Delete the unneeded <= relation from instances

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2023-07-15 12:18:50 -07:00
parent 09d2125aea
commit 7b993827bf

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@ -10,57 +10,24 @@ open import Agda.Primitive using (lsuc; Level)
open import NatMap using (NatMap) open import NatMap using (NatMap)
record IsPreorder {a} (A : Set a) (_≼_ : A A Set a) : Set a where record IsSemilattice {a} (A : Set a) (_⊔_ : A A A) : Set a where
field field
≼-refl : Reflexive (_≼_)
≼-trans : Transitive (_≼_)
≼-antisym : Antisymmetric (_≡_) (_≼_)
isPreorderFlip : {a : Level} {A : Set a} {_≼_ : A A Set a} IsPreorder A _≼_ IsPreorder A (λ x y y x)
isPreorderFlip isPreorder = record
{ ≼-refl = IsPreorder.≼-refl isPreorder
; ≼-trans = λ {x} {y} {z} x≽y y≽z IsPreorder.≼-trans isPreorder y≽z x≽y
; ≼-antisym = λ {x} {y} x≽y y≽x IsPreorder.≼-antisym isPreorder y≽x x≽y
}
record Preorder {a} (A : Set a) : Set (lsuc a) where
field
_≼_ : A A Set a
isPreorder : IsPreorder A _≼_
open IsPreorder isPreorder public
record IsSemilattice {a} (A : Set a) (_≼_ : A A Set a) (_⊔_ : A A A) : Set a where
field
isPreorder : IsPreorder A _≼_
⊔-assoc : (x y z : A) (x y) z x (y z) ⊔-assoc : (x y z : A) (x y) z x (y z)
⊔-comm : (x y : A) x y y x ⊔-comm : (x y : A) x y y x
⊔-idemp : (x : A) x x x ⊔-idemp : (x : A) x x x
⊔-bound : (x y z : A) x y z (x z × y z)
⊔-least : (x y z : A) x y z (z' : A) (x z' × y z') z z'
open IsPreorder isPreorder public
record Semilattice {a} (A : Set a) : Set (lsuc a) where record Semilattice {a} (A : Set a) : Set (lsuc a) where
field field
_≼_ : A A Set a
_⊔_ : A A A _⊔_ : A A A
isSemilattice : IsSemilattice A _≼_ _⊔_ isSemilattice : IsSemilattice A _⊔_
open IsSemilattice isSemilattice public open IsSemilattice isSemilattice public
record IsLattice {a} (A : Set a) (_≼_ : A A Set a) (_⊔_ : A A A) (_⊓_ : A A A) : Set a where record IsLattice {a} (A : Set a) (_⊔_ : A A A) (_⊓_ : A A A) : Set a where
_≽_ : A A Set a
a b = b a
field field
joinSemilattice : IsSemilattice A _≼_ _⊔_ joinSemilattice : IsSemilattice A _⊔_
meetSemilattice : IsSemilattice A _≽_ _⊓_ meetSemilattice : IsSemilattice A _⊓_
absorb-⊔-⊓ : (x y : A) x (x y) x absorb-⊔-⊓ : (x y : A) x (x y) x
absorb-⊓-⊔ : (x y : A) x (x y) x absorb-⊓-⊔ : (x y : A) x (x y) x
@ -70,139 +37,49 @@ record IsLattice {a} (A : Set a) (_≼_ : A → A → Set a) (_⊔_ : A → A
( ⊔-assoc to ⊓-assoc ( ⊔-assoc to ⊓-assoc
