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Leon Vatthauer 2023-11-21 16:58:46 +01:00
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28
src/Category/Construction/PreElgotMonads.lagda.md
View file

@ -71,12 +71,12 @@ PreElgotMonads = record
; α-η = identityˡ
; α-μ = sym (begin
T.μ.η _ ∘ T.F.₁ idC ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩
T.μ.η _ ∘ T.F.₁ idC ≈⟨ elimʳ T.F.identity ⟩
T.μ.η _ ≈⟨ sym identityˡ ⟩
idC ∘ T.μ.η _ ∎)
T.μ.η _ ∘ T.F.₁ idC ≈⟨ elimʳ T.F.identity ⟩
T.μ.η _ ≈⟨ sym identityˡ ⟩
idC ∘ T.μ.η _ ∎)
; preserves = λ f → begin
idC ∘ f # ≈⟨ identityˡ ⟩
f # ≈⟨ sym (#-resp-≈ elgotalgebras (elimˡ ([]-unique id-comm-sym id-comm-sym))) ⟩
idC ∘ f # ≈⟨ identityˡ ⟩
f # ≈⟨ sym (#-resp-≈ elgotalgebras (elimˡ ([]-unique id-comm-sym id-comm-sym))) ⟩
((idC +₁ idC) ∘ f) # ∎
}
where
@ -88,20 +88,20 @@ PreElgotMonads = record
{ α = αf ∘ᵥ αg
; α-η = λ {A} → begin
(αf.η A ∘ αg.η A) ∘ TX.η.η A ≈⟨ pullʳ (α-η g) ⟩
αf.η A ∘ TY.η.η A ≈⟨ α-η f ⟩
TZ.η.η A ∎
αf.η A ∘ TY.η.η A ≈⟨ α-η f ⟩
TZ.η.η A ∎
; α-μ = λ {A} → begin
(αf.η A ∘ αg.η A) ∘ TX.μ.η A ≈⟨ pullʳ (α-μ g) ⟩
αf.η A ∘ TY.μ.η A ∘ TY.F.₁ (αg.η A) ∘ αg.η (TX.F.₀ A) ≈⟨ pullˡ (α-μ f) ⟩
(αf.η A ∘ αg.η A) ∘ TX.μ.η A ≈⟨ pullʳ (α-μ g) ⟩
αf.η A ∘ TY.μ.η A ∘ TY.F.₁ (αg.η A) ∘ αg.η (TX.F.₀ A) ≈⟨ pullˡ (α-μ f) ⟩
(TZ.μ.η A ∘ TZ.F.₁ (αf.η A) ∘ αf.η (TY.F.₀ A)) ∘ TY.F.₁ (αg.η A) ∘ αg.η (TX.F.₀ A) ≈⟨ assoc ○ refl⟩∘⟨ pullʳ (pullˡ (NaturalTransformation.commute αf (αg.η A))) ⟩
TZ.μ.η A ∘ TZ.F.₁ (αf.η A) ∘ (TZ.F.₁ (αg.η A) ∘ αf.η (TX.F.₀ A)) ∘ αg.η (TX.F.₀ A) ≈⟨ refl⟩∘⟨ pullˡ (pullˡ (sym (Functor.homomorphism TZ.F))) ⟩
TZ.μ.η A ∘ (TZ.F.₁ (αf.η A ∘ αg.η A) ∘ αf.η (TX.F.₀ A)) ∘ αg.η (TX.F.₀ A) ≈⟨ refl⟩∘⟨ assoc ⟩
TZ.μ.η A ∘ TZ.F.₁ ((αf.η A ∘ αg.η A)) ∘ αf.η (TX.F.₀ A) ∘ αg.η (TX.F.₀ A) ∎
TZ.μ.η A ∘ (TZ.F.₁ (αf.η A ∘ αg.η A) ∘ αf.η (TX.F.₀ A)) ∘ αg.η (TX.F.₀ A) ≈⟨ refl⟩∘⟨ assoc ⟩
TZ.μ.η A ∘ TZ.F.₁ ((αf.η A ∘ αg.η A)) ∘ αf.η (TX.F.₀ A) ∘ αg.η (TX.F.₀ A) ∎
; preserves = λ {A} {B} h → begin
