Theory Lattices

(*  Title:      HOL/Lattices.thy
    Author:     Tobias Nipkow
*)

section Abstract lattices

theory Lattices
imports Groups
begin

subsection Abstract semilattice

text 
  These locales provide a basic structure for interpretation into
  bigger structures;  extensions require careful thinking, otherwise
  undesired effects may occur due to interpretation.


locale semilattice = abel_semigroup +
  assumes idem [simp]: "a * a = a"
begin

lemma left_idem [simp]: "a * (a * b) = a * b"
  by (simp add: assoc [symmetric])

lemma right_idem [simp]: "(a * b) * b = a * b"
  by (simp add: assoc)

end

locale semilattice_neutr = semilattice + comm_monoid

locale semilattice_order = semilattice +
  fixes less_eq :: "'a  'a  bool"  (infix "" 50)
    and less :: "'a  'a  bool"  (infix "<" 50)
  assumes order_iff: "a  b  a = a * b"
    and strict_order_iff: "a < b  a = a * b  a  b"
begin

lemma orderI: "a = a * b  a  b"
  by (simp add: order_iff)

lemma orderE:
  assumes "a  b"
  obtains "a = a * b"
  using assms by (unfold order_iff)

sublocale ordering less_eq less
proof
  show "a < b  a  b  a  b" for a b
    by (simp add: order_iff strict_order_iff)
next
  show "a  a" for a
    by (simp add: order_iff)
next
  fix a b
  assume "a  b" "b  a"
  then have "a = a * b" "a * b = b"
    by (simp_all add: order_iff commute)
  then show "a = b" by simp
next
  fix a b c
  assume "a  b" "b  c"
  then have "a = a * b" "b = b * c"
    by (simp_all add: order_iff commute)
  then have "a = a * (b * c)"
    by simp
  then have "a = (a * b) * c"
    by (simp add: assoc)
  with a = a * b [symmetric] have "a = a * c" by simp
  then show "a  c" by (rule orderI)
qed

lemma cobounded1 [simp]: "a * b  a"
  by (simp add: order_iff commute)

lemma cobounded2 [simp]: "a * b  b"
  by (simp add: order_iff)

lemma boundedI:
  assumes "a  b" and "a  c"
  shows "a  b * c"
proof (rule orderI)
  from assms obtain "a * b = a" and "a * c = a"
    by (auto elim!: orderE)
  then show "a = a * (b * c)"
    by (simp add: assoc [symmetric])
qed

lemma boundedE:
  assumes "a  b * c"
  obtains "a  b" and "a  c"
  using assms by (blast intro: trans cobounded1 cobounded2)

lemma bounded_iff [simp]: "a  b * c  a  b  a  c"
  by (blast intro: boundedI elim: boundedE)

lemma strict_boundedE:
  assumes "a < b * c"
  obtains "a < b" and "a < c"
  using assms by (auto simp add: commute strict_iff_order elim: orderE intro!: that)+

lemma coboundedI1: "a  c  a * b  c"
  by (rule trans) auto

lemma coboundedI2: "b  c  a * b  c"
  by (rule trans) auto

lemma strict_coboundedI1: "a < c  a * b < c"
  using irrefl
  by (auto intro: not_eq_order_implies_strict coboundedI1 strict_implies_order
      elim: strict_boundedE)

lemma strict_coboundedI2: "b < c  a * b < c"
  using strict_coboundedI1 [of b c a] by (simp add: commute)

lemma mono: "a  c  b  d  a * b  c * d"
  by (blast intro: boundedI coboundedI1 coboundedI2)

lemma absorb1: "a  b  a * b = a"
  by (rule antisym) (auto simp: refl)

lemma absorb2: "b  a  a * b = b"
  by (rule antisym) (auto simp: refl)

lemma absorb3: "a < b  a * b = a"
  by (rule absorb1) (rule strict_implies_order)

lemma absorb4: "b < a  a * b = b"
  by (rule absorb2) (rule strict_implies_order)

lemma absorb_iff1: "a  b  a * b = a"
  using order_iff by auto

lemma absorb_iff2: "b  a  a * b = b"
  using order_iff by (auto simp add: commute)

end

locale semilattice_neutr_order = semilattice_neutr + semilattice_order
begin

sublocale ordering_top less_eq less "1"
  by standard (simp add: order_iff)

lemma eq_neutr_iff [simp]: a * b = 1  a = 1  b = 1
  by (simp add: eq_iff)

lemma neutr_eq_iff [simp]: 1 = a * b  a = 1  b = 1
  by (simp add: eq_iff)

