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✨ Proof that K is initial strong pre-Elgot
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@ -34,7 +34,7 @@ open MR C
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open M C
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open M C
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```
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```
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# K is the initial (strong) pre-Elgot monad
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# K is the initial pre-Elgot monad
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```agda
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```agda
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isPreElgot : IsPreElgot monadK
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isPreElgot : IsPreElgot monadK
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@ -47,14 +47,8 @@ isPreElgot = record
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preElgot : PreElgotMonad
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preElgot : PreElgotMonad
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preElgot = record { T = monadK ; isPreElgot = isPreElgot }
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preElgot = record { T = monadK ; isPreElgot = isPreElgot }
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strongPreElgot : IsStrongPreElgot KStrong
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isInitialPreElgot : IsInitial PreElgotMonads preElgot
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strongPreElgot = record
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isInitialPreElgot = record
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{ preElgot = isPreElgot
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; strengthen-preserves = τ-comm
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}
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initialPreElgot : IsInitial PreElgotMonads preElgot
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initialPreElgot = record
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{ ! = !′
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{ ! = !′
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; !-unique = !-unique′
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; !-unique = !-unique′
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}
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}
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209
src/Monad/Instance/K/StrongPreElgot.lagda.md
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209
src/Monad/Instance/K/StrongPreElgot.lagda.md
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@ -0,0 +1,209 @@
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<!--
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```agda
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open import Level
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open import Category.Instance.AmbientCategory
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open import Categories.FreeObjects.Free
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open import Categories.Object.Initial
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open import Categories.NaturalTransformation
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open import Categories.NaturalTransformation.Equivalence
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open import Categories.Monad
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open import Categories.Monad.Strong
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open import Categories.Monad.Relative renaming (Monad to RMonad)
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open import Categories.Monad.Construction.Kleisli
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open import Data.Product using (_,_)
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open import Categories.Functor.Core
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import Monad.Instance.K as MIK
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```
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-->
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```agda
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module Monad.Instance.K.StrongPreElgot {o ℓ e} (ambient : Ambient o ℓ e) (MK : MIK.MonadK ambient) where
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open Ambient ambient
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open MIK ambient
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open MonadK MK
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open import Algebra.ElgotAlgebra ambient
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open import Algebra.UniformIterationAlgebra ambient
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open import Monad.PreElgot ambient
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open import Monad.Instance.K ambient
