Continued the proofs, will try to make a simpler account of proof substitution
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FFOLInitial.agda
156
FFOLInitial.agda
@ -2,15 +2,12 @@
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open import PropUtil
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module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
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module FFOLInitial where
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open import FinitaryFirstOrderLogic F R
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open import FinitaryFirstOrderLogic
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open import Agda.Primitive
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open import ListUtil
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variable
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n : Nat
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-- First definition of terms and term contexts --
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data Cont : Set₁ where
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◇t : Cont
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@ -23,11 +20,10 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
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data Tm : Cont → Set₁ where
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var : TmVar Γₜ → Tm Γₜ
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fun : F n → Array (Tm Γₜ) n → Tm Γₜ
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-- Now we can define formulæ
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data For : Cont → Set₁ where
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rel : R n → Array (Tm Γₜ) n → For Γₜ
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r : Tm Γₜ → Tm Γₜ → For Γₜ
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_⇒_ : For Γₜ → For Γₜ → For Γₜ
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∀∀ : For (Γₜ ▹t⁰) → For Γₜ
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@ -38,28 +34,15 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
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-- We subst on terms
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_[_]t : Tm Γₜ → Subt Δₜ Γₜ → Tm Δₜ
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_[_]tz : Array (Tm Γₜ) n → Subt Δₜ Γₜ → Array (Tm Δₜ) n
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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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fun f tz [ σ ]t = fun f (tz [ σ ]tz)
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zero [ σ ]tz = zero
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next t tz [ σ ]tz = next (t [ σ ]t) (tz [ σ ]tz)
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-- tz application is like mapping
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tzmap : {tz : Array (Tm Γₜ) n} {σ : Subt Δₜ Γₜ} → (tz [ σ ]tz) ≡ map (λ t → t [ σ ]t) tz
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tzmap {tz = zero} = refl
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tzmap {tz = next t tz} {σ = σ} = cong (next (t [ σ ]t)) tzmap
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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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lifttz : Array (Tm Γₜ) n → Array (Tm (Γₜ ▹t⁰)) n
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liftt (var tv) = var (tvnext tv)
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liftt (fun f tz) = fun f (lifttz tz)
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lifttz zero = zero
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lifttz (next t tz) = next (liftt t) (lifttz tz)
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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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@ -77,7 +60,7 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
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-- We subst on formulæ
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_[_]f : For Γₜ → Subt Δₜ Γₜ → For Δₜ
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(rel r tz) [ σ ]f = rel r ((map (λ t → t [ σ ]t) tz))
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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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@ -109,52 +92,30 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
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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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[]tz-id : {tz : Array (Tm Γₜ) n} → tz [ idₜ {Γₜ} ]tz ≡ tz
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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} {fun f tz} = substP (λ tz' → fun f tz' ≡ fun f tz) (≡sym []tz-id) refl
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[]tz-id {tz = zero} = refl
