Ok, commit before removing a lot of useless code (i thought it was useful, i swear)
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FFOLInitial.agda
217
FFOLInitial.agda
@ -30,57 +30,55 @@ module FFOLInitial where
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-- Then we define term substitutions, and the application of them on terms and formulæ
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data Subt : Cont → Cont → Set₁ where
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εₜ : Subt Γₜ ◇t
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wk▹t : Subt Δₜ Γₜ → Tm Δₜ → Subt Δₜ (Γₜ ▹t⁰)
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_,ₜ_ : Subt Δₜ Γₜ → Tm Δₜ → Subt Δₜ (Γₜ ▹t⁰)
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-- We subst on terms
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_[_]t : Tm Γₜ → Subt Δₜ Γₜ → Tm Δₜ
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var tvzero [ wk▹t σ t ]t = t
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var (tvnext tv) [ wk▹t σ t ]t = var tv [ σ ]t
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var tvzero [ σ ,ₜ t ]t = t
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var (tvnext tv) [ σ ,ₜ t ]t = var tv [ σ ]t
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-- We define liftings on term variables
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-- A term of n variables is a term of n+1 variables
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-- Same for a term array
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liftt : Tm Γₜ → Tm (Γₜ ▹t⁰)
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wkₜt : Tm Γₜ → Tm (Γₜ ▹t⁰)
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liftt (var tv) = var (tvnext tv)
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wkₜt (var tv) = var (tvnext tv)
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-- From a substition into n variables, we get a substitution into n+1 variables which don't use the last one
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llift : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) Γₜ
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llift εₜ = εₜ
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llift (wk▹t σ t) = wk▹t (llift σ) (liftt t)
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llift-liftt : {tv : TmVar Γₜ} → {σ : Subt Δₜ Γₜ} → liftt (var tv [ σ ]t) ≡ var tv [ llift σ ]t
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llift-liftt {tv = tvzero} {σ = wk▹t σ x} = refl
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llift-liftt {tv = tvnext tv} {σ = wk▹t σ x} = llift-liftt {tv = tv} {σ = σ}
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wkₜσt : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) Γₜ
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wkₜσt εₜ = εₜ
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wkₜσt (σ ,ₜ t) = (wkₜσt σ) ,ₜ (wkₜt t)
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wkₜσt-wkₜt : {tv : TmVar Γₜ} → {σ : Subt Δₜ Γₜ} → wkₜt (var tv [ σ ]t) ≡ var tv [ wkₜσt σ ]t
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wkₜσt-wkₜt {tv = tvzero} {σ = σ ,ₜ x} = refl
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wkₜσt-wkₜt {tv = tvnext tv} {σ = σ ,ₜ x} = wkₜσt-wkₜt {tv = tv} {σ = σ}
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-- From a substitution into n variables, we construct a substitution from n+1 variables to n+1 variables which maps it to itself
