Theory Finite_Lattice

(*  Title:      HOL/Library/Finite_Lattice.thy
    Author:     Alessandro Coglio
*)

section ‹Finite Lattices›

theory Finite_Lattice
imports Product_Order
begin

subsection ‹Finite Complete Lattices›

text ‹A non-empty finite lattice is a complete lattice.
Since types are never empty in Isabelle/HOL,
a type of classes \<^class>‹finite› and \<^class>‹lattice›
should also have class \<^class>‹complete_lattice›.
A type class is defined
that extends classes \<^class>‹finite› and \<^class>‹lattice›
with the operators \<^const>‹bot›, \<^const>‹top›, \<^const>‹Inf›, and \<^const>‹Sup›,
along with assumptions that define these operators
in terms of the ones of classes \<^class>‹finite› and \<^class>‹lattice›.
The resulting class is a subclass of \<^class>‹complete_lattice›.›

class finite_lattice_complete = finite + lattice + bot + top + Inf + Sup +
  assumes bot_def: "bot = Inf_fin UNIV"
  assumes top_def: "top = Sup_fin UNIV"
  assumes Inf_def: "Inf A = Finite_Set.fold inf top A"
  assumes Sup_def: "Sup A = Finite_Set.fold sup bot A"

text ‹The definitional assumptions
on the operators \<^const>‹bot› and \<^const>‹top›
of class \<^class>‹finite_lattice_complete›
ensure that they yield bottom and top.›

lemma finite_lattice_complete_bot_least: "(bot::'a::finite_lattice_complete) ≤ x"
  by (auto simp: bot_def intro: Inf_fin.coboundedI)

instance finite_lattice_complete ⊆ order_bot
  by standard (auto simp: finite_lattice_complete_bot_least)

lemma finite_lattice_complete_top_greatest: "(top::'a::finite_lattice_complete) ≥ x"
  by (auto simp: top_def Sup_fin.coboundedI)

instance finite_lattice_complete ⊆ order_top
  by standard (auto simp: finite_lattice_complete_top_greatest)

instance finite_lattice_complete ⊆ bounded_lattice ..

text ‹The definitional assumptions
on the operators \<^const>‹Inf› and \<^const>‹Sup›
of class \<^class>‹finite_lattice_complete›
ensure that they yield infimum and supremum.›

lemma finite_lattice_complete_Inf_empty: "Inf {} = (top :: 'a::finite_lattice_complete)"
  by (simp add: Inf_def)

lemma finite_lattice_complete_Sup_empty: "Sup {} = (bot :: 'a::finite_lattice_complete)"
  by (simp add: Sup_def)

lemma finite_lattice_complete_Inf_insert:
  fixes A :: "'a::finite_lattice_complete set"
  shows "Inf (insert x A) = inf x (Inf A)"
proof -
  interpret comp_fun_idem "inf :: 'a ⇒ _"
    by (fact comp_fun_idem_inf)
  show ?thesis by (simp add: Inf_def)
qed

lemma finite_lattice_complete_Sup_insert:
  fixes A :: "'a::finite_lattice_complete set"
  shows "Sup (insert x A) = sup x (Sup A)"
proof -
  interpret comp_fun_idem "sup :: 'a ⇒ _"
    by (fact comp_fun_idem_sup)
  show ?thesis by (simp add: Sup_def)
qed

lemma finite_lattice_complete_Inf_lower:
  "(x::'a::finite_lattice_complete) ∈ A ⟹ Inf A ≤ x"
  using finite [of A]
  by (induct A) (auto simp add: finite_lattice_complete_Inf_insert intro: le_infI2)

lemma finite_lattice_complete_Inf_greatest:
  "∀x::'a::finite_lattice_complete ∈ A. z ≤ x ⟹ z ≤ Inf A"
  using finite [of A]
  by (induct A) (auto simp add: finite_lattice_complete_Inf_empty finite_lattice_complete_Inf_insert)

lemma finite_lattice_complete_Sup_upper:
  "(x::'a::finite_lattice_complete) ∈ A ⟹ Sup A ≥ x"
  using finite [of A]
  by (induct A) (auto simp add: finite_lattice_complete_Sup_insert intro: le_supI2)

lemma finite_lattice_complete_Sup_least:
  "∀x::'a::finite_lattice_complete ∈ A. z ≥ x ⟹ z ≥ Sup A"
  using finite [of A]
  by (induct A) (auto simp add: finite_lattice_complete_Sup_empty finite_lattice_complete_Sup_insert)

