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Conor lecture experiments
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Smaug123
276
Experiments.agda
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276
Experiments.agda
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{-# OPTIONS --safe --warning=error #-}
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open import LogicalFormulae
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals
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open import Lists
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open import Vectors
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module Experiments where
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record One : Set where
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ListF : (A : Set) (T : Set) → Set
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ListF A T = One || A && T
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Choose : {I J : Set} -> (I -> Set) -> (J -> Set) -> (I || J) -> Set
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Choose X Y (inl i) = X i
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Choose X Y (inr j) = Y j
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data Poly (X : Set) : Set where
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var' : X -> Poly X
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konst' : Bool -> Poly X
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_+'_ _*'_ : Poly X -> Poly X -> Poly X
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Eval : {X : Set} -> (X -> Set) -> Poly X -> Set
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Eval elems (var' x) = elems x
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Eval elems (konst' BoolTrue) = One
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Eval elems (konst' BoolFalse) = False'
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Eval elems (poly +' poly1) = Eval elems poly || Eval elems poly1
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Eval elems (poly *' poly1) = Eval elems poly && Eval elems poly1
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data MU {elements nodes : Set} (nodeStructure : nodes → Poly (elements || nodes)) (elems : elements → Set) (topNode : nodes) : Set where
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<_> : Eval (Choose elems (MU nodeStructure elems)) (nodeStructure topNode) → MU nodeStructure elems topNode
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polyMap : {X Y : Set} → (f : X → Y) → Poly X → Poly Y
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polyMap f (var' x) = var' (f x)
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polyMap f (konst' x) = konst' x
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polyMap f (p +' q) = polyMap f p +' polyMap f q
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polyMap f (p *' q) = polyMap f p *' polyMap f q
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-- Naturals
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natF : Poly One
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natF = konst' BoolTrue +' var' (record {})
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natF' : Poly (False || One)
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natF' = polyMap (λ _ → inr (record {})) natF
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nats : Set
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nats = MU {False} {One} (λ _ → natF') (λ ()) (record {})
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f : nats → ℕ
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f < inl record {} > = 0
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f < inr x > = succ (f x)
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g : ℕ → nats
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g zero = < inl (record {}) >
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g (succ n) = < inr (g n) >
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fgId : {n : ℕ} → f (g n) ≡ n
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fgId {zero} = refl
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fgId {succ n} rewrite fgId {n} = refl
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gfId : {n : nats} → g (f n) ≡ n
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gfId {< inl record {} >} = refl
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gfId {< inr x >} rewrite gfId {x} = refl
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-- Lists
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listF : Poly (One || One)
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listF = konst' BoolTrue +' (var' (inl (record {})) *' var' (inr record {}))
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lists : (A : Set) → Set
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lists A = MU {One} {One} (λ _ → listF) (λ _ → A) record {}
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h : {A : Set} → lists A → List A
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h {A} < inl record {} > = []
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h {A} < inr (fst ,, snd) > = fst :: h snd
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i : {A : Set} → List A → lists A
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i {A} [] = < inl record {} >
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i {A} (x :: l) = < inr (x ,, i l) >
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hiId : {A : Set} → (r : List A) → h (i r) ≡ r
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hiId [] = refl
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hiId (x :: r) rewrite hiId r = refl
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ihId : {A : Set} → (r : lists A) → i (h r) ≡ r
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ihId < inl record {} > = refl
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ihId < inr (fst ,, snd) > rewrite ihId snd = refl
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-- Vectors
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vecF : ℕ → Poly (One || ℕ)
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vecF zero = konst' BoolTrue
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vecF (succ n) = var' (inl (record {})) *' var' (inr n)
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vecs : (A : Set) (n : ℕ) → Set
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vecs A n = MU vecF (λ _ → A) n
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j : {A : Set} → {n : ℕ} → vecs A n → Vec A n
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j {A} {zero} < record {} > = []
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j {A} {succ n} < fst ,, snd > = fst ,- j snd
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k : {A : Set} → {n : ℕ} → Vec A n → vecs A n
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k {A} {.0} [] = < record {} >
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k {A} {.(succ _)} (x ,- v) = < x ,, k v >
