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Some progress on left-id-law and such
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| {-# OPTIONS --without-K #-} | ||
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| open import 1-hits-pf.Core | ||
| open import Container | ||
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| -- Equality of algebra morphisms | ||
| module 1-hits-pf.Alg1.CatLaws (s : Spec) where | ||
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| open import Eq | ||
| open import lib.Basics | ||
| open import 1-hits-pf.Alg1.Core s | ||
| open Spec s | ||
| open import 1-hits-pf.Alg0 F₀ | ||
| open import 1-hits-pf.Alg0.FreeMonad F₀ | ||
| open import 1-hits-pf.Alg1.Eq s | ||
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| module _ | ||
| {𝓧 𝓨 : Alg₁-obj} | ||
| (𝓯 : Alg₁-hom 𝓧 𝓨) | ||
| where | ||
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| open Alg₁-obj 𝓧 | ||
| open Alg₁-obj 𝓨 renaming (𝓧' to 𝓨' ; X to Y ; θ₀ to ρ₀ ; θ₁ to ρ₁) | ||
| open Alg₁-hom 𝓯 | ||
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| left-id₀ = Ap-idf f₀ | ||
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| left-id-alg₁ : Eq (∘-alg₁ (id-alg₁ 𝓨) 𝓯) 𝓯 | ||
| left-id-alg₁ = alg₁-hom=' | ||
| f | ||
| (∘₀ {𝓧'} {𝓨'} (id-alg₀ 𝓨') 𝓯') | ||
| f₀ | ||
| (∘₁ (id-alg₁ 𝓨) 𝓯) | ||
| f₁ | ||
| left-id₀ | ||
| goal | ||
| where | ||
| T = Ap-∘ (idf Y ∘`) (f ∘`) θ₁ *v⊡ Ap-sq (idf _) f₁ ⊡v* sym (Ap-∘ (idf Y ∘`) (`∘ ⟦ F₁ ⟧₁ f) ρ₁) | ||
| B = Ap-∘ (`∘ ⟦ F₁ ⟧₁ f) (idf Y ∘`) ρ₁ *v⊡ Ap-sq (`∘ ⟦ F₁ ⟧₁ f) (id₁ 𝓨) ⊡v* sym (Ap-∘ (`∘ ⟦ F₁ ⟧₁ f) (`∘ idf (⟦ F₁ ⟧₀ Y)) ρ₁) | ||
| L = {!!} | ||
| R = {!!} | ||
| goal : Eq | ||
| ((L *h⊡ (T ⊡v B) ⊡h* R) ⊡h* Ap (λ h₀ → (star-hom₀ (alg₀-hom f h₀)) ₌∘ apply r X) left-id₀) | ||
| (Ap (λ h₀ → (star-hom₀ (alg₀-hom f h₀)) ₌∘ apply l X) left-id₀ *h⊡ f₁) | ||
| goal = {!!} | ||
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| right-id-alg₁ : Eq (∘-alg₁ 𝓯 (id-alg₁ 𝓧)) 𝓯 | ||
| right-id-alg₁ = {!!} --Ap (alg₁-hom f) (Ap-idf f₁) | ||
| open import 1-hits-pf.Alg1.CatLaws.LeftId public | ||
| open import 1-hits-pf.Alg1.CatLaws.RightId public | ||
| open import 1-hits-pf.Alg1.CatLaws.Assoc public |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,34 @@ | ||
| {-# OPTIONS --without-K #-} | ||
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| open import 1-hits-pf.Core | ||
| open import Container | ||
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| -- Equality of algebra morphisms | ||
| module 1-hits-pf.Alg1.CatLaws.Assoc (s : Spec) where | ||
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| open import Eq | ||
| open import lib.Basics | ||
| open import 1-hits-pf.Alg1.Core s | ||