end

text Interpretations for boolean operators

interpretation conj: semilattice_neutr (∧) True
  by standard auto

interpretation disj: semilattice_neutr (∨) False
  by standard auto

declare conj_assoc [ac_simps del] disj_assoc [ac_simps del] ― ‹already simp by default


subsection Syntactic infimum and supremum operations

class inf =
  fixes inf :: "'a  'a  'a" (infixl "" 70)

class sup =
  fixes sup :: "'a  'a  'a" (infixl "" 65)


subsection Concrete lattices

class semilattice_inf = order + inf +
  assumes inf_le1 [simp]: "x  y  x"
  and inf_le2 [simp]: "x  y  y"
  and inf_greatest: "x  y  x  z  x  y  z"

class semilattice_sup = order + sup +
  assumes sup_ge1 [simp]: "x  x  y"
  and sup_ge2 [simp]: "y  x  y"
  and sup_least: "y  x  z  x  y  z  x"
begin

text Dual lattice.
lemma dual_semilattice: "class.semilattice_inf sup greater_eq greater"
  by (rule class.semilattice_inf.intro, rule dual_order)
    (unfold_locales, simp_all add: sup_least)

end

class lattice = semilattice_inf + semilattice_sup


subsubsection Intro and elim rules

context semilattice_inf
begin

lemma le_infI1: "a  x  a  b  x"
  by (rule order_trans) auto

lemma le_infI2: "b  x  a  b  x"
  by (rule order_trans) auto

lemma le_infI: "x  a  x  b  x  a  b"
  by (fact inf_greatest) (* FIXME: duplicate lemma *)

lemma le_infE: "x  a  b  (x  a  x  b  P)  P"
  by (blast intro: order_trans inf_le1 inf_le2)

lemma le_inf_iff: "x  y  z  x  y  x  z"
  by (blast intro: le_infI elim: le_infE)

lemma le_iff_inf: "x  y  x  y = x"
  by (auto intro: le_infI1 order.antisym dest: order.eq_iff [THEN iffD1] simp add: le_inf_iff)

lemma inf_mono: "a  c  b  d  a  b  c  d"
  by (fast intro: inf_greatest le_infI1 le_infI2)

lemma mono_inf: "mono f  f (A  B)  f A  f B" for f :: "'a  'b::semilattice_inf"
  by (auto simp add: mono_def intro: Lattices.inf_greatest)

end

context semilattice_sup
begin

lemma le_supI1: "x  a  x  a  b"
  by (rule order_trans) auto

lemma le_supI2: "x  b  x  a  b"
  by (rule order_trans) auto

lemma le_supI: "a  x  b  x  a  b  x"
  by (fact sup_least) (* FIXME: duplicate lemma *)

lemma le_supE: "a  b  x  (a  x  b  x  P)  P"
  by (blast intro: order_trans sup_ge1 sup_ge2)

lemma le_sup_iff: "x  y  z  x  z  y  z"
  by (blast intro: le_supI elim: le_supE)

lemma le_iff_sup: "x  y  x  y = y"
  by (auto intro: le_supI2 order.antisym dest: order.eq_iff [THEN iffD1] simp add: le_sup_iff)

lemma sup_mono: "a  c  b  d  a  b  c  d"
  by (fast intro: sup_least le_supI1 le_supI2)

lemma mono_sup: "mono f  f A  f B  f (A  B)" for f :: "'a  'b::semilattice_sup"
  by (auto simp add: mono_def intro: Lattices.sup_least)

end


subsubsection Equational laws

context semilattice_inf
begin

sublocale inf: semilattice inf
proof
  fix a b c
  show "(a  b)  c = a  (b  c)"
    by (rule order.antisym) (auto intro: le_infI1 le_infI2 simp add: le_inf_iff)
  show "a  b = b  a"
    by (rule order.antisym) (auto simp add: le_inf_iff)
  show "a  a = a"
    by (rule order.antisym) (auto simp add: le_inf_iff)
qed

sublocale inf: semilattice_order inf less_eq less
  by standard (auto simp add: le_iff_inf less_le)

lemma inf_assoc: "(x  y)  z = x  (y  z)"
  by (fact inf.assoc)

lemma inf_commute: "(x  y) = (y  x)"
  by (fact inf.commute)

lemma inf_left_commute: "x  (y  z) = y  (x  z)"
  by (fact inf.left_commute)

lemma inf_idem: "x  x = x"
  by (fact inf.idem) (* already simp *)

lemma inf_left_idem: "x  (x  y) = x  y"
  by (fact inf.left_idem) (* already simp *)