; ⊔-comm to ⊓-comm ; ⊔-comm to ⊓-comm
; ⊔-idemp to ⊓-idemp ; ⊔-idemp to ⊓-idemp
; ⊔-bound to ⊓-bound
; ⊔-least to ⊓-least
) )
record Lattice {a} (A : Set a) : Set (lsuc a) where record Lattice {a} (A : Set a) : Set (lsuc a) where
field field
_≼_ : A A Set a
_⊔_ : A A A _⊔_ : A A A
_⊓_ : A A A _⊓_ : A A A
isLattice : IsLattice A _≼_ _⊔_ _⊓_ isLattice : IsLattice A _⊔_ _⊓_
open IsLattice isLattice public open IsLattice isLattice public
module PreorderInstances where
module ForNat where
open NatProps
NatPreorder : Preorder
NatPreorder = record
{ _≼_ = _≤_
; isPreorder = record
{ ≼-refl = ≤-refl
; ≼-trans = ≤-trans
; ≼-antisym = ≤-antisym
}
}
module ForProd {a} {A B : Set a} (pA : Preorder A) (pB : Preorder B) where
open Eq
private
_≼_ : A × B A × B Set a
(a₁ , b₁) (a₂ , b₂) = Preorder._≼_ pA a₁ a₂ × Preorder._≼_ pB b₁ b₂
≼-refl : {p : A × B} p p
≼-refl {(a , b)} = (Preorder.≼-refl pA {a}, Preorder.≼-refl pB {b})
≼-trans : {p₁ p₂ p₃ : A × B} p₁ p₂ p₂ p₃ p₁ p₃
≼-trans (a₁≼a₂ , b₁≼b₂) (a₂≼a₃ , b₂≼b₃) =
( Preorder.≼-trans pA a₁≼a₂ a₂≼a₃
, Preorder.≼-trans pB b₁≼b₂ b₂≼b₃
)
≼-antisym : {p₁ p₂ : A × B} p₁ p₂ p₂ p₁ p₁ p₂
≼-antisym (a₁≼a₂ , b₁≼b₂) (a₂≼a₁ , b₂≼b₁) = cong₂ (_,_) (Preorder.≼-antisym pA a₁≼a₂ a₂≼a₁) (Preorder.≼-antisym pB b₁≼b₂ b₂≼b₁)
ProdPreorder : Preorder (A × B)
ProdPreorder = record
{ _≼_ = _≼_
; isPreorder = record
{ ≼-refl = ≼-refl
; ≼-trans = ≼-trans
; ≼-antisym = ≼-antisym
}
}
module SemilatticeInstances where module SemilatticeInstances where
module ForNat where module ForNat where
open Nat open Nat
open NatProps open NatProps
open Eq open Eq
open PreorderInstances.ForNat
private
max-bound₁ : {x y z : } x y z x z
max-bound₁ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z rewrite ⊔-comm x y = m≤n⇒m≤o⊔n y (≤-refl)
max-bound₂ : {x y z : } x y z y z
max-bound₂ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z = m≤n⇒m≤o⊔n x (≤-refl)
max-least : (x y z : ) x y z (z' : ) (x z' × y z') z z'
max-least x y z x⊔y≡z z' (x≤z' , y≤z') with (⊔-sel x y)
... | inj₁ x⊔y≡x rewrite trans (sym x⊔y≡z) (x⊔y≡x) = x≤z'
... | inj₂ x⊔y≡y rewrite trans (sym x⊔y≡z) (x⊔y≡y) = y≤z'
NatMaxSemilattice : Semilattice NatMaxSemilattice : Semilattice
NatMaxSemilattice = record NatMaxSemilattice = record
{ _≼_ = _≤_ { _⊔_ = _⊔_
; _⊔_ = _⊔_
; isSemilattice = record ; isSemilattice = record
{ isPreorder = Preorder.isPreorder NatPreorder { ⊔-assoc = ⊔-assoc
; ⊔-assoc = ⊔-assoc
