125 lines
4.4 KiB
Agda
125 lines
4.4 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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_$_ : {A B : Set} → (A → B) → A → B
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f $ x = f x
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infixr 1 _$_
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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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-- A ≡ A [ wk ][ < t > ]
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wk-substt : {t : Term (l + n)} → subst-t l (wk-t l t) u ≡ t
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wk-substt {zero} = refl
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wk-substt {suc l} {t = zero} = refl
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wk-substt {suc l} {n} {t = suc t} = cong (λ t → suc t) (wk-substt {l})
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wk-substf : {A : Form (l + n)} → subst-F l (wk-F l A) u ≡ A
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wk-substf {A = A ⇒ A₁} = cong₂ (λ B B₁ → B ⇒ B₁) wk-substf wk-substf
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wk-substf {A = ∀F A} = cong (λ B → ∀F B) wk-substf
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wk-substf {l = l} {A = P x} = cong (λ t → P t) (wk-substt {l})
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-- (A ⇒ ∀ x . P x) ⇒ ∀ x . A → P x
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example : • ⊢ (A ⇒ (∀F (P zero))) ⇒ (∀F (wk-F 0 A) ⇒ P zero)
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example {A = A} = lam (lam (App (app (suc zero) (subst (λ Φ → (• ▷ A ⇒ ∀F (P zero)) ▷ ∀F (wk-F 0 A) ⊢ Φ) wk-substf (App zero zero))) zero))
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-- (∀ x ∀ y . A(x,y)) ⇒ ∀ y ∀ x . A(y,x)
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ex1 : {A : Form 2} → • ⊢ (∀F (∀F A)) ⇒ (∀F (∀F A))
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ex1 = lam zero
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-- (A ⇒ ∀ x . B(x)) ⇒ ∀ x . A ⇒ B(x)
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-- y → ∀x B(x) ==> y → ∀ x B(y)
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eq' : {l n : ℕ} → (l + suc n) ≡ (suc (l + n))
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eq : {l n : ℕ} → (1 + (l + (suc n))) ≡ (suc (suc (l + n)))
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--eq {zero} {zero} = refl
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--eq {zero} {suc n} = cong suc (eq {l = 0})
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--eq {suc l} = cong suc (eq {l = l})
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lm-t : {l n : ℕ} → {t : Term (l + suc n)} → subst-t {n = suc n} l (subst Term (sym eq) (wk-t (suc l) (subst Term eq' t)) ) zero ≡ t
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lm-F : {l n : ℕ} → {A : Form (l + suc n)} → subst-F {n = suc n} l (subst Form (sym eq) (wk-F (suc l) (subst Form eq' A)) ) zero ≡ A
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lm-t {t = t} = {!!}
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lm-F = {!!}
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{-
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lm-F : {A : Form 1} → subst-F 0 (wk-F 1 A) zero ≡ A
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lm-F {A ⇒ A₁} = cong₂ (λ B B₁ → B ⇒ B₁) lm-F lm-F
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lm-F {∀F A} = cong (λ B → ∀F B) {!lm-F!}
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lm-F {P x} = cong (λ t → P t) lm-t
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-}
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ex2 : {A : Form 0} → {B : Form 1} → • ⊢ ((A ⇒ (∀F B)))⇒(∀F ((wk-F 0 A) ⇒ B))
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ex2 {A = A} {B = B} = lam $ Lam $ lam $ subst (λ C → (• ▷ wk-F zero A ⇒ ∀F (wk-F 1 B)) ▷ wk-F 0 A ⊢ C) lm-F (((App (app (suc zero) zero) zero)))
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-- lam (Lam (lam ( subst (λ Φ → (• ▷ wk-F zero A ⇒ ∀F (wk-F 1 B)) ▷ wk-F 0 A ⊢ Φ) (wk-substf {l = 0}) (App (app (suc (suc lm)) (app (suc zero) zero)) zero))))
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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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-}
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