module rbt_t where

open import Data.Nat hiding (compare)
open import Data.Nat.Properties as NatProp -- <-cmp
open import Relation.Binary
open import Data.List

-- → t を適用するために必要だった
open import Level renaming ( suc to succ ; zero to Zero )
open import Level


data col : Set where
  black : col
  red : col

record node (A B : Set) : Set where
  field
    coler : A
    number : B

record tree (A B C : Set) : Set where
  field
    key : A
    ltree : B
    rtree : C

data rbt : Set where
  bt-empty : rbt
  bt-node  : (node : tree (node col ℕ) rbt rbt ) → rbt

record Env : Set (succ Zero) where
  field
    vart : rbt
    varn : ℕ
    varl : List rbt
open Env

whileTest_next : {l : Level} {t : Set l}  → (c10 : rbt) → (n : ℕ) → (list : List rbt) → (next : Env → t) → (exit : Env → t) → t
whileTest_next c10 n list next exit = next (record {vart = c10 ; varn = n ; varl = list} )

--whileTest_exit : {l : Level} {t : Set l}  → (c10 : bst) → (n : ℕ) → (list : List bst) → (next : Env → t) → (exit : Env → t) → t
--whileTest_exit c10 n list next exit = exit (record {vart = c10 ; varn = n ; varl = list} )

merge-tree :  {le : Level} {t : Set le} → Env → (next : Env → t) → (exit : Env → t) → t

-- 全てmerge-treeへ
split : {le : Level} {t : Set le} → Env → (next : Env → t) → (exit : Env → t) → t
split node@record { vart = bt-empty ; varn = varn ; varl = varl } next exit = exit node
split node@record { vart = (bt-node record { key = key ; ltree = bt-empty ; rtree = rtree }) ; varn = varn ; varl = varl } next exit = exit node
split node@record { vart = (bt-node record { key = key ; ltree = (bt-node node₁) ; rtree = bt-empty }) ; varn = varn ; varl = varl } next exit = exit node
split record { vart = (bt-node record { key = record { coler = coler₁ ; number = number₁ }
  ; ltree = (bt-node record { key = record { coler = ly ; number = ln } ; ltree = ll ; rtree = lr })
  ; rtree = (bt-node record { key = record { coler = ry ; number = rn } ; ltree = rl ; rtree = rr }) })
  ; varn = varn ; varl = varl } next exit
  = next record { vart = (bt-node record { key = record { coler = red ; number = number₁ }
    ; ltree = (bt-node record { key = record { coler = black ; number = ln } ; ltree = ll ; rtree = lr })
    ; rtree = (bt-node record { key = record { coler = black ; number = rn } ; ltree = rl ; rtree = rr }) })
    ; varn = varn ; varl = varl }

{-
split bt-empty n stack = bt-empty
split node@(bt-node (record { key = record { coler = y ; number = x }; ltree = l; rtree = bt-empty }) ) n stack = node

split node@(bt-node (record { key = record { coler = y ; number = x }; ltree = bt-empty; rtree = r }) ) n stack = node
split node@(bt-node (record { key = record { coler = y ; number = x }
  ; ltree = (bt-node (record { key = record { coler = ly ; number = lx }; ltree = ll; rtree = lr }) )
  ; rtree = (bt-node (record { key = record { coler = ry ; number = rx }; ltree = rl; rtree = rr }) )}) ) n stack
  = {!!}
-}
{-
merge-tree (bt-node (record { key = record { coler = red ; number = x }
      ; ltree = (bt-node (record { key = record { coler = black ; number = lx }; ltree = ll; rtree = lr }) )
      ; rtree = (bt-node (record { key = record { coler = black ; number = rx }; ltree = rl; rtree = rr }) )})) n stack
-}

-- 右回転
-- 実行時splitへ、それ以外はmerge-treeへ
merge-rotate_right : {le : Level} {t : Set le} → Env → (next : Env → t) → (exit : Env → t) → t
merge-rotate_right node@record { vart = bt-empty ; varn = varn ; varl = varl } next exit = exit node
merge-rotate_right node@record { vart = bt-node record { key = record { coler = coler ; number = number } ; ltree = bt-empty ; rtree = r } ; varn = varn ; varl = varl } next exit = exit node
merge-rotate_right record { vart = bt-node record { key = record { coler = y ; number = x }
  ; ltree = (bt-node record { key =  record { coler = ly ; number = lx } ; ltree = ll ; rtree = lr })
  ; rtree = r }
  ; varn = varn ; varl = varl } next exit
    = next record { vart = bt-node record { key = record { coler = y ; number = lx }
      ; ltree = ll
      ; rtree = (bt-node record { key = record { coler = red ; number = x } ; ltree = lr; rtree = r }) }
      ; varn = varn ; varl = varl }