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-- i.e. 0 -> 0 and for all i ->(old) σ(i) we get i+1 -> σ(i)+1
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lift : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) (Γₜ ▹t⁰)
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lift σ = wk▹t (llift σ) (var tvzero)
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liftₜσ : Subt Δₜ Γₜ → Subt (Δₜ ▹t⁰) (Γₜ ▹t⁰)
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liftₜσ σ = (wkₜσt σ) ,ₜ (var tvzero)
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-- We subst on formulæ
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_[_]f : For Γₜ → Subt Δₜ Γₜ → For Δₜ
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(r t u) [ σ ]f = r (t [ σ ]t) (u [ σ ]t)
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(A ⇒ B) [ σ ]f = (A [ σ ]f) ⇒ (B [ σ ]f)
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(∀∀ A) [ σ ]f = ∀∀ (A [ lift σ ]f)
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(∀∀ A) [ σ ]f = ∀∀ (A [ liftₜσ σ ]f)
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-- We now can define identity on term substitutions
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idₜ : Subt Γₜ Γₜ
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idₜ {◇t} = εₜ
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idₜ {Γₜ ▹t⁰} = lift (idₜ {Γₜ})
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idₜ {Γₜ ▹t⁰} = liftₜσ (idₜ {Γₜ})
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_∘ₜ_ : Subt Δₜ Γₜ → Subt Ξₜ Δₜ → Subt Ξₜ Γₜ
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εₜ ∘ₜ β = εₜ
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wk▹t α x ∘ₜ β = wk▹t (α ∘ₜ β) (x [ β ]t)
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(α ,ₜ x) ∘ₜ β = (α ∘ₜ β) ,ₜ (x [ β ]t)
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-- We have the access functions from the algebra, in restricted versions
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πₜ¹ : Subt Δₜ (Γₜ ▹t⁰) → Subt Δₜ Γₜ
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πₜ¹ (wk▹t σₜ t) = σₜ
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πₜ¹ (σₜ ,ₜ t) = σₜ
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πₜ² : Subt Δₜ (Γₜ ▹t⁰) → Tm Δₜ
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πₜ² (wk▹t σₜ t) = t
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_,ₜ_ : Subt Δₜ Γₜ → Tm Δₜ → Subt Δₜ (Γₜ ▹t⁰)
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σₜ ,ₜ t = wk▹t σₜ t
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πₜ² (σₜ ,ₜ t) = t
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-- And their equalities (the fact that there are reciprocical)
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πₜ²∘,ₜ : {σₜ : Subt Δₜ Γₜ} → {t : Tm Δₜ} → πₜ² (σₜ ,ₜ t) ≡ t
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@ -88,40 +86,52 @@ module FFOLInitial where
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πₜ¹∘,ₜ : {σₜ : Subt Δₜ Γₜ} → {t : Tm Δₜ} → πₜ¹ (σₜ ,ₜ t) ≡ σₜ
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πₜ¹∘,ₜ = refl
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,ₜ∘πₜ : {σₜ : Subt Δₜ (Γₜ ▹t⁰)} → (πₜ¹ σₜ) ,ₜ (πₜ² σₜ) ≡ σₜ
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,ₜ∘πₜ {σₜ = wk▹t σₜ t} = refl
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,ₜ∘πₜ {σₜ = σₜ ,ₜ t} = refl
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-- We can also prove the substitution equalities