| open Spec s | ||
| open import 1-hits-pf.Alg0 F₀ | ||
| open import FreeMonad renaming (_* to _⋆) | ||
| open import 1-hits-pf.Alg0.FreeMonad F₀ | ||
| open import 1-hits-pf.Alg1.Eq s | ||
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| module _ | ||
| {𝓧 𝓨 𝓩 𝓦 : Alg₁-obj} | ||
| (𝓱 : Alg₁-hom 𝓩 𝓦) | ||
| (𝓰 : Alg₁-hom 𝓨 𝓩) | ||
| (𝓯 : Alg₁-hom 𝓧 𝓨) | ||
| where | ||
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| open Alg₁-obj 𝓧 | ||
| open Alg₁-obj 𝓨 renaming (X to Y; θ₀ to ρ₀; θ₁ to ρ₁) | ||
| open Alg₁-obj 𝓩 renaming (X to Z; θ₀ to ζ₀; θ₁ to ζ₁) | ||
| open Alg₁-obj 𝓦 renaming (X to W; θ₀ to ω₀; θ₁ to ω₁) | ||
| open Alg₁-hom 𝓱 renaming (f to h; f₀ to h₀; f₁ to h₁) | ||
| open Alg₁-hom 𝓰 renaming (f to g; f₀ to g₀; f₁ to g₁) | ||
| open Alg₁-hom 𝓯 | ||
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| assoc-alg₁ : Eq (∘-alg₁ (∘-alg₁ 𝓱 𝓰) 𝓯) (∘-alg₁ 𝓱 (∘-alg₁ 𝓰 𝓯)) | ||
| assoc-alg₁ = {!!} |
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| Original file line number | Diff line number | Diff line change |
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| {-# OPTIONS --without-K #-} | ||
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| open import 1-hits-pf.Core | ||
| open import Container | ||
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| -- Equality of algebra morphisms | ||
| module 1-hits-pf.Alg1.CatLaws.LeftId (s : Spec) where | ||
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| open import Eq | ||
| open import lib.Basics | ||
| open import 1-hits-pf.Alg1.Core s | ||
| open Spec s | ||
| open import 1-hits-pf.Alg0 F₀ | ||
| open import FreeMonad renaming (_* to _⋆) | ||
| open import 1-hits-pf.Alg0.FreeMonad F₀ | ||
| open import 1-hits-pf.Alg1.Eq s | ||
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| module _ | ||
| {𝓧 𝓨 : Alg₁-obj} | ||
| (𝓯 : Alg₁-hom 𝓧 𝓨) | ||
| where | ||
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| open Alg₁-obj 𝓧 | ||
| open Alg₁-obj 𝓨 renaming (𝓧' to 𝓨' ; X to Y ; θ₀ to ρ₀ ; θ₁ to ρ₁) | ||
| open Alg₁-hom 𝓯 | ||
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| left-id₀ = Ap-idf f₀ | ||
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| left-id-alg₁ : Eq (∘-alg₁ (id-alg₁ 𝓨) 𝓯) 𝓯 | ||
| left-id-alg₁ = alg₁-hom=' | ||
| f | ||
| (∘₀ {𝓧'} {𝓨'} (id-alg₀ 𝓨') 𝓯') | ||
| f₀ | ||
| (∘₁ (id-alg₁ 𝓨) 𝓯) | ||
| f₁ | ||
| left-id₀ | ||
| goal | ||
| where | ||
| T : Square | ||
| (idf Y ∘₌ star-hom₀ 𝓯' ₌∘ apply l X) | ||
| (f ∘₌ θ₁) | ||
| (ρ₁ ₌∘ ⟦ F₁ ⟧₁ f) | ||
| (idf Y ∘₌ star-hom₀ 𝓯' ₌∘ apply r X) | ||
| T = Ap-∘ (idf Y ∘`) (f ∘`) θ₁ | ||
| *v⊡ Ap-sq (idf Y ∘`) f₁ | ||
| ⊡v* sym (Ap-∘ (idf Y ∘`) (`∘ ⟦ F₁ ⟧₁ f) ρ₁) | ||
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| B : Square | ||
