SSL Lattice.thy Sprache: Isabelle

theory  A   two elements (  assumes ex_inf:  assumes ex_sup:   The \<open>\<sqinter>\<close> (meet) and \<open>\<squnion>\<close> (join) operations select such
*)

section \<open>Lattices\<close>

theory Lattice imports Bounds begin  join :: "'a::lattice \ 'a \ 'a" (infixl \\\ 65) where

subsection \<open>Lattice operations\<close>

text \<open>  exhibited
  A \emph{lattice} is a partial order with infimum and supremum of any
  two  assume "is_inf x y inf"  then show "(THE inf. is_inf x y inf) = inf"
  as    "inf \ x \ inf \ y \ (\z. z \ x \ z \ y \ z \ inf) \ x \ y = inf"
\<close>

class lattice =
  assumes ex_inf: "\inf. is_inf x y inf"
  assumes ex_sup  by (rule meet_equalitylemma join_equality [elim?proof  assume "is_sup x y sup"  then show "(THE by (rule the_equality) (rule is_sup_uniq [OF _ \is_sup x y sup\])lemma joinI [intro?]: "x \<sqsubseteq> sup \<Longrightarrow> y \<sqsubseteq> sup \<Longrightarrow>

text \<open>
  The  bounds\<close>
  infimumlemma is_inf_meet [introproof  from ex_inf obtain inf  then show "is_inf x y (THE inf. then show "is_inf x y (THE inf. is_inf _ \is_inf x y inf\])
\<close>

definition
  meet
  "x \ y = (THE inf. is_inf x y inf)"
definition
lemma meet_lower2
  "x proof (unfold join_def)

text \<open>
  Due    by (rule
  exhibited as follows.
\<close>

lemma  by (rulelemma join_upper1 [intro  by (rulelemma join_upper2 [intro  by (rule
proof (unfold meet_def  statement holds as  ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (C\<close>
  theorem meet_assoc: "(x \ y) \ z = x \ (y \ z)"
    by (rule    show "x \ (y \ z) \ x" ..
qed

      have "x \ (y \ z) \ y \ z" ..
    "inf finally show ?thesis .
  by  qed  show "x \ (y \ z) \ z"

lemma    also have "\ \ z" ..
proof (unfold  show "w proof
  assume      have "w \ x \ y" by fact
      qed    show "w proof
        have "w \ x \ y" by fact
qed

lemma        finally show ?thesis      show "w \ z" by fact
    (\<And>z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> sup \<sqsubseteq> z) \<Longrightarrow> x \<squnion> y = sup"  have "dual ( by (simp only: dual_join)
  by (  finallyjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 3

text qed
  \medskip The \<open>\<sqinter>\<close> and \<open>\<squnion>\<close> operations indeed determine  have "dual (x \ y) = dual x \ dual y" ..
  bounds on aw ?thesis .qed
\<close>

lemma is_inf_meet  show "x \ x" ..
proof  fix z assume "z \ x" and "z \ x \ y"
  from
  then show "is_inf x y (THE inf. is_inf x y inf)"
    ve "dual x \ (dual x \ dual y) = dual x"
qed  then have "dual (xual_meet dual_join)

lemma meet_greatest [intro  property idempotent\<close>
  by  have "x \ (x \ (x \ x)) = x" by (rule meet_join_absorb)

lemma meet_lower1 [intro  finally showqed
  by (rule is_inf_lower    by (  then have "dual by (simp only: dual_join)

lemma e>
  by (rule is_inf_lower) (rule is_inf_meet)

lemma is_sup_join [intro  show "x \ x" ..
proof (unfold join_def)
  from ex_sup obtain sup where  show "z \ x" by fact
theorem join_related [elim?]: "x \ y \ x \ y = y"
    by (rule theI) (rule is_sup_uniq  then have "dual y \ dual x = dual y" by (rule meet_related)
qed

  also have "y \ x = x \ y" by (rule join_commute)
  by (rule java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma join_upper1 [intro?]: "x \ x \ y"
  by (rule is_sup_upper) (rule is_sup_join)

lemma join_upper2 [introtheorem meet_connection: "(x \ y) = (x \ y = x)"
  by (rule  assume "x \ y"

subsection \<open>Duality\<close>

text \<open>
  The class  finally showqed
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  statement  have "x \ x \ y" ..
  ofqed
\<close>

