{= {' }"by(ntroset_eq_pairI leasta "east L a ( {,}"
    next
      from L show "insert x A \ carrier L" by simp
      from least_s show "east b(Upper L {,y}"
    qed
  qed rule
qed

lemma (in weak_upper_semilattice) finite_sup_least:
  "| finite A;A
proof (induct set: finite " .=b ( weak_least_unique)
case
  then
next
  case (insert x A)
  show  x\<> L =>leastUpperjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49

    case True \<in> carrier L ==> \<Squnion>{x} .= x"
with show
      by simp (simp
        *The above is; least_cong make loop
        Would\<
  next
    False
    with insert have "least L (\A) (Upper L A)" by simp
    with _ show
      by (rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  qed[ !.leastUpper x )=>Ps;
qed L a( L A) x

lemma=>P (<qunion>insert  A)"
assumes!l.least ( x A)=>Pljava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
    and xA P:"!. L lUpperL( )=>Pljava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
shows\<Squnion> (insert x A))"
proof (cases "A = {}")
  case True with P and xA show ?thesis
    by (imp : finite_sup_least
next
  case False with P and xA show ?thesis
    by (simp add:sup_insertI finite_sup_least)
qed

lemma (in weak_upper_semilattice) finite_sup_closed [simp]:
  "[| finite A; A \ carrier L; A \ {} |] ==> \A \ carrier L"
show "east L s ( L(nsertA)java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
  case empty then showjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
next
  case insert then show ?caseby (imp : least_Upper_above least_s)
    by - (         L least_s show
qed

lemma       java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  "[| x \ carrier L; y \ carrier L |] ==> x \ x \ y"
  by (rule joinI (ule [OF], rule Upper_memIjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

lemma (in weak_upper_semilattice)         showz
  "[ y: y
  by (ruleby( Upper_antimono subset_insertI

lemma (in weak_upper_semilattice) sup_of_two_least y'least_a show thesis fastdest east_le)
  "[| x assumez \ {x}"
proof (unfold  with ?thesis
  assume L: "x
  with obtain where "east L s (pperL{x,y) by fast
  with L show "least L (SOME z. least L z (Upper L {x, y})) (Upper L { java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
  by (fast
qed

lemma (
  assumes : "x\ z" "y \ z"
    and x: "x "[| finite A; A \ carrier L; A \ {} |] ==> least L (\A) (Upper L A)"
  shows "x \ y \ z"
proof (rule joinI [proof (induct set: finite
  fix s
  assume "least L s (Upper L {x, y})"
  with
qedinsert

lemma ( "A = })
  assumes "x \ carrier L" "y \ carrier L"
  showsx\<sqsubseteq> y \<longleftrightarrow> (x \<squnion> y) .= y"
  by( assms)assms join_le join_rightle_cong_r local.e_refl)

lemma (in weak_upper_semilattice      by simp( add [OF] sup_of_singletonI
  assumes L: "x \ carrier L" "y \ carrier L" "z \ carrier L"
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof insert least
  \<comment> \<open>The textbook argument in Jacobson I, p 457\<close>_ ?thesis
  fix s
  assume sup: "least L s (Upper L {x, y, z})"
  show" \ (y \ z) .= s"
  proof (rule weak_le_antisym)
     sup " (y \ z) \ s"
      by (fastforce intro!: join_le elim: least_Upper_above P: !l    (pper==Pl
next
    from L show
     (cases " =})
     (blast!: Upper_memI intro:le_trans join_left  join_closed
  qedsimp_all add:L [OF])
qed (simp_all