| ((star-hom₀ (id-alg₀ 𝓨') ₌∘ apply l Y) ₌∘ ⟦ F₁ ⟧₁ f) | ||
| (ρ₁ ₌∘ ⟦ F₁ ⟧₁ f) | ||
| (ρ₁ ₌∘ ⟦ F₁ ⟧₁ f) | ||
| ((star-hom₀ (id-alg₀ 𝓨') ₌∘ apply r Y) ₌∘ ⟦ F₁ ⟧₁ f) | ||
| B = Ap-∘ (`∘ ⟦ F₁ ⟧₁ f) (idf Y ∘`) ρ₁ | ||
| *v⊡ Ap-sq (`∘ ⟦ F₁ ⟧₁ f) (id₁ 𝓨) | ||
| ⊡v* sym (Ap-∘ (`∘ ⟦ F₁ ⟧₁ f) (`∘ ⟦ F₁ ⟧₁ (idf Y)) ρ₁) | ||
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| lem-top : (α : ContHom F₁ (F₀ ⋆)) | ||
| → Eq ((idf Y ∘₌ star-hom₀ 𝓯') ₌∘ apply α X) (idf Y ∘₌ (star-hom₀ 𝓯' ₌∘ apply α X)) | ||
| lem-top α = Ap-swap (idf Y) (apply α X) (star-hom₀ 𝓯') | ||
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| lem-bot : (α : ContHom F₁ (F₀ ⋆)) | ||
| → Eq ((star-hom₀ (id-alg₀ 𝓨') ₌∘ ⟦ F₀ ⋆ ⟧₁ f) ₌∘ apply α X) | ||
| ((star-hom₀ (id-alg₀ 𝓨') ₌∘ apply α Y) ₌∘ ⟦ F₁ ⟧₁ f) | ||
| lem-bot α = sym (Ap-∘ (`∘ apply α X) (`∘ ⟦ F₀ ⋆ ⟧₁ f) (star-hom₀ (id-alg₀ 𝓨'))) | ||
| * Ap-∘ (`∘ ⟦ F₁ ⟧₁ f) (`∘ apply α Y) (star-hom₀ (id-alg₀ 𝓨')) | ||
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| lem : (α : ContHom F₁ (F₀ ⋆)) | ||
| → Eq (star-hom₀ (∘-alg₀ (id-alg₀ 𝓨') 𝓯') ₌∘ apply α X) | ||
| (((idf Y ∘₌ star-hom₀ 𝓯') ₌∘ apply α X) * | ||
| ((star-hom₀ (id-alg₀ 𝓨') ₌∘ ⟦ F₀ ⋆ ⟧₁ f) ₌∘ apply α X)) | ||
| lem α = | ||
| (star-hom₀ (∘-alg₀ (id-alg₀ 𝓨') 𝓯') ₌∘ apply α X) | ||
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| *⟨ Ap (λ P → Ap (`∘ apply α X) P) (*-∘ (id-alg₀ 𝓨') 𝓯') ⟩ -- *-∘ | ||
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| ((idf Y ∘₌ star-hom₀ 𝓯') * (star-hom₀ (id-alg₀ 𝓨') ₌∘ ⟦ F₀ ⋆ ⟧₁ f)) ₌∘ apply α X | ||
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| *⟨ Ap-* (`∘ apply α X) | ||
| (Ap ((idf Y) ∘`) (star-hom₀ 𝓯')) | ||
| (Ap (`∘ ⟦ F₀ ⋆ ⟧₁ f) (star-hom₀ (id-alg₀ 𝓨'))) | ||
| ⟩ -- ap-* | ||
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| ((idf Y ∘₌ star-hom₀ 𝓯') ₌∘ apply α X) * ((star-hom₀ (id-alg₀ 𝓨') ₌∘ ⟦ F₀ ⋆ ⟧₁ f) ₌∘ apply α X) ∎* | ||
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| id∘𝓯 : Square (star-hom₀ (∘-alg₀ (id-alg₀ 𝓨') 𝓯') ₌∘ apply l X) | ||
| (f ∘₌ θ₁) (ρ₁ ₌∘ ⟦ F₁ ⟧₁ f) | ||
| (star-hom₀ (∘-alg₀ (id-alg₀ 𝓨') 𝓯') ₌∘ apply r X) | ||
| id∘𝓯 = lem l *h⊡ ((lem-top l *h⊡ T ⊡h* sym (lem-top r)) | ||
| ⊡v (lem-bot l *h⊡ B ⊡h* sym (lem-bot r))) | ||
| ⊡h* sym (lem r) | ||
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| goal : Eq | ||
| (id∘𝓯 ⊡h* Ap (λ h₀ → star-hom₀ (alg₀-hom f h₀) ₌∘ apply r X) left-id₀) | ||
| (Ap (λ h₀ → star-hom₀ (alg₀-hom f h₀) ₌∘ apply l X) left-id₀ *h⊡ | ||
| f₁) | ||
| goal = {!!} |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,30 @@ | ||
| {-# OPTIONS --without-K #-} | ||
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| open import 1-hits-pf.Core | ||
| open import Container | ||