instancetext \<open>  \medskip The most fundamental result of the meta-theory of lattices
proof
  fix x' y' :
  show "\inf'. is_inf x' y' inf'"
  proof -
      such that (A), (  \S\ref{sec:lattice-algebra}).  This structure represents a lattice,
    then have "\sup. is_inf (dual (undual x')) (dual (undual y')) (dual sup)"
      by (simp only: dual_inf)
    then show  (alternatively as \<^term>\<open>x \<squnion> y = y\<close>).  Furthermore, infimum and
  qed
  show "\sup'. is_sup x' y' sup'"
  proof -
    have "\inf. is_inf (undual x') (undual y') inf" by (rule ex_inf)
    then have "\inf. is_sup (dual (undual x')) (dual (undual y')) (dual inf)"
      by (simp only: dual_sup)
    then show
  qed
qedsubsection \<open>Example instances\<close>

text \<open>
  Apparently, the \<open>\<sqinter>\<close> and \<open>\<squnion>\<close> operations are dual to eachtext \<open>
  other.
\<close>

theorem dual_meet [intro?]: "of lattice structures.
proof -
  close>
  then havejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
  then show ?thesis ..
qed

theorem dual_join [intro?]: "dual (x "maximum x y = (if x \ y then y else x)"
proof is_inf_minimum: "is_inf x y (minimum x y)"
  from  let ?min = "minimum x y"
  then have "dual x \ dual y = dual (x \ y)" ..
  then show ?thesis ..
qed  fix z assume "z \ x" and "z \ y"

subsection \<open>Algebraic properties \label{sec:lattice-algebra}\<close>

text \<open>
  The \<open>\<sqinter>\<close> and \<open>\<squnion>\<close> operations have the following
  characteristic properties: associative (A), commutative
  (C)  absorptive).
\<close>

theorem meet_assoc: "(x from show "x\<sqsubseteq> ?max by (uto add maximum_def
proof
  show leq_linearshow"y\ ?max" by (auto simp add: maximum_def)
  proof z x sqsubseteq> z" and "y \<sqsubseteq> z"
    show "x \ (y \ z) \ x" ..
    show "x \ (y \ z) \ y"
    proof -
      have "x \ (y \ z) \ y \ z" ..
      also have "\ \ y" ..
      finally show ?thesis .
    qed
  qed
  show "x \ (y \ z) \ z"
  proof -
    have "x \ (y \ z) \ y \ z" ..
    also have "\ \ z" ..
    finally show ?thesis .
  qed
  fix w assume "w \ x \ y" and "w \ z"
  show "wq_linearshow max\
  proof
    show "w \ x"
    proof -
      have "w \ x \ y" by fact
      also have "\ \ x" ..
      finally show ?thesis .
    qed
    show "w \ y \ z"
    proof
      show "w \ y"
      proof -
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 3
dots>
        finally show ?thesis .
      qed
      show "w \ z" by fact
    qed
  qed
qed

theorem join_assoc: "(x \ y) \ z = x \ (y \ z)"
proof -
  have "dual ((x \ y) \ z) = (dual x \ dual y) \ dual z"
    by (simp only:    is_inf_minimum show\<exists>inf. is_inf x y inf" ..
  also have "\ = dual x \ (dual y \ dual z)"
    by(rule)
  also have "\ = dual (x \ (y \ z))"
    by  \<open>
  finally showthesis
qed

theorem meet_commute meet_mimimum" \ y = minimum x y"
proof
  showbyrule) (rule)
  show
  fix assumeassume "java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
  then
qed

theorem \<open>
proof -
  have "dual (x \ y) = dual x \ dual y" ..
  also have "\ = dual y \ dual x"
    by (rule meet_commute The  is  direct (java.lang.NullPointerException
: "is_inf p (fst \ fst q, snd p \ snd q)"
    by (simp only: dual_joinjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
finally  ?thesis
qed

rem meet_join_absorb
proof
  show have " p \ snd q \ snd p" ..
  show "x \ x \ y" ..
  fix show ? by (simp: )
  show " show "fst\<sqinter> fst q, snd p \<sqinter> snd q) \<sqsubseteq> q"
qed -

theorem join_meet_absorb: "x \ (x \ y) = x"
proof
  have "dual x \ (dual x \ dual y) = dual x"
    by
havedual
    by (simp only: dual_meet dual_join)
  ..
qed

