Theory Lattice

(*  Title:      HOL/Lattice/Lattice.thy
    Author:     Markus Wenzel, TU Muenchen
*)

section ‹Lattices›

theory Lattice imports Bounds begin

subsection ‹Lattice operations›

text ‹
  A \emph{lattice} is a partial order with infimum and supremum of any
  two elements (thus any \emph{finite} number of elements have bounds
  as well).
›

class lattice =
  assumes ex_inf: "∃inf. is_inf x y inf"
  assumes ex_sup: "∃sup. is_sup x y sup"

text ‹
  The ‹⊓› (meet) and ‹⊔› (join) operations select such
  infimum and supremum elements.
›

definition
  meet :: "'a::lattice ⇒ 'a ⇒ 'a"  (infixl "⊓" 70) where
  "x ⊓ y = (THE inf. is_inf x y inf)"
definition
  join :: "'a::lattice ⇒ 'a ⇒ 'a"  (infixl "⊔" 65) where
  "x ⊔ y = (THE sup. is_sup x y sup)"

text ‹
  Due to unique existence of bounds, the lattice operations may be
  exhibited as follows.
›

lemma meet_equality [elim?]: "is_inf x y inf ⟹ x ⊓ y = inf"
proof (unfold meet_def)
  assume "is_inf x y inf"
  then show "(THE inf. is_inf x y inf) = inf"
    by (rule the_equality) (rule is_inf_uniq [OF _ ‹is_inf x y inf›])
qed

lemma meetI [intro?]:
    "inf ⊑ x ⟹ inf ⊑ y ⟹ (⋀z. z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ inf) ⟹ x ⊓ y = inf"
  by (rule meet_equality, rule is_infI) blast+

lemma join_equality [elim?]: "is_sup x y sup ⟹ x ⊔ y = sup"
proof (unfold join_def)
  assume "is_sup x y sup"
  then show "(THE sup. is_sup x y sup) = sup"
    by (rule the_equality) (rule is_sup_uniq [OF _ ‹is_sup x y sup›])
qed

lemma joinI [intro?]: "x ⊑ sup ⟹ y ⊑ sup ⟹
    (⋀z. x ⊑ z ⟹ y ⊑ z ⟹ sup ⊑ z) ⟹ x ⊔ y = sup"
  by (rule join_equality, rule is_supI) blast+


text ‹
  \medskip The ‹⊓› and ‹⊔› operations indeed determine
  bounds on a lattice structure.
›

lemma is_inf_meet [intro?]: "is_inf x y (x ⊓ y)"
proof (unfold meet_def)
  from ex_inf obtain inf where "is_inf x y inf" ..
  then show "is_inf x y (THE inf. is_inf x y inf)"
    by (rule theI) (rule is_inf_uniq [OF _ ‹is_inf x y inf›])
qed

lemma meet_greatest [intro?]: "z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ x ⊓ y"
  by (rule is_inf_greatest) (rule is_inf_meet)

lemma meet_lower1 [intro?]: "x ⊓ y ⊑ x"
  by (rule is_inf_lower) (rule is_inf_meet)

lemma meet_lower2 [intro?]: "x ⊓ y ⊑ y"
  by (rule is_inf_lower) (rule is_inf_meet)


lemma is_sup_join [intro?]: "is_sup x y (x ⊔ y)"
proof (unfold join_def)
  from ex_sup obtain sup where "is_sup x y sup" ..
  then show "is_sup x y (THE sup. is_sup x y sup)"
    by (rule theI) (rule is_sup_uniq [OF _ ‹is_sup x y sup›])
qed

lemma join_least [intro?]: "x ⊑ z ⟹ y ⊑ z ⟹ x ⊔ y ⊑ z"
  by (rule is_sup_least) (rule is_sup_join)