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

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  fixes L structure
  shows"|finite A;
  by (unfold join_def) (simp add: insert_commute set finitejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

lemma (in weak_upper_semilattice) weak_join_assoc
  assumes "
  shows "(x \ y) \ z .= x \ (y \ z)"
proof -
  (* 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)
  also from L have "... .= \{z, x, y}" by (simp add: weak_join_assoc_lemma)
  also from L lemma (in weak_upper_semilattice) join_right  "[| x \ carrier L; y \ carrier L |] ==> y \ x \ y"
lemma (in weak_upper_semilattice) sup_of_two_least  "[| x \ carrier L; y \ carrier L |] ==> least L (\{x, y}) (Upper L {x, y})"
  finally  assume L: "x \ carrier L" "y \ carrier L"
qed

subsubsection  with L show "least L (SOME z. least by (fast intro: someI2 weak_least_unique) (* blast fails *)

lemma (in weak_upper_semilattice)join_le
"[!!.greatest L i Lower L {,y} ==> P i;
  x \<in> carrier L; y \<in> carrier L |]
  ==> P (x \<sqinter> y)"
proof x: x \<in> carrier L" and y: "y \<in> carrier L" and z: "z \<in> carrier L"
  assume L: "x \ carrier L" "y \ carrier L"
    and "!.greatest Lg ( {,y)=>P "
  withjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  withshow " ( g. greatest Lower L x,y)"
  by (fastjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
qed

lemma (nweak_lower_semilattice [simp
  "[ x\ carrier L; y \ carrier L |] ==> x \ y \ carrier L"
  by (rule meetI) (rule greatest_closed)

lemma (in weak_lower_semilattice)lemmainw) weak_join_assoc_lemma
  assumes carr: "x shows "x \ (y \ z) .= \{x, y, z}"
    and xx': "x .= x'"
  shows "x \ y .= x' \ y"
proof (rule meetI, rule
  fix    sup:leastUpper }"
  from xx' carr
      have: "{x,y}.= x' y}  rule

  assume greatestaa (  x }java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
  assume "greatest L b (Lower L {x', y})"
  with
      have greatestb: "greatest L
by( add greatest_Lower_cong_r[F__seq]java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

  from(last!:Upper_memI:  join_left join_closed
      show =b  rule)
qed (rule carr(imp_all : L)

lemma (
  assumes carr: "x \ carrier L" "y \ carrier L" "y' \ carrier L"
    and join_comm
  shows \<sqinter> y .= x \<sqinter> y'"
(rule, rule)
     ( join_defsimp : insert_commute
  have "{x, y} = {ylemma (nweak_upper_semilattice) weak_join_assoc:
  also  carr'
         "(x \ y) \ z .= x \ (y \ z)"
  also have "{y', x} = {x, yp -
  finally
      have seq: "{x, y} {.=} {x, y'}"      omit

  assume greatesta  Lhave".=\{z, x, y}" by (simp add: weak_join_assoc_lemma)
  assumealsofrom  ". \{x, y, z}" by (simp add: insert_commute)
  with from L have ". = x \ (y \ z)" by (simp add: weak_join_assoc_lemma [symmetric])
      have greatestb: "greatest L b (Lower L show? by (simp add:Ljava.lang.StringIndexOutOfBoundsException: Range [39, 40) out of bounds for length 39
      by (simp[|!    Lower} =  ;

  from greatesta greatestb
      show=b  ( weak_greatest_unique
qed (rule (unfold inf_def

lemma (inwith obtain where "reatest L i LowerL{,y) fast
  "x\java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
  by(ule) auto

lemma
  "x\ carrier L ==> \{x} .= x"
  unfolding inf_def
java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71

lemma (in weak_partial_order) inf_of_singleton_closed:
\<in> carrier L ==> \<Sqinter>{x} \<in> carrier L"
  unfolding inf_def
ule someI2 intro)