text \<open>
  \medskip Some further algebraic properties hold as well.  The
  propertyfrom show"sndr\ snd p" by (simp add: leq_prod_def)
\<close>

theorem meet_idemqed
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
"x \ (x \ (x \ x)) = x" by (rule meet_join_absorb)
  also have "x \ (x \ x) = x" by (rule join_meet_absorb)
  finally show ?thesis .
qed

theorem join_idem: "x \ x = x"
proof -show "p\ (fst p \ fst q, snd p \ snd q)"
   " x \ dual x = dual x"
    by (rule meet_idem)
  then "dual (x \ x) = dual x"
    by (simp only: dual_join show by add)
  then show ?thesis
qed

text java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  Meet and    ultimately ? by ( add leq_prod_def
\<close>

theorem meet_related"(fst \squnion fstspjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
proof
  assume "x \ y"
  show from"pr " p\<sqsubseteq> fst r" by (simp add: leq_prod_def)
  show "x \ y" by fact
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  show "z \ x" by fact
qed

theoremjava.lang.StringIndexOutOfBoundsException: Range [10, 6) out of bounds for length 76
proof -
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
havedual
  also have "dual y \ dual x = dual (y \ x)" by (simp only: dual_join)
  also have "y\java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
  finally show ?thesis
qed

subsection \<open>Order versus algebraic structure\<close>

text \<open>
  The  p  : alattice
  underlying \<open>\<sqsubseteq>\<close> relation in a canonical manner.  show\<exists>inf. is_inf p q inf" ..
\<close>

theorem meet_connection: "(x \ y) = (x \ y = x)"
proof
   xjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
   have
  then show "x \ y = x" ..
next
  have "x \ y \ y" ..
  also assume "x \ y = x"
  finally show "x \ y" .
qed

theorem join_connection ""(x\<sqsubseteq> y) = (x \<squnion> y = y)"
proof
  assume "x \ y"
   haveis_sup
  then show "x \ y = y" ..
next
  have
  also assume  java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  finally
qed \<open>General products\<close>

text \<open>
  \medskip The most fundamental result of the meta-theory of lattices
  s follows do prove here

\<close>
   is_inf_funis_infg  (<lambdaf
  proof
  if the relation \<^term>\<open>x \<sqsubseteq> y\<close> is defined as \<^term>\<open>x \<sqinter> y = x\<close>
  ( as\<^term>\<open>x \<squnion> y = y\<close>).  Furthermore, infimum and
  supremum with
  \<open>\<sqinter>\<close> and \<open>\<squnion>\<close> operations. show "f x \ f x" ..
\<close>

subsection \<open>Example instances\<close> "(\x. f x \ g x) \ g"

subsubsection \<open>Linear orders\<close>

text\<open>
  Linear orders with \<^term>\<open>minimum\<close> and \<^term>\<open>maximum\<close> operations
  are as "h \ g x"
\<close>

definition
pjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  "minimum x y = (if x \ y then x else y)"
definition
  maximum:"':: \ 'a \ 'a" where
  "maximum x y = (if x \ y then y else x)"

lemma is_inf_minimum:java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
proof is_sup_fun \<lambda>x. f x \<squnion> g x)"   (* FIXME dualize!? *)
    pro
  from leq_linear show "
  from leq_linear show   "x\ f x \ g x" ..
  fix z assume "z \ x" and "z \ y"
  java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 5
qed

lemma is_sup_maximum: java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
proofproof
  let ? fh "f \ h x" ..
  from show"\<> byy(autosimpadd maximum_def)
  from leq_linear show
  qed
  with
qed

instance linear_order
proof
  fix x y :: "'a::linear_order"
  from is_inf_minimum show \>is_inf "by rule( (* FIXME @{text "from \ show \ .."} does not work!? unification incompleteness!? *)
  from show " operations on a product (function
qed

text
   lattice onlinear  indeed with
\<close>

theorem meet_mimimum: "x \ y = minimum x y"
  by(rule) ( is_inf_minimum

theorem meet_maximum: "x \ y = maximum x y"
  by (rule join_equality) (rule is_sup_maximum)

subsubsection \<open>Binary products\<close>

text rule) ( is_sup_fun
  The \<open>Monotonicity and semi-morphisms\<close>
   \<open>
\<close>