instance finite_lattice_complete ⊆ complete_lattice
proof
qed (auto simp:
  finite_lattice_complete_Inf_lower
  finite_lattice_complete_Inf_greatest
  finite_lattice_complete_Sup_upper
  finite_lattice_complete_Sup_least
  finite_lattice_complete_Inf_empty
  finite_lattice_complete_Sup_empty)

text ‹The product of two finite lattices is already a finite lattice.›

lemma finite_bot_prod:
  "(bot :: ('a::finite_lattice_complete × 'b::finite_lattice_complete)) =
    Inf_fin UNIV"
  by (metis Inf_fin.coboundedI UNIV_I bot.extremum_uniqueI finite_UNIV)

lemma finite_top_prod:
  "(top :: ('a::finite_lattice_complete × 'b::finite_lattice_complete)) =
    Sup_fin UNIV"
  by (metis Sup_fin.coboundedI UNIV_I top.extremum_uniqueI finite_UNIV)

lemma finite_Inf_prod:
  "Inf(A :: ('a::finite_lattice_complete × 'b::finite_lattice_complete) set) =
    Finite_Set.fold inf top A"
  by (metis Inf_fold_inf finite)

lemma finite_Sup_prod:
  "Sup (A :: ('a::finite_lattice_complete × 'b::finite_lattice_complete) set) =
    Finite_Set.fold sup bot A"
  by (metis Sup_fold_sup finite)

instance prod :: (finite_lattice_complete, finite_lattice_complete) finite_lattice_complete
  by standard (auto simp: finite_bot_prod finite_top_prod finite_Inf_prod finite_Sup_prod)

text ‹Functions with a finite domain and with a finite lattice as codomain
already form a finite lattice.›

lemma finite_bot_fun: "(bot :: ('a::finite ⇒ 'b::finite_lattice_complete)) = Inf_fin UNIV"
  by (metis Inf_UNIV Inf_fin_Inf empty_not_UNIV finite)

lemma finite_top_fun: "(top :: ('a::finite ⇒ 'b::finite_lattice_complete)) = Sup_fin UNIV"
  by (metis Sup_UNIV Sup_fin_Sup empty_not_UNIV finite)

lemma finite_Inf_fun:
  "Inf (A::('a::finite ⇒ 'b::finite_lattice_complete) set) =
    Finite_Set.fold inf top A"
  by (metis Inf_fold_inf finite)

lemma finite_Sup_fun:
  "Sup (A::('a::finite ⇒ 'b::finite_lattice_complete) set) =
    Finite_Set.fold sup bot A"
  by (metis Sup_fold_sup finite)

instance "fun" :: (finite, finite_lattice_complete) finite_lattice_complete
  by standard (auto simp: finite_bot_fun finite_top_fun finite_Inf_fun finite_Sup_fun)


subsection ‹Finite Distributive Lattices›

text ‹A finite distributive lattice is a complete lattice
whose \<^const>‹inf› and \<^const>‹sup› operators
distribute over \<^const>‹Sup› and \<^const>‹Inf›.›

class finite_distrib_lattice_complete =
  distrib_lattice + finite_lattice_complete

lemma finite_distrib_lattice_complete_sup_Inf:
  "sup (x::'a::finite_distrib_lattice_complete) (Inf A) = (INF y∈A. sup x y)"
  using finite
  by (induct A rule: finite_induct) (simp_all add: sup_inf_distrib1)

lemma finite_distrib_lattice_complete_inf_Sup:
  "inf (x::'a::finite_distrib_lattice_complete) (Sup A) = (SUP y∈A. inf x y)"
  using finite [of A] by induct (simp_all add: inf_sup_distrib1)

