\import Algebra.Monoid
\import Algebra.Semiring
\import Category.Limit
\import Data.Array
\import Data.Bool
\import Equiv
\import Function.Meta
\import HLevel
\import Homotopy.Fibration
\import Logic
\import Logic.Meta
\import Meta
\import Order.Directed
\import Paths
\import Order.PartialOrder
\import Paths.Meta
\import Set.Fin

\class MeetSemilattice \extends Poset, PrecatWithBprod (\lp, \lp) {
| meet \alias \infixl 7 ∧ : E -> E -> E
| meet-left {x y : E} : x ∧ y <= x
| meet-right {x y : E} : x ∧ y <= y
| meet-univ {x y z : E} : z <= x -> z <= y -> z <= x ∧ y
| Bprod x y => \new Product {
| apex => x ∧ y
| proj => \case \elim __ \with {
| 0 => meet-left
| 1 => meet-right
}
| tupleMap f => meet-univ (f 0) (f 1)
| tupleBeta => prop-pi
| tupleEq _ => prop-pi
}
\func Big_meet \alias Big_∧ {n : Nat} (l : Array E (suc n)) : E \elim l
| a :: l => Big (∧) a l

\lemma Big_meet-cond {n : Nat} {l : Array E (suc n)} {j : Fin (suc n)} : Big_∧ l <= l j \elim l, j
| a :: l, 0 => Big_<=_meet0
| a :: l, suc j => Big_<=_meet1 j

\lemma meet-idemp {x : E} : x ∧ x = x =>
<=-antisymmetric meet-left (meet-univ <=-refl <=-refl)

\lemma meet-comm {x y : E} : x ∧ y = y ∧ x =>
<=-antisymmetric (meet-univ meet-right meet-left)
(meet-univ meet-right meet-left)

\lemma meet-monotone {x y x' y' : E} (p : x <= x') (q : y <= y') : x ∧ y <= x' ∧ y'
=> meet-univ (meet-left <=∘ p) (meet-right <=∘ q)

\lemma meet-assoc {x y z : E} : (x ∧ y) ∧ z = x ∧ (y ∧ z) =>
<=-antisymmetric (meet-univ (meet-left <=∘ meet-left) (meet-monotone meet-right <=-refl))
(meet-univ (meet-monotone <=-refl meet-left) (meet-right <=∘ meet-right))

\lemma meet_<= {x y : E} (x<=y : x <= y) : x ∧ y = x
=> <=-antisymmetric meet-left (meet-univ <=-refl x<=y)

\lemma meet_<=' {x y : E} (p : x ∧ y = x) : x <= y
=> transport (<= y) p meet-right

\lemma Big_<=_meet0 {l : Array E} {x : E} : Big (∧) x l <= x \elim l
| nil => <=-refl
| :: a l => meet-right <=∘ Big_<=_meet0

\lemma Big_<=_meet1 {l : Array E} {x : E} (i : Fin l.len) : Big (∧) x l <= l i \elim l, i
| :: a l, 0 => meet-left
| :: a l, suc i => meet-right <=∘ Big_<=_meet1 i

\lemma Big_meet-univ {l : Array E} {x y : E} (p : y <= x) (f : \Pi (i : Fin l.len) -> y <= l i) : y <= Big (∧) x l \elim l
| nil => p
| :: a l => meet-univ (f 0) (Big_meet-univ p (\lam i => f (suc i)))
} \where {
-- | Meets x y is the type of elements which are meets of {x} and {y}.
\func Meets {E : Poset} (x y : E) =>
\Sigma (j : E) (j <= x) (j <= y) (\Pi (z : E) -> z <= x -> z <= y -> z <= j)

\lemma Meets-isProp {E : Poset} {x y : E} : isProp (Meets x y) => \lam p1 p2 =>
ext (<=-antisymmetric (p2.4 p1.1 p1.2 p1.3) (p1.4 p2.1 p2.2 p2.3))
}

\class JoinSemilattice \extends Poset {
| join \alias \infixl 6 ∨ : E -> E -> E
| join-left {x y : E} : x <= x ∨ y
| join-right {x y : E} : y <= x ∨ y
| join-univ {x y z : E} : x <= z -> y <= z -> x ∨ y <= z

