# Theory Finite_Lattice

theory Finite_Lattice
imports Product_Order
```(*  Title:      HOL/Library/Finite_Lattice.thy
Author:     Alessandro Coglio
*)

theory Finite_Lattice
imports Product_Order
begin

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)"

lemma finite_lattice_complete_Sup_empty: "Sup {} = (bot :: 'a::finite_lattice_complete)"

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

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)

instance finite_distrib_lattice_complete ⊆ complete_distrib_lattice
proof
qed (auto simp:
finite_distrib_lattice_complete_sup_Inf
finite_distrib_lattice_complete_inf_Sup)

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

```