Trying Presheaf model
This commit is contained in:
parent
91d7fd127d
commit
1d4cda091e
GPG Key ID:
7054D5D6A90F084F
@ -296,6 +296,45 @@ module FFOLCompleteness where
|
||||
; R[] = refl
|
||||
}
|
||||
|
||||
record Presheaf' : Set (lsuc (ℓ¹)) where
|
||||
field
|
||||
World : Set ℓ¹
|
||||
Arr : World → World → Set ℓ¹ -- arrows
|
||||
id-a : {w : World} → Arr w w -- id arrow
|
||||
_∘a_ : {w w' w'' : World} → Arr w w' → Arr w' w'' → Arr w w'' -- arrow composition
|
||||
∘a-ass : {w w' w'' w''' : World}{a : Arr w w'}{b : Arr w' w''}{c : Arr w'' w'''}
|
||||
→ ((a ∘a b) ∘a c) ≡ (a ∘a (b ∘a c))
|
||||
idl-a : {w w' : World} → {a : Arr w w'} → (id-a {w}) ∘a a ≡ a
|
||||
idr-a : {w w' : World} → {a : Arr w w'} → a ∘a (id-a {w'}) ≡ a
|
||||
TM : World → Set ℓ¹
|
||||
TM≤ : {w w' : World} → Arr w w' → TM w' → TM w
|
||||
REL : (w : World) → TM w → TM w → Set ℓ¹
|
||||
REL≤ : {w w' : World} → {t u : TM w'} → (eq : Arr w w') → REL w' t u → REL w (TM≤ eq t) (TM≤ eq u)
|
||||
|
||||
{-
|
||||
-- Now we try to interpret formulæ and contexts
|
||||
import FFOLInitial as I
|
||||
_⊩ᶠ_ : World → (I.For I.◇t) → Set ℓ¹
|
||||
w ⊩ᶠ I.R t u = REL w {!!} {!!}
|
||||
w ⊩ᶠ (A I.⇒ B) = ∀ (w' : World) → Arr w w' → w' ⊩ᶠ A → w' ⊩ᶠ B
|
||||
w ⊩ᶠ I.∀∀ A = ∀ (t : TM w) → {!!}
|
||||
-}
|
||||
|
||||
|
||||
|
||||
record Kripke : Set (lsuc (ℓ¹)) where
|
||||
field
|
||||
World : Set ℓ¹
|
||||
Arr : World → World → Prop ℓ¹ -- arrows
|
||||
id-a : {w : World} → Arr w w -- id arrow
|
||||
_∘a_ : {w w' w'' : World} → Arr w w' → Arr w' w'' → Arr w w'' -- arrow composition
|
||||
-- associativity and id rules are trivial cause arrows are in prop
|
||||
TM : World → Set ℓ¹
|
||||
TM≤ : {w w' : World} → Arr w w' → TM w' → TM w
|
||||
REL : (w : World) → TM w → TM w → Prop ℓ¹
|
||||
REL≤ : {w w' : World} → {t u : TM w'} → (eq : Arr w w') → REL w' t u → REL w (TM≤ eq t) (TM≤ eq u)
|
||||
|
||||
|
||||
-- Completeness proof
|
||||
|
||||
-- We first build our universal Kripke model
|
||||
@ -305,26 +344,67 @@ module FFOLCompleteness where
|
||||
-- We have a model, we construct the Universal Presheaf model of this model
|
||||
import FFOLInitial as I
|
||||
|
||||
UniversalPresheaf : Presheaf
|
||||
UniversalPresheaf : Kripke
|
||||
UniversalPresheaf = record
|
||||
{ World = I.Con
|
||||
; Arr = I.Sub
|
||||
; id-a = I.id
|
||||
; _∘a_ = λ σ σ' → σ' I.∘ σ
|
||||
; ∘a-ass = ≡sym I.∘-ass
|
||||
; idl-a = I.idr
|
||||
; idr-a = I.idl
|
||||
; TM = λ Γ → I.Tm (I.Con.t Γ)
|
||||
; TM≤ = λ σ t → t I.[ I.Sub.t σ ]t
|
||||
; REL = λ Γ t u → I.Pf Γ (I.r t u)
|
||||