lemma inf_right_idem: "(x  y)  y = x  y"
  by (fact inf.right_idem) (* already simp *)

lemma inf_absorb1: "x  y  x  y = x"
  by (rule order.antisym) auto

lemma inf_absorb2: "y  x  x  y = y"
  by (rule order.antisym) auto

lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem

end

context semilattice_sup
begin

sublocale sup: semilattice sup
proof
  fix a b c
  show "(a  b)  c = a  (b  c)"
    by (rule order.antisym) (auto intro: le_supI1 le_supI2 simp add: le_sup_iff)
  show "a  b = b  a"
    by (rule order.antisym) (auto simp add: le_sup_iff)
  show "a  a = a"
    by (rule order.antisym) (auto simp add: le_sup_iff)
qed

sublocale sup: semilattice_order sup greater_eq greater
  by standard (auto simp add: le_iff_sup sup.commute less_le)

lemma sup_assoc: "(x  y)  z = x  (y  z)"
  by (fact sup.assoc)

lemma sup_commute: "(x  y) = (y  x)"
  by (fact sup.commute)

lemma sup_left_commute: "x  (y  z) = y  (x  z)"
  by (fact sup.left_commute)

lemma sup_idem: "x  x = x"
  by (fact sup.idem) (* already simp *)

lemma sup_left_idem [simp]: "x  (x  y) = x  y"
  by (fact sup.left_idem)

lemma sup_absorb1: "y  x  x  y = x"
  by (rule order.antisym) auto

lemma sup_absorb2: "x  y  x  y = y"
  by (rule order.antisym) auto

lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem

end

context lattice
begin

lemma dual_lattice: "class.lattice sup (≥) (>) inf"
  by (rule class.lattice.intro,
      rule dual_semilattice,
      rule class.semilattice_sup.intro,
      rule dual_order)
    (unfold_locales, auto)

lemma inf_sup_absorb [simp]: "x  (x  y) = x"
  by (blast intro: order.antisym inf_le1 inf_greatest sup_ge1)

lemma sup_inf_absorb [simp]: "x  (x  y) = x"
  by (blast intro: order.antisym sup_ge1 sup_least inf_le1)

lemmas inf_sup_aci = inf_aci sup_aci

lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2

text Towards distributivity.

lemma distrib_sup_le: "x  (y  z)  (x  y)  (x  z)"
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)

lemma distrib_inf_le: "(x  y)  (x  z)  x  (y  z)"
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)

text If you have one of them, you have them all.

lemma distrib_imp1:
  assumes distrib: "x y z. x  (y  z) = (x  y)  (x  z)"
  shows "x  (y  z) = (x  y)  (x  z)"
proof-
  have "x  (y  z) = (x  (x  z))  (y  z)"
    by simp
  also have " = x  (z  (x  y))"
    by (simp add: distrib inf_commute sup_assoc del: sup_inf_absorb)
  also have " = ((x  y)  x)  ((x  y)  z)"
    by (simp add: inf_commute)
  also have " = (x  y)  (x  z)" by(simp add:distrib)
  finally show ?thesis .
qed

lemma distrib_imp2:
  assumes distrib: "x y z. x  (y  z) = (x  y)  (x  z)"
  shows "x  (y  z) = (x  y)  (x  z)"
proof-
  have "x  (y  z) = (x  (x  z))  (y  z)"
    by simp
  also have " = x  (z  (x  y))"
    by (simp add: distrib sup_commute inf_assoc del: inf_sup_absorb)
  also have " = ((x  y)  x)  ((x  y)  z)"
    by (simp add: sup_commute)
  also have " = (x  y)  (x  z)" by (simp add:distrib)
  finally show ?thesis .
qed

end


subsubsection Strict order

context semilattice_inf
begin

lemma less_infI1: "a < x  a  b < x"
  by (auto simp add: less_le inf_absorb1 intro: le_infI1)

lemma less_infI2: "b < x  a  b < x"
  by (auto simp add: less_le inf_absorb2 intro: le_infI2)

end

context semilattice_sup
begin

lemma less_supI1: "x < a  x < a  b"
  using dual_semilattice
  by (rule semilattice_inf.less_infI1)

lemma less_supI2: "x < b  x < a  b"
  using dual_semilattice
  by (rule semilattice_inf.less_infI2)

end


subsection Distributive lattices

class distrib_lattice = lattice +
  assumes sup_inf_distrib1: "x  (y  z) = (x  y)  (x  z)"

context distrib_lattice
begin

lemma sup_inf_distrib2: "(y  z)  x = (y  x)  (z  x)"
  by (simp add: sup_commute sup_inf_distrib1)