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

lemma join_upper2 [intro?]: "y ⊑ x ⊔ y"
  by (rule is_sup_upper) (rule is_sup_join)


subsection ‹Duality›

text ‹
  The class of lattices is closed under formation of dual structures.
  This means that for any theorem of lattice theory, the dualized
  statement holds as well; this important fact simplifies many proofs
  of lattice theory.
›

instance dual :: (lattice) lattice
proof
  fix x' y' :: "'a::lattice dual"
  show "∃inf'. is_inf x' y' inf'"
  proof -
    have "∃sup. is_sup (undual x') (undual y') sup" by (rule ex_sup)
    then have "∃sup. is_inf (dual (undual x')) (dual (undual y')) (dual sup)"
      by (simp only: dual_inf)
    then show ?thesis by (simp add: dual_ex [symmetric])
  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 ?thesis by (simp add: dual_ex [symmetric])
  qed
qed

text ‹
  Apparently, the ‹⊓› and ‹⊔› operations are dual to each
  other.
›

theorem dual_meet [intro?]: "dual (x ⊓ y) = dual x ⊔ dual y"
proof -
  from is_inf_meet have "is_sup (dual x) (dual y) (dual (x ⊓ y))" ..
  then have "dual x ⊔ dual y = dual (x ⊓ y)" ..
  then show ?thesis ..
qed

theorem dual_join [intro?]: "dual (x ⊔ y) = dual x ⊓ dual y"
proof -
  from is_sup_join have "is_inf (dual x) (dual y) (dual (x ⊔ y))" ..
  then have "dual x ⊓ dual y = dual (x ⊔ y)" ..
  then show ?thesis ..
qed


subsection ‹Algebraic properties \label{sec:lattice-algebra}›

text ‹
  The ‹⊓› and ‹⊔› operations have the following
  characteristic algebraic properties: associative (A), commutative
  (C), and absorptive (AB).
›

theorem meet_assoc: "(x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)"
proof
  show "x ⊓ (y ⊓ z) ⊑ x ⊓ y"
  proof
    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 "w ⊑ x ⊓ (y ⊓ z)"
  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 -
        have "w ⊑ x ⊓ y" by fact
        also have "… ⊑ y" ..
        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: dual_join)
  also have "… = dual x ⊓ (dual y ⊓ dual z)"
    by (rule meet_assoc)
  also have "… = dual (x ⊔ (y ⊔ z))"
    by (simp only: dual_join)
  finally show ?thesis ..
qed

theorem meet_commute: "x ⊓ y = y ⊓ x"
proof
  show "y ⊓ x ⊑ x" ..
  show "y ⊓ x ⊑ y" ..
  fix z assume "z ⊑ y" and "z ⊑ x"
  then show "z ⊑ y ⊓ x" ..
qed

theorem join_commute: "x ⊔ y = y ⊔ x"
proof -
  have "dual (x ⊔ y) = dual x ⊓ dual y" ..
  also have "… = dual y ⊓ dual x"
    by (rule meet_commute)
  also have "… = dual (y ⊔ x)"
    by (simp only: dual_join)
  finally show ?thesis ..
qed

theorem meet_join_absorb: "x ⊓ (x ⊔ y) = x"
proof
  show "x ⊑ x" ..
  show "x ⊑ x ⊔ y" ..
  fix z assume "z ⊑ x" and "z ⊑ x ⊔ y"
  show "z ⊑ x" by fact
qed

theorem join_meet_absorb: "x ⊔ (x ⊓ y) = x"
proof -
  have "dual x ⊓ (dual x ⊔ dual y) = dual x"
    by (rule meet_join_absorb)
  then have "dual (x ⊔ (x ⊓ y)) = dual x"
    by (simp only: dual_meet dual_join)
  then show ?thesis ..
qed

text ‹
  \medskip Some further algebraic properties hold as well.  The
  property idempotent (I) is a basic algebraic consequence of (AB).
›

theorem meet_idem: "x ⊓ x = x"
proof -
  have "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 -
  have "dual x ⊓ dual x = dual x"
    by (rule meet_idem)
  then have "dual (x ⊔ x) = dual x"
    by (simp only: dual_join)
  then show ?thesis ..
qed

text ‹
  Meet and join are trivial for related elements.
›

theorem meet_related [elim?]: "x ⊑ y ⟹ x ⊓ y = x"
proof
  assume "x ⊑ y"
  show "x ⊑ x" ..
  show "x ⊑ y" by fact
  fix z assume "z ⊑ x" and "z ⊑ y"
  show "z ⊑ x" by fact
qed

