Syntax is now clean (most of it)
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FFOLInitial.agda
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FFOLInitial.agda
@ -190,18 +190,12 @@ module FFOLInitial where
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-- We can also add a term that will not be used in the formulæ already present
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-- (that's why we use wkₜσₜ)
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_▹tp : Conp Γₜ → Conp (Γₜ ▹t⁰)
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◇p ▹tp = ◇p
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(Γₚ ▹p⁰ A) ▹tp = (Γₚ ▹tp) ▹p⁰ (A [ wkₜσₜ idₜ ]f)
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Γ ▹tp = Γ [ wkₜσₜ idₜ ]c
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-- We show how it interacts with ,ₜ and lfₜσₜ
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lem9 : (Δₚ [ wkₜσₜ idₜ ]c) ≡ (Δₚ ▹tp)
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lem9 {Δₚ = ◇p} = refl
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lem9 {Δₚ = Δₚ ▹p⁰ x} = cong₂ _▹p⁰_ lem9 refl
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▹tp,ₜ : {σₜ : Subt Γₜ Δₜ}{t : Tm Γₜ} → (Γₚ ▹tp) [ σₜ ,ₜ t ]c ≡ Γₚ [ σₜ ]c
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▹tp,ₜ {Γₚ = ◇p} = refl
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▹tp,ₜ {Γₚ = Γₚ ▹p⁰ t} = cong₂ _▹p⁰_ ▹tp,ₜ (≡tran (≡sym []f-∘) (cong (λ σ → t [ σ ]f) (≡tran wkₜ∘ₜ,ₜ idlₜ)))
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▹tp,ₜ {Γₚ = Γₚ} = ≡tran (≡sym []c-∘) (cong (λ ξ → Γₚ [ ξ ]c) (≡tran wkₜ∘ₜ,ₜ idlₜ))
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▹tp-lfₜ : {σ : Subt Δₜ Γₜ} → ((Δₚ ▹tp) [ lfₜσₜ σ ]c) ≡ ((Δₚ [ σ ]c) ▹tp)
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▹tp-lfₜ {Δₚ = ◇p} = refl
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▹tp-lfₜ {Δₚ = Δₚ ▹p⁰ A} = cong₂ _▹p⁰_ ▹tp-lfₜ (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (≡tran² (≡sym wkₜσₜ-∘ₜl) (cong wkₜσₜ (≡tran idlₜ (≡sym idrₜ))) (≡sym wkₜσₜ-∘ₜr))) []f-∘)
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▹tp-lfₜ {Δₚ = Δₚ} = ≡tran² (≡sym []c-∘) (cong (λ ξ → Δₚ [ ξ ]c) (≡tran² (≡sym wkₜσₜ-∘ₜl) (cong wkₜσₜ (≡tran idlₜ (≡sym idrₜ))) (≡sym wkₜσₜ-∘ₜr))) []c-∘
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@ -310,7 +304,10 @@ module FFOLInitial where
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wkₜσₚ : Subp {Δₜ} Δₚ' Δₚ → Subp {Δₜ ▹t⁰} (Δₚ' ▹tp) (Δₚ ▹tp)
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wkₜσₚ εₚ = εₚ
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wkₜσₚ {Δₜ = Δₜ} (_,ₚ_ {A = A} σₚ pf) = (wkₜσₚ σₚ) ,ₚ substP (λ Ξₚ → Pf (Δₜ ▹t⁰) Ξₚ (A [ wkₜσₜ idₜ ]f)) lem9 (_[_]pₜ {Γₜ = Δₜ ▹t⁰} pf (wkₜσₜ idₜ))
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wkₜσₚ {Δₜ = Δₜ} (_,ₚ_ {A = A} σₚ pf) = (wkₜσₚ σₚ) ,ₚ substP (λ Ξₚ → Pf (Δₜ ▹t⁰) Ξₚ (A [ wkₜσₜ idₜ ]f)) refl (_[_]pₜ {Γₜ = Δₜ ▹t⁰} pf (wkₜσₜ idₜ))
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wkₚ[] : {σₜ : Subt Γₜ Δₜ} {σₚ : Subp Δₚ Δₚ'} {A : For Δₜ} → (wkₚσₚ {A = A} σₚ) [ σₜ ]σₚ ≡ wkₚσₚ (σₚ [ σₜ ]σₚ)
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wkₚ[] {σₚ = εₚ} = refl
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wkₚ[] {σₚ = σₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ _) (wkₚ[] {σₚ = σₚ})
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_[_]p : {A : For Δₜ} → Pf Δₜ Δₚ A → (σ : Subp {Δₜ} Δₚ' Δₚ) → Pf Δₜ Δₚ' A
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var pvzero [ σ ,ₚ pf ]p = pf
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@ -326,26 +323,85 @@ module FFOLInitial where
