\import Algebra.Meta
\import Algebra.Monoid
\import Arith.Nat
\import Data.Bool
\import Data.Or
\import Function
\import Function.Meta
\import Logic
\import Logic.Meta
\import Meta
\import Order.LinearOrder
\import Order.PartialOrder
\import Order.StrictOrder
\import Paths
\import Paths.Meta
\import Set

\data List (A : \Type)
| nil
| \infixr 5 :: A (List A)

\func length {A : \Type} (l : List A) : Nat \elim l
| nil => 0
| :: a l => suc (length l)

\func \infixl 9 !! {A : \Type} (l : List A) (i : Fin (length l)) : A \elim l, i
| :: a l, 0 => a
| :: a l, suc i => l !! i

\func \infixr 5 ++ {A : \Type} (xs ys : List A) : List A \elim xs
| nil => ys
| :: a xs => a :: xs ++ ys

\func ++-assoc {A : \Type} {xs ys zs : List A} : (xs ++ ys) ++ zs = xs ++ (ys ++ zs) \elim xs
| nil => idp
| :: x xs => pmap (x ::) ++-assoc

\func ++_nil {A : \Type} {l : List A} : l ++ nil = l \elim l
| nil => idp
| :: a l => pmap (a ::) ++_nil

\lemma length_++ {A : \Type} {l l' : List A} : length (l ++ l') = length l + length l' \elim l
| nil => idp
| :: a l => pmap suc length_++

\func replicate {A : \Type} (n : Nat) (a : A) : List A \elim n
| 0 => nil
| suc n => a :: replicate n a

\lemma length_replicate {A : \Type} {n : Nat} {a : A} : length (replicate n a) = n \elim n
| 0 => idp
| suc n => pmap suc length_replicate

\func map {A B : \Type} (f : A -> B) (l : List A) : List B \elim l
| nil => nil
| :: a l => f a :: map f l

\func length_map {A B : \Type} (f : A -> B) (l : List A) : length (map f l) = length l \elim l
| nil => idp
| :: a l => pmap suc (length_map f l)

\func map_comp {A B C : \Type} (g : B -> C) (f : A -> B) (l : List A) : map (\lam x => g (f x)) l = map g (map f l) \elim l
| nil => idp
| :: a l => pmap (_ ::) (map_comp g f l)

\func headDef {A : \Type} (x : A) (xs : List A) : A \elim xs
| nil => x
| :: a _ => a

\func tail {A : \Type} (l : List A) : List A \elim l
| nil => nil
| :: a l => l

\instance ListMonoid {A : \Set} : Monoid (List A)
| ide => nil
| * => ++
| ide-left => idp
| ide-right => ++_nil
| *-assoc => ++-assoc

\func splitAt {A : \Type} (n : Nat) (l : List A) : \Sigma (List A) (List A) \elim n, l
| 0, l => (nil, l)
| suc _, nil => (nil, nil)
| suc n, :: a l =>
\let! (l1, l2) => splitAt n l
\in (a :: l1, l2)
\where
\func appendLem {A : \Type} (n : Nat) (l : List A) : take n l ++ drop n l = l \elim n, l
| 0, l => idp
| suc n, nil => idp
| suc n, :: a l => pmap (a ::) (appendLem n l)

\func take {A : \Type} (n : Nat) (l : List A) => (splitAt n l).1

\func drop {A : \Type} (n : Nat) (l : List A) => (splitAt n l).2

\func replace {A : \Type} (l : List A) (i s : Nat) (r : List A) =>
\let! (l1, l2) => splitAt i l
\in l1 ++ r ++ drop s l2

\func slice {A : \Type} (l : List A) (i s : Nat) => take s (drop i l)
\where
\func appendLem {A : \Type} (l : List A) (i s : Nat) : take i l ++ slice l i s ++ drop s (drop i l) = l \elim l, i, s
| l, 0, s => splitAt.appendLem s l
| nil, suc i, 0 => idp
| nil, suc i, suc s => idp
| :: a l, suc i, s => pmap (a ::) (appendLem l i s)

\data All {A : \Type} (P : A -> \Prop) (l : List A) \elim l
| nil => all-nil
| :: x l => all-cons (P x) (All P l)