theorem join_related [elim?]: "x ⊑ y ⟹ x ⊔ y = y"
proof -
  assume "x ⊑ y" then have "dual y ⊑ dual x" ..
  then have "dual y ⊓ dual x = dual y" by (rule meet_related)
  also have "dual y ⊓ dual x = dual (y ⊔ x)" by (simp only: dual_join)
  also have "y ⊔ x = x ⊔ y" by (rule join_commute)
  finally show ?thesis ..
qed


subsection ‹Order versus algebraic structure›

text ‹
  The ‹⊓› and ‹⊔› operations are connected with the
  underlying ‹⊑› relation in a canonical manner.
›

theorem meet_connection: "(x ⊑ y) = (x ⊓ y = x)"
proof
  assume "x ⊑ y"
  then have "is_inf x y x" ..
  then show "x ⊓ y = x" ..
next
  have "x ⊓ y ⊑ y" ..
  also assume "x ⊓ y = x"
  finally show "x ⊑ y" .
qed

theorem join_connection: "(x ⊑ y) = (x ⊔ y = y)"
proof
  assume "x ⊑ y"
  then have "is_sup x y y" ..
  then show "x ⊔ y = y" ..
next
  have "x ⊑ x ⊔ y" ..
  also assume "x ⊔ y = y"
  finally show "x ⊑ y" .
qed

text ‹
  \medskip The most fundamental result of the meta-theory of lattices
  is as follows (we do not prove it here).

  Given a structure with binary operations ‹⊓› and ‹⊔›
  such that (A), (C), and (AB) hold (cf.\
  \S\ref{sec:lattice-algebra}).  This structure represents a lattice,
  if the relation \<^term>‹x ⊑ y› is defined as \<^term>‹x ⊓ y = x›
  (alternatively as \<^term>‹x ⊔ y = y›).  Furthermore, infimum and
  supremum with respect to this ordering coincide with the original
  ‹⊓› and ‹⊔› operations.
›


subsection ‹Example instances›

subsubsection ‹Linear orders›

text ‹
  Linear orders with \<^term>‹minimum› and \<^term>‹maximum› operations
  are a (degenerate) example of lattice structures.
›

definition
  minimum :: "'a::linear_order ⇒ 'a ⇒ 'a" where
  "minimum x y = (if x ⊑ y then x else y)"
definition
  maximum :: "'a::linear_order ⇒ 'a ⇒ 'a" where
  "maximum x y = (if x ⊑ y then y else x)"

lemma is_inf_minimum: "is_inf x y (minimum x y)"
proof
  let ?min = "minimum x y"
  from leq_linear show "?min ⊑ x" by (auto simp add: minimum_def)
  from leq_linear show "?min ⊑ y" by (auto simp add: minimum_def)
  fix z assume "z ⊑ x" and "z ⊑ y"
  with leq_linear show "z ⊑ ?min" by (auto simp add: minimum_def)
qed

lemma is_sup_maximum: "is_sup x y (maximum x y)"      (* FIXME dualize!? *)
proof
  let ?max = "maximum x y"
  from leq_linear show "x ⊑ ?max" by (auto simp add: maximum_def)
  from leq_linear show "y ⊑ ?max" by (auto simp add: maximum_def)
  fix z assume "x ⊑ z" and "y ⊑ z"
  with leq_linear show "?max ⊑ z" by (auto simp add: maximum_def)
qed

instance linear_order ⊆ lattice
proof
  fix x y :: "'a::linear_order"
  from is_inf_minimum show "∃inf. is_inf x y inf" ..
  from is_sup_maximum show "∃sup. is_sup x y sup" ..
qed

text ‹
  The lattice operations on linear orders indeed coincide with \<^term>‹minimum› and \<^term>‹maximum›.
›