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[]t-id : {t : Tm Γₜ} → t [ idₜ {Γₜ} ]t ≡ t
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[]t-id {Γₜ ▹t⁰} {var tvzero} = refl
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[]t-id {Γₜ ▹t⁰} {var (tvnext tv)} = substP (λ t → t ≡ var (tvnext tv)) (llift-liftt {tv = tv} {σ = idₜ}) (substP (λ t → liftt t ≡ var (tvnext tv)) (≡sym ([]t-id {t = var tv})) refl)
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[]t-id {Γₜ ▹t⁰} {var (tvnext tv)} = substP (λ t → t ≡ var (tvnext tv)) (wkₜσt-wkₜt {tv = tv} {σ = idₜ}) (substP (λ t → wkₜt t ≡ var (tvnext tv)) (≡sym ([]t-id {t = var tv})) refl)
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[]t-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → {t : Tm Γₜ} → t [ β ∘ₜ α ]t ≡ (t [ β ]t) [ α ]t
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[]t-∘ {α = α} {β = wk▹t β t} {t = var tvzero} = refl
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[]t-∘ {α = α} {β = wk▹t β t} {t = var (tvnext tv)} = []t-∘ {t = var tv}
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[]t-∘ {α = α} {β = β ,ₜ t} {t = var tvzero} = refl
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[]t-∘ {α = α} {β = β ,ₜ t} {t = var (tvnext tv)} = []t-∘ {t = var tv}
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[]f-id : {F : For Γₜ} → F [ idₜ {Γₜ} ]f ≡ F
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[]f-id {F = r t u} = cong₂ r []t-id []t-id
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[]f-id {F = F ⇒ G} = cong₂ _⇒_ []f-id []f-id
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[]f-id {F = ∀∀ F} = cong ∀∀ []f-id
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llift-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → llift (β ∘ₜ α) ≡ (llift β ∘ₜ lift α)
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liftt[] : {α : Subt Δₜ Γₜ} → {t : Tm Γₜ} → liftt (t [ α ]t) ≡ (liftt t [ lift α ]t)
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llift-∘ {β = εₜ} = refl
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llift-∘ {β = wk▹t β t} = cong₂ wk▹t llift-∘ (liftt[] {t = t})
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liftt[] {α = wk▹t α t} {var tvzero} = refl
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liftt[] {α = wk▹t α t} {var (tvnext tv)} = liftt[] {t = var tv}
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lift-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → lift (β ∘ₜ α) ≡ (lift β) ∘ₜ (lift α)
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lift-∘ {α = α} {β = εₜ} = refl
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lift-∘ {α = α} {β = wk▹t β t} = cong₂ wk▹t (cong₂ wk▹t llift-∘ (liftt[] {t = t})) refl
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wkₜσt-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → wkₜσt (β ∘ₜ α) ≡ (wkₜσt β ∘ₜ liftₜσ α)
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wkₜt[] : {α : Subt Δₜ Γₜ} → {t : Tm Γₜ} → wkₜt (t [ α ]t) ≡ (wkₜt t [ liftₜσ α ]t)
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wkₜσt-∘ {β = εₜ} = refl
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wkₜσt-∘ {β = β ,ₜ t} = cong₂ _,ₜ_ wkₜσt-∘ (wkₜt[] {t = t})
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wkₜt[] {α = α ,ₜ t} {var tvzero} = refl
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wkₜt[] {α = α ,ₜ t} {var (tvnext tv)} = wkₜt[] {t = var tv}
