125 lines
3.5 KiB
Agda
125 lines
3.5 KiB
Agda
data _≡_ {A : Set} : A → A → Set where
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refl : {a : A} → a ≡ a
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context : {A B : Set} → {a b : A} → (f : A → B) → a ≡ b → f a ≡ f b
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context f refl = refl
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trans : {A : Set} → {a b c : A} → a ≡ b → b ≡ c → a ≡ c
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trans refl refl = refl
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data ⊥ : Set where
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¬ : Set → Set
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¬ A = A → ⊥
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_≢_ : {A : Set} → A -> A -> Set
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_≢_ a b = ¬ (a ≡ b)
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data _∧_ (A B : Set) : Set where
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_∧ᵢ_ : A → B → A ∧ B
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infixr 19 _∧_
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infixr 19 _∧ᵢ_
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data ℕ : Set where
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zero : ℕ
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succ : ℕ → ℕ
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{-# BUILTIN NATURAL ℕ #-}
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_+_ : ℕ → ℕ → ℕ
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n + 0 = n
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n + succ m = succ (n + m)
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data _<_ : ℕ → ℕ → Set where
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z<sn : (n : ℕ) → 0 < succ n
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s<s : (n m : ℕ) → n < m → succ n < succ m
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data _≤_ : ℕ → ℕ → Set where
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z≤n : ∀ (n : ℕ) -> 0 ≤ n
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s≤s : ∀ (n m : ℕ) -> n ≤ m -> succ n ≤ succ m
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-- Task 2
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data ℤ : Set where
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pos : ℕ → ℤ
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negsucc : ℕ → ℤ -- -n - 1 alternatively - (n + 1)
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recN : {A : Set} → ℕ → A → (ℕ → A → A) → A
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recN 0 d e = d
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recN (succ x) d e = e x (recN x d e)
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addℕ : ℕ → ℕ → ℕ
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addℕ n m = recN m n (λ _ x → succ x)
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prfadd : (n m : ℕ) → (n + m) ≡ (addℕ n m)
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prfadd n 0 = refl
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prfadd n (succ m) = context succ (prfadd n m)
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-- Task 3
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zrec : {A : Set} → ℤ → (ℕ → A) → (ℕ → A) → A
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zrec (pos n) f _ = f n
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zrec (negsucc n) _ g = g n
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addOne : ℤ → ℤ
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addOne a = zrec a (λ x → pos (succ x)) (λ x → recN x (pos x) (λ n _ → negsucc n))
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subOne : ℤ → ℤ
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subOne a = zrec a (λ x → recN x (negsucc x) (λ n _ → pos n)) (λ x → negsucc (succ x))
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prfℤ₁ : (z : ℤ) → subOne (addOne z) ≡ z
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prfℤ₁ (pos x) = refl
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prfℤ₁ (negsucc 0) = refl
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prfℤ₁ (negsucc (succ x)) = refl
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prfℤ₂ : (z : ℤ) → addOne (subOne z) ≡ z
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prfℤ₂ (pos 0) = refl
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prfℤ₂ (pos (succ x)) = refl
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prfℤ₂ (negsucc x) = refl
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-- Task 4
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data Vec (A : Set) : ℕ → Set where
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nil : Vec A 0
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_,_ : {n : ℕ} → Vec A n → (a : A) → Vec A (succ n)
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infixl 20 _,_
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index : {A : Set} {n : ℕ} → Vec A n → (i : ℕ) → (0 ≤ i ∧ i < n) → A
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index (xs , a) 0 (z≤n n ∧ᵢ z<sn n₁) = a
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index (xs , a) (succ i) (z≤n n ∧ᵢ s<s i m i<m) = index xs i (z≤n i ∧ᵢ i<m)
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data PRF : ℕ → Set where
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zero : PRF 0
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succ : PRF 1
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proj : {n : ℕ} → (i : ℕ) → 0 ≤ i ∧ i < n → PRF n
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comp : {n m : ℕ} → (f : PRF m) → (gs : Vec (PRF n) m) → PRF n
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rec : {n : ℕ} → (f : PRF n) → (g : PRF (succ (succ n))) → PRF (succ n)
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⟦_⟧ : {n : ℕ} → PRF n → (Vec ℕ n) → ℕ
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⟦ zero ⟧ nil = 0
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⟦ succ ⟧ (nil , n) = succ n
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⟦ proj i prf ⟧ ρ = index ρ i prf
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⟦ rec f g ⟧ (ρ , 0) = ⟦ f ⟧ ρ
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⟦ rec f g ⟧ (ρ , succ n) = ⟦ g ⟧ ((ρ , n , ⟦ rec f g ⟧ (ρ , n)))
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⟦ comp f gs ⟧ ρ = ⟦ f ⟧ (⟦ gs ⟧* ρ)
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where
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⟦_⟧* : {n m : ℕ} → Vec (PRF m) n → (Vec ℕ m) → Vec ℕ n
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⟦ nil ⟧* ρ = nil
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⟦ fs , f ⟧* ρ = (⟦ fs ⟧* ρ , ⟦ f ⟧ ρ)
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addProg : PRF 2
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addProg =
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rec
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(proj 0 (z≤n 0 ∧ᵢ z<sn 0))
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(comp succ (nil , proj 0 (z≤n 0 ∧ᵢ z<sn 2)))
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add : ℕ → ℕ → ℕ
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add n m = ⟦ addProg ⟧ (nil , n , m)
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prfProg : (n m : ℕ) → (add n m) ≡ (n + m)
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prfProg n 0 = refl
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prfProg n (succ m) = context succ (prfProg n m)
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prfProg2 : (n m : ℕ) → (add n m) ≡ (addℕ n m)
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prfProg2 n m = trans (prfProg n m) (prfadd n m)
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