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| -- Equality of algebra morphisms | ||
| module 1-hits-pf.Alg1.CatLaws.RightId (s : Spec) where | ||
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| open import Eq | ||
| open import lib.Basics | ||
| open import 1-hits-pf.Alg1.Core s | ||
| open Spec s | ||
| open import 1-hits-pf.Alg0 F₀ | ||
| open import FreeMonad renaming (_* to _⋆) | ||
| open import 1-hits-pf.Alg0.FreeMonad F₀ | ||
| open import 1-hits-pf.Alg1.Eq s | ||
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| module _ | ||
| {𝓧 𝓨 : Alg₁-obj} | ||
| (𝓯 : Alg₁-hom 𝓧 𝓨) | ||
| where | ||
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| open Alg₁-obj 𝓧 | ||
| open Alg₁-obj 𝓨 renaming (𝓧' to 𝓨' ; X to Y ; θ₀ to ρ₀ ; θ₁ to ρ₁) | ||
| open Alg₁-hom 𝓯 | ||
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| right-id₀ = Ap-idf f₀ | ||
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| right-id-alg₁ : Eq (∘-alg₁ (id-alg₁ 𝓨) 𝓯) 𝓯 | ||
| right-id-alg₁ = {!!} |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,44 @@ | ||
| {-# OPTIONS --without-K #-} | ||
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| open import 1-hits-pf.Core | ||
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| module 1-hits-pf.Alg1.Limits (s : Spec) where | ||
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| open Spec s | ||
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| open import Container | ||
| open import lib.Basics | ||
| open import lib.types.Sigma | ||
| open import Cat | ||
| open import Eq | ||
| open import 1-hits-pf.Alg1.Core s | ||
| open import 1-hits-pf.Alg0.Core F₀ | ||
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| module _ | ||
| (𝓧 𝓨 : Alg₁-obj) | ||
| where | ||
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| open Alg₁-obj 𝓧 | ||
| open Alg₁-obj 𝓨 renaming (X to Y ; θ₀ to ρ₀ ; θ₁ to ρ₁) | ||
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| product-alg₁ : Product Alg₁ 𝓧 𝓨 | ||
| product-alg₁ = record | ||
| { prod = ×-alg₁ | ||
| ; π₁ = π₁-alg₁ | ||
| ; π₂ = π₂-alg₁ | ||
| } | ||
| where | ||
| ×₀ : has-alg₀ (X × Y) | ||
| ×₀ = λ x → θ₀ (⟦ F₀ ⟧₁ fst x) , ρ₀ (⟦ F₀ ⟧₁ snd x) | ||
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| ×₁ : has-alg₁ (alg₀ (X × Y) ×₀) --has-alg₁ | ||
| ×₁ = {!!} | ||
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| ×-alg₁ : Alg₁-obj | ||
| ×-alg₁ = alg₁ (alg₀ (X × Y) ×₀) ×₁ | ||
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| π₁-alg₁ : Alg₁-hom ×-alg₁ 𝓧 | ||
| π₁-alg₁ = alg₁-hom (alg₀-hom fst refl) {!!} | ||
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| π₂-alg₁ : Alg₁-hom ×-alg₁ 𝓨 | ||
| π₂-alg₁ = alg₁-hom (alg₀-hom snd refl) {!!} |