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jkId : {A : Set} {n : ℕ} → {v : Vec A n} → j (k v) ≡ v
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jkId {A} {zero} {[]} = refl
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jkId {A} {succ n} {x ,- v} rewrite jkId {v = v} = refl
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kjId : {A : Set} {n : ℕ} {v : vecs A n} → k (j v) ≡ v
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kjId {n = zero} {< record {} >} = refl
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kjId {n = succ n} {< fst ,, snd >} rewrite kjId {v = snd} = refl
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-- Vectors where even-numbered places are pairs
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evenQ : ℕ → Bool
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evenQ 0 = BoolTrue
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evenQ (succ n) = not (evenQ n)
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data VecWeird {a : _} (A : Set a) : ℕ → Set a where
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empty : VecWeird A 0
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consEven : {n : ℕ} → VecWeird A n → (evenQ n ≡ BoolFalse) → (A && A) → (VecWeird A (succ n))
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consOdd : {n : ℕ} → VecWeird A n → (evenQ n ≡ BoolTrue) → A → (VecWeird A (succ n))
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example : VecWeird ℕ 4
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example = consEven {n = 3} (consOdd (consEven (consOdd empty refl 4) refl (3 ,, 3)) refl 2) refl (1 ,, 1)
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vecWeirdF : (ℕ && Bool) → Poly (One || (ℕ && Bool))
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vecWeirdF (zero ,, BoolTrue) = konst' BoolTrue
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vecWeirdF (zero ,, BoolFalse) = konst' BoolFalse
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vecWeirdF ((succ n) ,, BoolTrue) = (var' (inl (record {})) *' var' (inl (record {}))) *' var' (inr (n ,, BoolFalse))
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vecWeirdF ((succ n) ,, BoolFalse) = var' (inl (record {})) *' var' (inr (n ,, BoolTrue))
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vecWeirds : (A : Set) (n : ℕ) → Set
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vecWeirds A n = MU vecWeirdF (λ _ → A) (n ,, evenQ n)
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vecWeirdsEq : {A : Set} {n : ℕ} → VecWeird A n → vecWeirds A n
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vecWeirdsEq empty = < record {} >
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vecWeirdsEq {n = .succ n} (consEven v pr (x ,, y)) with inspect (evenQ n)
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vecWeirdsEq {_} {.(succ _)} (consEven v s (x ,, y)) | BoolTrue with≡ p rewrite p = exFalso (q s)
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where
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q : (BoolTrue ≡ BoolFalse) → False
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q ()
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vecWeirdsEq {A} {succ n} (consEven v _ (x ,, y)) | BoolFalse with≡ p rewrite p = < (x ,, y) ,, typeCast (vecWeirdsEq v) (applyEquality (MU vecWeirdF (λ _ → A)) b) >
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where
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b : (n ,, evenQ n) ≡ (n ,, BoolFalse)
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b rewrite p = refl
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vecWeirdsEq {_} {succ n} (consOdd v pr x) with inspect (evenQ n)
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vecWeirdsEq {A} {succ n} (consOdd v pr x) | BoolTrue with≡ _ rewrite pr = < x ,, typeCast (vecWeirdsEq v) (applyEquality (MU vecWeirdF (λ _ → A)) b) >
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where
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b : (n ,, evenQ n) ≡ (n ,, BoolTrue)
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b rewrite pr = refl
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vecWeirdsEq {_} {succ n} (consOdd v pr x) | BoolFalse with≡ q rewrite q = exFalso (r pr)
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where
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r : (BoolFalse ≡ BoolTrue) → False
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r ()
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vecWeirdsEq' : {A : Set} {n : ℕ} → vecWeirds A n → VecWeird A n
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vecWeirdsEq' {n = zero} < x > = empty
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vecWeirdsEq' {n = succ n} < x > with inspect (evenQ n)
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vecWeirdsEq' {A} {succ n} < x > | BoolTrue with≡ p = ans
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where
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y : Eval (Choose (λ _ → A) (MU vecWeirdF (λ _ → A))) (vecWeirdF (succ n ,, BoolFalse))
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y = typeCast x (applyEquality (λ i → Eval (Choose (λ _ → A) (MU vecWeirdF (λ _ → A))) (vecWeirdF (succ n ,, not i))) p)
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ans : VecWeird A (succ n)
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ans with y
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ans | fst ,, snd = consOdd (vecWeirdsEq' insert) p fst
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where
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insert : vecWeirds A n
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insert = typeCast snd (applyEquality (λ i → MU vecWeirdF (λ _ → A) (n ,, i)) (equalityCommutative p))
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vecWeirdsEq' {A} {succ n} < x > | BoolFalse with≡ p = ans
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where
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y : Eval (Choose (λ _ → A) (MU vecWeirdF (λ _ → A))) (vecWeirdF (succ n ,, BoolTrue))
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y = typeCast x (applyEquality (λ i → Eval (Choose (λ _ → A) (MU vecWeirdF (λ _ → A))) (vecWeirdF (succ n ,, not i))) p)
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ans : VecWeird A (succ n)
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ans with y
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ans | fst ,, snd = consEven (vecWeirdsEq' insert) p fst
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where
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insert : vecWeirds A n
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insert = typeCast snd (applyEquality (λ i → MU vecWeirdF (λ _ → A) (n ,, i)) (equalityCommutative p))
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vecWeirdsEquiv : {A : Set} {n : ℕ} {v : vecWeirds A n} → vecWeirdsEq (vecWeirdsEq' v) ≡ v
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vecWeirdsEquiv {A} {zero} {< record {} >} = refl
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vecWeirdsEquiv {A} {succ n} {< x >} with inspect (evenQ n)