text

lemma (in weak_lower_semilattice) java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 30
  "[| !!i. xx'carr
  greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |] : "{x } {= {' }"by( java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
== P(\<Sqinter>(insert x A))"
proof inf_def
  ssume
    andby simp: greatest_Lower_cong_r[F   ])
  from greatestb
  fromL greatest_a: "a carrierL"byjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
  from L inf_of_two_exists in) meet_cong_r
  java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  show "P (SOME g. greatest L "x \<sqinter> y .= x \<sqinter> y'"
  proof (ule someI2)
    show "greatest L i (Lower L ab
  have{ y} y " fast
       z
      assume "z \ insert x A"
       "\ z"
      proof "{' x ={x,y} ast
        assume "z = x" then show ?thesis
           simp:  [OF] L La
      next
        assume "z \ A"
withgreatest_a
          by (rule_tacassume"reatest L b ( x,y'}"
      qed       greatestbL b(Lower}"
    next
      fix y
      assume y: "y \ Lower L (insert x A)"
      show "y \ i"
proof greatest_leOF],  Lower_memI)
        fix z
        assume zqed rule)java.lang.StringIndexOutOfBoundsException: Range [16, 17) out of bounds for length 16
        then show "y \ z"
        proof
          have y': "y \ Lower L A"
            by (meson Lower_antimono   ( greatest_LowerI
          assume "z =lemma (in weak_partialorderweak_inf_of_singleton [simp]:
          with y' greatest_a show ?thesis inf_def
        next
          assume" \ {x}"
withshow last
qed
      qed Lower_closed [THEN subsetD, OF y])
    next
fromL show insert
      from greatest_i show "i
qed
  qed (rule P)
qed

lemma (in weak_lower_semilattice) finite_inf_greatest:
  "[ finite A; A \ carrier L; A \ {} |] ==> greatest L (\A) (Lower L A)"
proof induct finitejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  case    L:" \ carrier L" "A \ carrier L"
next
     greatest_a   Lower
  show ?case
proof ( "A = {})
    case True
    with show?java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
bysimpaddOF
        inf_of_singleton_closed inf_of_singletonI)

    case Falseproof( someI2
from  thesis
    proof (rule_tac inf_insertI)
      fromproof (ule greatest_LowerI
    qed simp_allz\<in> insert x A"
     then show
qed

lemmain) finite_inf_insertI
  assumes P: "!!i. greatest (imp: greatest_Lower_below [OF greatest_i] L La)
     xA"inite " " \ carrier L" "A \ carrier L"
  shows P(<> (nsert
proof (cases "A = {}")
  caseTrue with  and xA thesis
    by (simp
next
       y
add:  finite_inf_greatest
qedshow

lemma(in) finite_inf_closed]:
  "[ z
nduct: finite)
  casethenshow" \ z"
next
insertshow?
    by (rule_tac finite_inf_insertI) (simp_all)
qed

lemma (in           "z="
  "[| x \ carrier L; y \ carrier L |] ==> x \ y \ x"
  by (rule meetI [folded

lower_semilattice:
  "[| x java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 11
  by (rule meetI

lemma ( weak_lower_semilattice:
  "[| x \ carrier L; y \ carrier L |] ==>
         greatest_i " \ carrier L" by simp
proof    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  assume L: "xlemma (in weak_) finite_inf_greatest:
withobtainwhere L s (ower L {,}) byfast
  with
  show "reatest (.greatest L {x, y})))java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
show
qed

lemma (in    withi show thesis
  assumes sub: "z \ x" "z \ y"
    and"x \<in> carrier L" and y: "y \<in> carrier L" and z: "z \<in> carrier L"
  shows          inf_of_singletonI
proof False
  fix i
  assume "greatest L i ( insert ?thesis
with sub z  "z \ i" by (fast elim: greatest_le intro: Lower_memI)
qed

lemma  insert "greatest L (\A) (Lower L A)" by simp
  assumes simp_all
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  by (meson assms(1) assms P:"!. greatest L i Lower L (insert x A) =>Pi"

lemma (in weak_lower_semilattice) weak_meet_assoc_lemma:
  assumes L: "x \ carrier L" "y \ carrier L" "z \ carrier L"
  shows "x \ (y \ z) .= \{x, y, z}"
proofrule)
  txt \<open>The textbook argument in Jacobson I, p 457\<close>
  fix i
  assume inf: "greatest L i (Lower L {x, y, z})"
  show "x \ (y \ z) .= i"
  proof (rule weak_le_antisym)
by(imp add: finite_inf_greatest)
      by (astforce!:  elim)
  next False  P and show
    frominf "x\ (y \ z) \ i"
    by (erule_tac
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qedsimp_all greatest_closed)
qed (simp_all induct:finite)

lemma meet_comm:
  fixes L (structure)
  showsjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  bynext

lemma ) meet_left
    "[| x \  L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
   "(x
proof -
  (* FIXME: improved simp, see weak_join_assoc above *)
have ( \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
  also from L    (ule meetI meet_defblast: )
  also from L have (in ) inf_of_two_greatest
  also from L have ".e greatest L (\{x, y}) (Lower L {x, y})"
  finally ?thesis add L)
qed