context finite_distrib_lattice_complete
begin
subclass finite_distrib_lattice
proof -
  show "class.finite_distrib_lattice Inf Sup inf (≤) (<) sup bot top"
  proof
    show "bot = Inf UNIV"
      unfolding bot_def top_def Inf_def
      using Inf_fin.eq_fold Inf_fin.insert inf.absorb2 by force
  next
    show "top = Sup UNIV"
      unfolding bot_def top_def Sup_def
      using Sup_fin.eq_fold Sup_fin.insert by force
  next
    show "Inf {} = Sup UNIV"
      unfolding Inf_def Sup_def bot_def top_def
      using Sup_fin.eq_fold Sup_fin.insert by force
  next
    show "Sup {} = Inf UNIV"
      unfolding Inf_def Sup_def bot_def top_def
      using Inf_fin.eq_fold Inf_fin.insert inf.absorb2 by force
  next
    interpret comp_fun_idem_inf: comp_fun_idem inf
      by (fact comp_fun_idem_inf)
    show "Inf (insert a A) = inf a (Inf A)" for a A
      using comp_fun_idem_inf.fold_insert_idem Inf_def finite by simp
  next
    interpret comp_fun_idem_sup: comp_fun_idem sup
      by (fact comp_fun_idem_sup)
    show "Sup (insert a A) = sup a (Sup A)" for a A
      using comp_fun_idem_sup.fold_insert_idem Sup_def finite by simp
  qed
qed
end

instance finite_distrib_lattice_complete ⊆ complete_distrib_lattice ..

text ‹The product of two finite distributive lattices
is already a finite distributive lattice.›

instance prod ::
  (finite_distrib_lattice_complete, finite_distrib_lattice_complete)
  finite_distrib_lattice_complete
  ..

text ‹Functions with a finite domain
and with a finite distributive lattice as codomain
already form a finite distributive lattice.›

instance "fun" ::
  (finite, finite_distrib_lattice_complete) finite_distrib_lattice_complete
  ..

subsection ‹Linear Orders›

text ‹A linear order is a distributive lattice.
A type class is defined
that extends class \<^class>‹linorder›
with the operators \<^const>‹inf› and \<^const>‹sup›,
along with assumptions that define these operators
in terms of the ones of class \<^class>‹linorder›.
The resulting class is a subclass of \<^class>‹distrib_lattice›.›

class linorder_lattice = linorder + inf + sup +
  assumes inf_def: "inf x y = (if x ≤ y then x else y)"
  assumes sup_def: "sup x y = (if x ≥ y then x else y)"

text ‹The definitional assumptions
on the operators \<^const>‹inf› and \<^const>‹sup›
of class \<^class>‹linorder_lattice›
ensure that they yield infimum and supremum
and that they distribute over each other.›

lemma linorder_lattice_inf_le1: "inf (x::'a::linorder_lattice) y ≤ x"
  unfolding inf_def by (metis (full_types) linorder_linear)

lemma linorder_lattice_inf_le2: "inf (x::'a::linorder_lattice) y ≤ y"
  unfolding inf_def by (metis (full_types) linorder_linear)

lemma linorder_lattice_inf_greatest:
  "(x::'a::linorder_lattice) ≤ y ⟹ x ≤ z ⟹ x ≤ inf y z"
  unfolding inf_def by (metis (full_types))

lemma linorder_lattice_sup_ge1: "sup (x::'a::linorder_lattice) y ≥ x"
  unfolding sup_def by (metis (full_types) linorder_linear)

lemma linorder_lattice_sup_ge2: "sup (x::'a::linorder_lattice) y ≥ y"
  unfolding sup_def by (metis (full_types) linorder_linear)

lemma linorder_lattice_sup_least:
  "(x::'a::linorder_lattice) ≥ y ⟹ x ≥ z ⟹ x ≥ sup y z"
  by (auto simp: sup_def)

lemma linorder_lattice_sup_inf_distrib1:
  "sup (x::'a::linorder_lattice) (inf y z) = inf (sup x y) (sup x z)"
  by (auto simp: inf_def sup_def)

instance linorder_lattice ⊆ distrib_lattice
proof
qed (auto simp:
  linorder_lattice_inf_le1
  linorder_lattice_inf_le2
  linorder_lattice_inf_greatest
  linorder_lattice_sup_ge1
  linorder_lattice_sup_ge2
  linorder_lattice_sup_least
  linorder_lattice_sup_inf_distrib1)


subsection ‹Finite Linear Orders›

text ‹A (non-empty) finite linear order is a complete linear order.›

class finite_linorder_complete = linorder_lattice + finite_lattice_complete

instance finite_linorder_complete ⊆ complete_linorder ..

text ‹A (non-empty) finite linear order is a complete lattice
whose \<^const>‹inf› and \<^const>‹sup› operators
distribute over \<^const>‹Sup› and \<^const>‹Inf›.›

instance finite_linorder_complete ⊆ finite_distrib_lattice_complete ..

end