lemma inf_sup_distrib1: "x  (y  z) = (x  y)  (x  z)"
  by (rule distrib_imp2 [OF sup_inf_distrib1])

lemma inf_sup_distrib2: "(y  z)  x = (y  x)  (z  x)"
  by (simp add: inf_commute inf_sup_distrib1)

lemma dual_distrib_lattice: "class.distrib_lattice sup (≥) (>) inf"
  by (rule class.distrib_lattice.intro, rule dual_lattice)
    (unfold_locales, fact inf_sup_distrib1)

lemmas sup_inf_distrib = sup_inf_distrib1 sup_inf_distrib2

lemmas inf_sup_distrib = inf_sup_distrib1 inf_sup_distrib2

lemmas distrib = sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2

end


subsection Bounded lattices

class bounded_semilattice_inf_top = semilattice_inf + order_top
begin

sublocale inf_top: semilattice_neutr inf top
  + inf_top: semilattice_neutr_order inf top less_eq less
proof
  show "x   = x" for x
    by (rule inf_absorb1) simp
qed

lemma inf_top_left: "  x = x"
  by (fact inf_top.left_neutral)

lemma inf_top_right: "x   = x"
  by (fact inf_top.right_neutral)

lemma inf_eq_top_iff: "x  y =   x =   y = "
  by (fact inf_top.eq_neutr_iff)

lemma top_eq_inf_iff: " = x  y  x =   y = "
  by (fact inf_top.neutr_eq_iff)

end

class bounded_semilattice_sup_bot = semilattice_sup + order_bot
begin

sublocale sup_bot: semilattice_neutr sup bot
  + sup_bot: semilattice_neutr_order sup bot greater_eq greater
proof
  show "x   = x" for x
    by (rule sup_absorb1) simp
qed

lemma sup_bot_left: "  x = x"
  by (fact sup_bot.left_neutral)

lemma sup_bot_right: "x   = x"
  by (fact sup_bot.right_neutral)

lemma sup_eq_bot_iff: "x  y =   x =   y = "
  by (fact sup_bot.eq_neutr_iff)

lemma bot_eq_sup_iff: " = x  y  x =   y = "
  by (fact sup_bot.neutr_eq_iff)

end

class bounded_lattice_bot = lattice + order_bot
begin

subclass bounded_semilattice_sup_bot ..

lemma inf_bot_left [simp]: "  x = "
  by (rule inf_absorb1) simp

lemma inf_bot_right [simp]: "x   = "
  by (rule inf_absorb2) simp

end

class bounded_lattice_top = lattice + order_top
begin

subclass bounded_semilattice_inf_top ..

lemma sup_top_left [simp]: "  x = "
  by (rule sup_absorb1) simp

lemma sup_top_right [simp]: "x   = "
  by (rule sup_absorb2) simp

end

class bounded_lattice = lattice + order_bot + order_top
begin

subclass bounded_lattice_bot ..
subclass bounded_lattice_top ..

lemma dual_bounded_lattice: "class.bounded_lattice sup greater_eq greater inf  "
  by unfold_locales (auto simp add: less_le_not_le)

end


subsection min/max› as special case of lattice

context linorder
begin

sublocale min: semilattice_order min less_eq less
  + max: semilattice_order max greater_eq greater
  by standard (auto simp add: min_def max_def)

declare min.absorb1 [simp] min.absorb2 [simp]
  min.absorb3 [simp] min.absorb4 [simp]
  max.absorb1 [simp] max.absorb2 [simp]
  max.absorb3 [simp] max.absorb4 [simp]

lemma min_le_iff_disj: "min x y  z  x  z  y  z"
  unfolding min_def using linear by (auto intro: order_trans)

lemma le_max_iff_disj: "z  max x y  z  x  z  y"
  unfolding max_def using linear by (auto intro: order_trans)

lemma min_less_iff_disj: "min x y < z  x < z  y < z"
  unfolding min_def le_less using less_linear by (auto intro: less_trans)

lemma less_max_iff_disj: "z < max x y  z < x  z < y"
  unfolding max_def le_less using less_linear by (auto intro: less_trans)

lemma min_less_iff_conj [simp]: "z < min x y  z < x  z < y"
  unfolding min_def le_less using less_linear by (auto intro: less_trans)

lemma max_less_iff_conj [simp]: "max x y < z  x < z  y < z"
  unfolding max_def le_less using less_linear by (auto intro: less_trans)