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[]tz-id {tz = next x tz} = substP (λ tz' → (next (x [ idₜ ]t) tz') ≡ next x tz) (≡sym []tz-id) (substP (λ x' → next x' tz ≡ next x tz) (≡sym []t-id) 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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[]tz-∘ : {α : Subt Ξₜ Δₜ} → {β : Subt Δₜ Γₜ} → {tz : Array (Tm Γₜ) n} → tz [ β ∘ₜ α ]tz ≡ (tz [ β ]tz) [ α ]tz
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[]tz-∘ {tz = zero} = refl
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[]tz-∘ {tz = next t tz} = cong₂ next ([]t-∘ {t = t}) []tz-∘
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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 = fun f tz} = cong (fun f) ([]tz-∘ {tz = tz})
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fun[] : {σ : Subt Δₜ Γₜ} → {f : F n} → {tz : Array (Tm Γₜ) n} → (fun f tz) [ σ ]t ≡ fun f (map (λ t → t [ σ ]t) tz)
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fun[] {tz = zero} = refl
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fun[] {σ = σ} {f = f} {tz = next t tz} = cong (fun f) (cong (next (t [ σ ]t)) tzmap)
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[]f-id : {F : For Γₜ} → F [ idₜ {Γₜ} ]f ≡ F
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[]f-id {F = rel r tz} = cong (rel r) (≡tran (≡sym tzmap) []tz-id)
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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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lifttz[] : {α : Subt Δₜ Γₜ} → {tz : Array (Tm Γₜ) n} → lifttz (tz [ α ]tz) ≡ (lifttz tz [ lift α ]tz)
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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[] {t = fun f tz} = cong (fun f) lifttz[]
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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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lifttz[] {tz = zero} = refl
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lifttz[] {tz = next t tz} = cong₂ next (liftt[] {t = t}) lifttz[]
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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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[]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 = rel r tz} = cong (rel r) (≡tran (≡tran (≡sym tzmap) (substP (λ tzz → (tz [ β ∘ₜ α ]tz) ≡ (tzz [ α ]tz)) tzmap []tz-∘)) tzmap)
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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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rel[] : {σ : Subt Δₜ Γₜ} → {r : R n} → {tz : Array (Tm Γₜ) n} → (rel r tz) [ σ ]f ≡ rel r (map (λ t → t [ σ ]t) tz)
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rel[] {r = r} = cong (rel r) refl
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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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data Conp : Cont → Set₁ -- pu tit in Prop
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@ -166,6 +127,7 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
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data Conp where
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◇p : Conp Γₜ
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_▹p⁰_ : Conp Γₜ → For Γₜ → Conp Γₜ
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record Con : Set₁ where
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constructor con
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field
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@ -201,18 +163,96 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) 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∀∀i : {A : For (Con.t (Γ ▹t))} → Pf (Γ ▹t) A → Pf Γ (∀∀ A)
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--p∀∀e : {A : For Γ} → Pf Γ (∀∀ A) → Pf Γ (A [ t , id ])
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--p∀∀i : {A : For (Γ ▹t)} → Pf (Γ [?]) 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) Γₜ → Sub Δ (con Γₜ ◇p) -- Γₜ → Δₜ ≡≡> (Γₜ,◇p) → (Δₜ,Δₚ)
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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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wk▹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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_,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 (wk▹p σ pf) = subt σ
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subt (σ ,ₚ pf) = subt σ
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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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-- An order on contexts, where we can only change positions
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infixr 5 _∈ₚ_ _∈ₚ*_
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data _∈ₚ_ : For Γₜ → Conp Γₜ → Set₁ where
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zero∈ₚ : {A : For Γₜ} → A ∈ₚ Γₚ ▹p⁰ A
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next∈ₚ : {A B : For Γₜ} → A ∈ₚ Γₚ → A ∈ₚ Γₚ ▹p⁰ B