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-- We can now define identity and composition on proof substitutions
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idₚ : Subp {Δₜ} Δₚ Δₚ
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idₚ {Δₚ = ◇p} = εₚ
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idₚ {Δₚ = Δₚ ▹p⁰ x} = lfₚσₚ (idₚ {Δₚ = Δₚ})
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_∘ₚ_ : {Γₚ Δₚ Ξₚ : Conp Δₜ} → Subp {Δₜ} Δₚ Ξₚ → Subp {Δₜ} Γₚ Δₚ → Subp {Δₜ} Γₚ Ξₚ
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εₚ ∘ₚ β = εₚ
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(α ,ₚ pf) ∘ₚ β = (α ∘ₚ β) ,ₚ (pf [ β ]p)
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-- And now we have to show all their equalities
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idₚ[] : {σₜ : Subt Γₜ Δₜ} → ((idₚ {Δₜ} {Δₚ}) [ σₜ ]σₚ) ≡ idₚ {Γₜ} {Δₚ [ σₜ ]c}
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idₚ[] {Δₚ = ◇p} = refl
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idₚ[] {Δₚ = Δₚ ▹p⁰ A} = cong (λ ξ → ξ ,ₚ var pvzero) (≡tran wkₚ[] (cong wkₚσₚ idₚ[]))
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-- Cancelling a wkₚσₚ with a ,ₚ
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wkₚ∘, : {Δₜ : Cont}{Γₚ Δₚ Ξₚ : Conp Δₜ}{α : Subp {Δₜ} Γₚ Δₚ}{β : Subp {Δₜ} Ξₚ Γₚ}{A : For Δₜ}{pf : Pf Δₜ Ξₚ A} → (wkₚσₚ α) ∘ₚ (β ,ₚ pf) ≡ (α ∘ₚ β)
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wkₚ∘, {α = εₚ} = refl
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wkₚ∘, {α = α ,ₚ pf} {β = β} {pf = pf'} = cong (λ ξ → ξ ,ₚ (pf [ β ]p)) wkₚ∘,
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-- Categorical rules
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idlₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → (idₚ {Δₚ = Δₚ}) ∘ₚ σₚ ≡ σₚ
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idlₚ {Δₚ = ◇p} {εₚ} = refl
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idlₚ {Δₚ = Δₚ ▹p⁰ pf} {σₚ ,ₚ pf'} = cong (λ ξ → ξ ,ₚ pf') (≡tran wkₚ∘, (idlₚ {σₚ = σₚ}))
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idrₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → σₚ ∘ₚ (idₚ {Δₚ = Γₚ}) ≡ σₚ
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idrₚ {σₚ = εₚ} = refl
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idrₚ {σₚ = σₚ ,ₚ prf} = cong (λ X → X ,ₚ prf) (idrₚ {σₚ = σₚ})
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∘ₚ-ass : {Γₚ Δₚ Ξₚ Ψₚ : Conp Γₜ}{αₚ : Subp Γₚ Δₚ}{βₚ : Subp Δₚ Ξₚ}{γₚ : Subp Ξₚ Ψₚ} → (γₚ ∘ₚ βₚ) ∘ₚ αₚ ≡ γₚ ∘ₚ (βₚ ∘ₚ αₚ)
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∘ₚ-ass {γₚ = εₚ} = refl
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∘ₚ-ass {αₚ = αₚ} {βₚ} {γₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ (x [ βₚ ∘ₚ αₚ ]p)) ∘ₚ-ass
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lemG' :
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-- Unicity of the terminal morphism
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εₚ-u : {Γₚ : Conp Γₜ} → {σ : Subp Γₚ ◇p} → σ ≡ εₚ {Δₚ = Γₚ}
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εₚ-u {σ = εₚ} = refl
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-- Term substitution for proof substitutions
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[]σₚ-id : {σₚ : Subp {Δₜ} Δₚ Δₚ'} → coe (cong₂ Subp []c-id []c-id) (σₚ [ idₜ ]σₚ) ≡ σₚ
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[]σₚ-id {σₚ = εₚ} = εₚ-u
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[]σₚ-id {Δₜ}{Δₚ}{Δₚ' ▹p⁰ A} {σₚ = σₚ ,ₚ x} =