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liftₜσ-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → liftₜσ (β ∘ₜ α) ≡ (liftₜσ β) ∘ₜ (liftₜσ α)
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liftₜσ-∘ {α = α} {β = εₜ} = refl
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liftₜσ-∘ {α = α} {β = β ,ₜ t} = cong₂ _,ₜ_ (cong₂ _,ₜ_ wkₜσt-∘ (wkₜt[] {t = t})) refl
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[]f-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → {F : For Γₜ} → F [ β ∘ₜ α ]f ≡ (F [ β ]f) [ α ]f
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[]f-∘ {α = α} {β = β} {F = r t u} = cong₂ r ([]t-∘ {α = α} {β = β} {t = t}) ([]t-∘ {α = α} {β = β} {t = u})
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[]f-∘ {F = F ⇒ G} = cong₂ _⇒_ []f-∘ []f-∘
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[]f-∘ {F = ∀∀ F} = cong ∀∀ (≡tran (cong (λ σ → F [ σ ]f) lift-∘) []f-∘)
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[]f-∘ {F = ∀∀ F} = cong ∀∀ (≡tran (cong (λ σ → F [ σ ]f) liftₜσ-∘) []f-∘)
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R[] : {σ : Subt Δₜ Γₜ} → {t u : Tm Γₜ} → (r t u) [ σ ]f ≡ r (t [ σ ]t) (u [ σ ]t)
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R[] = refl
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wk[,] : {t : Tm Γₜ}{u : Tm Δₜ}{β : Subt Δₜ Γₜ} → (wkₜt t) [ β ,ₜ u ]t ≡ t [ β ]t
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wk[,] {t = var tvzero} = refl
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wk[,] {t = var (tvnext tv)} = refl
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wk∘, : {α : Subt Γₜ Δₜ}{β : Subt Ξₜ Γₜ}{t : Tm Ξₜ} → (wkₜσt α) ∘ₜ (β ,ₜ t) ≡ (α ∘ₜ β)
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wk∘, {α = εₜ} = refl
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wk∘, {α = α ,ₜ t} {β = β} = cong₂ _,ₜ_ (wk∘, {α = α}) (wk[,] {t = t} {β = β})
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σ-idl : {α : Subt Δₜ Γₜ} → idₜ ∘ₜ α ≡ α
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σ-idl {α = εₜ} = refl
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σ-idl {α = α ,ₜ x} = cong₂ _,ₜ_ (≡tran wk∘, σ-idl) refl
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σ-idr : {α : Subt Δₜ Γₜ} → α ∘ₜ idₜ ≡ α
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σ-idr {α = εₜ} = refl
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σ-idr {α = α ,ₜ x} = cong₂ _,ₜ_ σ-idr []t-id
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data Conp : Cont → Set₁ -- pu tit in Prop
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variable
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Γₚ : Conp Γₜ
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Δₚ : Conp Δₜ
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Γₚ Γₚ' : Conp Γₜ
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Δₚ Δₚ' : Conp Δₜ
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Ξₚ : Conp Ξₜ
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data Conp where
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@ -147,10 +157,10 @@ module FFOLInitial where
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-- We can add term, that will not be used in the formulæ already present
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-- (that's why we use llift)
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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 [ llift idₜ ]f)