\func Big_join \alias Big_∨ {n : Nat} (l : Array E (suc n)) : E \elim l
| a :: l => Big (∨) a l

\lemma Big_join-cond {n : Nat} {l : Array E (suc n)} {j : Fin (suc n)} : l j <= Big_∨ l \elim l, j
| a :: l, 0 => Big_<=_join0
| a :: l, suc j => Big_<=_join1 j

\lemma join-monotone {x y x' y' : E} (p : x <= x') (q : y <= y') : x ∨ y <= x' ∨ y'
=> join-univ (p <=∘ join-left) (q <=∘ join-right)

\lemma join-idemp {x : E} : x ∨ x = x =>
<=-antisymmetric (join-univ <=-refl <=-refl) join-left

\lemma join-comm {x y : E} : x ∨ y = y ∨ x =>
<=-antisymmetric (join-univ join-right join-left)
(join-univ join-right join-left)

\lemma join-assoc {x y z : E} : (x ∨ y) ∨ z = x ∨ (y ∨ z) =>
<=-antisymmetric (join-univ (join-univ join-left (join-left >> join-right))
(join-right >> join-right))
(join-univ (join-left >> join-left)
(join-univ (join-right >> join-left) join-right))

\lemma join_<= {x y : E} (x<=y : x <= y) : x ∨ y = y
=> <=-antisymmetric (join-univ x<=y <=-refl) join-right

\lemma join_<=' {x y : E} (p : x ∨ y = y) : x <= y
=> transport (x <=) p join-left

\lemma Big_<=_join0 {l : Array E} {x : E} : x <= Big (∨) x l \elim l
| nil => <=-refl
| :: a l => Big_<=_join0 <=∘ join-right

\lemma Big_<=_join1 {l : Array E} {x : E} (i : Fin l.len) : l i <= Big (∨) x l \elim l, i
| :: a l, 0 => join-left
| :: a l, suc i => Big_<=_join1 i <=∘ join-right

\lemma Big_join-univ {l : Array E} {x y : E} (p : x <= y) (f : \Pi (i : Fin l.len) -> l i <= y) : Big (∨) x l <= y \elim l
| nil => p
| :: a l => join-univ (f 0) (Big_join-univ p (\lam i => f (suc i)))
} \where {
-- | Joins x y is the type of elements which are joins of {x} and {y}.
\func Joins {E : Poset} (x y : E) =>
\Sigma (m : E) (x <= m) (y <= m) (\Pi (z : E) -> x <= z -> y <= z -> m <= z)

\lemma Joins-isProp {E : Poset} {x y : E} : isProp (Joins x y) => \lam p1 p2 =>
ext (<=-antisymmetric (p1.4 p2.1 p2.2 p2.3) (p2.4 p1.1 p1.2 p1.3))
}

\class Lattice \extends MeetSemilattice, JoinSemilattice {
\lemma ldistr<= {x y z : E} : (x ∧ y) ∨ (x ∧ z) <= x ∧ (y ∨ z)
=> join-univ (meet-monotone <=-refl join-left) (meet-monotone <=-refl join-right)
}

\class DistributiveLattice \extends Lattice {
| ldistr>= {x y z : E} : x ∧ (y ∨ z) <= (x ∧ y) ∨ (x ∧ z)

\lemma ldistr {x y z : E} : x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
=> <=-antisymmetric ldistr>= ldistr<=

\lemma rdistr {x y z : E} : (y ∨ z) ∧ x = (y ∧ x) ∨ (z ∧ x)
=> meet-comm *> ldistr *> pmap2 (∨) meet-comm meet-comm