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open import Monad.Instance.K.ElgotAlgebra ambient MK
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open import Monad.Instance.K.Commutative ambient MK
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open import Monad.Instance.K.Strong ambient MK
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open import Monad.Instance.K.PreElgot ambient MK
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open import Category.Construction.StrongPreElgotMonads ambient
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open import Category.Construction.ElgotAlgebras ambient
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open import Algebra.Properties ambient
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open Equiv
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open HomReasoning
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open MR C
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open M C
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```
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# K is the initial strong pre-Elgot monad
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```agda
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isStrongPreElgot : IsStrongPreElgot KStrong
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isStrongPreElgot = record
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{ preElgot = isPreElgot
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; strengthen-preserves = τ-comm
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}
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strongPreElgot : StrongPreElgotMonad
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strongPreElgot = record
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{ SM = KStrong
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; isStrongPreElgot = isStrongPreElgot
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}
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isInitialStrongPreElgot : IsInitial StrongPreElgotMonads strongPreElgot
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isInitialStrongPreElgot = record { ! = !′ ; !-unique = !-unique′ }
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where
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!′ : ∀ {A : StrongPreElgotMonad} → StrongPreElgotMonad-Morphism strongPreElgot A
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!′ {A} = record
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{ α = ntHelper (record { η = η' ; commute = commute })
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; α-η = α-η
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; α-μ = α-μ
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; α-strength = α-strength
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; α-preserves = λ {X} {B} f → Elgot-Algebra-Morphism.preserves (((freeElgot B) FreeObject.*) {A = record { A = T.F.F₀ B ; algebra = StrongPreElgotMonad.elgotalgebras A }} (T.η.η B))
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}
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where
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open StrongPreElgotMonad A using (SM)
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module SM = StrongMonad SM
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open SM using (strengthen) renaming (M to T)
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open RMonad (Monad⇒Kleisli C T) using (extend)
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open monadK using () renaming (η to ηK; μ to μK)
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open strongK using () renaming (strengthen to strengthenK)
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open Elgot-Algebra-on using (#-resp-≈)
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T-Alg : ∀ (X : Obj) → Elgot-Algebra
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T-Alg X = record { A = T.F.₀ X ; algebra = StrongPreElgotMonad.elgotalgebras A }
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K-Alg : ∀ (X : Obj) → Elgot-Algebra
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K-Alg X = record { A = K.₀ X ; algebra = elgot X }
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η' : ∀ (X : Obj) → K.₀ X ⇒ T.F.₀ X
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η' X = Elgot-Algebra-Morphism.h (_* {A = T-Alg X} (T.η.η X))
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where open FreeObject (freeElgot X)
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_#K = λ {B} {C} f → Elgot-Algebra._# (FreeObject.FX (freeElgot C)) {B} f
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_#T = λ {B} {C} f → StrongPreElgotMonad.elgotalgebras._# A {B} {C} f
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-- some preservation facts that follow immediately, since these things are elgot-algebra-morphisms.
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K₁-preserves : ∀ {X Y Z : Obj} (f : X ⇒ Y) (g : Z ⇒ K.₀ X + Z) → K.₁ f ∘ (g #K) ≈ ((K.₁ f +₁ idC) ∘ g) #K