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(Γₚ ▹p⁰ A) ▹tp = (Γₚ ▹tp) ▹p⁰ (A [ wkₜσt idₜ ]f)
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_▹t : Con → Con
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Γ ▹t = con ((Con.t Γ) ▹t⁰) (Con.p Γ ▹tp)
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@ -163,24 +173,24 @@ module FFOLInitial where
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var : {A : For (Con.t Γ)} → PfVar Γ A → Pf Γ A
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app : {A B : For (Con.t Γ)} → Pf Γ (A ⇒ B) → Pf Γ A → Pf Γ B
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lam : {A B : For (Con.t Γ)} → Pf (Γ ▹p A) B → Pf Γ (A ⇒ B)
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p∀∀e : {A : For ((Con.t Γ) ▹t⁰)} → {t : Tm (Con.t Γ)} → Pf Γ (∀∀ A) → Pf Γ (A [ wk▹t idₜ t ]f)
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p∀∀e : {A : For ((Con.t Γ) ▹t⁰)} → {t : Tm (Con.t Γ)} → Pf Γ (∀∀ A) → Pf Γ (A [ idₜ ,ₜ t ]f)
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p∀∀i : {A : For (Con.t (Γ ▹t))} → Pf (Γ ▹t) A → Pf Γ (∀∀ A)
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data Sub : Con → Con → Set₁
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subt : Sub Δ Γ → Subt (Con.t Δ) (Con.t Γ)
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data Sub where
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εₚ : Subt (Con.t Δ) (Con.t Γ) → Sub Δ (con (Con.t Γ) ◇p) -- Γₜ → Δₜ ≡≡> (Γₜ,◇p) → (Δₜ,Δₚ)
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-- If i tell you by what you should replace a missing proof of A, then you can
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-- prove anything that uses a proof of A
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_,ₚ_ : {A : For (Con.t Γ)} → (σ : Sub Δ Γ) → Pf Δ (A [ subt σ ]f) → Sub Δ (Γ ▹p A)
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subt (εₚ σₜ) = σₜ
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subt (σ ,ₚ pf) = subt σ
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data Subp : {Δₜ : Cont} → Conp Δₜ → Conp Δₜ → Set₁ where
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εₚ : Subp Δₚ ◇p
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_,ₚ_ : {A : For Δₜ} → (σ : Subp Δₚ Δₚ') → Pf (con Δₜ Δₚ) A → Subp Δₚ (Δₚ' ▹p⁰ A)
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_[_]c : Conp Γₜ → Subt Δₜ Γₜ → Conp Δₜ
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◇p [ σₜ ]c = ◇p
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(Γₚ ▹p⁰ A) [ σₜ ]c = (Γₚ [ σₜ ]c) ▹p⁰ (A [ σₜ ]f)
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πₚ¹ : {Γ Δ : Con} → {F : For (Con.t Γ)} → Sub Δ (Γ ▹p F) → Sub Δ Γ
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πₚ¹ (σ ,ₚ pf) = σ
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πₚ² : {Γ Δ : Con} → {F : For (Con.t Γ)} → (σ : Sub Δ (Γ ▹p F)) → Pf Δ (F [ subt (πₚ¹ σ) ]f)
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πₚ² (σ ,ₚ pf) = pf
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record Sub (Γ : Con) (Δ : Con) : Set₁ where
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constructor sub
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field
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t : Subt (Con.t Γ) (Con.t Δ)
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p : Subp {Con.t Γ} (Con.p Γ) ((Con.p Δ) [ t ]c)
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-- An order on contexts, where we can only change positions
