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The insertTT has a problem with the implicit {n : ℕ}
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module MergeSort where | |
open import Function | |
open import Data.Nat | |
open import Data.Product | |
open import Relation.Binary.PropositionalEquality | |
data Order : Set where | |
le : Order | |
ge : Order | |
data List (X : Set) : Set where | |
nil : List X | |
_∷_ : (x : X) -> (xs : List X) -> List X | |
order : (x : ℕ) (y : ℕ) -> Order | |
order zero y = le | |
order (suc n) zero = ge | |
order (suc n) (suc n') = order n n' | |
merge : List ℕ -> List ℕ -> List ℕ | |
merge nil ys = ys | |
merge xs nil = xs | |
merge (x ∷ xs') (y ∷ ys') with order x y | merge xs' (y ∷ ys') | merge (x ∷ xs') ys' | |
merge (x ∷ xs') (y ∷ ys') | le | m1 | m2 = x ∷ m1 | |
merge (x ∷ xs') (y ∷ ys') | ge | m1 | m2 = y ∷ m2 | |
deal : {X : Set} (xs : List X) -> List X × List X | |
deal nil = nil , nil | |
deal (y ∷ nil) = y ∷ nil , nil | |
deal (y ∷ (y' ∷ y0)) with deal y0 | |
deal (y' ∷ (y1 ∷ y0)) | proj₁ , y = y' ∷ proj₁ , y1 ∷ y | |
data Parity : Set where | |
p₀ : Parity | |
p₁ : Parity | |
data DealT (X : Set) : Set where | |
empT : DealT X | |
leafT : (x : X) -> DealT X | |
nodeT : (p : Parity) -> (l : DealT X) -> (r : DealT X) -> DealT X | |
insertT : {X : Set} (x : X) -> (t : DealT X) -> DealT X | |
insertT x empT = leafT x | |
insertT x (leafT x') = nodeT p₀ (leafT x') (leafT x) | |
insertT x (nodeT p₀ l r) = nodeT p₁ (insertT x l) r | |
insertT x (nodeT p₁ l r) = nodeT p₀ l (insertT x r) | |
dealT : {X : Set} (xs : List X) -> DealT X | |
dealT nil = empT | |
dealT (x ∷ xs) = insertT x (dealT xs) | |
mergeT : DealT ℕ -> List ℕ | |
mergeT empT = nil | |
mergeT (leafT x) = x ∷ nil | |
mergeT (nodeT p l r) = merge (mergeT l) (mergeT r) | |
sort : List ℕ -> List ℕ | |
sort = mergeT ∘ dealT | |
data Vec (X : Set) : ℕ -> Set where | |
vnil : Vec X 0 | |
vcons : ∀ {n : ℕ} -> (x : X) -> (v : Vec X n) -> Vec X (suc n) | |
vtail : {n : ℕ} {X : Set} -> (Vec X (suc n)) -> Vec X n | |
vtail (vcons x v) = v | |
infixl 5 _<+>_ | |
_<+>_ : {S T : Set} {n : ℕ} -> Vec (S -> T) n -> Vec S n -> Vec T n | |
vnil <+> vnil = vnil | |
vcons f fv <+> vcons x xs = vcons (f x) (fv <+> xs) | |
infixl 5 _++_ | |
_++_ : {n m : ℕ} {X : Set} -> Vec X n -> Vec X m -> Vec X (n + m) | |
vnil ++ ys = ys | |
vcons x v ++ ys = vcons x (v ++ ys) | |
vec : {X : Set} {n : ℕ} (x : X) -> Vec X n | |
vec {_} {0} x = vnil | |
vec {_} {suc n} x = vcons x (vec x) | |
xpose : {i j : ℕ} {X : Set} -> Vec (Vec X j) i -> Vec (Vec X i) j | |
xpose vnil = vec vnil | |
xpose (vcons x v) = vec vcons <+> x <+> xpose v | |
plusSuc : (k : ℕ) -> (n : ℕ) -> k + (suc n) ≡ suc (k + n) | |
plusSuc zero n = refl | |
plusSuc (suc n) n' with (plusSuc n n') | |
... | p = cong suc p | |
vrevacc : {X : Set}{n m : ℕ} -> Vec X n -> Vec X m -> Vec X (n + m) | |
vrevacc vnil ys = ys | |
vrevacc {X} {suc k} {m} (vcons x v) ys = subst (Vec X) (plusSuc k m) (vrevacc v (vcons x ys)) | |
plusZero : ∀ (n : ℕ) -> n + 0 ≡ n | |
plusZero 0 = refl | |
plusZero (suc k) = cong suc (plusZero k) | |
vrev : {X : Set} {n : ℕ} -> (xs : Vec X n) -> Vec X n | |
vrev {X} {n} xs = subst (Vec X) (plusZero n) (vrevacc xs vnil) | |
vmerge : {m n : ℕ} -> (xs : Vec ℕ m) -> (ys : Vec ℕ n) -> Vec ℕ (m + n) | |
vmerge vnil ys = ys | |
vmerge {suc m'} {0} (vcons x v) vnil = subst (Vec ℕ) (cong suc (sym (plusZero m'))) (vcons x v) | |
vmerge {suc m'} {suc n'} (vcons x v) (vcons x' v') with order x x' | vmerge v (vcons x' v') | vmerge (vcons x v) v' | |
vmerge {suc m'} {suc n'} (vcons x v) (vcons x' v') | le | m1 | m2 = vcons x m1 | |
vmerge {suc m'} {suc n'} (vcons x v) (vcons x' v') | ge | m1 | m2 = | |
vcons x' (subst (Vec ℕ) (sym (plusSuc m' n')) m2) | |
num : Parity -> ℕ | |
num p₀ = 0 | |
num p₁ = 1 | |
data DealTT (X : Set) : ℕ -> Set where | |
empT : DealTT X 0 | |
leafT : (x : X) -> DealTT X 1 | |
nodeT : {n : ℕ} -> (p : Parity) -> (l : DealTT X (num p + n)) -> (r : DealTT X n) -> | |
DealTT X ((num p + n) + n) | |
mergeTT : {n : ℕ} -> (t : DealTT ℕ n) -> Vec ℕ n | |
mergeTT empT = vnil | |
mergeTT (leafT x) = vcons x vnil | |
mergeTT (nodeT p l r) = vmerge (mergeTT l) (mergeTT r) | |
insertTT : {X : Set}{n : ℕ} -> (x : X) -> (t : DealTT X n) -> DealTT X (suc n) | |
insertTT x empT = leafT x | |
insertTT x (leafT x') = nodeT p₀ (leafT x') (leafT x) | |
insertTT x (nodeT p₀ l r) = nodeT p₁ (insertTT x l) r | |
insertTT {X} x (nodeT p₁ l r) = subst (DealTT X) (cong suc (plusSuc {!!} {!!})) (nodeT p₀ l (insertTT x r)) |
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