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

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



subsubsection ‹Binary products›

text ‹
  The class of lattices is closed under direct binary products (cf.\
  \S\ref{sec:prod-order}).
›

lemma is_inf_prod: "is_inf p q (fst p ⊓ fst q, snd p ⊓ snd q)"
proof
  show "(fst p ⊓ fst q, snd p ⊓ snd q) ⊑ p"
  proof -
    have "fst p ⊓ fst q ⊑ fst p" ..
    moreover have "snd p ⊓ snd q ⊑ snd p" ..
    ultimately show ?thesis by (simp add: leq_prod_def)
  qed
  show "(fst p ⊓ fst q, snd p ⊓ snd q) ⊑ q"
  proof -
    have "fst p ⊓ fst q ⊑ fst q" ..
    moreover have "snd p ⊓ snd q ⊑ snd q" ..
    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 -
    have "fst r ⊑ fst p ⊓ fst q"
    proof
      from rp show "fst r ⊑ fst p" by (simp add: leq_prod_def)
      from rq show "fst r ⊑ fst q" by (simp add: leq_prod_def)
    qed
    moreover have "snd r ⊑ snd p ⊓ snd q"
    proof
      from rp show "snd r ⊑ snd p" by (simp add: leq_prod_def)
      from rq show "snd r ⊑ snd q" by (simp add: leq_prod_def)
    qed
    ultimately show ?thesis by (simp add: leq_prod_def)
  qed
qed

lemma is_sup_prod: "is_sup p q (fst p ⊔ fst q, snd p ⊔ snd q)"  (* FIXME dualize!? *)
proof
  show "p ⊑ (fst p ⊔ fst q, snd p ⊔ snd q)"
  proof -
    have "fst p ⊑ fst p ⊔ fst q" ..
    moreover have "snd p ⊑ snd p ⊔ snd q" ..
    ultimately show ?thesis by (simp add: leq_prod_def)
  qed
  show "q ⊑ (fst p ⊔ fst q, snd p ⊔ snd q)"
  proof -
    have "fst q ⊑ fst p ⊔ fst q" ..
    moreover have "snd q ⊑ snd p ⊔ snd q" ..
    ultimately show ?thesis by (simp add: leq_prod_def)
  qed
  fix r assume "pr": "p ⊑ r" and qr: "q ⊑ r"
  show "(fst p ⊔ 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 qr show "fst q ⊑ fst r" by (simp add: leq_prod_def)
    qed
    moreover have "snd p ⊔ snd q ⊑ snd r"
    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
    ultimately show ?thesis by (simp add: leq_prod_def)
  qed
qed

instance prod :: (lattice, lattice) lattice
proof
  fix p q :: "'a::lattice × 'b::lattice"
  from is_inf_prod show "∃inf. is_inf p q inf" ..
  from is_sup_prod show "∃sup. is_sup p q sup" ..
qed

text ‹
  The lattice operations on a binary product structure indeed coincide
  with the products of the original ones.
›

theorem meet_prod: "p ⊓ q = (fst p ⊓ fst q, snd p ⊓ snd q)"
  by (rule meet_equality) (rule is_inf_prod)

theorem join_prod: "p ⊔ q = (fst p ⊔ fst q, snd p ⊔ snd q)"
  by (rule join_equality) (rule is_sup_prod)


subsubsection ‹General products›

text ‹
  The class of lattices is closed under general products (function
  spaces) as well (cf.\ \S\ref{sec:fun-order}).
›

lemma is_inf_fun: "is_inf f g (λx. f x ⊓ g x)"
proof
  show "(λx. f x ⊓ g x) ⊑ f"
  proof
    fix x show "f x ⊓ g x ⊑ f x" ..
  qed
  show "(λx. 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 ⊑ (λx. 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 (λx. f x ⊔ g x)"   (* FIXME dualize!? *)
proof
  show "f ⊑ (λx. f x ⊔ g x)"
  proof
    fix x show "f x ⊑ f x ⊔ g x" ..
  qed
  show "g ⊑ (λx. 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 "(λx. 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 ‹
  The lattice operations on a general product structure (function
  space) indeed emerge by point-wise lifting of the original ones.
›

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

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


subsection ‹Monotonicity and semi-morphisms›

text ‹
  The lattice operations are monotone in both argument positions.  In
  fact, monotonicity of the second position is trivial due to
  commutativity.
›

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 ‹
  \medskip A semi-morphisms is a function ‹f› that preserves the
  lattice operations in the following manner: \<^term>‹f (x ⊓ y) ⊑ f x
  ⊓ f y› and \<^term>‹f x ⊔ f y ⊑ f (x ⊔ y)›, respectively.  Any of
  these properties is equivalent with monotonicity.
›

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 ‹x ⊑ y› have "x ⊔ 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