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infixr 5 _∈ₚ_ _∈ₚ*_
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@ -201,7 +211,7 @@ module FFOLInitial where
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refl∈ₚ* {Γₚ = ◇p} = zero∈ₚ*
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refl∈ₚ* {Γₚ = Γₚ ▹p⁰ x} = both∈ₚ* refl∈ₚ*
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∈ₚ▹tp : {A : For Δₜ} → A ∈ₚ Δₚ → A [ llift idₜ ]f ∈ₚ (Δₚ ▹tp)
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∈ₚ▹tp : {A : For Δₜ} → A ∈ₚ Δₚ → A [ wkₜσt idₜ ]f ∈ₚ (Δₚ ▹tp)
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∈ₚ▹tp zero∈ₚ = zero∈ₚ
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∈ₚ▹tp (next∈ₚ x) = next∈ₚ (∈ₚ▹tp x)
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∈ₚ*▹tp : Γₚ ∈ₚ* Δₚ → (Γₚ ▹tp) ∈ₚ* (Δₚ ▹tp)
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@ -218,40 +228,73 @@ module FFOLInitial where
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mon∈ₚ∈ₚ* : {A : For Γₜ} → A ∈ₚ Γₚ → Γₚ ∈ₚ* Δₚ → A ∈ₚ Δₚ
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mon∈ₚ∈ₚ* zero∈ₚ (next∈ₚ* x x₁) = x
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mon∈ₚ∈ₚ* (next∈ₚ s) (next∈ₚ* x x₁) = mon∈ₚ∈ₚ* s x₁
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liftpₚ : {Δₚ Ξₚ : Conp Δₜ} {A : For Δₜ} → Δₚ ∈ₚ* Ξₚ → Pf (con Δₜ Δₚ) A → Pf (con Δₜ Ξₚ) A
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liftpₚ s (var x) = var (∈ₚ→var (mon∈ₚ∈ₚ* (var→∈ₚ x) s))
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liftpₚ s (app pf pf₁) = app (liftpₚ s pf) (liftpₚ s pf₁)
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liftpₚ s (lam pf) = lam (liftpₚ (both∈ₚ* s) pf)
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liftpₚ s (p∀∀e pf) = p∀∀e (liftpₚ s pf)
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liftpₚ s (p∀∀i pf) = p∀∀i (liftpₚ (∈ₚ*▹tp s) pf)
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lliftₚ : {Δₚ Ξₚ : Conp Δₜ} → Δₚ ∈ₚ* Ξₚ → Sub (con Δₜ Δₚ) Γ → Sub (con Δₜ Ξₚ) Γ
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lliftₚ≡subt : {σ : Sub (con Δₜ Δₚ) Γ} → {s : Δₚ ∈ₚ* Ξₚ} → subt (lliftₚ s σ) ≡ subt σ
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lliftₚ≡subt {σ = εₚ x} = {!refl!}
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lliftₚ≡subt {σ = σ ,ₚ x} = {!lliftₚ≡subt {σ = σ}!}
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lliftₚ {Γ = Γ} _ (εₚ σₜ) = εₚ {Γ = Γ} σₜ
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lliftₚ {Δₜ = Δₜ} {Δₚ = Δₚ} s (_,ₚ_ {A = A} σ pf) = lliftₚ s σ ,ₚ liftpₚ s (substP (λ σₜ → Pf (con Δₜ Δₚ) (A [ σₜ ]f)) (≡sym (lliftₚ≡subt {σ = σ} {s = s})) pf)
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llift' : {A : For (Con.t Δ)} → Sub Δ Γ → Sub (Δ ▹p A) Γ
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llift' s = lliftₚ (right∈ₚ* refl∈ₚ*) s
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_[_]p : {Γ Δ : Con} → {F : For (Con.t Γ)} → Pf Γ F → (σ : Sub Δ Γ) → Pf Δ (F [ subt σ ]f) -- The functor's action on morphisms
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∈ₚ*→Sub : Δₚ ∈ₚ* Δₚ' → Subp {Δₜ} Δₚ' Δₚ
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∈ₚ*→Sub zero∈ₚ* = εₚ
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∈ₚ*→Sub (next∈ₚ* x s) = ∈ₚ*→Sub s ,ₚ var (∈ₚ→var x)
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idₚ : Subp {Δₜ} Δₚ Δₚ