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vecWeirdsEquiv {A} {succ n} {< x >} | BoolTrue with≡ pr = {!!}
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vecWeirdsEquiv {A} {succ n} {< x >} | BoolFalse with≡ pr = {!!}
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vecWeirdsEquiv' : {A : Set} {n : ℕ} {v : VecWeird A n} → vecWeirdsEq' (vecWeirdsEq v) ≡ v
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vecWeirdsEquiv' {A} {zero} {empty} = refl
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vecWeirdsEquiv' {A} {succ n} {consEven v x pr} with inspect (evenQ n)
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vecWeirdsEquiv' {A} {succ n} {consEven v x pr} | BoolTrue with≡ pr' = exFalso (fa x)
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where
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fa : (evenQ n ≡ BoolFalse) → False
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fa a rewrite pr' = b a
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where
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b : BoolTrue ≡ BoolFalse → False
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b ()
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vecWeirdsEquiv' {A} {succ n} {consEven v pr (fst ,, snd)} | BoolFalse with≡ pr' with vecWeirdsEquiv' {A} {n} {v}
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... | bl = {!!}
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vecWeirdsEquiv' {A} {succ n} {consOdd v x pr} = {!!}
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-- Trees
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treeF : Poly (One || One)
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treeF = konst' BoolTrue +' (var' (inl (record {})) *' (var' (inr record {}) *' var' (inr record {})))
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trees : (A : Set) → Set
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trees A = MU (λ _ → treeF) (λ _ → A) record {}
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data Tree {a : _} (A : Set a) : Set a where
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emptyTree : Tree A
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branch : Tree A → Tree A → A → Tree A
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treeEq : {A : Set} → trees A → Tree A
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treeEq < inl x > = emptyTree
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treeEq < inr (x ,, (branch1 ,, branch2)) > = branch (treeEq branch1) (treeEq branch2) x
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treeEq' : {A : Set} → Tree A → trees A
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treeEq' emptyTree = < inl (record {}) >
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treeEq' (branch left right x) = < inr (x ,, (treeEq' left ,, treeEq' right)) >
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treeEqId : {A : Set} → {t : Tree A} → treeEq (treeEq' t) ≡ t
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treeEqId {t = emptyTree} = refl
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treeEqId {t = branch left right x} rewrite treeEqId {t = left} | treeEqId {t = right} = refl
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treeEqId' : {A : Set} → {t : trees A} → treeEq' (treeEq t) ≡ t
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treeEqId' {t = < inl record {} >} = refl
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treeEqId' {t = < inr (a ,, (left ,, right)) >} rewrite treeEqId' {t = left} | treeEqId' {t = right} = refl
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-- Expressions
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data Expr : (A : Set) → Set where
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constant : (A : Set) → A → Expr A
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plus : (a b : Expr ℕ) → Expr ℕ
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negate : (a : Expr Bool) → Expr Bool
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data ExprType : Set where
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nat : ExprType
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bool : ExprType
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a : ExprType
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data NodeType : Set where
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const' : NodeType
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plus' : NodeType
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negate' : NodeType
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exprF : NodeType → Poly (ExprType || NodeType)
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exprF const' = var' (inl a)
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exprF plus' = var' (inr {!!}) *' var' (inr {!!})
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exprF negate' = var' (inr {!!})
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exprShapes : (A : Set) → ExprType → Set
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exprShapes A nat = A
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exprShapes _ bool = ℕ && ℕ
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exprShapes _ a = Bool
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exprs : (A : Set) → Set
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exprs A = MU {ExprType} {NodeType} exprF (exprShapes A) const'
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exprsEq : {A : Set} → Expr A → exprs A
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exprsEq (constant A x) = < {!!} >
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exprsEq (plus e1 e2) with exprsEq e1
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exprsEq (plus e1 e2) | a1 with exprsEq e2
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exprsEq (plus e1 e2) | < x > | < y > = < {!!} >
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exprsEq (negate e) with exprsEq e
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... | < toNegate > = < not toNegate >
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exprsEq' : {A : Set} → exprs A → Expr A
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exprsEq' < x > = {!!}
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exprsEqId : {A : Set} {e : Expr A} → exprsEq' (exprsEq e) ≡ e
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exprsEqId {e = constant A x} = {!!}
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exprsEqId {e = plus x y} with exprsEq x
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exprsEqId {_} {plus x y} | < x1 > with exprsEq y
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exprsEqId {_} {plus x y} | < x1 > | < y1 > = {!!}
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exprsEqId {e = negate e} = {!!}
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exprsEqId' : {A : Set} {e : exprs A} → exprsEq (exprsEq' e) ≡ e
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exprsEqId' {e = e} = {!!}
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