Second steps, but i guess there is a problem
This commit is contained in:
parent
a2fc828d8f
commit
ab7e77e833
GPG Key ID:
7054D5D6A90F084F
@ -8,17 +8,17 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
|
||||
open import ListUtil
|
||||
|
||||
variable
|
||||
ℓ¹ ℓ² ℓ³ : Level
|
||||
ℓ¹ ℓ² ℓ³ ℓ⁴̂ ℓ⁵ : Level
|
||||
|
||||
record FFOL (F : Nat → Set) (R : Nat → Set) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³)) where
|
||||
record FFOL (F : Nat → Set) (R : Nat → Set) : Set (lsuc (ℓ¹ ⊔ ℓ² ⊔ ℓ³ ⊔ ℓ⁴̂ ⊔ ℓ⁵)) where
|
||||
infixr 10 _∘_
|
||||
field
|
||||
Con : Set ℓ¹
|
||||
Sub : Con → Con → Set -- It makes a posetal category
|
||||
Sub : Con → Con → Set ℓ⁵ -- It makes a posetal category
|
||||
_∘_ : {Γ Δ Ξ : Con} → Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
|
||||
id : {Γ : Con} → Sub Γ Γ
|
||||
◇ : Con -- The initial object of the category
|
||||
ε : {Γ : Con} → Sub ◇ Γ -- The morphism from the initial to any object
|
||||
ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
|
||||
|
||||
-- Functor Con → Set called Tm
|
||||
Tm : Con → Set ℓ²
|
||||
@ -50,7 +50,7 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
|
||||
rel[] : {Γ Δ : Con} → {σ : Sub Δ Γ} → {n : Nat} → {r : R n} → {tz : Array (Tm Γ) n} → (rel r tz) [ σ ]f ≡ rel r (map (λ t → t [ σ ]t) tz)
|
||||
|
||||
-- Proofs
|
||||
_⊢_ : (Γ : Con) → For Γ → Prop
|
||||
_⊢_ : (Γ : Con) → For Γ → Prop ℓ⁴̂
|
||||
--_[_]p : {Γ Δ : Con} → {F : For Γ} → Γ ⊢ F → (σ : Sub Δ Γ) → Δ ⊢ (F [ σ ]f) -- The functor's action on morphisms
|
||||
-- Equalities below are useless because Γ ⊢ F is in prop
|
||||
-- []p-id : {Γ : Con} → {F : For Γ} → {prf : Γ ⊢ F} → prf [ id {Γ} ]p ≡ prf
|
||||
@ -86,90 +86,146 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
|
||||
|
||||
module Initial where
|
||||
|
||||
data TmCon : Set₁ where -- isom integer ≡ number of terms in the context
|
||||
◇t : TmCon
|
||||
_▹t : TmCon → TmCon
|
||||
data Con : Set₁
|
||||
data For : Con → Set₁
|
||||
data Con where -- isom integer ≡ number of terms in the context
|
||||
◇ : Con
|
||||
_▹t : Con → Con
|
||||
_▹p_ : (Γ : Con) → For Γ → Con
|
||||
|
||||
variable
|
||||
Γ : TmCon
|
||||
Γ Δ Ξ : Con
|
||||
n : Nat
|
||||
A : For Γ
|
||||
|
||||
data TmVar : TmCon → Set₁ where -- if Γ ≡ k, then TmVar Γ ≡ ⟦ 0 , k-1 ⟧
|
||||
data TmVar : Con → Set₁ where
|
||||
tvzero : TmVar (Γ ▹t)
|
||||
tvnext : TmVar Γ → TmVar (Γ ▹t)
|
||||
tvdisc : TmVar Γ → TmVar (Γ ▹p A)
|
||||
|
||||
data Tm : TmCon → Set₁ where
|
||||
tvar : TmVar Γ → Tm Γ
|
||||
tfun : F n → Array (Tm Γ) n → Tm Γ
|
||||
data Tm : Con → Set₁ where
|
||||
var : TmVar Γ → Tm Γ
|
||||
fun : F n → Array (Tm Γ) n → Tm Γ
|
||||
|
||||
data For : TmCon → Set₁ where
|
||||
data For where
|
||||
rel : R n → Array (Tm Γ) n → For Γ
|
||||
_⇒_ : For Γ → For Γ → For Γ
|
||||
∀∀ : For (Γ ▹t) → For Γ
|
||||
|
||||
