110 lines
2.2 KiB
Agda
110 lines
2.2 KiB
Agda
open import Data.Nat
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open import Relation.Binary.PropositionalEquality
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variable m n : 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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data Weak : ℕ → Set where
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wk : Weak (suc n)
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suc : Weak n → Weak (suc n)
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data Subst : ℕ → Set where
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<_> : Term n → Subst n
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suc : Subst n → Subst (suc n)
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_[_]t : Term n → Weak n → Term (suc n)
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t [ wk ]t = suc t
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zero [ suc w ]t = zero
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suc t [ suc w ]t = suc (t [ w ]t)
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{-
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0 -> 0
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1 -> 2
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2 -> 3
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suc wk
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wk
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0 -> 1
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1 -> 2
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2 -> 3
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-}
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_s[_]t : Term (suc n) → Subst n → Term n
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zero s[ < u > ]t = u
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suc t s[ < u > ]t = t
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zero s[ suc s ]t = zero
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suc t s[ suc s ]t = suc (t s[ s ]t)
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{-
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x,y,z --> x,z
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0,1,2 y => x,z ⊢ t
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x => x 0 => 0
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y => t 1 => t
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z => z 2 => 1
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-}
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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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_[_]F : Form n → Weak n → Form (suc n)
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(A ⇒ B) [ w ]F = (A [ w ]F) ⇒ (B [ w ]F)
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∀F A [ w ]F = ∀F (A [ suc w ]F)
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P a [ w ]F = P (a [ w ]t )
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_s[_]F : Form (suc n) → Subst n → Form n
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(A ⇒ B) s[ s ]F = (A s[ s ]F) ⇒ (B s[ s ]F)
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∀F A s[ s ]F = ∀F (A s[ suc s ]F)
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P a s[ s ]F = P (a s[ s ]t )
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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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_[_]C : Con n → Weak n → Con (suc n)
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• [ w ]C = •
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(Γ ▷ A) [ w ]C = (Γ [ w ]C) ▷ (A [ w ]F)
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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 : Γ [ wk ]C ⊢ A → Γ ⊢ ∀F A
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App : Γ ⊢ ∀F A → (t : Term _) → Γ ⊢ A s[ < t > ]F
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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 : (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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