\lemma all-map {A B : \Type} (f : A -> B) {P : B -> \Prop} {l : List A} (a : All P (map f l)) : All (\lam a => P (f a)) l \elim l, a
| nil, all-nil => all-nil
| :: x l, all-cons p a => all-cons p (all-map f a)

\lemma all-implies {A : \Type} {P Q : A -> \Prop} {l : List A} (a : All P l) (p : All (\lam a => P a -> Q a) l) : All Q l \elim l, a, p
| nil, all-nil, all-nil => all-nil
| :: x l, all-cons c a, all-cons f p => all-cons (f c) (all-implies a p)

\lemma all-forall {A : \Type} {P : A -> \Prop} {l : List A} (p : \Pi (a : A) -> P a) : All P l \elim l
| nil => all-nil
| :: a l => all-cons (p a) (all-forall p)

\data All2 {A : \Type} (P : A -> A -> \Prop) (l : List A) \elim l
| nil => all2-nil
| :: x l => all2-cons (All (P x) l) (All2 P l)

\data AllC {A : \Type} (P : A -> A -> \Prop) (l : List A) \elim l
| nil => allC-nil
| :: x nil => allC-single
| :: x (:: y l) => allC-cons (P x y) (AllC P (y :: l))

\lemma allC-tail {A : \Type} {a : A} {l : List A} {P : A -> A -> \Prop} (a : AllC P (a :: l)) : AllC P l \elim l, a
| nil, allC-single => allC-nil
| :: y l, allC-cons p a => a

\func count {A : DecSet} (l : List A) (a : A) : Nat \elim l
| nil => 0
| :: x l => \case decideEq x a \with {
| yes _ => suc (count l a)
| no _ => count l a
}
\where {
\lemma all-diff {A : DecSet} {l : List A} {a : A} (d : All (a /=) l) : count l a = 0 \elim l, d
| nil, all-nil => idp
| :: x l, all-cons p d => mcases \with {
| yes e => \case p (inv e)
| no _ => all-diff d
}
}

\lemma count_perm {A : DecSet} {l l' : List A} (p : Perm l l') (a : A) : count l a = count l' a \elim l, l', p
| nil, nil, perm-nil => idp
| :: x l, :: _ l', perm-:: idp p => rewrite (count_perm p) idp
| :: x (:: x' l), :: _ (:: _ _), perm-swap idp idp idp => mcases \with {
| yes e, yes e' => rewrite (decideEq=_reduce e, decideEq=_reduce e') idp
| yes e, no q => rewrite (decideEq=_reduce e, decideEq/=_reduce q) idp
| no q, yes e => rewrite (decideEq/=_reduce q, decideEq=_reduce e) idp
| no q, no q' => rewrite (decideEq/=_reduce q, decideEq/=_reduce q') idp
}
| l, l', perm-trans p1 p2 => count_perm p1 a *> count_perm p2 a

\lemma count_++ {A : DecSet} {l l' : List A} {a : A} : count (l ++ l') a = count l a Nat.+ count l' a \elim l
| nil => idp
| :: x l => mcases \with {
| yes e => pmap suc count_++
| no q => count_++
}

\func group {A : DecSet} (l : List A) : List (\Sigma A Nat) \elim l
| nil => nil
| :: a l => \case group l \with {
| nil => (a,1) :: nil
| :: (a',n) l' => \case decideEq a a' \with {
| yes _ => (a', suc n) :: l'
| no _ => (a,1) :: (a',n) :: l'
}
}

\open Sort

\lemma group-sorted {A : DecSet} {P : Preorder A} {l : List A} (s : Sorted l) : Sorted (map __.1 (group l)) \elim l, s
| nil, sorted-nil => sorted-nil
| :: x l, sorted-cons x<=l s => mcases {1} {arg addPath} \with {
| nil, _ => sorted-cons <=-refl sorted-nil
| :: (a',n) l', p =>
\have s' => rewrite p in group-sorted s
\in mcases \with {
| yes _ => s'
| no _ => sorted-cons (transport (x <=) (head-lem p) x<=l) s'
}
}
\where {
\lemma head-lem {A : DecSet} {l : List A} {a x : A} {n : Nat} {l' : List (\Sigma A Nat)} (p : group l = (a,n) :: l') : headDef x l = a \elim l
| nil => \case p
| :: y l => mcases {1} p _ \with {
| nil, p' => pmap (\lam s => (headDef (a,n) s).1) p'
| :: z l'', p' => mcases p' _ \with {
| yes e, p'' => e *> pmap (\lam s => (headDef (a,n) s).1) p''
| no q, p'' => pmap (\lam s => (headDef (a,n) s).1) p''
}
}
}