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K₁-preserves {X} {Y} {Z} f g = Elgot-Algebra-Morphism.preserves (((freeElgot X) FreeObject.*) {A = K-Alg Y} (ηK.η _ ∘ f))
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μK-preserves : ∀ {X Y : Obj} (g : Y ⇒ K.₀ (K.₀ X) + Y) → μK.η X ∘ g #K ≈ ((μK.η X +₁ idC) ∘ g) #K
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μK-preserves {X} g = Elgot-Algebra-Morphism.preserves (((freeElgot (K.₀ X)) FreeObject.*) {A = K-Alg X} idC)
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η'-preserves : ∀ {X Y : Obj} (g : Y ⇒ K.₀ X + Y) → η' X ∘ g #K ≈ ((η' X +₁ idC) ∘ g) #T
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η'-preserves {X} g = Elgot-Algebra-Morphism.preserves (((freeElgot X) FreeObject.*) {A = T-Alg X} (T.η.η X))
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commute : ∀ {X Y : Obj} (f : X ⇒ Y) → η' Y ∘ K.₁ f ≈ T.F.₁ f ∘ η' X
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commute {X} {Y} f = begin
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η' Y ∘ K.₁ f ≈⟨ FreeObject.*-uniq
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(freeElgot X)
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{A = T-Alg Y}
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(T.F.₁ f ∘ T.η.η X)
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(record { h = η' Y ∘ K.₁ f ; preserves = pres₁ })
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comm₁ ⟩
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Elgot-Algebra-Morphism.h (FreeObject._* (freeElgot X) {A = T-Alg Y} (T.F.₁ f ∘ T.η.η _)) ≈⟨ sym (FreeObject.*-uniq
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(freeElgot X)
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{A = T-Alg Y}
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(T.F.₁ f ∘ T.η.η X)
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(record { h = T.F.₁ f ∘ η' X ; preserves = pres₂ })
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(pullʳ (FreeObject.*-lift (freealgebras X) (T.η.η X)))) ⟩
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T.F.₁ f ∘ η' X ∎
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where
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pres₁ : ∀ {Z} {g : Z ⇒ K.₀ X + Z} → (η' Y ∘ K.₁ f) ∘ g #K ≈ ((η' Y ∘ K.₁ f +₁ idC) ∘ g) #T
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pres₁ {Z} {g} = begin
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(η' Y ∘ K.₁ f) ∘ (g #K) ≈⟨ pullʳ (K₁-preserves f g) ⟩
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η' Y ∘ (((K.₁ f +₁ idC) ∘ g) #K) ≈⟨ η'-preserves ((K.₁ f +₁ idC) ∘ g) ⟩
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(((η' Y +₁ idC) ∘ (K.₁ f +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
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((η' Y ∘ K.₁ f +₁ idC) ∘ g) #T ∎
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pres₂ : ∀ {Z} {g : Z ⇒ K.₀ X + Z} → (T.F.₁ f ∘ η' X) ∘ g #K ≈ ((T.F.₁ f ∘ η' X +₁ idC) ∘ g) #T
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pres₂ {Z} {g} = begin
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(T.F.₁ f ∘ η' X) ∘ g #K ≈⟨ pullʳ (η'-preserves g) ⟩
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T.F.₁ f ∘ ((η' X +₁ idC) ∘ g) #T ≈⟨ (sym (F₁⇒extend T f)) ⟩∘⟨refl ⟩
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extend (T.η.η Y ∘ f) ∘ ((η' X +₁ idC) ∘ g) #T ≈⟨ sym (StrongPreElgotMonad.extend-preserves A ((η' X +₁ idC) ∘ g) (T.η.η Y ∘ f)) ⟩
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(((extend (T.η.η Y ∘ f) +₁ idC) ∘ (η' X +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ ((F₁⇒extend T f) ⟩∘⟨refl) identity²)) ⟩
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((T.F.₁ f ∘ η' X +₁ idC) ∘ g) #T ∎
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comm₁ : (η' Y ∘ K.₁ f) ∘ _ ≈ T.F.₁ f ∘ T.η.η X
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comm₁ = begin
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(η' Y ∘ K.₁ f) ∘ _ ≈⟨ pullʳ (K₁η f) ⟩
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η' Y ∘ ηK.η _ ∘ f ≈⟨ pullˡ (FreeObject.*-lift (freealgebras Y) (T.η.η Y)) ⟩
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T.η.η Y ∘ f ≈⟨ NaturalTransformation.commute T.η f ⟩
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T.F.₁ f ∘ T.η.η X ∎
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α-η : ∀ {X : Obj} → η' X ∘ ηK.η X ≈ T.η.η X
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α-η {X} = FreeObject.*-lift (freealgebras X) (T.η.η X)
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α-μ : ∀ {X : Obj} → η' X ∘ μK.η X ≈ T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)
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α-μ {X} = begin
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η' X ∘ μK.η X ≈⟨ FreeObject.*-uniq
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(freeElgot (K.₀ X))
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{A = T-Alg X}
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(η' X)