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idₚ = ∈ₚ*→Sub refl∈ₚ*
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wkₚp : {A : For Δₜ} → Δₚ ∈ₚ* Δₚ' → Pf (con Δₜ Δₚ) A → Pf (con Δₜ Δₚ') A
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wkₚp s (var pv) = var (∈ₚ→var (mon∈ₚ∈ₚ* (var→∈ₚ pv) s))
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wkₚp s (app pf pf₁) = app (wkₚp s pf) (wkₚp s pf₁)
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wkₚp s (lam {A = A} pf) = lam (wkₚp (both∈ₚ* s) pf)
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wkₚp s (p∀∀e pf) = p∀∀e (wkₚp s pf)
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wkₚp s (p∀∀i pf) = p∀∀i (wkₚp (∈ₚ*▹tp s) pf)
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lliftₚ : {Γₚ : Conp Δₜ} → Δₚ ∈ₚ* Δₚ' → Subp {Δₜ} Δₚ Γₚ → Subp {Δₜ} Δₚ' Γₚ
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lliftₚ s εₚ = εₚ
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lliftₚ s (σₚ ,ₚ pf) = lliftₚ s σₚ ,ₚ wkₚp s pf
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lem3 : {α : Subt Γₜ Δₜ} → {β : Subt Ξₜ Γₜ} → α ∘ₜ (wkₜσt β) ≡ wkₜσt (α ∘ₜ β)
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lem3 {α = εₜ} = refl
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lem3 {α = α ,ₜ var tv} = cong₂ _,ₜ_ (lem3 {α = α}) (≡sym (wkₜσt-wkₜt {tv = tv}))
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lem7 : {σ : Subt Δₜ Γₜ} → ((Δₚ ▹tp) [ liftₜσ σ ]c) ≡ ((Δₚ [ σ ]c) ▹tp)
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lem7 {Δₚ = ◇p} = refl
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lem7 {Δₚ = Δₚ ▹p⁰ A} = cong₂ _▹p⁰_ lem7 (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (≡tran² (≡sym wkₜσt-∘) (cong wkₜσt (≡tran σ-idl (≡sym σ-idr))) (≡sym lem3))) []f-∘)
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lem8 : {σ : Subt Δₜ Γₜ} {t : Tm Γₜ} → ((wkₜσt σ ∘ₜ (idₜ ,ₜ (t [ σ ]t))) ,ₜ (t [ σ ]t)) ≡ ((idₜ ∘ₜ σ) ,ₜ (t [ σ ]t))
|
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lem8 = cong₂ _,ₜ_ (≡tran² wk∘, σ-idr (≡sym σ-idl)) refl
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_[_]pvₜ : {A : For Δₜ} → PfVar (con Δₜ Δₚ) A → (σ : Subt Γₜ Δₜ) → PfVar (con Γₜ (Δₚ [ σ ]c)) (A [ σ ]f)
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||||
_[_]pₜ : {A : For Δₜ} → Pf (con Δₜ Δₚ) A → (σ : Subt Γₜ Δₜ) → Pf (con Γₜ (Δₚ [ σ ]c)) (A [ σ ]f)
|
||||
pvzero [ σ ]pvₜ = pvzero
|
||||
pvnext pv [ σ ]pvₜ = pvnext (pv [ σ ]pvₜ)
|
||||
var pv [ σ ]pₜ = var (pv [ σ ]pvₜ)
|
||||
app pf pf' [ σ ]pₜ = app (pf [ σ ]pₜ) (pf' [ σ ]pₜ)
|
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lam pf [ σ ]pₜ = lam (pf [ σ ]pₜ)
|
||||
_[_]pₜ {Δₚ = Δₚ} {Γₜ = Γₜ} (p∀∀e {A = A} {t = t} pf) σ = substP (λ F → Pf (con Γₜ (Δₚ [ σ ]c)) F) (≡tran² (≡sym []f-∘) (cong (λ σ → A [ σ ]f) (lem8 {t = t})) ([]f-∘)) (p∀∀e {t = t [ σ ]t} (pf [ σ ]pₜ))
|
||||
_[_]pₜ {Γₜ = Γₜ} (p∀∀i pf) σ = p∀∀i (substP (λ Ξₚ → Pf (con (Γₜ ▹t⁰) (Ξₚ)) _) lem7 (pf [ liftₜσ σ ]pₜ))
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
lem9 : (Δₚ [ wkₜσt idₜ ]c) ≡ (Δₚ ▹tp)
|
||||
lem9 {Δₚ = ◇p} = refl
|
||||
lem9 {Δₚ = Δₚ ▹p⁰ x} = cong₂ _▹p⁰_ lem9 refl
|
||||
wkₜσₚ : Subp {Δₜ} Δₚ' Δₚ → Subp {Δₜ ▹t⁰} (Δₚ' ▹tp) (Δₚ ▹tp)
|
||||
wkₜσₚ εₚ = εₚ
|
||||
wkₜσₚ {Δₜ = Δₜ} (_,ₚ_ {A = A} σₚ pf) = (wkₜσₚ σₚ) ,ₚ substP (λ Ξₚ → Pf (con (Δₜ ▹t⁰) Ξₚ) (A [ wkₜσt idₜ ]f)) lem9 (_[_]pₜ {Γₜ = Δₜ ▹t⁰} pf (wkₜσt idₜ))
|
||||
|
||||
_[_]p : {A : For Δₜ} → Pf (con Δₜ Δₚ) A → (σ : Subp {Δₜ} Δₚ' Δₚ) → Pf (con Δₜ Δₚ') A
|
||||