(αf.η B ∘ αg.η B) ∘ (h #X) ≈⟨ pullʳ (preserves g h) ⟩
αf.η B ∘ ((αg.η B +₁ idC) ∘ h) #Y ≈⟨ preserves f ((αg.η B +₁ idC) ∘ h) ⟩
(αf.η B ∘ αg.η B) ∘ (h #X) ≈⟨ pullʳ (preserves g h) ⟩
αf.η B ∘ ((αg.η B +₁ idC) ∘ h) #Y ≈⟨ preserves f ((αg.η B +₁ idC) ∘ h) ⟩
(((αf.η B +₁ idC) ∘ (αg.η B +₁ idC) ∘ h) #Z) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras Z) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
(((αf.η B ∘ αg.η B +₁ idC) ∘ h) #Z) ∎
(((αf.η B ∘ αg.η B +₁ idC) ∘ h) #Z) ∎
}
where
module TX = Monad (PreElgotMonad.T X)

38
src/Category/Construction/StrongPreElgotMonads.lagda.md
View file

@ -72,17 +72,17 @@ StrongPreElgotMonads = record
; α-η = identityˡ
; α-μ = sym (begin
M.μ.η _ ∘ M.F.₁ idC ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩
M.μ.η _ ∘ M.F.₁ idC ≈⟨ elimʳ M.F.identity ⟩
M.μ.η _ ≈⟨ sym identityˡ ⟩
idC ∘ M.μ.η _ ∎)
M.μ.η _ ∘ M.F.₁ idC ≈⟨ elimʳ M.F.identity ⟩
M.μ.η _ ≈⟨ sym identityˡ ⟩
idC ∘ M.μ.η _ ∎)
; α-strength = λ {X} {Y} → sym (begin
strengthen.η (X , Y) ∘ (idC ⁂ idC) ≈⟨ refl⟩∘⟨ (⁂-cong₂ refl (sym M.F.identity)) ⟩
strengthen.η (X , Y) ∘ (idC ⁂ idC) ≈⟨ refl⟩∘⟨ (⁂-cong₂ refl (sym M.F.identity)) ⟩
strengthen.η (X , Y) ∘ (idC ⁂ M.F.₁ idC) ≈⟨ strengthen.commute (idC , idC) ⟩
M.F.₁ (idC ⁂ idC) ∘ strengthen.η (X , Y) ≈⟨ (M.F.F-resp-≈ (⟨⟩-unique id-comm id-comm) ○ M.F.identity) ⟩∘⟨refl ⟩
idC ∘ strengthen.η (X , Y) ∎)
idC ∘ strengthen.η (X , Y) ∎)
; α-preserves = λ f → begin
idC ∘ f # ≈⟨ identityˡ ⟩
f # ≈⟨ sym (#-resp-≈ elgotalgebras (elimˡ ([]-unique id-comm-sym id-comm-sym))) ⟩
idC ∘ f # ≈⟨ identityˡ ⟩
f # ≈⟨ sym (#-resp-≈ elgotalgebras (elimˡ ([]-unique id-comm-sym id-comm-sym))) ⟩
((idC +₁ idC) ∘ f) # ∎
}
where
@ -94,25 +94,25 @@ StrongPreElgotMonads = record
{ α = αf ∘ᵥ αg
; α-η = λ {A} → begin
(αf.η A ∘ αg.η A) ∘ MX.η.η A ≈⟨ pullʳ (α-η g) ⟩
αf.η A ∘ MY.η.η A ≈⟨ α-η f ⟩
MZ.η.η A ∎
αf.η A ∘ MY.η.η A ≈⟨ α-η f ⟩
MZ.η.η A ∎
; α-μ = λ {A} → begin
(αf.η A ∘ αg.η A) ∘ MX.μ.η A ≈⟨ pullʳ (α-μ g) ⟩
αf.η A ∘ MY.μ.η A ∘ MY.F.₁ (αg.η A) ∘ αg.η (MX.F.₀ A) ≈⟨ pullˡ (α-μ f) ⟩