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substP (λ ξ → coe (cong₂ Subp []c-id []c-id) (ξ ,ₚ (x [ idₜ ]pₜ)) ≡ (σₚ ,ₚ x))
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(≡sym (coeshift []σₚ-id))
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(≡sym (coeshift {eq = cong₂ Subp (≡sym []c-id) (≡sym []c-id)}
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(substfpoly⁴
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{A = ((Conp Δₜ) × (Conp Δₜ)) × (For Δₜ)}
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{P = λ W → Subp (proj×₁ (proj×₁ W)) ((proj×₂ (proj×₁ W)) ▹p⁰ (proj×₂ W))}
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{R = λ W → Subp (proj×₁ (proj×₁ W)) (proj×₂ (proj×₁ W))}
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{Q = λ W → Pf Δₜ (proj×₁ (proj×₁ W)) (proj×₂ W)}
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{α = (Δₚ ,× Δₚ') ,× A}
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{eq = ×≡ (×≡ (≡sym []c-id) (≡sym []c-id)) (≡sym []f-id)}
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{f = λ {W} ξ pf → _,ₚ_ ξ pf}{x = σₚ}{y = x}
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)))
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[]σₚ-∘ : {Ξₚ Ξₚ' : Conp Ξₜ}{α : Subt Δₜ Ξₜ} {β : Subt Γₜ Δₜ} {σₚ : Subp {Ξₜ} Ξₚ Ξₚ'}
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→ coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) ((σₚ [ α ]σₚ) [ β ]σₚ) ≡ σₚ [ α ∘ₜ β ]σₚ
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[]σₚ-∘ {σₚ = εₚ} = εₚ-u
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[]σₚ-∘ {Ξₜ}{Δₜ}{Γₜ}{Ξₚ}{Δₚ' ▹p⁰ A}{α}{β}{σₚ = σₚ ,ₚ pf} =
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substP (λ ξ → coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) (ξ ,ₚ ((pf [ α ]pₜ) [ β ]pₜ)) ≡ ((σₚ [ α ∘ₜ β ]σₚ) ,ₚ (pf [ α ∘ₜ β ]pₜ))) (≡sym (coeshift []σₚ-∘))
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(≡sym (coeshift {eq = cong₂ Subp []c-∘ []c-∘}
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(substfpoly⁴
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{A = ((Conp Γₜ) × (Conp Γₜ)) × (For Γₜ)}
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{P = λ W → Subp (proj×₁ (proj×₁ W)) ((proj×₂ (proj×₁ W)) ▹p⁰ (proj×₂ W))}
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{R = λ W → Subp (proj×₁ (proj×₁ W)) (proj×₂ (proj×₁ W))}
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{Q = λ W → Pf Γₜ (proj×₁ (proj×₁ W)) (proj×₂ W)}
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{eq = ×≡ (×≡ []c-∘ []c-∘) []f-∘}
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{f = λ {W} ξ pf → _,ₚ_ ξ pf}{x = σₚ [ α ∘ₜ β ]σₚ}{y = pf [ α ∘ₜ β ]pₜ})
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))
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-- How ∘ₚ and ∘ₜ interact to make associativity (to be proven later for Sub)
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∘ₚₜ-ass⁰ :
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{Γₜ Δₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Δₜ}{Ψₚ : Conp Δₜ}
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{αₜ : Subt Γₜ Δₜ}{γₚ : Subp Ξₚ Ψₚ}{βₚ : Subp Δₚ Ξₚ}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
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→ ((γₚ ∘ₚ βₚ) [ αₜ ]σₚ) ∘ₚ αₚ ≡ (γₚ [ αₜ ]σₚ) ∘ₚ ((βₚ [ αₜ ]σₚ) ∘ₚ αₚ)
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lemG' {γₚ = εₚ} = refl
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lemG' {αₜ = αₜ}{γₚ ,ₚ x}{βₚ}{αₚ} = cong (λ ξ → ξ ,ₚ (((x [ βₚ ]p) [ αₜ ]pₜ) [ αₚ ]p)) (lemG' {γₚ = γₚ})
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ε-u : {Γₚ : Conp Γₜ} → {σ : Subp Γₚ ◇p} → σ ≡ εₚ {Δₚ = Γₚ}
