139 lines
3.6 KiB
Agda
139 lines
3.6 KiB
Agda
open import Data.Nat
|
||
open import Relation.Binary.PropositionalEquality
|
||
|
||
variable m n l : ℕ
|
||
|
||
data Term : ℕ → Set where
|
||
zero : Term (suc n)
|
||
suc : Term n → Term (suc n)
|
||
|
||
variable t u : Term n
|
||
|
||
data Subst : ℕ → ℕ → Set where
|
||
ε : Subst n 0
|
||
_,_ : Subst m n → Term m → Subst m (suc n)
|
||
|
||
suc-subst : Subst m n → Subst (suc m) n
|
||
suc-subst ε = ε
|
||
suc-subst (ts , t) = suc-subst ts , suc t
|
||
|
||
id : Subst m m
|
||
id {zero} = ε
|
||
id {suc m} = (suc-subst (id {m})) , zero
|
||
|
||
subst-t : Term m → Subst n m → Term n
|
||
subst-t zero (us , u) = u
|
||
subst-t (suc t) (us , u) = subst-t t us
|
||
|
||
wk-t1 : Term n → Term (suc n)
|
||
wk-t1 t = subst-t t (suc-subst id)
|
||
|
||
subst-t1 : Term (suc n) → Term n → Term n
|
||
subst-t1 t u = subst-t t (id , u)
|
||
|
||
comp : Subst m l → Subst n m → Subst n l
|
||
comp ε us = ε
|
||
comp (ts , u) us = (comp ts us) , (subst-t u us)
|
||
|
||
{-
|
||
t [ suc-subst vs ] ≡ suc (t [vs ])
|
||
suc-subst ts ∘ (us , t) ≡ ts ∘ us
|
||
|
||
t [ id ] ≡ t
|
||
t [ us ∘ vs ] ≡ t [ us ] [ vs ]
|
||
|
||
ts ∘ id ≡ ts
|
||
id ∘ ts ≡ ts
|
||
(ts ∘ us) ∘ vs ≡ ts ∘ (us ∘ vs)
|
||
|
||
-}
|
||
|
||
|
||
{-
|
||
wk-t : (l : ℕ) → Term (l + n) → Term (suc (l + n))
|
||
wk-t zero t = suc t
|
||
wk-t (suc l) zero = zero
|
||
wk-t (suc l) (suc t) = suc (wk-t l t)
|
||
|
||
subst-t : (l : ℕ) → Term (suc (l + n)) → Term n → Term (l + n)
|
||
subst-t zero zero u = u
|
||
subst-t zero (suc t) u = t
|
||
subst-t (suc l) zero u = zero
|
||
subst-t (suc l) (suc t) u = suc (subst-t l t u)
|
||
|
||
infix 15 _⇒_
|
||
|
||
data Form : ℕ → Set where
|
||
_⇒_ : Form n → Form n → Form n
|
||
∀F : Form (suc n) → Form n
|
||
P : Term n → Form n
|
||
-- R : Term n → Term n → Form n
|
||
|
||
wk-F : (l : ℕ) → Form (l + n) → Form (suc (l + n))
|
||
wk-F l (A ⇒ B) = wk-F l A ⇒ wk-F l B
|
||
wk-F l (∀F A) = ∀F (wk-F (suc l) A)
|
||
wk-F l (P t) = P (wk-t l t)
|
||
|
||
subst-F : (l : ℕ) → Form (suc (l + n)) → Term n → Form (l + n)
|
||
subst-F l (A ⇒ B) u = subst-F l A u ⇒ subst-F l B u
|
||
subst-F l (∀F A) u = ∀F (subst-F (suc l) A u)
|
||
subst-F l (P t) u = P (subst-t l t u)
|
||
|
||
infix 10 _▷_
|
||
|
||
data Con : ℕ → Set where
|
||
• : Con n
|
||
_▷_ : Con n → Form n → Con n
|
||
|
||
wk-C : (l : ℕ) → Con (l + n) → Con (suc (l + n))
|
||
wk-C l • = •
|
||
wk-C l (Γ ▷ A) = wk-C l Γ ▷ wk-F l A
|
||
|
||
variable Γ Δ : Con n
|
||
variable A B C : Form n
|
||
|
||
infix 5 _⊢_
|
||
|
||
data _⊢_ : Con n → Form n → Set where
|
||
zero : Γ ▷ A ⊢ A
|
||
suc : Γ ⊢ A → Γ ▷ B ⊢ A
|
||
lam : Γ ▷ A ⊢ B → Γ ⊢ A ⇒ B
|
||
app : Γ ⊢ A ⇒ B → Γ ⊢ A → Γ ⊢ B
|
||
Lam : (wk-C zero Γ) ⊢ A → Γ ⊢ ∀F A
|
||
App : Γ ⊢ ∀F A → (t : Term _) → Γ ⊢ subst-F zero A t
|
||
|
||
{-
|
||
-- (A ⇒ ∀ x . P x) ⇒ ∀ x . A → P x
|
||
|
||
-- A ≡ A [ wk ][ < t > ]
|
||
|
||
wk-subst : subst l (wk l A) t ≡ t
|
||
|
||
wk-subst : (A [ wk ]F) s[ < t > ]F ≡ A
|
||
wk-subst = {!!}
|
||
|
||
example : • ⊢ (A ⇒ (∀F (P zero))) ⇒ (∀F (A [ wk ]F) ⇒ P zero)
|
||
example {A = A} = lam (lam (App (app (suc zero)
|
||
(subst (λ X → (• ▷ A ⇒ ∀F (P zero)) ▷ ∀F (A [ wk ]F) ⊢ X)
|
||
(wk-subst {A = A}) (App zero zero))) zero))
|
||
|
||
|
||
-- (∀ x ∀ y . A(x,y)) ⇒ ∀ y ∀ x . A(y,x)
|
||
-- (A ⇒ ∀ x . B(x)) ⇒ ∀ x . A ⇒ B(x)
|
||
-- ∀ x y . A(x,y) ⇒ ∀ x . A(x,x)
|
||
-- ∀ x . A (x) ⇒ ∀ x y . A(x)
|
||
-- (((∀ x . A (x)) ⇒ B)⇒ B) ⇒ ∀ x . ((A (x) ⇒ B) ⇒ B)
|
||
|
||
-}
|
||
|
||
--eqq : (suc (suc (l + n))) ≡ suc l + suc n
|
||
eqq : (suc (l + n)) ≡ l + suc n
|
||
eqq = {!!}
|
||
|
||
eq : ∀ {t : Term (suc (l + n))}{u : Term n} →
|
||
-- subst-t {n = suc n} l (subst Term (eqq {l = l}{n = n})(wk-t {n = n} (suc l) t)) (wk-t zero u)
|
||
subst-t {n = suc n} l (subst Term (eqq {l = (suc l)}{n = n})(wk-t {n = n} (suc l) t)) (wk-t zero u)
|
||
≡ {! subst-t l t !} -- subst-t l (wk-t (suc l) t) u ≡ subst-t l t u
|
||
eq = {!!}
|
||
-}
|