\lemma group-diff {A : DecSet} {P : Poset A} {l : List A} (s : Sorted l) : All2 (__.1 /= __.1) (group l) \elim l, s
| nil, sorted-nil => all2-nil
| :: x l, sorted-cons x<=l s => mcases {1} {arg addPath} \with {
| nil, _ => all2-cons all-nil all2-nil
| :: ((a',n)) l', p =>
\have (all2-cons t t') => rewrite p in group-diff s
\in mcases \with {
| yes _ => all2-cons t t'
| no q =>
\have | s' => rewrite p in group-sorted s
| (all-cons _ r) => diff-lem s' a' a' (transport (x <=) (group-sorted.head-lem p) x<=l) q
\in all2-cons (all-cons q (all-map __.1 r)) (all2-cons t t')
}
}
\where {
\lemma diff-lem {A : DecSet} {P : Poset A} {l : List A} (s : Sorted l) {x : A} (y z : A) (p : x <= headDef y l) (q : x /= headDef z l) : All (x /=) l \elim l, s
| nil, sorted-nil => all-nil
| :: a l, sorted-cons a<=l s => all-cons q (diff-lem s _ _ (p <=âˆ˜ a<=l) (\lam t => q (<=-antisymmetric p (transportInv (a <=) t a<=l))))
}