(αf.η A ∘ αg.η A) ∘ MX.μ.η A ≈⟨ pullʳ (α-μ g) ⟩
αf.η A ∘ MY.μ.η A ∘ MY.F.₁ (αg.η A) ∘ αg.η (MX.F.₀ A) ≈⟨ pullˡ (α-μ f) ⟩
(MZ.μ.η A ∘ MZ.F.₁ (αf.η A) ∘ αf.η (MY.F.₀ A)) ∘ MY.F.₁ (αg.η A) ∘ αg.η (MX.F.₀ A) ≈⟨ assoc ○ refl⟩∘⟨ pullʳ (pullˡ (NaturalTransformation.commute αf (αg.η A))) ⟩
MZ.μ.η A ∘ MZ.F.₁ (αf.η A) ∘ (MZ.F.₁ (αg.η A) ∘ αf.η (MX.F.₀ A)) ∘ αg.η (MX.F.₀ A) ≈⟨ refl⟩∘⟨ pullˡ (pullˡ (sym (Functor.homomorphism MZ.F))) ⟩
MZ.μ.η A ∘ (MZ.F.₁ (αf.η A ∘ αg.η A) ∘ αf.η (MX.F.₀ A)) ∘ αg.η (MX.F.₀ A) ≈⟨ refl⟩∘⟨ assoc ⟩
MZ.μ.η A ∘ MZ.F.₁ ((αf.η A ∘ αg.η A)) ∘ αf.η (MX.F.₀ A) ∘ αg.η (MX.F.₀ A) ∎
MZ.μ.η A ∘ (MZ.F.₁ (αf.η A ∘ αg.η A) ∘ αf.η (MX.F.₀ A)) ∘ αg.η (MX.F.₀ A) ≈⟨ refl⟩∘⟨ assoc ⟩
MZ.μ.η A ∘ MZ.F.₁ ((αf.η A ∘ αg.η A)) ∘ αf.η (MX.F.₀ A) ∘ αg.η (MX.F.₀ A) ∎
; α-strength = λ {A} {B} → begin
(αf.η (A × B) ∘ αg.η (A × B)) ∘ strengthenX.η (A , B) ≈⟨ pullʳ (α-strength g) ⟩
αf.η (A × B) ∘ strengthenY.η (A , B) ∘ (idC ⁂ αg.η B) ≈⟨ pullˡ (α-strength f) ⟩
(αf.η (A × B) ∘ αg.η (A × B)) ∘ strengthenX.η (A , B) ≈⟨ pullʳ (α-strength g) ⟩
αf.η (A × B) ∘ strengthenY.η (A , B) ∘ (idC ⁂ αg.η B) ≈⟨ pullˡ (α-strength f) ⟩
(strengthenZ.η (A , B) ∘ (idC ⁂ αf.η B)) ∘ (idC ⁂ αg.η B) ≈⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² refl) ⟩
strengthenZ.η (A , B) ∘ (idC ⁂ (αf.η B ∘ αg.η B)) ∎
strengthenZ.η (A , B) ∘ (idC ⁂ (αf.η B ∘ αg.η B)) ∎
; α-preserves = λ {A} {B} h → begin
(αf.η B ∘ αg.η B) ∘ (h #X) ≈⟨ pullʳ (α-preserves g h) ⟩
αf.η B ∘ ((αg.η B +₁ idC) ∘ h) #Y ≈⟨ α-preserves f ((αg.η B +₁ idC) ∘ h) ⟩
(αf.η B ∘ αg.η B) ∘ (h #X) ≈⟨ pullʳ (α-preserves g h) ⟩
αf.η B ∘ ((αg.η B +₁ idC) ∘ h) #Y ≈⟨ α-preserves f ((αg.η B +₁ idC) ∘ h) ⟩
(((αf.η B +₁ idC) ∘ (αg.η B +₁ idC) ∘ h) #Z) ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras Z) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
(((αf.η B ∘ αg.η B +₁ idC) ∘ h) #Z) ∎
(((αf.η B ∘ αg.η B +₁ idC) ∘ h) #Z) ∎
}
where
open StrongPreElgotMonad X using () renaming (SM to SMX)