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ε-u {σ = εₚ} = refl
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lemG :
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∘ₚₜ-ass⁰ {γₚ = εₚ} = refl
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∘ₚₜ-ass⁰ {αₜ = αₜ}{γₚ ,ₚ x}{βₚ}{αₚ} = cong (λ ξ → ξ ,ₚ (((x [ βₚ ]p) [ αₜ ]pₜ) [ αₚ ]p)) (∘ₚₜ-ass⁰ {γₚ = γₚ})
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∘ₚₜ-ass :
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{Γₜ Δₜ Ξₜ Ψₜ : Cont}{Γₚ : Conp Γₜ}{Δₚ : Conp Δₜ}{Ξₚ : Conp Ξₜ}{Ψₚ : Conp Ψₜ}
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{αₜ : Subt Γₜ Δₜ}{βₜ : Subt Δₜ Ξₜ}{γₜ : Subt Ξₜ Ψₜ}{γₚ : Subp Ξₚ (Ψₚ [ γₜ ]c)}{βₚ : Subp Δₚ (Ξₚ [ βₜ ]c)}{αₚ : Subp Γₚ (Δₚ [ αₜ ]c)}
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{eq₁ : Subp Γₚ (Ψₚ [ (γₜ ∘ₜ βₜ) ∘ₜ αₜ ]c) ≡ Subp Γₚ (Ψₚ [ γₜ ∘ₜ (βₜ ∘ₜ αₜ) ]c)}
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@ -354,37 +410,7 @@ module FFOLInitial where
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{eq₄ : Subp (Δₚ [ αₜ ]c) ((Ξₚ [ βₜ ]c) [ αₜ ]c) ≡ Subp (Δₚ [ αₜ ]c) (Ξₚ [ βₜ ∘ₜ αₜ ]c)}
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{eq₅ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]c)}
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→ coe eq₁ ((coe eq₂ (((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ)) ∘ₚ αₚ) ≡ (coe eq₃ (γₚ [ βₜ ∘ₜ αₜ ]σₚ)) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ)
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lemK : {σₜ : Subt Γₜ Δₜ}{σₚ : Subp (Γₚ [ idₜ ]c) ((Δₚ [ σₜ ]c)[ idₜ ]c)}
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{eq1 : Subp Γₚ ((Δₚ [ σₜ ]c) [ idₜ ]c) ≡ Subp Γₚ (Δₚ [ σₜ ]c)}
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{eq2 : Subp Γₚ Γₚ ≡ Subp Γₚ (Γₚ [ idₜ ]c)}
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{eq3 : Subp (Γₚ [ idₜ ]c) ((Δₚ [ σₜ ]c)[ idₜ ]c) ≡ Subp Γₚ (Δₚ [ σₜ ]c)}
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→ coe eq1 (σₚ ∘ₚ coe eq2 idₚ)
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≡ (coe eq3 σₚ ∘ₚ idₚ)
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lemK {σₚ = σₚ}{eq1}{eq2}{eq3} = substP (λ X → coe eq1 (X ∘ₚ coe eq2 idₚ) ≡ (coe eq3 σₚ ∘ₚ idₚ)) (coeaba {eq1 = eq3}{eq2 = ≡sym eq3}) (coep∘ {p = λ {Γₚ}{Δₚ}{Ξₚ} x y → _∘ₚ_ {Δₚ = Γₚ} x y} {eq1 = refl}{eq2 = ≡sym []c-id}{eq3 = ≡sym []c-id})
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lemJ : {Δₜ : Cont}{Δₚ : Conp Δₜ}{A : For Δₜ} → Pf Δₜ Δₚ A → (Pf Δₜ (Δₚ [ idₜ ]c) (A [ idₜ ]f))
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lemJ {Δₜ}{Δₚ}{A} pf = substP (Pf Δₜ (Δₚ [ idₜ ]c)) (≡sym []f-id) (substP (λ Ξₚ → Pf Δₜ Ξₚ A) (≡sym []c-id) pf)
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[]σₚ-id : {σₚ : Subp {Δₜ} Δₚ Δₚ'} → coe (cong₂ Subp []c-id []c-id) (σₚ [ idₜ ]σₚ) ≡ σₚ
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[]σₚ-id {σₚ = εₚ} = ε-u
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[]σₚ-id {Δₜ}{Δₚ}{Δₚ' ▹p⁰ A} {σₚ = σₚ ,ₚ x} = substP (λ ξ → coe (cong₂ Subp []c-id []c-id) (ξ ,ₚ (x [ idₜ ]pₜ)) ≡ (σₚ ,ₚ x)) (≡sym (coeshift ([]σₚ-id)))
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(≡sym (coeshift {eq = cong₂ Subp (≡sym []c-id) (≡sym []c-id)}
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(substfpoly'' {A = (Conp Δₜ) × (Conp Δₜ)}{P = λ W F → Subp (proj×₁ W) ((proj×₂ W) ▹p⁰ F)}{R = λ W → Subp (proj×₁ W) (proj×₂ W)}{Q = λ W F → Pf Δₜ (proj×₁ W) F}{α = Δₚ ,× Δₚ'}{ε = A}{eq = ×≡ (≡sym []c-id) (≡sym []c-id)}{eq'' = ≡sym []f-id}{f = λ {W} {F} ξ pf → _,ₚ_ ξ pf}{x = σₚ}{y = x})))