\lemma rdistr>= {x y z : E} : (y ∨ z) ∧ x <= (y ∧ x) ∨ (z ∧ x)
=> =_<= rdistr

\lemma lcodistr {x y z : E} : x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) => inv $ldistr *> pmap2 (∨) (meet-comm *> meet_<= join-left) (meet-comm *> ldistr) *> inv join-assoc *> pmap2 (∨) (join-comm *> join_<= meet-right) meet-comm } \module Bounded \where { \class MeetSemilattice \extends Order.Lattice.MeetSemilattice, CartesianPrecat (\lp,\lp) { | top : E | top-univ {x : E} : x <= top | terminal => \new Product { | apex => top | proj => \case __ | tupleMap _ => top-univ | tupleBeta {_} {_} {j} => \case j | tupleEq _ => prop-pi } \lemma top-left {x : E} : top ∧ x = x => <=-antisymmetric meet-right (meet-univ top-univ <=-refl) \lemma top-right {x : E} : x ∧ top = x => <=-antisymmetric meet-left (meet-univ <=-refl top-univ) \func BigMeet (l : Array E) : E => Big (∧) top l \lemma BigMeet-cond {l : Array E} (j : Fin l.len) : BigMeet l <= l j \elim l, j | a :: l, 0 => meet-left | a :: l, suc j => meet-right <=∘ BigMeet-cond j \lemma BigMeet-univ {l : Array E} {x : E} (p : \Pi (j : Fin l.len) -> x <= l j) : x <= BigMeet l \elim l | nil => top-univ | a :: l => meet-univ (p 0)$ BigMeet-univ \lam j => p (suc j)
} \where {
\use \coerce toMonoid (L : MeetSemilattice) : CMonoid L \cowith
| ide => top
| * => meet
| ide-left => top-left
| *-assoc => meet-assoc
| *-comm => meet-comm
}

\class JoinSemilattice \extends Order.Lattice.JoinSemilattice, DirectedSet {
| bottom : E
| bottom-univ {x : E} : bottom <= x

| isInhabitted => inP bottom
| isDirected x y => inP (x ∨ y, join-left, join-right)

\func BigJoin (l : Array E) : E
=> Big (∨) bottom l

\lemma BigJoin-cond {l : Array E} (j : Fin l.len) : l j <= BigJoin l \elim l, j
| a :: l, 0 => join-left
| a :: l, suc j => BigJoin-cond j <=∘ join-right

\lemma BigJoin-univ {l : Array E} {x : E} (p : \Pi (j : Fin l.len) -> l j <= x) : BigJoin l <= x \elim l
| nil => bottom-univ
| a :: l => join-univ (p 0) $BigJoin-univ \lam j => p (suc j) \sfunc FinJoin {J : FinSet} (a : J -> E) : E => AbMonoid.FinSum {\this} a \lemma FinJoin-cond {J : FinSet} (j : J) {a : J -> E} : a j <= FinJoin a => \case AbMonoid.FinSum.aux.2 \with { | inP (e : Equiv, q) => \have t => (\peval FinJoin a) *> (\peval AbMonoid.FinSum a) *> inv q \in transport2 (<=) (pmap a (e.f_ret j)) (inv t) (BigJoin-cond (e.ret j)) } \lemma FinJoin-univ {J : FinSet} {a : J -> E} {x : E} (p : \Pi (j : J) -> a j <= x) : FinJoin a <= x => \case AbMonoid.FinSum.aux.2 \with { | inP (e,q) => \have t => (\peval FinJoin a) *> (\peval AbMonoid.FinSum a) *> inv q \in transportInv (<= x) t$ BigJoin-univ \lam j => p (e j)
}
} \where {
\use \coerce toMonoid (L : JoinSemilattice) : AbMonoid L \cowith
| zro => bottom
| + => join
| zro-left {x} => <=-antisymmetric (join-univ bottom-univ <=-refl) join-right
| +-assoc => join-assoc
| +-comm => join-comm
}

\class Lattice \extends Order.Lattice.Lattice, MeetSemilattice, JoinSemilattice

\class DistributiveLattice \extends Lattice, Order.Lattice.DistributiveLattice
\where {
\use \coerce toSemiring (L : DistributiveLattice) : CSemiring L \cowith
| zro => bottom
| + => join
| zro-left {x} => <=-antisymmetric (join-univ bottom-univ <=-refl) join-right
| +-assoc => join-assoc
| +-comm => join-comm
| ide => top
| * => meet
| ide-left {x} => <=-antisymmetric meet-right (meet-univ top-univ <=-refl)
| *-assoc => meet-assoc
| ldistr => Order.Lattice.DistributiveLattice.ldistr
| zro_*-left {x} => <=-antisymmetric meet-left bottom-univ
| *-comm => meet-comm
}
}