; REL≤ = λ σ pf → (pf I.[ I.Sub.t σ ]pₜ) I.[ I.Sub.p σ ]p
|
||||
{ World = (Γₜ : I.Cont) → I.Conp Γₜ
|
||||
; Arr = λ w₁ w₂ → (Γₜ : I.Cont) → (I.Pf* Γₜ (w₂ Γₜ) (w₁ Γₜ))
|
||||
; id-a = λ {w} Γₜ → I.Pf*-id
|
||||
; _∘a_ = λ σ₁ σ₂ Γₜ → I.Pf*-∘ (σ₁ Γₜ) (σ₂ Γₜ)
|
||||
--; ∘a-ass = λ {w} → ≡fun' (λ Γₜ → ≡sym (I.∘ₚ-ass {Γₚ = w Γₜ}))
|
||||
--; idl-a = λ {w} {w'} → ≡fun' (λ Γₜ → I.idrₚ {Γₚ = w Γₜ} {Δₚ = w' Γₜ})
|
||||
--; idr-a = λ {w} {w'} → ≡fun' (λ Γₜ → I.idlₚ {Γₚ = w Γₜ} {Δₚ = w' Γₜ})
|
||||
; TM = λ w → (Γₜ : I.Cont) → (I.Tm Γₜ)
|
||||
; TM≤ = λ σ t → t
|
||||
; REL = λ w t u → (Γₜ : I.Cont) → I.Pf Γₜ (w Γₜ) (I.R (t Γₜ) (u Γₜ))
|
||||
; REL≤ = λ σ pf → λ Γₜ → I.Pf*Pf {!!} (pf Γₜ)
|
||||
}
|
||||
|
||||
-- I.xx are from initial, xx are from up
|
||||
open Presheaf UniversalPresheaf
|
||||
open Kripke UniversalPresheaf
|
||||
|
||||
-- We now create the forcing relation for our Universal presheaf
|
||||
-- We need the world to depend of a term context (i guess), so i think i cannot make it so
|
||||
-- the forcing relation works for every Kripke Model.
|
||||
_⊩f_ : (w : World) → {Γₜ : I.Cont} → I.For Γₜ → Prop₁
|
||||
_⊩f_ w {Γₜ} (I.R t v) = I.Pf Γₜ (w Γₜ) (I.R t v)
|
||||
w ⊩f (A I.⇒ B) = ∀ w' → Arr w w' → w' ⊩f A → w' ⊩f B
|
||||
w ⊩f I.∀∀ A = w ⊩f A
|
||||
|
||||
⊩f-mon : {w w' : World} → Arr w w' → {Γₜ : I.Cont} → {A : I.For Γₜ} → w ⊩f A → w' ⊩f A
|
||||
⊩f-mon s {Γₜ} {I.R t v} wh = I.Pf*Pf (s Γₜ) wh
|
||||
⊩f-mon s {A = A I.⇒ B} wh w'' s' w''h = wh w'' (s ∘a s' ) w''h
|
||||
⊩f-mon s {A = I.∀∀ A} wh = ⊩f-mon s {A = A} wh
|
||||
|
||||
⊩fPf : {Γₜ : I.Cont}{w : World}{A : I.For Γₜ} → w ⊩f A → I.Pf Γₜ (w Γₜ) A
|
||||
⊩fPf {A = I.R t v} pf = pf
|
||||
⊩fPf {A = A I.⇒ A₁} pf = {!I.app!}
|
||||
⊩fPf {A = I.∀∀ A} pf = I.p∀∀i (substP (λ X → I.Pf _ X A) {!!} (⊩fPf pf))
|
||||
|
||||
|
||||
_⊩c_ : (w : World) → {Γₜ : I.Cont} → I.Conp Γₜ → Prop₁
|
||||
w ⊩c I.◇p = ⊤
|
||||
w ⊩c (Γₚ I.▹p⁰ A) = (w ⊩c Γₚ) ∧ (w ⊩f A)
|
||||
|
||||
⊩c-mon : {w w' : World} → Arr w w' → {Γₜ : I.Cont} → {Γₚ : I.Conp Γₜ} → w ⊩c Γₚ → w' ⊩c Γₚ
|
||||
⊩c-mon s {Γₚ = I.◇p} wh = tt
|
||||
⊩c-mon s {Γₜ} {Γₚ = Γₚ I.▹p⁰ A} wh = ⟨ (⊩c-mon s (proj₁ wh)) , ⊩f-mon s {Γₜ} {A} (proj₂ wh) ⟩
|
||||
|
||||
⊩cPf* : {Γₜ : I.Cont}{w : World}{Γₚ : I.Conp Γₜ} → w ⊩c Γₚ → I.Pf* Γₜ (w Γₜ) Γₚ
|
||||
⊩cPf* {Γₚ = I.◇p} h = tt
|
||||
⊩cPf* {Γₚ = Γₚ I.▹p⁰ x} h = ⟨ (⊩cPf* (proj₁ h)) , {!proj₂ h!} ⟩
|
||||
|
||||
_⊫_ : {Γₜ : I.Cont} → (I.Conp Γₜ) → I.For Γₜ → Prop₁
|
||||
Γₚ ⊫ A = ∀ w → w ⊩c Γₚ → w ⊩f A
|
||||
|
||||
-- Now we want to show universality of this model, that is
|
||||
-- if you have a proof in UP, you have the same in I.