\lemma group_count-lem {A : DecSet} {P : Poset A} {l : List A} (s : Sorted l) : All (\lam p => p.2 = count l p.1) (group l) \elim l, s
| nil, sorted-nil => all-nil
| :: a l, sorted-cons a<=l s => mcases {2} {arg addPath} \with {
| nil, p => all-cons (rewrite (decideEq=_reduce idp, group_nil-lem p) idp) all-nil
| :: (a',n) l', p =>
\have (all-cons t t') => rewrite p in group_count-lem s
\in mcases {2} \with {
| yes e =>
\have (all2-cons t'' _) => rewrite p in group-diff s
\in all-cons (rewrite (decideEq=_reduce e) $pmap suc t)$ all-implies t' $all-implies t''$ all-forall \lam q c d => rewrite (decideEq/=_reduce (\lam e' => c (inv e *> e'))) d
| no q =>
\have (all-cons _ t'') : All (\lam x => a /= x.1) ((a',n) :: l') => all-map __.1 (group-diff.diff-lem (rewrite p in group-sorted s) a a (transport (a <=) (group-sorted.head-lem p) a<=l) q)
\in all-cons (rewrite (decideEq=_reduce idp) $pmap suc$ inv $count.all-diff$ group-diff.diff-lem s a a a<=l $rewrite (group-sorted.head-lem p) q)$
all-cons (rewrite (decideEq/=_reduce q) t) $all-implies t'$ all-implies t'' $all-forall \lam y c d => rewrite (decideEq/=_reduce c) d } } \where { \lemma group_nil-lem {A : DecSet} {l : List A} (p : group l = nil) : l = nil \elim l | nil => idp | :: a l => mcases {1} p _ \with { | nil, () | :: ((a',n)) l', p' => mcases p' _ \with { | yes p1, () | no n1, () } } } \module Sort \where { \data Perm {A : \Type} (xs ys : List A) \elim xs, ys | nil, nil => perm-nil | :: x xs, :: y ys => perm-:: (x = y) (Perm xs ys) | :: x (:: x' xs), :: y (:: y' ys) => perm-swap (x = y') (x' = y) (xs = ys) | xs, ys => perm-trans {zs : List A} (Perm xs zs) (Perm zs ys) \where { \func perm-refl {A : \Type} {xs : List A} : Perm xs xs \elim xs | nil => perm-nil | :: a l => perm-:: idp perm-refl \func perm-sym {A : \Type} {xs ys : List A} (p : Perm xs ys) : Perm ys xs \elim xs, ys, p | nil, nil, perm-nil => perm-nil | :: x xs, :: y ys, perm-:: p p1 => perm-:: (inv p) (perm-sym p1) | :: x (:: x' xs), :: y (:: y' ys), perm-swap p p1 p2 => perm-swap (inv p1) (inv p) (inv p2) | xs, ys, perm-trans p1 p2 => perm-trans (perm-sym p2) (perm-sym p1) \func perm-head {A : \Type} {a : A} {xs ys : List A} : Perm (a :: xs ++ ys) (xs ++ a :: ys) \elim xs | nil => perm-refl | :: a1 xs => perm-trans (perm-swap idp idp idp) (perm-:: idp perm-head) \lemma perm_length {A : \Type} {xs ys : List A} (p : Perm xs ys) : length xs = length ys \elim xs, ys, p | nil, nil, perm-nil => idp | x :: xs, y :: ys, perm-:: _ q => pmap suc (perm_length q) | x :: (x' :: xs), y :: (y' :: _), perm-swap _ _ idp => idp | xs, ys, perm-trans p1 p2 => perm_length p1 *> perm_length p2 } \data Sorted {A : Preorder} (xs : List A) \elim xs | nil => sorted-nil | :: x xs => sorted-cons (x A.<= headDef x xs) (Sorted xs) \where { \lemma allSorted {A : Preorder} {a1 a2 : A} (l1 l2 : List A) (s : Sorted (a1 :: l1 ++ a2 :: l2)) : a1 A.<= a2 => aux a1 (a1 :: l1) l2 s \where \lemma aux {A : Preorder} (a' : A) {a : A} (l1 l2 : List A) (s : Sorted (l1 ++ a :: l2)) : headDef a' (l1 ++ a :: l2) A.<= a \elim l1, s | nil, sorted-cons _ s => <=-refl | :: a1 l1, sorted-cons a1<=h s => <=-transitive a1<=h (aux a1 l1 l2 s) \lemma headSorted {A : Preorder} {l1 l2 : List A} (s : Sorted (l1 ++ l2)) : Sorted l1 \elim l1, s | nil, _ => sorted-nil | :: a l1, sorted-cons e s => sorted-cons (\case \elim l1, \elim e \with { | nil, _ => <=-refl | :: a1 l1, e => e }) (headSorted s) \lemma tailSorted {A : Preorder} (l1 l2 : List A) (s : Sorted (l1 ++ l2)) : Sorted l2 \elim l1, s | nil, s => s | :: a l1, sorted-cons _ s => tailSorted l1 l2 s } \open LinearOrder \module Insertion \where { \func sort {A : Dec} (xs : List A) : List A | nil => nil | :: a xs => insert a (sort xs) \where { \func insert {A : Dec} (a : A) (xs : List A) : List A \elim xs | nil => a :: nil | :: x xs => \case dec<_<= x a \with { | inl x<a => x :: insert a xs | inr a<=x => a :: x :: xs } } \open sort \lemma sort-sorted {A : Dec} (xs : List A) : Sorted (sort xs) \elim xs | nil => sorted-nil | :: a xs => insert-sorted a (sort-sorted xs) \where { \lemma insert-sorted {A : Dec} (a : A) {xs : List A} (s : Sorted xs) : Sorted (sort.insert a xs) \elim xs, s | nil, _ => sorted-cons <=-refl sorted-nil | :: a1 xs, sorted-cons p s1 \as s2 => mcases \with { | inl a1<a => sorted-cons (\case \elim xs, \elim p \with { | nil, _ => <=-less a1<a | :: a2 xs, a1<=a2 => mcases \with { | inl _ => a1<=a2 | inr _ => <=-less a1<a } }) (insert-sorted a s1) | inr a<=a1 => sorted-cons a<=a1 s2 } } \func sort-perm {A : Dec} (xs : List A) : Perm xs (sort xs) \elim xs | nil => perm-nil | :: a l => perm-trans (perm-:: idp (sort-perm l)) (insert-perm a (sort l)) \where { \func insert-perm {A : Dec} (a : A) (xs : List A) : Perm (a :: xs) (insert a xs) \elim xs | nil => perm-:: idp perm-nil | :: b xs => mcases \with { | inl b<=a => perm-trans (perm-swap idp idp idp) (perm-:: idp (insert-perm a xs)) | inr a<=b => Perm.perm-refl } \lemma insert-comm {A : Dec} (a a' : A) (l : List A) : insert a (insert a' l) = insert a' (insert a l) => \case trichotomy a a' \with { | less a<a' => aux a<a' l | equals a=a' => rewrite a=a' idp | greater a>a' => inv (aux a>a' l) } \where { \lemma aux {A : Dec} {a a' : A} (a<a' : a < a') (l : List A) : insert a (insert a' l) = insert a' (insert a l) \elim l | nil => mcases (idp <|> contradiction) | :: a1 l => repeat {3} mcases (idp <|> contradiction <|> pmap (a1 ::) (aux a<a' l)) } } \lemma perm_sort {A : Dec} {xs ys : List A} (p : Perm xs ys) : sort xs = sort ys \elim xs, ys, p | nil, nil, perm-nil => idp | x :: xs, y :: ys, perm-:: p q => pmap2 insert p (perm_sort q) | x :: (x' :: xs), _ :: (_ :: _), perm-swap idp idp idp => sort-perm.insert-comm x x' (sort xs) | xs, ys, perm-trans p1 p2 => perm_sort p1 *> perm_sort p2 \lemma sorted_sort {A : Dec} {xs : List A} (s : Sorted xs) : sort xs = xs \elim xs, s | nil, sorted-nil => idp | x :: xs, sorted-cons p s => pmap (insert x) (sorted_sort s) *> \case \elim xs, \elim p \with { | nil, p => idp | a :: xs, x<=a => mcases \with { | inl a<x => absurd$ <-irreflexive (a<x <âˆ˜l x<=a)
| inr x<=a => idp
}
}
}