; ⊔-comm = ⊔-comm ; ⊔-comm = ⊔-comm
; ⊔-idemp = ⊔-idem ; ⊔-idemp = ⊔-idem
; ⊔-bound = λ x y z x⊔y≡z (max-bound₁ x⊔y≡z , max-bound₂ x⊔y≡z)
; ⊔-least = max-least
} }
} }
private
min-bound₁ : {x y z : } x y z z x
min-bound₁ {x} {y} {z} x⊓y≡z rewrite sym x⊓y≡z = m≤n⇒m⊓o≤n y (≤-refl)
min-bound₂ : {x y z : } x y z z y
min-bound₂ {x} {y} {z} x⊓y≡z rewrite sym x⊓y≡z rewrite ⊓-comm x y = m≤n⇒m⊓o≤n x (≤-refl)
min-greatest : (x y z : ) x y z (z' : ) (z' x × z' y) z' z
min-greatest x y z x⊓y≡z z' (z'≤x , z'≤y) with (⊓-sel x y)
... | inj₁ x⊓y≡x rewrite trans (sym x⊓y≡z) (x⊓y≡x) = z'≤x
... | inj₂ x⊓y≡y rewrite trans (sym x⊓y≡z) (x⊓y≡y) = z'≤y
NatMinSemilattice : Semilattice NatMinSemilattice : Semilattice
NatMinSemilattice = record NatMinSemilattice = record
{ _≼_ = _≥_ { _⊔_ = _⊓_
; _⊔_ = _⊓_
; isSemilattice = record ; isSemilattice = record
{ isPreorder = isPreorderFlip (Preorder.isPreorder NatPreorder) { ⊔-assoc = ⊓-assoc
; ⊔-assoc = ⊓-assoc
; ⊔-comm = ⊓-comm ; ⊔-comm = ⊓-comm
; ⊔-idemp = ⊓-idem ; ⊔-idemp = ⊓-idem
; ⊔-bound = λ x y z x⊓y≡z (min-bound₁ x⊓y≡z , min-bound₂ x⊓y≡z)
; ⊔-least = min-greatest
} }
} }
module ForProd {a} {A B : Set a} (sA : Semilattice A) (sB : Semilattice B) where module ForProd {a} {A B : Set a} (sA : Semilattice A) (sB : Semilattice B) where
private
_≼₁_ = Semilattice._≼_ sA
_≼₂_ = Semilattice._≼_ sB
pA = record { _≼_ = _≼₁_; isPreorder = Semilattice.isPreorder sA }
pB = record { _≼_ = _≼₂_; isPreorder = Semilattice.isPreorder sB }
open PreorderInstances.ForProd pA pB
open Eq open Eq
open Data.Product open Data.Product
private private
_≼_ = Preorder._≼_ ProdPreorder
_⊔_ : A × B A × B A × B _⊔_ : A × B A × B A × B
(a₁ , b₁) (a₂ , b₂) = (Semilattice._⊔_ sA a₁ a₂ , Semilattice._⊔_ sB b₁ b₂) (a₁ , b₁) (a₂ , b₂) = (Semilattice._⊔_ sA a₁ a₂ , Semilattice._⊔_ sB b₁ b₂)
@ -221,38 +98,13 @@ module SemilatticeInstances where
rewrite Semilattice.⊔-idemp sA a rewrite Semilattice.⊔-idemp sA a
rewrite Semilattice.⊔-idemp sB b = refl rewrite Semilattice.⊔-idemp sB b = refl
⊔-bound₁ : {p₁ p₂ p₃ : A × B} p₁ p₂ p₃ p₁ p₃
⊔-bound₁ {(a₁ , b₁)} {(a₂ , b₂)} {(a₃ , b₃)} p₁⊔p₂≡p₃ = (⊔-bound-a , ⊔-bound-b)
where
⊔-bound-a = proj₁ (Semilattice.⊔-bound sA a₁ a₂ a₃ (cong proj₁ p₁⊔p₂≡p₃))
⊔-bound-b = proj₁ (Semilattice.⊔-bound sB b₁ b₂ b₃ (cong proj₂ p₁⊔p₂≡p₃))
⊔-bound₂ : {p₁ p₂ p₃ : A × B} p₁ p₂ p₃ p₂ p₃
⊔-bound₂ {(a₁ , b₁)} {(a₂ , b₂)} {(a₃ , b₃)} p₁⊔p₂≡p₃ = (⊔-bound-a , ⊔-bound-b)
where