data PfCon : TmCon → Set₁ where
|
||||
◇p : PfCon Γ
|
||||
_▹p_ : PfCon Γ → For Γ → PfCon Γ
|
||||
|
||||
variable
|
||||
Ψ : PfCon Γ
|
||||
data PfVar : Con → For Γ → Set₁ where
|
||||
pvzero : {A : For Γ} → PfVar (Γ ▹p A) A
|
||||
pvnext : {A : For Δ} → {B : For Γ} → PfVar Γ A → PfVar (Γ ▹p B) A
|
||||
pvdisc : {A : For Δ} → PfVar Γ A → PfVar (Γ ▹t) A
|
||||
|
||||
data PfVar : PfCon Γ → For Γ → Set₁ where
|
||||
pfzero : {A : For Γ} → PfVar (Ψ ▹p A) A
|
||||
pfnext : {A B : For Γ} → PfVar Ψ A → PfVar (Ψ ▹p B) A
|
||||
data Pf : Con → For Γ → Prop₁ where
|
||||
var : {A : For Δ} → PfVar Γ A → Pf Γ A
|
||||
app : {A B : For Δ} → Pf Γ (A ⇒ B) → Pf Γ A → Pf Γ B
|
||||
lam : {A B : For Γ} → Pf (Γ ▹p A) B → Pf Γ (A ⇒ B)
|
||||
--p∀∀e : {A : For Γ} → Pf Γ (∀∀ A) → Pf Γ (A [ t , id ])
|
||||
--p∀∀i : {A : For (Γ ▹t)} → Pf (Γ [?]) A → Pf Γ (∀∀ A)
|
||||
|
||||
data Pf : PfCon Γ → For Γ → Set₁ where
|
||||
pvar : {A : For Γ} → PfVar Ψ A → Pf Ψ A
|
||||
papp : {A B : For Γ} → Pf Ψ (A ⇒ B) → Pf Ψ A → Pf Ψ B
|
||||
plam : {A B : For Γ} → Pf (Ψ ▹p A) B → Pf Ψ (A ⇒ B)
|
||||
--p∀∀e : {A : For Γ} → Pf Ψ (∀∀ A) → Pf Ψ (A [ t , id ])
|
||||
--p∀∀i : {A : For (Γ ▹t)} → Pf (Ψ [?]) A → Pf Ψ (∀∀ A)
|
||||
data Sub : Con → Con → Set₁ where -- TODO replace with prop
|
||||
ε : Sub Γ ◇
|
||||
wk▹t : Sub Δ Γ → Tm Δ → Sub Δ (Γ ▹t)
|
||||
wk▹p : Sub Δ Γ → Pf Δ A → Sub Δ (Γ ▹p A)
|
||||
|
||||
record Con : Set₁ where
|
||||
constructor _,_
|
||||
field
|
||||
tc : TmCon
|
||||
pc : PfCon tc
|
||||
-- We subst on terms
|
||||
_[_]t : Tm Γ → Sub Δ Γ → Tm Δ
|
||||
_[_]tz : Array (Tm Γ) n → Sub Δ Γ → Array (Tm Δ) n
|
||||
|
||||
imod : FFOL {lsuc lzero} {lzero} {lzero} F R
|
||||
var tvzero [ wk▹t σ t ]t = t
|
||||
var (tvnext tv) [ wk▹t σ x ]t = var tv [ σ ]t
|
||||
var (tvdisc tv) [ wk▹p σ x ]t = var tv [ σ ]t
|
||||
fun f tz [ σ ]t = fun f (tz [ σ ]tz)
|
||||
zero [ σ ]tz = zero
|
||||
next t tz [ σ ]tz = next (t [ σ ]t) (tz [ σ ]tz)
|
||||
|
||||
-- We subst on proofs
|
||||
_[_]p : Pf Γ A → Sub Δ Γ → Pf Δ A
|
||||
_[_]p = {!!}
|
||||
|
||||
-- We subst on formulæ
|
||||
_[_]f : For Γ → Sub Δ Γ → For Δ
|
||||
_[_]f = {!!}
|
||||
|
||||
|
||||
_∘_ : Sub Δ Ξ → Sub Γ Δ → Sub Γ Ξ
|
||||
ε ∘ β = ε
|
||||
wk▹t α t ∘ β = wk▹t (α ∘ β) (t [ β ]t)
|
||||
wk▹p α pf ∘ β = wk▹p (α ∘ β) (pf [ β ]p)
|
||||
|
||||
pgcd : Con → Con → Con
|
||||
pgcd ◇ Δ = ◇
|
||||
pgcd (Γ ▹t) ◇ = ◇
|
||||
pgcd (Γ ▹t) (Δ ▹t) = pgcd Γ Δ
|
||||
pgcd (Γ ▹t) (Δ ▹p x) = pgcd Γ Δ
|
||||
pgcd (Γ ▹p x) ◇ = ◇
|
||||
pgcd (Γ ▹p x) (Δ ▹t) = pgcd Γ Δ
|
||||
pgcd (Γ ▹p x) (Δ ▹p x₁) = pgcd Γ Δ
|
||||
|
||||
|
||||
len : Con → Nat
|
||||
len ◇ = 0
|
||||
len (Γ ▹t) = succ (len Γ)
|
||||
len (Γ ▹p A) = succ (len Γ)
|
||||
|
||||
lift▹tPf : Pf Γ A → Pf (Γ ▹t) A
|
||||
lift▹tPf (var x) = var (pvdisc x)
|
||||
lift▹tPf (app p p₁) = app (lift▹tPf p) (lift▹tPf p₁)
|
||||