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(record { h = η' X ∘ μK.η X ; preserves = pres₁ })
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(cancelʳ monadK.identityʳ) ⟩
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Elgot-Algebra-Morphism.h (((freeElgot (K.₀ X)) FreeObject.*) {A = T-Alg X} (η' X)) ≈⟨ sym (FreeObject.*-uniq
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(freeElgot (K.₀ X))
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{A = T-Alg X}
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(η' X)
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(record { h = T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) ; preserves = pres₂ })
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comm) ⟩
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T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) ∎
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where
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pres₁ : ∀ {Z} {g : Z ⇒ K.₀ (K.₀ X) + Z} → (η' X ∘ μK.η X) ∘ g #K ≈ ((η' X ∘ μK.η X +₁ idC) ∘ g) #T
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pres₁ {Z} {g} = begin
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(η' X ∘ μK.η X) ∘ (g #K) ≈⟨ pullʳ (μK-preserves g) ⟩
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η' X ∘ ((μK.η X +₁ idC) ∘ g) #K ≈⟨ η'-preserves ((μK.η X +₁ idC) ∘ g) ⟩
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(((η' X +₁ idC) ∘ (μK.η X +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
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(((η' X ∘ μK.η X +₁ idC) ∘ g) #T) ∎
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pres₂ : ∀ {Z} {g : Z ⇒ K.₀ (K.₀ X) + Z} → (T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ g #K ≈ ((T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) +₁ idC) ∘ g) #T
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pres₂ {Z} {g} = begin
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(T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ (g #K) ≈⟨ pullʳ (pullʳ (η'-preserves g)) ⟩
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T.μ.η X ∘ T.F.₁ (η' X) ∘ (((η' (K.₀ X) +₁ idC) ∘ g) #T) ≈⟨ refl⟩∘⟨ ((sym (F₁⇒extend T (η' X))) ⟩∘⟨refl ○ sym (StrongPreElgotMonad.extend-preserves A ((η' (K.₀ X) +₁ idC) ∘ g) (T.η.η (T.F.F₀ X) ∘ η' X)) )⟩
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T.μ.η X ∘ ((extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T ≈⟨ (sym (elimʳ T.F.identity)) ⟩∘⟨refl ⟩
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extend idC ∘ ((extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T ≈⟨ sym (StrongPreElgotMonad.extend-preserves A ((extend (T.η.η (T.F.F₀ X) ∘ η' X) +₁ idC) ∘ (η' (K.₀ X) +₁ idC) ∘ g) idC) ⟩
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(((extend idC +₁ idC) ∘ (extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T) ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ ((elimʳ T.F.identity) ⟩∘⟨ (F₁⇒extend T (η' X))) identity²)) ⟩
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(((T.μ.η X ∘ T.F.₁ (η' X) +₁ idC) ∘ (η' _ +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ assoc identity²)) ⟩
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(((T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) +₁ idC) ∘ g) #T) ∎
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comm : (T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ ηK.η (K.₀ X) ≈ η' X
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comm = begin
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(T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ ηK.η (K.₀ X) ≈⟨ (refl⟩∘⟨ sym (commute (η' X))) ⟩∘⟨refl ⟩
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(T.μ.η X ∘ η' _ ∘ K.₁ (η' X)) ∘ ηK.η (K.₀ X) ≈⟨ assoc ○ refl⟩∘⟨ (assoc ○ refl⟩∘⟨ sym (monadK.η.commute (η' X))) ⟩
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T.μ.η X ∘ η' _ ∘ ηK.η (T.F.F₀ X) ∘ η' X ≈⟨ refl⟩∘⟨ (pullˡ (FreeObject.*-lift (freealgebras _) (T.η.η _))) ⟩
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T.μ.η X ∘ T.η.η _ ∘ η' X ≈⟨ cancelˡ (Monad.identityʳ T) ⟩
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η' X ∎
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α-strength : ∀ {X Y : Obj} → η' (X × Y) ∘ strengthenK.η (X , Y) ≈ strengthen.η (X , Y) ∘ (idC ⁂ η' Y)
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α-strength {X} {Y} = begin
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η' (X × Y) ∘ strengthenK.η (X , Y) ≈⟨ IsStableFreeUniformIterationAlgebra.♯-unique (stable Y) (T.η.η (X × Y)) (η' (X × Y) ∘ strengthenK.η (X , Y)) (sym pres₁) pres₃ ⟩
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IsStableFreeUniformIterationAlgebra.[ (stable Y) , Functor.₀ elgot-to-uniformF (T-Alg (X × Y)) ]♯ (T.η.η (X × Y)) ≈⟨ sym (IsStableFreeUniformIterationAlgebra.♯-unique (stable Y) (T.η.η (X × Y)) (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) (sym pres₂) pres₄) ⟩
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strengthen.η (X , Y) ∘ (idC ⁂ η' Y) ∎
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where