⊔-bound-a = proj₂ (Semilattice.⊔-bound sA a₁ a₂ a₃ (cong proj₁ p₁⊔p₂≡p₃))
⊔-bound-b = proj₂ (Semilattice.⊔-bound sB b₁ b₂ b₃ (cong proj₂ p₁⊔p₂≡p₃))
⊔-least : (p₁ p₂ p₃ : A × B) p₁ p₂ p₃ (p₃' : A × B) (p₁ p₃' × p₂ p₃') p₃ p₃'
⊔-least (a₁ , b₁) (a₂ , b₂) (a₃ , b₃) p₁⊔p₂≡p₃ (a₃' , b₃') (p₁≼p₃' , p₂≼p₃') = (⊔-least-a , ⊔-least-b)
where
⊔-least-a : a₃ ≼₁ a₃'
⊔-least-a = Semilattice.⊔-least sA a₁ a₂ a₃ (cong proj₁ p₁⊔p₂≡p₃) a₃' (proj₁ p₁≼p₃' , proj₁ p₂≼p₃')
⊔-least-b : b₃ ≼₂ b₃'
⊔-least-b = Semilattice.⊔-least sB b₁ b₂ b₃ (cong proj₂ p₁⊔p₂≡p₃) b₃' (proj₂ p₁≼p₃' , proj₂ p₂≼p₃')
ProdSemilattice : Semilattice (A × B) ProdSemilattice : Semilattice (A × B)
ProdSemilattice = record ProdSemilattice = record
{ _≼_ = _≼_ { _⊔_ = _⊔_
; _⊔_ = _⊔_
; isSemilattice = record ; isSemilattice = record
{ isPreorder = Preorder.isPreorder ProdPreorder { ⊔-assoc = ⊔-assoc
; ⊔-assoc = ⊔-assoc
; ⊔-comm = ⊔-comm ; ⊔-comm = ⊔-comm
; ⊔-idemp = ⊔-idemp ; ⊔-idemp = ⊔-idemp
; ⊔-bound = λ x y z x⊓y≡z (⊔-bound₁ x⊓y≡z , ⊔-bound₂ x⊓y≡z)
; ⊔-least = ⊔-least
} }
} }
@ -266,11 +118,17 @@ module LatticeInstances where
private private
max-bound₁ : {x y z : } x y z x z
max-bound₁ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z rewrite ⊔-comm x y = m≤n⇒m≤o⊔n y (≤-refl)
min-bound₁ : {x y z : } x y z z x
min-bound₁ {x} {y} {z} x⊓y≡z rewrite sym x⊓y≡z = m≤n⇒m⊓o≤n y (≤-refl)
minmax-absorb : {x y : } x (x y) x minmax-absorb : {x y : } x (x y) x
minmax-absorb {x} {y} = ≤-antisym x⊓x⊔y≤x (helper x⊓x≤x⊓x⊔y (⊓-idem x)) minmax-absorb {x} {y} = ≤-antisym x⊓x⊔y≤x (helper x⊓x≤x⊓x⊔y (⊓-idem x))
where where
x⊓x⊔y≤x = proj₁ (Semilattice.⊔-bound NatMinSemilattice x (x y) (x (x y)) refl) x⊓x⊔y≤x = min-bound₁ {x} {x y} {x (x y)} refl
x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMaxSemilattice x y (x y) refl)) x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (max-bound₁ {x} {y} {x y} refl)
-- >:( -- >:(
helper : x x x (x y) x x x x x (x y) helper : x x x (x y) x x x x x (x y)
@ -279,8 +137,8 @@ module LatticeInstances where
maxmin-absorb : {x y : } x (x y) x maxmin-absorb : {x y : } x (x y) x
maxmin-absorb {x} {y} = ≤-antisym (helper x⊔x⊓y≤x⊔x (⊔-idem x)) x≤x⊔x⊓y maxmin-absorb {x} {y} = ≤-antisym (helper x⊔x⊓y≤x⊔x (⊔-idem x)) x≤x⊔x⊓y
where where
x≤x⊔x⊓y = proj₁ (Semilattice.⊔-bound NatMaxSemilattice x (x y) (x (x y)) refl) x≤x⊔x⊓y = max-bound₁ {x} {x y} {x (x y)} refl