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[]σₚ-∘ : {Ξₚ Ξₚ' : Conp Ξₜ}{α : Subt Δₜ Ξₜ} {β : Subt Γₜ Δₜ} {σₚ : Subp {Ξₜ} Ξₚ Ξₚ'}
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--{eq₅ : Subp (Ξₚ [ βₜ ]c) ((Ψₚ [ γₜ ]c) [ βₜ ]c) ≡ Subp (Ξₚ [ βₜ ]c) (Ψₚ [ γₜ ∘ₜ βₜ ]c)}
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→ coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) ((σₚ [ α ]σₚ) [ β ]σₚ) ≡ σₚ [ α ∘ₜ β ]σₚ
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[]σₚ-∘ {σₚ = εₚ} = ε-u
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[]σₚ-∘ {Ξₜ}{Δₜ}{Γₜ}{Ξₚ}{Δₚ' ▹p⁰ A}{α}{β}{σₚ = σₚ ,ₚ pf} =
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substP (λ ξ → coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) (ξ ,ₚ ((pf [ α ]pₜ) [ β ]pₜ)) ≡ ((σₚ [ α ∘ₜ β ]σₚ) ,ₚ (pf [ α ∘ₜ β ]pₜ))) (≡sym (coeshift []σₚ-∘))
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(≡sym (coeshift {eq = cong₂ Subp []c-∘ []c-∘}
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(substfpoly''
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{A = (Conp Γₜ) × (Conp Γₜ)}
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{P = λ W F → Subp (proj×₁ W) ((proj×₂ W) ▹p⁰ F)}
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{R = λ W → Subp (proj×₁ W) (proj×₂ W)}
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{Q = λ W F → Pf Γₜ (proj×₁ W) F}
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{eq = cong₂ _,×_ []c-∘ []c-∘}
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{eq'' = []f-∘ {α = β} {β = α} {F = A}}
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{f = λ {W} {F} ξ pf → _,ₚ_ ξ pf}{x = σₚ [ α ∘ₜ β ]σₚ}{y = pf [ α ∘ₜ β ]pₜ})
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))
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lemG {Γₜ}{Δₜ}{Ξₜ}{Ψₜ}{Γₚ}{Δₚ}{Ξₚ}{Ψₚ}{αₜ = αₜ}{βₜ}{γₜ}{γₚ}{βₚ}{αₚ}{eq₁}{eq₂}{eq₃}{eq₄}{eq₅} =
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∘ₚₜ-ass {Γₜ}{Δₜ}{Ξₜ}{Ψₜ}{Γₚ}{Δₚ}{Ξₚ}{Ψₚ}{αₜ = αₜ}{βₜ}{γₜ}{γₚ}{βₚ}{αₚ}{eq₁}{eq₂}{eq₃}{eq₄}{eq₅} =
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substP (λ X → coe eq₁ ((coe eq₂ (((coe eq₅ (γₚ [ βₜ ]σₚ)) ∘ₚ βₚ) [ αₜ ]σₚ)) ∘ₚ αₚ) ≡ (coe eq₃ X) ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ)) []σₚ-∘ (
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≡tran⁵
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(cong (coe eq₁) (≡tran (
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@ -402,7 +428,7 @@ module FFOLInitial where
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)))
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))
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(≡tran coecoe-coe coecoe-coe)
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(cong (coe (≡tran (cong (λ z → Subp Γₚ (z [ αₜ ]c)) (≡sym []c-∘)) (≡tran (cong (λ z → Subp Γₚ z) (≡sym []c-∘)) eq₁))) lemG')
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(cong (coe (≡tran (cong (λ z → Subp Γₚ (z [ αₜ ]c)) (≡sym []c-∘)) (≡tran (cong (λ z → Subp Γₚ z) (≡sym []c-∘)) eq₁))) ∘ₚₜ-ass⁰)
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(≡sym coecoe-coe)
|
||||
(cong (coe (cong (λ z → Subp Γₚ z) (≡sym []c-∘))) (substppoly
|
||||
{A = (Conp Γₜ) × (Conp Γₜ)}
|
||||
@ -418,63 +444,106 @@ module FFOLInitial where
|
||||
{f = λ {τ} ξ → (ξ ∘ₚ ((coe eq₄ (βₚ [ αₜ ]σₚ)) ∘ₚ αₚ))}
|
||||
{x = (coe (cong₂ Subp (≡sym []c-∘) (≡sym []c-∘)) ((γₚ [ βₜ ]σₚ) [ αₜ ]σₚ))}
|
||||
))
|
||||
wkₚ∘, : {Δₜ : Cont}{Γₚ Δₚ Ξₚ : Conp Δₜ}{α : Subp {Δₜ} Γₚ Δₚ}{β : Subp {Δₜ} Ξₚ Γₚ}{A : For Δₜ}{pf : Pf Δₜ Ξₚ A} → (wkₚσₚ α) ∘ₚ (β ,ₚ pf) ≡ (α ∘ₚ β)
|
||||