lemma min_max_distrib1: "min (max b c) a = max (min b a) (min c a)"
  by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemma min_max_distrib2: "min a (max b c) = max (min a b) (min a c)"
  by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemma max_min_distrib1: "max (min b c) a = min (max b a) (max c a)"
  by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemma max_min_distrib2: "max a (min b c) = min (max a b) (max a c)"
  by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemmas min_max_distribs = min_max_distrib1 min_max_distrib2 max_min_distrib1 max_min_distrib2

lemma split_min [no_atp]: "P (min i j)  (i  j  P i)  (¬ i  j  P j)"
  by (simp add: min_def)

lemma split_max [no_atp]: "P (max i j)  (i  j  P j)  (¬ i  j  P i)"
  by (simp add: max_def)

lemma split_min_lin [no_atp]:
  P (min a b)  (b = a  P a)  (a < b  P a)  (b < a  P b)
  by (cases a b rule: linorder_cases) auto

lemma split_max_lin [no_atp]:
  P (max a b)  (b = a  P a)  (a < b  P b)  (b < a  P a)
  by (cases a b rule: linorder_cases) auto

lemma min_of_mono: "mono f  min (f m) (f n) = f (min m n)" for f :: "'a  'b::linorder"
  by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)

lemma max_of_mono: "mono f  max (f m) (f n) = f (max m n)" for f :: "'a  'b::linorder"
  by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)

end

lemma max_of_antimono: "antimono f  max (f x) (f y) = f (min x y)"
  and min_of_antimono: "antimono f  min (f x) (f y) = f (max x y)"
  for f::"'a::linorder  'b::linorder"
  by (auto simp: antimono_def Orderings.max_def min_def intro!: antisym)

lemma inf_min: "inf = (min :: 'a::{semilattice_inf,linorder}  'a  'a)"
  by (auto intro: antisym simp add: min_def fun_eq_iff)

lemma sup_max: "sup = (max :: 'a::{semilattice_sup,linorder}  'a  'a)"
  by (auto intro: antisym simp add: max_def fun_eq_iff)


subsection Uniqueness of inf and sup

lemma (in semilattice_inf) inf_unique:
  fixes f  (infixl "" 70)
  assumes le1: "x y. x  y  x"
    and le2: "x y. x  y  y"
    and greatest: "x y z. x  y  x  z  x  y  z"
  shows "x  y = x  y"
proof (rule order.antisym)
  show "x  y  x  y"
    by (rule le_infI) (rule le1, rule le2)
  have leI: "x y z. x  y  x  z  x  y  z"
    by (blast intro: greatest)
  show "x  y  x  y"
    by (rule leI) simp_all
qed

lemma (in semilattice_sup) sup_unique:
  fixes f  (infixl "" 70)
  assumes ge1 [simp]: "x y. x  x  y"
    and ge2: "x y. y  x  y"
    and least: "x y z. y  x  z  x  y  z  x"
  shows "x  y = x  y"
proof (rule order.antisym)
  show "x  y  x  y"
    by (rule le_supI) (rule ge1, rule ge2)
  have leI: "x y z. x  z  y  z  x  y  z"
    by (blast intro: least)
  show "x  y  x  y"
    by (rule leI) simp_all
qed


subsection Lattice on typ_  _

instantiation "fun" :: (type, semilattice_sup) semilattice_sup
begin

definition "f  g = (λx. f x  g x)"

lemma sup_apply [simp, code]: "(f  g) x = f x  g x"
  by (simp add: sup_fun_def)

instance
  by standard (simp_all add: le_fun_def)

end

instantiation "fun" :: (type, semilattice_inf) semilattice_inf
begin

definition "f  g = (λx. f x  g x)"

lemma inf_apply [simp, code]: "(f  g) x = f x  g x"
  by (simp add: inf_fun_def)

instance by standard (simp_all add: le_fun_def)

end

instance "fun" :: (type, lattice) lattice ..

instance "fun" :: (type, distrib_lattice) distrib_lattice
  by standard (rule ext, simp add: sup_inf_distrib1)

instance "fun" :: (type, bounded_lattice) bounded_lattice ..

instantiation "fun" :: (type, uminus) uminus
begin

definition fun_Compl_def: "- A = (λx. - A x)"

lemma uminus_apply [simp, code]: "(- A) x = - (A x)"
  by (simp add: fun_Compl_def)

instance ..

end

instantiation "fun" :: (type, minus) minus
begin

definition fun_diff_def: "A - B = (λx. A x - B x)"

lemma minus_apply [simp, code]: "(A - B) x = A x - B x"
  by (simp add: fun_diff_def)

instance ..

end

end