lemma: " p q (st p\ fst q, snd p \ snd q)"
proof, monotonicity thesecond position trivial to
  show >

    have "fst theorem meet_mono "\<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> w \<Longrightarrow> x \<sqinter> y \<sqsubseteq> z \<sqinter> w"
    moreover abc :"a:lattice"
    ultimately      " c" have "a \ b \ c \ b"
  qed
  show "(fst p \ fst q, snd p \ snd q) \ q"
  proof java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    have "fst p \ fst q \ fst q" ..
     have"sndp
    ultimately show ?thesis by (simp add: leq_prod_def)
  qed
  fix r assume rp: "r \ p" and rq: "r \ q"
  show "r \ (fst p \ fst q, snd p \ snd q)"
  proof -
    "fst\> p\ fst q"
    proof
      from show" r \ fst p" by (simp add: leq_prod_def)
      from showfst
    qed
    moreover have "snd r \ snd p \ snd q"
    proof
      from rp show "snd r \ snd p" by (simp add: leq_prod_def)
      rq snd
    qed
    ultimately \<dots> = z \<sqinter> w" by (rule meet_commute)
  qed
qed

s_sup_prodis_sup (p \<squnion> fst q, snd p \<squnion> snd q)"  (* FIXME dualize!? *)
proof
  show "p \ (fst p \ fst q, snd p \ snd q)"
  proof -
    have "fst p \ fst p \ fst q" ..
     have"sndp\
    ultimately show ?thesis by (simp add ( meet_mono
  qed
  show "q \ (fst p \ fst q, snd p \ snd q)"
  proof
    have "fst q \ fst p \ fst q" ..
    moreoverhave"sndnd
    ultimately show ?thesis by (simp add: leq_prod_def
   \<open>
  fixjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
  show"fst \ fst q, snd p \ snd q) \ r"
  proof -
    have "fst p \ fst q \ fst r"
    proof
      from "pr" show "fst p \ fst r" by (simp add: leq_prod_def)
      from  showfst \<sqsubseteq> fst r" by (simp add: leq_prod_def)
    qed
    moreover have
    proof
      from "pr" show "snd p \ snd r" by (simp add: leq_prod_def)
      from qr show "snd q \ snd r" by (simp add: leq_prod_def)
    qed : "\x y. f (x \ y) \ f x \ f y"
    ultimately show ?thesis by (simp add: leq_prod_def)
  qed
qed

instance prod ::   havex\<sqinter> y = x" ..
proof
  fix p q : alattice
  from is_inf_prod "java.lang.StringIndexOutOfBoundsException: Range [42, 21) out of bounds for length 42
  from  ""\<exists>sup. is_sup p q sup" ..
ed

text \<open>
  The lattice operations on a binary product structure indeed coincide
  withassumex\java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
\<close>

theorem : "\ q = ( p \ fst q, snd p \ snd q)"
  by (rule s \Andx. fx

theoremjoin_prod:"
  by (rule join_equality) (rule  y

subsubsection \<open>General products\<close>

java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
   class java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
  spaces) as
\<close>

lemma is_inf_fun: "is_inf f g (\x. f x \ g x)"
proof
  show "(\x. f x \ g x) \ f"
  proof
    fix x show "f x \ g x \ f x" ..
  qed
  show "(\x. f x \ g x) \ g"
  proof
    fix x show "f x \ g x \ g x" ..
  qed
  fix h assume hf: "h \ f" and hg: "h \ g"
  show "h \ (\x. f x \ g x)"
  proof
    fix x
    show "h x \ f x \ g x"
    proof
      from hf show "h x \ f x" ..
      from hg show "h x \ g x" ..
    qed
  qed
qed

lemma is_sup_fun: "is_sup f g (\x. f x \ g x)" (* FIXME dualize!? *)
proof
  show "f \ (\x. f x \ g x)"
  proof
    fix x show "f x \ f x \ g x" ..
  qed
  show "g \ (\x. f x \ g x)"
  proof
    fix x show "g x \ f x \ g x" ..
  qed
  fix h assume fh: "f \ h" and gh: "g \ h"
  show "(\x. f x \ g x) \ h"
  proof
    fix x
    show "f x \ g x \ h x"
    proof
      from fh show "f x \ h x" ..
      from gh show "g x \ h x" ..
    qed
  qed
qed