\module RedBlack \where {
\func sort {A : Dec} (l : List A) => rbTreeToList (aux l rbLeaf) nil
\where {
\data Color | red | black

\data RBTree (A : \Type) | rbLeaf | rbBranch Color (RBTree A) A (RBTree A)

\func rbTreeToList {A : \Type} (t : RBTree A) (r : List A) : List A \elim t
| rbLeaf => r
| rbBranch _ t1 a t2 => rbTreeToList t1 (a :: rbTreeToList t2 r)

\func aux {A : Dec} (l : List A) (\strict r : RBTree A) : RBTree A \elim l
| nil => r
| :: a l => aux l (repaint (insert a r))

\func repaint {A : \Type} (t : RBTree A) : RBTree A
| rbBranch red l a r => rbBranch black l a r
| t => t

\func insert {A : Dec} (a : A) (t : RBTree A) : RBTree A \elim t
| rbLeaf => rbBranch red rbLeaf a rbLeaf
| rbBranch c l1 a1 l2 => \case dec<_<= a1 a \with {
| inl a1<a => balanceRight c l1 a1 (insert a l2)
| inr a<=a1 => balanceLeft c (insert a l1) a1 l2
}

\func balanceLeft {A : \Type} (c : Color) (\strict l : RBTree A) (v : A) (r : RBTree A) : RBTree A \elim c, l
| black, rbBranch red (rbBranch red a x b) y c => rbBranch red (rbBranch black a x b) y (rbBranch black c v r)
| black, rbBranch red a x (rbBranch red b y c) => rbBranch red (rbBranch black a x b) y (rbBranch black c v r)
| c, a => rbBranch c a v r

\func balanceRight {A : \Type} (c : Color) (l : RBTree A) (v : A) (\strict r : RBTree A) : RBTree A \elim c, r
| black, rbBranch red (rbBranch red b y c) z d => rbBranch red (rbBranch black l v b) y (rbBranch black c z d)
| black, rbBranch red b y (rbBranch red c z d) => rbBranch red (rbBranch black l v b) y (rbBranch black c z d)
| c, b => rbBranch c l v b
}