|
||||
|
||||
u : {Γₜ : I.Cont}{Γₚ : I.Conp Γₜ}{A : I.For Γₜ} → I.Pf Γₜ Γₚ A → Γₚ ⊫ A
|
||||
q : {Γₜ : I.Cont}{Γₚ : I.Conp Γₜ}{A : I.For Γₜ} → Γₚ ⊫ A → I.Pf Γₜ Γₚ A
|
||||
|
||||
q : {Γ : Con}{A : For Γ} → Γ ⊢ A → I.Pf {!!} {!!}
|
||||
u : {Γ : Con}{A : For Γ} → I.Pf {!!} {!!} → Γ ⊢ A
|
||||
u {A = I.R t s} pf w wh = {!!}
|
||||
u {A = A I.⇒ B} pf w wh w' s w'h = u {A = B} (I.app pf (q λ w'' w''h → {!!})) w' (⊩c-mon s wh)
|
||||
u {A = I.∀∀ A} pf w wh = {!!}
|
||||
q {A = I.R t s} h = {!!}
|
||||
q {A = A I.⇒ B} h = {!!}
|
||||
q {A = I.∀∀ A} h = {!!}
|
||||
|
||||
@ -200,7 +200,7 @@ module FFOLInitial where
|
||||
|
||||
|
||||
-- With those contexts, we have everything to define proofs
|
||||
data PfVar : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Set₁ where
|
||||
data PfVar : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Prop₁ where
|
||||
pvzero : {A : For Γₜ} → PfVar Γₜ (Γₚ ▹p⁰ A) A
|
||||
pvnext : {A B : For Γₜ} → PfVar Γₜ Γₚ A → PfVar Γₜ (Γₚ ▹p⁰ B) A
|
||||
|
||||
@ -389,9 +389,6 @@ module FFOLInitial where
|
||||
{f = λ {W} ξ pf → _,ₚ_ ξ pf}{x = σₚ [ α ∘ₜ β ]σₚ}{y = pf [ α ∘ₜ β ]pₜ})
|
||||
))
|
||||
|
||||
|
||||
|
||||
|
||||
-- How ∘ₚ and ∘ₜ interact to make associativity (to be proven later for Sub)
|
||||
|
||||
∘ₚₜ-ass⁰ :
|
||||
@ -645,3 +642,45 @@ module FFOLInitial where
|
||||
; ∀e = λ {Γ} {F} pf {t} → p∀∀e pf
|
||||
}
|
||||
|
||||
|
||||
-- We define normal and neutral forms
|
||||
data Ne : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Prop₁
|
||||
data Nf : (Γₜ : Cont) → (Γₚ : Conp Γₜ) → For Γₜ → Prop₁
|
||||
data Ne where
|
||||
var : {A : For Γₜ} → PfVar Γₜ Γₚ A → Ne Γₜ Γₚ A
|
||||
app : {A B : For Γₜ} → Ne Γₜ Γₚ (A ⇒ B) → Nf Γₜ Γₚ A → Ne Γₜ Γₚ B
|
||||
p∀∀e : {A : For (Γₜ ▹t⁰)} → {t : Tm Γₜ} → Ne Γₜ Γₚ (∀∀ A) → Ne Γₜ Γₚ (A [ idₜ ,ₜ t ]f)
|
||||
data Nf where
|
||||
R : {t u : Tm Γₜ} → Ne Γₜ Γₚ (R t u) → Nf Γₜ Γₚ (R t u)
|
||||
lam : {A B : For Γₜ} → Nf Γₜ (Γₚ ▹p⁰ A) B → Nf Γₜ Γₚ (A ⇒ B)
|
||||
p∀∀i : {A : For (Γₜ ▹t⁰)} → Nf (Γₜ ▹t⁰) (Γₚ ▹tp) A → Nf Γₜ Γₚ (∀∀ A)
|
||||
|
||||
|
||||
Pf* : (Γₜ : Cont) → Conp Γₜ → Conp Γₜ → Prop₁
|
||||
Pf* Γₜ Γₚ ◇p = ⊤
|
||||
Pf* Γₜ Γₚ (Γₚ' ▹p⁰ A) = (Pf* Γₜ Γₚ Γₚ') ∧ (Pf Γₜ Γₚ A)
|
||||
|
||||