\class CompleteLattice \extends Bounded.Lattice, CompleteCat (\lp,\lp) {
| Join {J : \Set} : (J -> E) -> E
| Join-cond {J : \Set} (j : J) {f : J -> E} : f j <= Join f
| Join-univ {J : \Set} {f : J -> E} {e : E} : (\Pi (j : J) -> f j <= e) -> Join f <= e

| bottom => Join absurd
| bottom-univ {x} => Join-univ \case __

| Meet {J : \Set} : (J -> E) -> E
| Meet-cond {J : \Set} (j : J) {f : J -> E} : Meet f <= f j
| Meet-univ {J : \Set} {f : J -> E} {e : E} : (\Pi (j : J) -> e <= f j) -> e <= Meet f

\default join \as join-impl x y => Join (if __ x y)
\default join-left \as join-left-impl {x} {y} : x <= join-impl x y => Join-cond true
\default join-right \as join-right-impl {x} {y} : y <= join-impl x y => Join-cond false
\default join-univ \as join-univ-impl {x} {y} {z} x<=z y<=z : join-impl x y <= z => Join-univ \case \elim __ \with {
| true => x<=z
| false => y<=z
}

\default Meet \as Meet-impl {J} g => Join (\lam (t : Total (\Pi (j : J) -> __ <= g j)) => t.1)
\default Meet-cond \as Meet-cond-impl {J} j {f} : Meet-impl f <= f j => Join-univ (__.2 j)
\default Meet-univ \as Meet-univ-impl {J} {f} {e} p : e <= Meet-impl f => Join-cond (later (e,p))

\default meet \as meet-impl x y => Meet (if __ x y)
\default meet-left \as meet-left-impl {x} {y} : meet-impl x y <= x => Meet-cond true
\default meet-right \as meet-right-impl {x} {y} : meet-impl x y <= y => Meet-cond false
\default meet-univ \as meet-univ-impl {x} {y} {z} z<=x z<=y : z <= meet-impl x y => Meet-univ \case \elim __ \with {
| true => z<=x
| false => z<=y
}

\default top \as top-impl => Meet absurd
\default top-univ \as top-univ-impl {x} : x <= top-impl => Meet-univ \case __

| limit {J} G => \new Limit {
| apex => Meet (Total.proj (\lam x => ∃ (j : J) (G j = x)))
| coneMap j => Meet-cond (later (G j, inP (j, idp)))
| coneCoh _ => prop-pi
| isLimit x => \new QEquiv {
| ret (c : Cone) => Meet-univ (\lam (y, inP (j,p)) => transport (x <=) p (c.coneMap j))
| ret_f h => prop-pi
| f_sec c => exts (\lam j => prop-pi)
}
}
| pullback {x} {y} f g => \new Pullback {
| apex => x ∧ y
| pbProj1 => meet-left
| pbProj2 => meet-right
| pbCoh => prop-pi
| pbMap p1 p2 _ => meet-univ p1 p2
| pbBeta1 => prop-pi
| pbBeta2 => prop-pi
| pbEta _ _ => prop-pi
}

\lemma Join-double {I J : \Set} {f : I -> J -> E} : Join (\lam i => Join (\lam j => f i j)) = Join (\lam (p : \Sigma I J) => f p.1 p.2)
=> <=-antisymmetric (Join-univ (\lam i => Join-univ (\lam j => Join-cond (i,j)))) (Join-univ (\lam p => Join-cond p.2 <=∘ Join-cond p.1))

\lemma Join_= {J : \Set} {f g : J -> E} (p : \Pi (j : J) -> f j = g j) : Join f = Join g
=> path (\lam i => Join (p __ i))

\func SJoin (U : E -> \Prop) => Join (\lam (t : Total U) => t.1)

\lemma SJoin-cond {U : E -> \Prop} {x : E} (Ux : U x) : x <= SJoin U
=> Join-cond {_} {Total U} (x,Ux)

\lemma SJoin-univ {U : E -> \Prop} {e : E} (c : \Pi {x : E} -> U x -> x <= e) : SJoin U <= e
=> Join-univ \lam s => c s.2
}