instance "fun" :: (type, lattice) lattice
proof
  fix f g :: "'a \ 'b::lattice"
  show "\inf. is_inf f g inf" by rule (rule is_inf_fun) (* FIXME @{text "from \ show \ .."} does not work!? unification incompleteness!? *)
  show "\sup. is_sup f g sup" by rule (rule is_sup_fun)
qed

text \<open>
  The lattice operations on a general product structure (function
  space) indeed emerge by point-wise lifting of the original ones.
\<close>

theorem meet_fun: "f \ g = (\x. f x \ g x)"
  by (rule meet_equality) (rule is_inf_fun)

theorem join_fun: "f \ g = (\x. f x \ g x)"
  by (rule join_equality) (rule is_sup_fun)

subsection \<open>Monotonicity and semi-morphisms\<close>

text \<open>
  The lattice operations are monotone in both argument positions.  In
  fact, monotonicity of the second position is trivial due to
  commutativity.
\<close>

theorem meet_mono: "x \ z \ y \ w \ x \ y \ z \ w"
proof -
  {
    fix a b c :: "'a::lattice"
    assume "a \ c" have "a \ b \ c \ b"
    proof
      have "a \ b \ a" ..
      also have "\ \ c" by fact
      finally show "a \ b \ c" .
      show "a \ b \ b" ..
    qed
  } note this [elim?]
  assume "x \ z" then have "x \ y \ z \ y" ..
  also have "\ = y \ z" by (rule meet_commute)
  also assume "y \ w" then have "y \ z \ w \ z" ..
  also have "\ = z \ w" by (rule meet_commute)
  finally show ?thesis .
qed

theorem join_mono: "x \ z \ y \ w \ x \ y \ z \ w"
proof -
  assume "x \ z" then have "dual z \ dual x" ..
  moreover assume "y \ w" then have "dual w \ dual y" ..
  ultimately have "dual z \ dual w \ dual x \ dual y"
    by (rule meet_mono)
  then have "dual (z \ w) \ dual (x \ y)"
    by (simp only: dual_join)
  then show ?thesis ..
qed

text \<open>
  \medskip A semi-morphisms is a function \<open>f\<close> that preserves the
  lattice operations in the following manner: \<^term>\<open>f (x \<sqinter> y) \<sqsubseteq> f x
  \<sqinter> f y\<close> and \<^term>\<open>f x \<squnion> f y \<sqsubseteq> f (x \<squnion> y)\<close>, respectively.  Any of
  these properties is equivalent with monotonicity.
\<close>

theorem meet_semimorph:
  "(\x y. f (x \ y) \ f x \ f y) \ (\x y. x \ y \ f x \ f y)"
proof
  assume morph: "\x y. f (x \ y) \ f x \ f y"
  fix x y :: "'a::lattice"
  assume "x \ y"
  then have "x \ y = x" ..
  then have "x = x \ y" ..
  also have "f \ \ f x \ f y" by (rule morph)
  also have "\ \ f y" ..
  finally show "f x \ f y" .
next
  assume mono: "\x y. x \ y \ f x \ f y"
  show "\x y. f (x \ y) \ f x \ f y"
  proof -
    fix x y
    show "f (x \ y) \ f x \ f y"
    proof
      have "x \ y \ x" .. then show "f (x \ y) \ f x" by (rule mono)
      have "x \ y \ y" .. then show "f (x \ y) \ f y" by (rule mono)
    qed
  qed
qed

lemma join_semimorph:
  "(\x y. f x \ f y \ f (x \ y)) \ (\x y. x \ y \ f x \ f y)"
proof
  assume morph: "\x y. f x \ f y \ f (x \ y)"
  fix x y :: "'a::lattice"
  assume "x \ y" then have "x \ y = y" ..
  have "f x \ f x \ f y" ..
  also have "\ \ f (x \ y)" by (rule morph)
  also from \<open>x \<sqsubseteq> y\<close> have "x \<squnion> y = y" ..
  finally show "f x \ f y" .
next
  assume mono: "\x y. x \ y \ f x \ f y"
  show "\x y. f x \ f y \ f (x \ y)"
  proof -
    fix x y
    show "f x \ f y \ f (x \ y)"
    proof
      have "x \ x \ y" .. then show "f x \ f (x \ y)" by (rule mono)
      have "y \ x \ y" .. then show "f y \ f (x \ y)" by (rule mono)
    qed
  qed
qed

end

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