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pres₁ : (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ ηK.η Y) ≈ T.η.η (X × Y)
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pres₁ = begin
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(η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ ηK.η Y) ≈⟨ pullʳ (τ-η (X , Y)) ⟩
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η' (X × Y) ∘ ηK.η (X × Y) ≈⟨ α-η ⟩
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T.η.η (X × Y) ∎
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pres₂ : (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ ηK.η Y) ≈ T.η.η (X × Y)
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pres₂ = begin
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(strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ ηK.η Y) ≈⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² α-η) ⟩
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strengthen.η (X , Y) ∘ (idC ⁂ T.η.η Y) ≈⟨ SM.η-comm ⟩
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T.η.η (X × Y) ∎
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pres₃ : ∀ {Z : Obj} (h : Z ⇒ K.₀ Y + Z) → (η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ h #K) ≈ ((η' (X × Y) ∘ strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T
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pres₃ {Z} h = begin
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(η' (X × Y) ∘ strengthenK.η (X , Y)) ∘ (idC ⁂ h #K) ≈⟨ pullʳ (τ-comm h) ⟩
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η' (X × Y) ∘ ((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #K ≈⟨ η'-preserves ((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ⟩
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((η' (X × Y) +₁ idC) ∘ (strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
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((η' (X × Y) ∘ strengthenK.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ∎
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pres₄ : ∀ {Z : Obj} (h : Z ⇒ K.₀ Y + Z) → (strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ h #K) ≈ ((strengthen.η (X , Y) ∘ (idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T
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pres₄ {Z} h = begin
|
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(strengthen.η (X , Y) ∘ (idC ⁂ η' Y)) ∘ (idC ⁂ h #K) ≈⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² (η'-preserves h)) ⟩
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strengthen.η (X , Y) ∘ (idC ⁂ ((η' Y +₁ idC) ∘ h) #T) ≈⟨ StrongPreElgotMonad.strengthen-preserves A ((η' Y +₁ idC) ∘ h) ⟩
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((strengthen.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ (η' Y +₁ idC) ∘ h)) #T ≈⟨ sym (#-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (refl⟩∘⟨ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² refl)))) ⟩
|
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(((strengthen.η (X , Y) +₁ idC) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (η' Y +₁ idC))) ∘ (idC ⁂ h)) #T) ≈⟨ sym (#-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (refl⟩∘⟨ (pullˡ ((+₁-cong₂ refl (sym (⟨⟩-unique id-comm id-comm))) ⟩∘⟨refl ○ distribute₁ idC (η' Y) idC)))) ⟩
|
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-- ((strengthen.η (X , Y) +₁ idC) ∘ ((idC ⁂ η' Y) +₁ (idC ⁂ idC)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ {! !} ⟩
|
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((strengthen.η (X , Y) +₁ idC) ∘ ((idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ≈⟨ #-resp-≈ (StrongPreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
|
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((strengthen.η (X , Y) ∘ (idC ⁂ η' Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #T ∎
|
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!-unique′ : ∀ {A : StrongPreElgotMonad} (f : StrongPreElgotMonad-Morphism strongPreElgot A) → StrongPreElgotMonad-Morphism.α (!′ {A = A}) ≃ StrongPreElgotMonad-Morphism.α f
|
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!-unique′ {A} f {X} = sym (FreeObject.*-uniq
|
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(freeElgot X)
|
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{A = record { A = T.F.F₀ X ; algebra = StrongPreElgotMonad.elgotalgebras A }}
|
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|
(T.η.η X)
|
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(record { h = α.η X ; preserves = α-preserves _ })
|
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|
α-η)
|
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|
where
|
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|
open StrongPreElgotMonad-Morphism f using (α; α-η; α-preserves)
|
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|
open StrongPreElgotMonad A using (SM)
|
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|
open StrongMonad SM using () renaming (M to T)
|
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|
```
|
||||||
Loading…
Reference in a new issue