Sub→Pf* : {Γₜ : Cont} {Γₚ Γₚ' : Conp Γₜ} → Subp {Γₜ} Γₚ Γₚ' → Pf* Γₜ Γₚ Γₚ'
|
||||
Sub→Pf* εₚ = tt
|
||||
Sub→Pf* (σₚ ,ₚ pf) = ⟨ (Sub→Pf* σₚ) , pf ⟩
|
||||
Pf*-id : {Γₜ : Cont} {Γₚ : Conp Γₜ} → Pf* Γₜ Γₚ Γₚ
|
||||
Pf*-id = Sub→Pf* idₚ
|
||||
|
||||
Pf*▹p : {Γₜ : Cont}{Γₚ Γₚ' : Conp Γₜ}{A : For Γₜ} → Pf* Γₜ Γₚ Γₚ' → Pf* Γₜ (Γₚ ▹p⁰ A) Γₚ'
|
||||
Pf*▹p {Γₚ' = ◇p} s = tt
|
||||
Pf*▹p {Γₚ' = Γₚ' ▹p⁰ x} s = ⟨ (Pf*▹p (proj₁ s)) , (wkᵣp (rightRen reflRen) (proj₂ s)) ⟩
|
||||
Pf*▹tp : {Γₜ : Cont}{Γₚ Γₚ' : Conp Γₜ} → Pf* Γₜ Γₚ Γₚ' → Pf* (Γₜ ▹t⁰) (Γₚ ▹tp) (Γₚ' ▹tp)
|
||||
Pf*▹tp {Γₚ' = ◇p} s = tt
|
||||
Pf*▹tp {Γₚ' = Γₚ' ▹p⁰ A} s = ⟨ Pf*▹tp (proj₁ s) , (proj₂ s) [ wkₜσₜ idₜ ]pₜ ⟩
|
||||
|
||||
Pf*Pf : {Γₜ : Cont} {Γₚ Γₚ' : Conp Γₜ} {A : For Γₜ} → Pf* Γₜ Γₚ Γₚ' → Pf Γₜ Γₚ' A → Pf Γₜ Γₚ A
|
||||
Pf*Pf s (var pvzero) = proj₂ s
|
||||
Pf*Pf s (var (pvnext pv)) = Pf*Pf (proj₁ s) (var pv)
|
||||
Pf*Pf s (app p p') = app (Pf*Pf s p) (Pf*Pf s p')
|
||||
Pf*Pf s (lam p) = lam (Pf*Pf (⟨ (Pf*▹p s) , (var pvzero) ⟩) p)
|
||||
Pf*Pf s (p∀∀e p) = p∀∀e (Pf*Pf s p)
|
||||
Pf*Pf s (p∀∀i p) = p∀∀i (Pf*Pf (Pf*▹tp s) p)
|
||||
|
||||
Pf*-∘ : {Γₜ : Cont} {Γₚ Δₚ Ξₚ : Conp Γₜ} → Pf* Γₜ Δₚ Ξₚ → Pf* Γₜ Γₚ Δₚ → Pf* Γₜ Γₚ Ξₚ
|
||||
Pf*-∘ {Ξₚ = ◇p} α β = tt
|
||||
Pf*-∘ {Ξₚ = Ξₚ ▹p⁰ A} α β = ⟨ Pf*-∘ (proj₁ α) β , Pf*Pf β (proj₂ α) ⟩
|
||||
|
||||
@ -13,7 +13,7 @@ module PropUtil where
|
||||
-- ⊥ is a data with no constructor
|
||||
-- ⊤ is a record with one always-available constructor
|
||||
data ⊥ : Prop where
|
||||
record ⊤ : Prop where
|
||||
record ⊤ : Prop ℓ where
|
||||
constructor tt
|
||||
|
||||
|
||||
@ -21,7 +21,7 @@ module PropUtil where
|
||||
inj₁ : {P Q : Prop} → P → P ∨ Q
|
||||
inj₂ : {P Q : Prop} → Q → P ∨ Q
|
||||
|
||||
record _∧_ (P Q : Prop) : Prop where
|
||||
record _∧_ (P Q : Prop ℓ) : Prop ℓ where
|
||||
constructor ⟨_,_⟩
|
||||
field
|
||||
p : P
|
||||
@ -31,9 +31,9 @@ module PropUtil where
|
||||
infixr 11 _∨_
|
||||
|
||||
-- ∧ elimination
|
||||
proj₁ : {P Q : Prop} → P ∧ Q → P
|
||||
proj₁ : {P Q : Prop ℓ} → P ∧ Q → P
|
||||
proj₁ pq = _∧_.p pq
|
||||
proj₂ : {P Q : Prop} → P ∧ Q → Q
|
||||
proj₂ : {P Q : Prop ℓ} → P ∧ Q → Q
|
||||
proj₂ pq = _∧_.q pq
|
||||
|
||||
-- ∨ elimination
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user