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data _∈ₚ*_ : Conp Γₜ → Conp Γₜ → Set₁ where
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zero∈ₚ* : ◇p ∈ₚ* Γₚ
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next∈ₚ* : {A : For Γₜ} → A ∈ₚ Δₚ → Γₚ ∈ₚ* Δₚ → (Γₚ ▹p⁰ A) ∈ₚ* Δₚ
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-- Allows to grow ∈ₚ* to the right
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right∈ₚ* :{A : For Δₜ} → Γₚ ∈ₚ* Δₚ → Γₚ ∈ₚ* (Δₚ ▹p⁰ A)
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right∈ₚ* zero∈ₚ* = zero∈ₚ*
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right∈ₚ* (next∈ₚ* x h) = next∈ₚ* (next∈ₚ x) (right∈ₚ* h)
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both∈ₚ* : {A : For Γₜ} → Γₚ ∈ₚ* Δₚ → (Γₚ ▹p⁰ A) ∈ₚ* (Δₚ ▹p⁰ A)
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both∈ₚ* zero∈ₚ* = next∈ₚ* zero∈ₚ zero∈ₚ*
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both∈ₚ* (next∈ₚ* x h) = next∈ₚ* zero∈ₚ (next∈ₚ* (next∈ₚ x) (right∈ₚ* h))
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refl∈ₚ* : Γₚ ∈ₚ* Γₚ
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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 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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∈ₚ*▹tp zero∈ₚ* = zero∈ₚ*
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∈ₚ*▹tp (next∈ₚ* x s) = next∈ₚ* (∈ₚ▹tp x) (∈ₚ*▹tp s)
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-- Todo fuse both concepts (remove ∈ₚ)
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var→∈ₚ : {A : For Γₜ} → (x : PfVar (con Γₜ Γₚ) A) → A ∈ₚ Γₚ
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∈ₚ→var : {A : For Γₜ} → A ∈ₚ Γₚ → PfVar (con Γₜ Γₚ) A
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var→∈ₚ pvzero = zero∈ₚ
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var→∈ₚ (pvnext x) = next∈ₚ (var→∈ₚ x)
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∈ₚ→var zero∈ₚ = pvzero
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∈ₚ→var (next∈ₚ s) = pvnext (∈ₚ→var s)
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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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var pvzero [ σ ,ₚ pf ]p = pf
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var (pvnext pv) [ σ ,ₚ pf ]p = var pv [ σ ]p
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app pf pf₁ [ σ ]p = app (pf [ σ ]p) (pf₁ [ σ ]p)
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lam pf [ σ ]p = lam (pf [ llift' {!σ!} ,ₚ var pvzero ]p)
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p∀∀e pf [ σ ]p = {!p∀∀e!}
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p∀∀i pf [ σ ]p = p∀∀i {!!}
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_∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
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εₚ σₜ ∘ β = {!!}
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(α ,ₚ pf) ∘ β = {!!}
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-- Equalities below are useless because Γ ⊢ F is in Prop
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,ₚ∘πₚ : {Γ Δ : Con} → {F : For (Con.t Γ)} → {σ : Sub Δ (Γ ▹p F)} → (πₚ¹ σ) ,ₚ (πₚ² σ) ≡ σ
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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
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-- πₚ²∘,ₚ : {Γ Δ : Con} → {σ : Sub Δ Γ} → {F : For Γ} → {prf : Δ ⊢ (F [ σ ]f)} → πₚ² (σ ,ₚ prf) ≡ prf
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,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))
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-- lifts
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--liftpt : Pf Δ (A [ subt σ ]f) → Pf Δ ((A [ llift idₜ ]f) [ subt (σ ,ₜ t) ]f)
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@ -277,7 +317,7 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
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sub αₜ αₚ ∘ (sub βₜ βₚ) = sub (αₜ ∘ₜ βₜ) {!!}
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-}
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imod : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero} F R
|
||||
imod : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero}
|
||||
imod = record
|
||||
{ Con = Con
|
||||
; Sub = Sub
|
||||
@ -289,8 +329,6 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
|
||||
; _[_]t = λ t σ → t [ subt σ ]t
|
||||
; []t-id = {!!}
|
||||
; []t-∘ = {!!}
|
||||
; fun = fun
|
||||
; fun[] = {!!}
|
||||
; _▹ₜ = _▹t
|
||||
; πₜ¹ = {!!}
|
||||
; πₜ² = {!!}
|
||||
@ -303,8 +341,8 @@ module FFOLInitial (F : Nat → Set) (R : Nat → Set) where
|
||||
; _[_]f = λ A σ → A [ subt σ ]f
|
||||
; []f-id = λ {Γ} {F} → []f-id {Con.t 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}!}
|
||||
; rel = rel
|
||||
; rel[] = rel[]
|
||||
; R = r
|
||||
; R[] = {!!}
|
||||
; _⊢_ = λ Γ A → Pf Γ A
|
||||
; _[_]p = {!!}
|
||||
; _▹ₚ_ = _▹p_
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user