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