wkₚ∘, {α = εₚ} = refl
|
||||
wkₚ∘, {α = α ,ₚ pf} {β = β} {pf = pf'} = cong (λ ξ → ξ ,ₚ (pf [ β ]p)) wkₚ∘,
|
||||
idlₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → (idₚ {Δₚ = Δₚ}) ∘ₚ σₚ ≡ σₚ
|
||||
idlₚ {Δₚ = ◇p} {εₚ} = refl
|
||||
idlₚ {Δₚ = Δₚ ▹p⁰ pf} {σₚ ,ₚ pf'} = cong (λ ξ → ξ ,ₚ pf') (≡tran wkₚ∘, (idlₚ {σₚ = σₚ}))
|
||||
idrₚ : {Γₚ Δₚ : Conp Γₜ} {σₚ : Subp Γₚ Δₚ} → σₚ ∘ₚ (idₚ {Δₚ = Γₚ}) ≡ σₚ
|
||||
idrₚ {σₚ = εₚ} = refl
|
||||
idrₚ {σₚ = σₚ ,ₚ prf} = cong (λ X → X ,ₚ prf) (idrₚ {σₚ = σₚ})
|
||||
wkₚ[] : {σₜ : Subt Γₜ Δₜ} {σₚ : Subp Δₚ Δₚ'} {A : For Δₜ} → (wkₚσₚ {A = A} σₚ) [ σₜ ]σₚ ≡ wkₚσₚ (σₚ [ σₜ ]σₚ)
|
||||
wkₚ[] {σₚ = εₚ} = refl
|
||||
wkₚ[] {σₚ = σₚ ,ₚ x} = cong (λ ξ → ξ ,ₚ _) (wkₚ[] {σₚ = σₚ})
|
||||
idₚ[] : {σₜ : Subt Γₜ Δₜ} → ((idₚ {Δₜ} {Δₚ}) [ σₜ ]σₚ) ≡ idₚ {Γₜ} {Δₚ [ σₜ ]c}
|
||||
idₚ[] {Δₚ = ◇p} = refl
|
||||
idₚ[] {Δₚ = Δₚ ▹p⁰ A} = cong (λ ξ → ξ ,ₚ var pvzero) (≡tran wkₚ[] (cong wkₚσₚ idₚ[]))
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
-- We can now merge the two notions of contexts, substitutions, and everything
|
||||
record Con : Set₁ where
|
||||
constructor con
|
||||
field
|
||||
t : Cont
|
||||
p : Conp t
|
||||
|
||||
◇ : Con
|
||||
◇ = con ◇t ◇p
|
||||
_▹p_ : (Γ : Con) → For (Con.t Γ) → Con
|
||||
Γ ▹p A = con (Con.t Γ) (Con.p Γ ▹p⁰ A)
|
||||
|
||||
|
||||
variable
|
||||
Γ Δ Ξ : Con
|
||||
_▹t : Con → Con
|
||||
Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp)
|
||||
|
||||
record Sub (Γ : Con) (Δ : Con) : Set₁ where
|
||||
constructor sub
|
||||
field
|
||||
t : Subt (Con.t Γ) (Con.t Δ)
|
||||
p : Subp {Con.t Γ} (Con.p Γ) ((Con.p Δ) [ t ]c)
|
||||
|
||||
-- (Con,Sub) is a category with an initial object
|
||||
id : Sub Γ Γ
|
||||
id {Γ} = sub idₜ (subst (Subp _) (≡sym []c-id) idₚ)
|
||||
_∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
|
||||
sub αₜ αₚ ∘ sub βₜ βₚ = sub (αₜ ∘ₜ βₜ) (subst (Subp _) (≡sym []c-∘) (αₚ [ βₜ ]σₚ) ∘ₚ βₚ)
|
||||
idl : {Γ Δ : Con} {σ : Sub Γ Δ} → (id {Δ}) ∘ σ ≡ σ
|
||||
idl {Δ = Δ} {σ = sub σₜ σₚ} = cong₂' sub idlₜ (≡tran (substfpoly {α = ((Con.p Δ) [ idₜ ∘ₜ σₜ ]c)} {β = ((Con.p Δ) [ σₜ ]c)} {eq = cong (λ ξ → Con.p Δ [ ξ ]c) idlₜ} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ σₚ}) (≡tran (cong₂ _∘ₚ_ (≡tran³ coecoe-coe (substfpoly {eq = []c-id} {f = λ {Ξₚ} ξ → _[_]σₚ {Δₚ = Con.p Δ} {Δₚ' = Ξₚ} ξ σₜ}) (cong (λ ξ → ξ [ σₜ ]σₚ) coeaba) idₚ[]) refl) idlₚ))
|
||||
idl {Δ = Δ} {σ = sub σₜ σₚ} =
|
||||
cong₂' sub idlₜ (≡tran²
|
||||
(substfpoly
|
||||
{α = ((Con.p Δ) [ idₜ ∘ₜ σₜ ]c)}
|
||||
{β = ((Con.p Δ) [ σₜ ]c)}
|
||||
{eq = cong (λ ξ → Con.p Δ [ ξ ]c) idlₜ}
|
||||
{f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ σₚ}
|
||||
) (
|
||||
cong₂ _∘ₚ_ (≡tran³
|
||||
coecoe-coe
|
||||
(substfpoly
|
||||
{eq = []c-id}
|
||||
{f = λ {Ξₚ} ξ → _[_]σₚ {Δₚ = Con.p Δ} {Δₚ' = Ξₚ} ξ σₜ}
|
||||
)
|
||||
(cong (λ ξ → ξ [ σₜ ]σₚ) coeaba)
|
||||
idₚ[]
|
||||
) refl)
|
||||
idlₚ)
|
||||
idr : {Γ Δ : Con} {σ : Sub Γ Δ} → σ ∘ (id {Γ}) ≡ σ
|
||||
idr {Γ} {Δ} {σ = sub σₜ σₚ} = cong₂' sub idrₜ (≡tran⁴ (cong (coe _) (≡sym (substfpoly {eq = ≡sym ([]c-∘ {α = σₜ} {β = idₜ}{Ξₚ = Con.p Δ})} {f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ (coe (cong (Subp (Con.p Γ)) (≡sym []c-id)) idₚ)} {x = σₚ [ idₜ ]σₚ}))) coecoe-coe lemK idrₚ []σₚ-id)
|
||||
|
||||
idr {Γ} {Δ} {σ = sub σₜ σₚ} =
|
||||
cong₂' sub idrₜ (≡tran⁴
|
||||
(cong (coe _) (≡sym (
|
||||
substfpoly
|
||||
{eq = ≡sym ([]c-∘ {α = σₜ} {β = idₜ}{Ξₚ = Con.p Δ})}
|
||||
{f = λ {Ξₚ} ξ → _∘ₚ_ {Ξₚ = Ξₚ} ξ (coe (cong (Subp (Con.p Γ)) (≡sym []c-id)) idₚ)}
|
||||
{x = σₚ [ idₜ ]σₚ})))
|
||||
coecoe-coe
|
||||
(substP