lift▹tPf (lam p) = {!!}
|
||||
lift▹t : Sub Γ Δ → Sub (Γ ▹t) Δ
|
||||
lift▹t ε = ε
|
||||
lift▹t (wk▹t σ t) = wk▹t (lift▹t σ) (var tvzero)
|
||||
lift▹t (wk▹p {A = A} σ x) = wk▹p (lift▹t σ) (var (pvdisc {!x!}))
|
||||
|
||||
id : Sub Γ Γ
|
||||
id {◇} = ε
|
||||
id {Γ ▹t} = wk▹t {!!} (var tvzero)
|
||||
id {◇ ▹p A} = wk▹p ε (var pvzero)
|
||||
id {(Γ ▹t) ▹p A} = wk▹p (wk▹t {!!} (var (tvdisc tvzero))) (var pvzero)
|
||||
id {(Γ ▹p x) ▹p A} = wk▹p {!!} (var pvzero)
|
||||
|
||||
|
||||
imod : FFOL {lsuc lzero} {lsuc lzero} {lsuc lzero} {lsuc lzero} F R
|
||||
imod = record
|
||||
{ Con = Con
|
||||
; Sub = {!!}
|
||||
; _∘_ = {!!}
|
||||
; id = {!!}
|
||||
; ◇ = {!!}
|
||||
; ε = {!!}
|
||||
; Tm = {!!}
|
||||
; _[_]t = {!!}
|
||||
; Sub = Sub
|
||||
; _∘_ = _∘_
|
||||
; id = id
|
||||
; ◇ = ◇
|
||||
; ε = ε
|
||||
; Tm = Tm
|
||||
; _[_]t = _[_]t
|
||||
; []t-id = {!!}
|
||||
; []t-∘ = {!!}
|
||||
; fun = {!!}
|
||||
; fun = fun
|
||||
; fun[] = {!!}
|
||||
; _▹ₜ = {!!}
|
||||
; _▹ₜ = _▹t
|
||||
; πₜ¹ = {!!}
|
||||
; πₜ² = {!!}
|
||||
; _,ₜ_ = {!!}
|
||||
; πₜ²∘,ₜ = {!!}
|
||||
; πₜ¹∘,ₜ = {!!}
|
||||
; ,ₜ∘πₜ = {!!}
|
||||
; For = {!!}
|
||||
; For = For
|
||||
; _[_]f = {!!}
|
||||
; []f-id = {!!}
|
||||
; []f-∘ = {!!}
|
||||
; rel = {!!}
|
||||
; rel = rel
|
||||
; rel[] = {!!}
|
||||
; _⊢_ = {!!}
|
||||
; _▹ₚ_ = {!!}
|
||||
; _⊢_ = λ (Γ : Con) (A : For Γ) → Pf Γ A
|
||||
; _▹ₚ_ = _▹p_
|
||||
; πₚ¹ = {!!}
|
||||
; πₚ² = {!!}
|
||||
; _,ₚ_ = {!!}
|
||||
; ,ₚ∘πₚ = {!!}
|
||||
; πₚ¹∘,ₚ = {!!}
|
||||
; _⇒_ = {!!}
|
||||
; _⇒_ = _⇒_
|
||||
; []f-⇒ = {!!}
|
||||
; ∀∀ = {!!}
|
||||
; ∀∀ = ∀∀
|
||||
; []f-∀∀ = {!!}
|
||||
; lam = {!!}
|
||||
; app = {!!}
|
||||
; app = app
|
||||
; ∀i = {!!}
|
||||
; ∀e = {!!}
|
||||
}
|
||||
@ -186,9 +242,10 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
|
||||
f ∘ g = λ x → f (g x)
|
||||
id : {Γ : Con} → Sub Γ Γ
|
||||
id = λ x → x
|
||||
data ◇ : Con where
|
||||
ε : {Γ : Con} → Sub ◇ Γ -- The morphism from the initial to any object
|
||||
ε ()
|
||||
record ◇ : Con where
|
||||
constructor ◇◇
|
||||
ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
|
||||
ε Γ = ◇◇
|
||||
|
||||
-- Functor Con → Set called Tm
|
||||
Tm : Con → Set
|
||||
@ -358,11 +415,12 @@ module FinitaryFirstOrderLogic (F : Nat → Set) (R : Nat → Set) where
|
||||
α ∘ β = λ w γ → α w (β w γ)
|
||||
id : {Γ : Con} → Sub Γ Γ
|
||||
id = λ w γ → γ
|
||||
data ◇⁰ : Set where
|
||||
record ◇⁰ : Set where
|
||||
constructor ◇◇⁰
|
||||
◇ : Con -- The initial object of the category
|
||||
◇ = λ w → ◇⁰
|
||||
ε : {Γ : Con} → Sub ◇ Γ -- The morphism from the initial to any object
|
||||
ε w ()
|
||||
ε : {Γ : Con} → Sub Γ ◇ -- The morphism from the initial to any object
|
||||
ε w Γ = ◇◇⁰
|
||||
|
||||
-- Functor Con → Set called Tm
|
||||
Tm : Con → Set
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user