x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMinSemilattice x y (x y) refl)) x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (min-bound₁ {x} {y} {x y} refl)
-- >:( -- >:(
helper : x (x y) x x x x x x (x y) x helper : x (x y) x x x x x x (x y) x
@ -288,8 +146,7 @@ module LatticeInstances where
NatLattice : Lattice NatLattice : Lattice
NatLattice = record NatLattice = record
{ _≼_ = _≤_ { _⊔_ = _⊔_
; _⊔_ = _⊔_
; _⊓_ = _⊓_ ; _⊓_ = _⊓_
; isLattice = record ; isLattice = record
{ joinSemilattice = Semilattice.isSemilattice NatMaxSemilattice { joinSemilattice = Semilattice.isSemilattice NatMaxSemilattice
@ -301,27 +158,13 @@ module LatticeInstances where
module ForProd {a} {A B : Set a} (lA : Lattice A) (lB : Lattice B) where module ForProd {a} {A B : Set a} (lA : Lattice A) (lB : Lattice B) where
private private
_≼₁_ = Lattice._≼_ lA module ProdJoin = SemilatticeInstances.ForProd
_≼₂_ = Lattice._≼_ lB record { _⊔_ = Lattice._⊔_ lA; isSemilattice = Lattice.joinSemilattice lA }
record { _⊔_ = Lattice._⊔_ lB; isSemilattice = Lattice.joinSemilattice lB }
module ProdMeet = SemilatticeInstances.ForProd
record { _⊔_ = Lattice._⊓_ lA; isSemilattice = Lattice.meetSemilattice lA }
record { _⊔_ = Lattice._⊓_ lB; isSemilattice = Lattice.meetSemilattice lB }
_⊔₁_ = Lattice._⊔_ lA
_⊔₂_ = Lattice._⊔_ lB
_⊓₁_ = Lattice._⊓_ lA
_⊓₂_ = Lattice._⊓_ lB
joinA = record { _≼_ = _≼₁_; _⊔_ = _⊔₁_; isSemilattice = Lattice.joinSemilattice lA }
joinB = record { _≼_ = _≼₂_; _⊔_ = _⊔₂_; isSemilattice = Lattice.joinSemilattice lB }
meetA = record { _≼_ = λ a b b ≼₁ a; _⊔_ = _⊓₁_; isSemilattice = Lattice.meetSemilattice lA }
meetB = record { _≼_ = λ a b b ≼₂ a; _⊔_ = _⊓₂_; isSemilattice = Lattice.meetSemilattice lB }
module ProdJoin = SemilatticeInstances.ForProd joinA joinB
module ProdMeet = SemilatticeInstances.ForProd meetA meetB
_≼_ = Semilattice._≼_ ProdJoin.ProdSemilattice
_⊔_ = Semilattice._⊔_ ProdJoin.ProdSemilattice _⊔_ = Semilattice._⊔_ ProdJoin.ProdSemilattice
_⊓_ = Semilattice._⊔_ ProdMeet.ProdSemilattice _⊓_ = Semilattice._⊔_ ProdMeet.ProdSemilattice
@ -341,8 +184,7 @@ module LatticeInstances where
ProdLattice : Lattice (A × B) ProdLattice : Lattice (A × B)
ProdLattice = record ProdLattice = record
{ _≼_ = _≼_ { _⊔_ = _⊔_
; _⊔_ = _⊔_
; _⊓_ = _⊓_ ; _⊓_ = _⊓_
; isLattice = record ; isLattice = record
{ joinSemilattice = Semilattice.isSemilattice ProdJoin.ProdSemilattice { joinSemilattice = Semilattice.isSemilattice ProdJoin.ProdSemilattice