\open sort

\func toList {A : \Type} (t : RBTree A) : List A \elim t
| rbLeaf => nil
| rbBranch _ t1 a t2 => toList t1 ++ a :: toList t2
\where {
\lemma =rbTreeToList {A : \Set} (t : RBTree A) : rbTreeToList t nil = toList t
=> aux t nil *> ++_nil
\where
\lemma aux {A : \Set} (t : RBTree A) (l : List A) : rbTreeToList t l = toList t ++ l \elim t
| rbLeaf => idp
| rbBranch c t1 a t2 => aux t1 (a :: rbTreeToList t2 l) *> pmap (toList t1 ++ a :: __) (aux t2 l) *> inv ++-assoc
}

\lemma sort=insert {A : Dec} (l : List A) : sort l = Insertion.sort l
=> toList.=rbTreeToList _ *> aux=makeTree l nil *> pmap (\lam s => toList (makeTree s)) ++_nil *> toList_mkTree l
\where {
\func makeTree {A : Dec} (l : List A) : RBTree A \elim l
| nil => rbLeaf
| :: a l => repaint (insert a (makeTree l))

\lemma toList_repaint {A : \Set} (t : RBTree A) : toList (repaint t) = toList t \elim t
| rbLeaf => idp
| rbBranch red t1 a t2 => idp
| rbBranch black t1 a t2 => idp

\lemma toList_balanceLeft {A : \Set} (c : Color) (l : RBTree A) (v : A) (r : RBTree A)
: toList (balanceLeft c l v r) = toList (rbBranch c l v r)
=> mcases {balanceLeft} equation

\lemma toList_balanceRight {A : \Set} (c : Color) (l : RBTree A) (v : A) (r : RBTree A)
: toList (balanceRight c l v r) = toList (rbBranch c l v r)
=> mcases {balanceRight} equation

\lemma insert_++-left {A : Dec} {a a1 : A} (a<=a1 : a <= a1) {l1 l2 : List A} (s : Sorted (l1 ++ a1 :: l2))
: Insertion.sort.insert a (l1 ++ a1 :: l2) = Insertion.sort.insert a l1 ++ a1 :: l2 \elim l1, s
| nil, _ => rewrite (dec<=_reduce a<=a1) idp
| :: a2 l1, sorted-cons _ s => mcases \with {
| inl a2<a => pmap (a2 ::) (insert_++-left a<=a1 s)
| inr a<=a2 => idp
}

\lemma insert_++-right {A : Dec} {a a1 : A} (a1<a : a1 < a) {l1 l2 : List A} (s : Sorted (l1 ++ a1 :: l2))
: Insertion.sort.insert a (l1 ++ a1 :: l2) = l1 ++ a1 :: Insertion.sort.insert a l2 \elim l1, s
| nil, _ => rewrite (dec<_reduce a1<a) idp
| :: a' l1, sorted-cons _ s1 \as s2 => rewrite (dec<_reduce (Sorted.allSorted l1 l2 s2 <âˆ˜r a1<a)) (pmap (a' ::) (insert_++-right a1<a s1))

\lemma toList_insert' {A : Dec} (a : A) (t : RBTree A) (s : Sorted (toList t)) : toList (insert a t) = Insertion.sort.insert a (toList t) \elim t
| rbLeaf => idp
| rbBranch c t1 a1 t2 => mcases \with {
| inl a1<a => toList_balanceRight _ _ _ _ *> pmap (toList t1 ++ a1 :: __) (toList_insert' a t2 (Sorted.tailSorted (a1 :: nil) _ (Sorted.tailSorted _ _ s))) *> inv (insert_++-right a1<a s)
| inr a<=a1 => toList_balanceLeft _ _ _ _ *> pmap (__ ++ a1 :: toList t2) (toList_insert' a t1 (Sorted.headSorted s)) *> inv (insert_++-left a<=a1 s)
}

\lemma toList_mkTree {A : Dec} (l : List A) : toList (makeTree l) = Insertion.sort l \elim l
| nil => idp
| :: a l => toList_repaint (insert a (makeTree l)) *> toList_insert' a (makeTree l) (rewrite toList_mkTree (Insertion.sort-sorted l)) *> pmap (Insertion.sort.insert a) (toList_mkTree l)