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

   L "\ carrier L" "y \ carrier L"
proof
   x java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
assume:" \ carrier L" "y \ carrier L"
  show "\s. least L s (Upper L {x, y})"
   -
    note total L
    moreover
    {
assume" <> y"
      with L have "least L y (Upper L {x, y})"
        by   subz\sqsubseteq x" " \<sqsubseteq> y"
    }
  shows
    {
      assumey\<sqsubseteq> x"
with " L x (Upper {,y)"
        by (rule_tac least_UpperI   Lower)
    }
  ultimately ?thesis
  qed
next
  fix x y
  assume L: l (inweak_lattice:
  show".greatest L i LowerL{x y)java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
  -
    note total L
lemma weak_lower_semilattice:
    {
mejava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
      with L have "greatest L yproof( finite_inf_insertI)
         rule_tac) auto
     i
    moreover
    {
      assume "x "x
      with    (rule)
        by ( greatest_LowerI
    }
     show? by blast
  qed inf "
qed

subsection      (blast intro! Lower_memI intro meet_left meet_closed

locale (imp_all add L)
java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0
     "x \ y = y \ x"
  weak_partial_order_top
begin

lemma bottom_meet\in L \<Longrightarrow> \<bottom> \<sqinter> x .= \<bottom>"
  by

lemma bottom_join: "x \ carrier L \ \ \ x .= x"
  by (metis join_closed join_right least_def)

bottom_weak_eq
  "\ b \ carrier L; \ x. x \ carrier L \ b \ x \ \ b .= \"
  by (show "\s. least L s (Upper L {x, y})"

lemma     total
    java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

lemma top_meet: "x java.lang.StringIndexOutOfBoundsException: Range [0, 69) out of bounds for length 39
by (etis le_refl meet_closed meet_le meet_right top_higher)

next
     x java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

end

sublocale weak_bounded_lattice \<subseteq> weak_partial_order ..

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

locale upper_semilattice = partial_order +
  assumes
    "[| x \ carrier L; y \ carrier L |] ==> \s. least L s (Upper L {x, y})"

sublocale upper_semilattice \<subseteq> weak?: weak_upper_semilattice
  by unfold_locales

locale lower_semilattice =  L xLower}java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
  assumes inf_of_two_exists:
    "ultimately show ?thesisbyblast

sublocale
  by unfold_locales inf_of_two_exists

locale \<open>Weak Bounded Lattices\<close>

> weak_lattice

lemmain) dual_lattice
  "lattice (inv_gorder L)"
proof -
  interpret dual: weak_lattice
    by (metis dual_weak_lattice)

  show ?thesis
    apply (unfold_locales)
    apply (simp_all add: inf_of_two_exists sup_of_two_exists)
    apply (rule eq_is_equal : "x \ carrier L \ \ \ x .= \"
  done
qed

(n lattice
  assumes   metis join_closed join_right least_def)
  shows "x \ y \ x = (x \ y)"
    "<>b \ carrier L; \ x. x \ carrier L \ b \ x \ \ b .= \"

lemma (in lattice) le_iff_meet:
  assumes top_join:" \ carrier L \ \ \ x .= \"
  shows join_closed top_closed weak_le_antisym
  java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

text

sublocale
l top_weak_eq"<>

text \<open>Functions that preserve joins and meets\<close>

definition join_pres : "a,')gorder_scheme
"join_pres X java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

definition meet_pres :: "('a, 'c) gorder_scheme \ ('b, 'd) gorder_scheme \ ('a \ 'b) \ bool" where
"meet_presjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma join_pres_isotone sup_of_two_exists
assumes \<in> carrier X \<rightarrow> carrier Y" "join_pres X Y f"
  showsX Yf
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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