var pvzero [ σ ,ₚ pf ]p = pf
|
||||
var (pvnext pv) [ σ ,ₚ pf ]p = var pv [ σ ]p
|
||||
app pf pf₁ [ σ ]p = app (pf [ σ ]p) (pf₁ [ σ ]p)
|
||||
lam pf [ σ ]p = lam (pf [ llift' {!σ!} ,ₚ var pvzero ]p)
|
||||
p∀∀e pf [ σ ]p = {!p∀∀e!}
|
||||
p∀∀i pf [ σ ]p = p∀∀i {!!}
|
||||
_∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
|
||||
εₚ σₜ ∘ β = {!!}
|
||||
(α ,ₚ pf) ∘ β = {!!}
|
||||
|
||||
-- Equalities below are useless because Γ ⊢ F is in Prop
|
||||
,ₚ∘πₚ : {Γ Δ : Con} → {F : For (Con.t Γ)} → {σ : Sub Δ (Γ ▹p F)} → (πₚ¹ σ) ,ₚ (πₚ² σ) ≡ σ
|
||||
πₚ¹∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For (Con.t Γ)} → {prf : Pf Δ (F [ subt σ ]f)} → πₚ¹ (σ ,ₚ prf) ≡ σ
|
||||
-- πₚ²∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ² (σ ,ₚ prf) ≡ prf
|
||||
,ₚ∘ : {Γ Δ Ξ : Con}{σ : Sub Γ Ξ}{δ : Sub Δ Γ}{F : For (Con.t Ξ)}{prf : Pf Γ (F [ subt σ ]f)} → (σ ,ₚ prf) ∘ δ ≡ (σ ∘ δ) ,ₚ (substP (λ F → Pf Δ F) (≡sym {!!}) (prf [ δ ]p))
|
||||
|
||||
|
||||
lam pf [ σ ]p = lam (pf [ lliftₚ (right∈ₚ* refl∈ₚ*) σ ,ₚ var pvzero ]p)
|
||||
p∀∀e pf [ σ ]p = p∀∀e (pf [ σ ]p)
|
||||
p∀∀i pf [ σ ]p = p∀∀i (pf [ wkₜσₚ σ ]p)
|
||||
|
||||
|
||||
-- lifts
|
||||
@ -326,7 +369,7 @@ module FFOLInitial where
|
||||
; ◇ = ◇
|
||||
; ε = {!!}
|
||||
; Tm = λ Γ → Tm (Con.t Γ)
|
||||
; _[_]t = λ t σ → t [ subt σ ]t
|
||||
; _[_]t = λ t σ → t [ {!!} ]t
|
||||
; []t-id = {!!}
|
||||
; []t-∘ = {!!}
|
||||
; _▹ₜ = _▹t
|
||||
@ -338,8 +381,8 @@ module FFOLInitial where
|
||||
; ,ₜ∘πₜ = {!!}
|
||||
; ,ₜ∘ = {!!}
|
||||
; For = λ Γ → For (Con.t Γ)
|
||||
; _[_]f = λ A σ → A [ subt σ ]f
|
||||
; []f-id = λ {Γ} {F} → []f-id {Con.t Γ} {F}
|
||||
; _[_]f = λ A σ → A [ {!!} ]f
|
||||
; []f-id = λ {Γ} {F} → {!!}
|
||||
; []f-∘ = {!λ {Γ Δ Ξ} {α} {β} {F} → []f-∘ {Con.t Γ} {Con.t Δ} {Con.t Ξ} {Sub.t α} {Sub.t β} {F}!}
|
||||
; R = r
|
||||
; R[] = {!!}
|
||||
|
||||
@ -61,14 +61,24 @@ module PropUtil where
|
||||
≡sym : {ℓ : Level} → {A : Set ℓ}→ {a a' : A} → a ≡ a' → a' ≡ a
|
||||
≡sym refl = refl
|
||||
|
||||
|
||||
≡tran : {ℓ : Level} {A : Set ℓ} → {a a' a'' : A} → a ≡ a' → a' ≡ a'' → a ≡ a''
|
||||
≡tran refl refl = refl
|
||||
≡tran² : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₀ ≡ a₃
|
||||
≡tran³ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₀ ≡ a₄
|
||||
≡tran⁴ : {ℓ : Level} {A : Set ℓ} → {a₀ a₁ a₂ a₃ a₄ a₅ : A} → a₀ ≡ a₁ → a₁ ≡ a₂ → a₂ ≡ a₃ → a₃ ≡ a₄ → a₄ ≡ a₅ → a₀ ≡ a₅
|
||||
≡tran refl refl = refl
|
||||
≡tran² refl refl refl = refl
|
||||
≡tran³ refl refl refl refl = refl
|
||||
≡tran⁴ refl refl refl refl refl = refl
|
||||
|
||||
-- We can make a proof-irrelevant substitution
|
||||
substP : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Prop ℓ'){a a' : A} → a ≡ a' → P a → P a'
|
||||
substP P refl h = h
|
||||
|
||||
postulate ≡fun : {ℓ ℓ' : Level} → {A : Set ℓ} → {B : Set ℓ'} → {f g : A → B} → ((x : A) → (f x ≡ g x)) → f ≡ g
|
||||
postulate ≡fun' : {ℓ ℓ' : Level} → {A : Set ℓ} → {B : A → Set ℓ'} → {f g : (a : A) → B a} → ((x : A) → (f x ≡ g x)) → f ≡ g
|
||||
|
||||
postulate subst : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Set ℓ'){a a' : A} → a ≡ a' → P a → P a'
|
||||
postulate substP : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Prop ℓ'){a a' : A} → a ≡ a' → P a → P a'
|
||||
|
||||
postulate substrefl : ∀{ℓ}{A : Set ℓ}{ℓ'}{P : A → Set ℓ'}{a : A}{e : a ≡ a}{p : P a} → subst P e p ≈ p
|
||||
{-# REWRITE substrefl #-}
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user