\lemma makeTree-sorted {A : Dec} (l : List A) : Sorted (toList (makeTree l))
=> rewrite toList_mkTree (Insertion.sort-sorted l)

\lemma toList_insert {A : Dec} (a : A) (l : List A) : toList (insert a (makeTree l)) = Insertion.sort.insert a (toList (makeTree l))
=> toList_insert' a (makeTree l) (makeTree-sorted l)

\lemma makeTree_insert {A : Dec} (a : A) (l l' : List A) : toList (makeTree (l ++ a :: l')) = toList (repaint (insert a (makeTree (l ++ l')))) \elim l
| nil => idp
| :: a' l => toList_repaint _ *> toList_insert a' _ *> pmap (Insertion.sort.insert a') (makeTree_insert a l l' *> toList_repaint _ *> toList_insert a _) *> Insertion.sort-perm.insert-comm a' a _ *> inv (
toList_repaint _ *>
toList_insert' a _ (transportInv Sorted (toList_repaint _ *> toList_insert a' _) (Insertion.sort-sorted.insert-sorted a' (makeTree-sorted _))) *>
pmap (Insertion.sort.insert a) (toList_repaint _ *> toList_insert a' _))

\lemma aux=makeTree {A : Dec} (l l' : List A) : toList (aux l (makeTree l')) = toList (makeTree (l ++ l')) \elim l
| nil => idp
| :: a l => aux=makeTree l (a :: l') *> makeTree_insert a l l'
}
}
}

\func contains {A : DecSet} (l : List A) (a : A) : Bool \elim l
| nil => false
| b :: l => \case decideEq b a \with {
| yes _ => true
| no _ => contains l a
}

\func union {A : DecSet} (l l' : List A) : List A \elim l
| nil => l'
| a :: l => if (contains l' a) (union l l') (a :: union l l')

\truncated \data InList {A : \Type} (a : A) (l : List A) : \Prop \elim l
| a' :: l => { here (a = a') | there (InList a l) }
\where {
\func \infix 4 ~ {A : \Type} (l l' : List A) => \Pi (a : A) -> InList a l <-> InList a l'

\lemma ~_:: {A : \Type} {a : A} {l l' : List A} (p : l ~ l') : a :: l ~ a :: l'
=> \lam b => (InList_:: (p b).1, InList_:: (p b).2)
\where
\lemma InList_:: {A : \Type} {a b : A} {l l' : List A} (p : InList b l -> InList b l') (c : InList b (a :: l)) : InList b (a :: l') \elim c
| here q => here q
| there c => there (p c)
}

\lemma InList_++ {A : \Type} {a : A} {l l' : List A} (p : InList a (l ++ l')) : InList a l || InList a l' \elim l, p
| nil, p => byRight p
| a' :: l, here p => byLeft (here p)
| a' :: l, there p => ||.map there id (InList_++ p)

\lemma InList_++-left {A : \Type} {a : A} {l l' : List A} (p : InList a l) : InList a (l ++ l') \elim l, p
| a' :: l, here p => here p
| a' :: l, there p => there (InList_++-left p)

\lemma InList_++-right {A : \Type} {a : A} {l l' : List A} (p : InList a l') : InList a (l ++ l') \elim l
| nil => p
| a' :: l => there (InList_++-right p)

\open InList

\lemma contains_InList {A : DecSet} {a : A} {l : List A} : (contains l a = true) <-> InList a l \elim l
| nil => ((\case __), \case __)
| a' :: l => (\lam p => cases (decideEq a' a, p) \with {
| yes e, _ => here (inv e)
| no _, p => there (contains_InList.1 p)
}, \lam p => mcases \with {
| yes _ => idp
| no q => \case \elim p \with {
| here p => absurd $q (inv p) | there p => contains_InList.2 p } }) \lemma union~++ {A : DecSet} {l l' : List A} : union l l' ~ l ++ l' \elim l | nil => \lam a => (id,id) | a :: l => cases (contains l' a arg addPath) \with { | false, _ => ~_:: union~++ | true, q => \lam b => (\lam c => there ((union~++ b).1 c), \case __ \with { | here p => (union~++ b).2$ InList_++-right \$ rewrite p (contains_InList.1 q)
| there p => (union~++ b).2 p
})
}