|
||||
(λ X → coe (≡tran (cong (Subp (Con.p Γ)) (≡sym []c-∘))
|
||||
(cong (λ z → Subp (Con.p Γ) (Con.p Δ [ z ]c)) idrₜ))
|
||||
(X ∘ₚ coe (cong (Subp (Con.p Γ)) (≡sym []c-id)) idₚ)
|
||||
≡ (coe (cong₂ Subp []c-id []c-id) (σₚ [ idₜ ]σₚ) ∘ₚ idₚ))
|
||||
((coeaba {eq1 = (cong₂ Subp []c-id []c-id)}{eq2 = ≡sym (cong₂ Subp []c-id []c-id)}))
|
||||
((coep∘
|
||||
{p = λ {Γₚ}{Δₚ}{Ξₚ} x y → _∘ₚ_ {Δₚ = Γₚ} x y}
|
||||
{eq1 = refl}
|
||||
{eq2 = ≡sym []c-id}
|
||||
{eq3 = ≡sym []c-id}
|
||||
)))
|
||||
idrₚ
|
||||
[]σₚ-id)
|
||||
∘-ass : {Γ Δ Ξ Ψ : Con}{α : Sub Γ Δ}{β : Sub Δ Ξ}{γ : Sub Ξ Ψ} → (γ ∘ β) ∘ α ≡ γ ∘ (β ∘ α)
|
||||
∘-ass {Γ}{Δ}{Ξ}{Ψ}{α = sub αₜ αₚ} {β = sub βₜ βₚ} {γ = sub γₜ γₚ} = cong₂' sub ∘ₜ-ass lemG
|
||||
∘-ass {Γ}{Δ}{Ξ}{Ψ}{α = sub αₜ αₚ} {β = sub βₜ βₚ} {γ = sub γₜ γₚ} = cong₂' sub ∘ₜ-ass ∘ₚₜ-ass
|
||||
|
||||
-- SUB-ization
|
||||
◇ : Con
|
||||
◇ = con ◇t ◇p
|
||||
|
||||
|
||||
-- We have our two context extension operators
|
||||
_▹t : Con → Con
|
||||
Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp)
|
||||
_▹p_ : (Γ : Con) → For (Con.t Γ) → Con
|
||||
Γ ▹p A = con (Con.t Γ) (Con.p Γ ▹p⁰ A)
|
||||
|
||||
|
||||
|
||||
|
||||
-- We define the access function from the algebra, but defined for fully-featured substitutions
|
||||
-- For term substitutions
|
||||
πₜ¹* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Sub Δ Γ
|
||||
πₜ¹* (sub (σₜ ,ₜ t) σₚ) = sub σₜ (subst (Subp _) ▹tp,ₜ σₚ)
|
||||
πₜ²* : {Γ Δ : Con} → Sub Δ (Γ ▹t) → Tm (Con.t Δ)
|
||||
πₜ²* (sub (σₜ ,ₜ t) σₚ) = t
|
||||
_,ₜ*_ : {Γ Δ : Con} → Sub Δ Γ → Tm (Con.t Δ) → Sub Δ (Γ ▹t)
|
||||
(sub σₜ σₚ) ,ₜ* t = sub (σₜ ,ₜ t) (subst (Subp _) (≡sym ▹tp,ₜ) σₚ)
|
||||
-- And the equations
|
||||
πₜ²∘,ₜ* : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm (Con.t Δ)} → πₜ²* (σ ,ₜ* t) ≡ t
|
||||
πₜ²∘,ₜ* = refl
|
||||
πₜ¹∘,ₜ* : {Γ Δ : Con} → {σ : Sub Δ Γ} → {t : Tm (Con.t Δ)} → πₜ¹* (σ ,ₜ* t) ≡ σ
|
||||
@ -496,24 +565,36 @@ module FFOLInitial where
|
||||
{f = λ σₚ → σₚ [ δₜ ]σₚ}
|
||||
{x = σₚ})
|
||||
|
||||
-- And for proof substitutions
|
||||
πₚ¹* : {Γ Δ : Con} {A : For (Con.t Γ)} → Sub Δ (Γ ▹p A) → Sub Δ Γ
|
||||
πₚ¹* (sub σₜ (σₚ ,ₚ pf)) = sub σₜ σₚ
|
||||
πₚ² : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ (Γ ▹p F)) → Pf (Con.t Δ) (Con.p Δ) (F [ Sub.t (πₚ¹* σ) ]f)
|
||||
πₚ² (sub σₜ (σₚ ,ₚ pf)) = pf
|
||||
πₚ²* : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ (Γ ▹p F)) → Pf (Con.t Δ) (Con.p Δ) (F [ Sub.t (πₚ¹* σ) ]f)
|
||||
πₚ²* (sub σₜ (σₚ ,ₚ pf)) = pf
|
||||
_,ₚ*_ : {Γ Δ : Con} {F : For (Con.t Γ)} (σ : Sub Δ Γ) → Pf (Con.t Δ) (Con.p Δ) (F [ Sub.t σ ]f) → Sub Δ (Γ ▹p F)
|
||||
sub σₜ σₚ ,ₚ* pf = sub σₜ (σₚ ,ₚ pf)
|
||||
|
||||
,ₚ∘πₚ : {Γ Δ : Con} → {F : For (Con.t Γ)} → {σ : Sub Δ (Γ ▹p F)} → (πₚ¹* σ) ,ₚ* (πₚ² σ) ≡ σ
|
||||
-- And the equations
|
||||
,ₚ∘πₚ : {Γ Δ : Con} → {F : For (Con.t Γ)} → {σ : Sub Δ (Γ ▹p F)} → (πₚ¹* σ) ,ₚ* (πₚ²* σ) ≡ σ
|
||||
,ₚ∘πₚ {σ = sub σₜ (σₚ ,ₚ p)} = refl
|
||||
,ₚ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{F : For (Con.t Ξ)}{prf : Pf (Con.t Γ) (Con.p Γ) (F [ Sub.t σ ]f)}
|
||||
→ (σ ,ₚ* prf) ∘ δ ≡ (σ ∘ δ) ,ₚ* (substP (λ F → Pf (Con.t Δ) (Con.p Δ) F) (≡sym []f-∘) ((prf [ Sub.t δ ]pₜ) [ Sub.p δ ]p))
|
||||
,ₚ∘ {Γ}{Δ}{Ξ}{σ = sub σₜ σₚ} {sub δₜ δₚ} {F = A} {prf} = cong (sub (σₜ ∘ₜ δₜ)) (cong (λ ξ → ξ ∘ₚ δₚ)
|
||||
(substfpoly⁴ {P = λ W → Subp (Con.p Γ [ δₜ ]c) ((proj×₁ W) ▹p⁰ (proj×₂ W))}{R = λ W → Subp (Con.p Γ [ δₜ ]c) (proj×₁ W)}{Q = λ W → Pf (Con.t Δ) (Con.p Γ [ δₜ ]c) (proj×₂ W)}{α = ((Con.p Ξ [ σₜ ]c) [ δₜ ]c) ,× ((A [ σₜ ]f) [ δₜ ]f)}{eq = cong₂ _,×_ (≡sym []c-∘) (≡sym []f-∘)}{f = λ ξ p → ξ ,ₚ p} {x = σₚ [ δₜ ]σₚ}{y = prf [ δₜ ]pₜ})) --
|
||||
,ₚ∘ {Γ}{Δ}{Ξ}{σ = sub σₜ σₚ} {sub δₜ δₚ} {F = A} {prf} =
|
||||
cong (λ ξ → sub (σₜ ∘ₜ δₜ) (ξ ∘ₚ δₚ)) (
|
||||
substfpoly⁴
|
||||
{P = λ W → Subp (Con.p Γ [ δₜ ]c) ((proj×₁ W) ▹p⁰ (proj×₂ W))}
|
||||
{R = λ W → Subp (Con.p Γ [ δₜ ]c) (proj×₁ W)}
|
||||
{Q = λ W → Pf (Con.t Δ) (Con.p Γ [ δₜ ]c) (proj×₂ W)}
|
||||
{α = ((Con.p Ξ [ σₜ ]c) [ δₜ ]c) ,× ((A [ σₜ ]f) [ δₜ ]f)}
|
||||
{eq = cong₂ _,×_ (≡sym []c-∘) (≡sym []f-∘)}
|
||||
{f = λ ξ p → ξ ,ₚ p}
|
||||
{x = σₚ [ δₜ ]σₚ}{y = prf [ δₜ ]pₜ}
|
||||
)
|
||||
|
||||
|
||||
lemD : {A : For (Con.t Γ)}{σ : Sub Δ (Γ ▹p A)} → Sub.t (πₚ¹* σ) ≡ Sub.t σ
|
||||
lemD {σ = sub σₜ (σₚ ,ₚ pf)} = refl
|
||||
|
||||
|
||||
-- and FINALLY, we compile everything into an implementation of the FFOL record
|
||||
|
||||
imod : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero}
|
||||
imod = record
|
||||
@ -526,7 +607,7 @@ module FFOLInitial where
|
||||
; idr = idr
|
||||
; ◇ = ◇
|
||||
; ε = sub εₜ εₚ
|
||||
; ε-u = cong₂' sub εₜ-u ε-u
|
||||
; ε-u = cong₂' sub εₜ-u εₚ-u
|
||||
; Tm = λ Γ → Tm (Con.t Γ)
|
||||
; _[_]t = λ t σ → t [ Sub.t σ ]t
|
||||
; []t-id = []t-id
|
||||
@ -549,7 +630,7 @@ module FFOLInitial where
|
||||
; _[_]p = λ pf σ → (pf [ Sub.t σ ]pₜ) [ Sub.p σ ]p
|
||||
; _▹ₚ_ = _▹p_
|
||||
; πₚ¹ = πₚ¹*
|
||||
; πₚ² = πₚ²
|
||||
; πₚ² = πₚ²*
|
||||
; _,ₚ_ = _,ₚ*_
|
||||
; ,ₚ∘πₚ = ,ₚ∘πₚ
|
||||
; πₚ¹∘,ₚ = refl
|
||||
|
||||
@ -168,23 +168,10 @@ module PropUtil where
|
||||
→ coe (cong P eq) (f {α} x y) ≡ f {β} (coe (cong R eq) x) (coe (cong Q eq) y)
|
||||
substppoly {eq = refl} {f}{x}{y} = ≡tran coerefl (cong₂ f (≡sym coerefl) (≡sym coerefl))
|
||||
|
||||
substfpoly' : {ℓ ℓ' ℓ'' : Level}{A B : Set ℓ}{P R : A → Set ℓ'}{Q : B → Prop ℓ''}{α β : A}{γ δ : B}
|
||||
{eq : α ≡ β}{eq' : γ ≡ δ} {f : {ξ : A}{ι : B} → R ξ → Q ι → P ξ} {x : R α} {y : Q γ}
|
||||
→ coe (cong P eq) (f {α} {γ} x y) ≡ f {β} {δ} (coe (cong R eq) x) (substP Q eq' y)
|
||||
substfpoly' {eq = refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
substfpoly⁴ : {ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{P R : A → Set ℓ'}{Q : A → Prop ℓ''}{α β : A}
|
||||
{eq : α ≡ β} {f : {ξ : A} → R ξ → Q ξ → P ξ} {x : R α} {y : Q α}
|
||||
→ coe (cong P eq) (f {α} x y) ≡ f {β} (coe (cong R eq) x) (substP Q eq y)
|
||||
substfpoly⁴ {eq = refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
substfpoly³ : {ℓ ℓ' ℓ'' ℓ''' : Level}{A B C : Set ℓ}{R : A → Set ℓ'}{Q : B → Prop ℓ''}{P : C → Set ℓ'''}{α β : A}{γ δ : B}{ε φ : C}
|
||||
{eq : α ≡ β}{eq' : γ ≡ δ}{eq'' : ε ≡ φ} {f : {ξ : A}{ι : B}{τ : C} → R ξ → Q ι → P τ} {x : R α} {y : Q γ}
|
||||
→ coe (cong P eq'') (f {α} {γ} {ε} x y) ≡ f {β} {δ} {φ} (coe (cong R eq) x) (substP Q eq' y)
|
||||
substfpoly³ {eq = refl} {refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
substfpoly'' : {ℓ ℓ' ℓ'' : Level}{A C : Set ℓ}{P : A → C → Set ℓ'}{R : A → Set ℓ'}{Q : A → C → Prop ℓ''}{α β : A}{ε φ : C}
|
||||
{eq : α ≡ β}{eq'' : ε ≡ φ} {f : {ξ : A}{κ : C} → R ξ → Q ξ κ → P ξ κ} {x : R α} {y : Q α ε}
|
||||
→ coe (cong₂ P eq eq'') (f {α} {ε} x y) ≡ f {β} {φ} (coe (cong R eq) x) (substP (λ X → Q X φ) eq (substP (Q α) eq'' y))
|
||||
substfpoly'' {eq = refl} {refl} {f}{x}{y} = ≡tran² coerefl (cong (λ x → f x y) (≡sym coerefl)) refl
|
||||
|
||||
substfgpoly : {ℓ ℓ' : Level}{A B : Set ℓ} {P Q : A → Set ℓ'} {R : B → Set ℓ'} {F : B → A} {α β : A} {ε φ : B}
|
||||
{eq₁ : α ≡ β} {eq₂ : F ε ≡ α} {eq₃ : F φ ≡ β} {eq₄ : ε ≡ φ}
|
||||
{g : {a : A} → Q a → P a} {f : {b : B} → R b → Q (F b)} {x : R ε}
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user