{-
merge-rotate_right bt-empty n stack = bt-empty
merge-rotate_right node@(bt-node (record { key = _; ltree = bt-empty; rtree = _ }) ) n stack = node
merge-rotate_right (bt-node (record { key = record { coler = y ; number = x }
  ; ltree = (bt-node (record { key = record { coler = ly ; number = lx }; ltree = ll; rtree = lr }) )
  ; rtree = r }) ) n stack
    = {!!}
-}
{-
split (bt-node (record { key = record { coler = y ; number = lx }
      ; ltree = ll
      ; rtree = (bt-node (record { key = record { coler = red ; number = x }; ltree = lr; rtree = r }) ) }) ) n stack
-}

-- split
split_branch : {le : Level} {t : Set le} → Env → (next : Env → t) → (exit : Env → t) → t
split_branch node@record{ vart = bt-empty ; varn = varn ; varl = varl } next exit = exit node
split_branch node@record{ vart = bt-node record { key = key ; ltree = bt-empty ; rtree = rtree } ; varn = varn ; varl = varl } next exit = exit node
split_branch node@record{ vart = bt-node record { key = key₁ ; ltree = (bt-node record { key = record { coler = lc ; number = number } ; ltree = bt-empty ; rtree = lr }) ; rtree = rtree } ; varn = varn ; varl = varl } next exit = exit node
split_branch node@record{ vart = bt-node record { key = key₁ ; ltree = (bt-node record { key = record { coler = black ; number = number } ; ltree = (bt-node node₁) ; rtree = lr }) ; rtree = rtree } ; varn = varn ; varl = varl } next exit = exit node
split_branch node@record{ vart = bt-node record { key = key₁ ; ltree = (bt-node record { key = record { coler = red ; number = number₁ } ; ltree = (bt-node record { key = record { coler = black ; number = number } ; ltree = ltree ; rtree = rtree₁ }) ; rtree = lr }) ; rtree = rtree } ; varn = varn ; varl = varl } next exit = exit node
split_branch node@record{ vart = bt-node record { key = key₁
  ; ltree = (bt-node record { key = record { coler = red ; number = number₁ }
    ; ltree = (bt-node record { key = record { coler = red ; number = number } ; ltree = ltree ; rtree = rtree₁ })
    ; rtree = lr })
  ; rtree = rtree }
  ; varn = varn ; varl = varl } next exit
  = next node


{-
split_branch bt-empty n stack = bt-empty
split_branch node@(bt-node (record { key = _; ltree = bt-empty; rtree = _}) ) n stack = node
split_branch node@(bt-node (record { key = record { coler = y ; number = x }
  ; ltree = (bt-node (record { key = record { coler = red ; number = rx }
    ; ltree = (bt-node (record { key = record { coler = red ; number = rrx }; ltree = rrl; rtree = rrr }) )
    ; rtree = lr }))
  ; rtree = r }) ) n stack
    = {!!}
    --merge-rotate_right node n stack

split_branch node@(bt-node (record { key = record { coler = y ; number = x }
  ; ltree = (bt-node (record { key = record { coler = red ; number = rx }
    ; ltree = (bt-node (record { key = record { coler = black ; number = rrx }; ltree = lll; rtree = rrr }))
    ; rtree = lr  }) )
  ; rtree = r }) ) n stack
    = {!!}
    --merge-tree node n stack

split_branch node@(bt-node (record { key = record { coler = y ; number = x }
  ; ltree = (bt-node (record { key = record { coler = red ; number = rx }
    ; ltree =  bt-empty
    ; rtree = rr }) )
  ; rtree = r}) ) n stack
    = {!!}
    --merge-tree node n stack

split_branch node@(bt-node (record { key = record { coler = y ; number = x }
  ; ltree = (bt-node (record { key = record { coler = black ; number = rx }; ltree = rl; rtree = rr }) )
  ; rtree = r }) ) n stack
    = {!!}
    --merge-tree node n stack
-}
-- 左回転、exitはsplit_branchへ　nextもsplit_branchへ
merge-rotate_left : {le : Level} {t : Set le} → Env → (next : Env → t) → (exit : Env → t) → t
merge-rotate_left node@record { vart = bt-empty ; varn = varn ; varl = varl }  next exit = exit node
merge-rotate_left node@record { vart = bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = bt-empty } ; varn = varn ; varl = varl } next exit = exit node
merge-rotate_left record { vart = bt-node record { key = record { coler = y ; number = x } ; ltree = l ; rtree = (bt-node record { key = record { coler = ry ; number = rx } ; ltree = rl ; rtree = rr }) } ; varn = varn ; varl = varl } next exit
  = next record { vart = bt-node record { key = record { coler = y ; number = rx }
  　　; ltree = (bt-node record { key = record { coler = red ; number = x } ; ltree = l ; rtree = rl })
  　　; rtree = rr}
  　　; varn = varn ; varl = varl }

{-
merge-rotate_left bt-empty n stack = bt-empty
merge-rotate_left node@(bt-node (record { key = _; ltree = _; rtree = bt-empty }) ) n stack = node
merge-rotate_left (bt-node (record { key = record { coler = y ; number = x }
  ; ltree = l
  ; rtree = (bt-node (record { key = record { coler = ry ; number = rx }; ltree = rl; rtree = rr }) ) }) ) n stack
  = split_branch (bt-node (record { key = record { coler = y ; number = rx }
    ; ltree = (bt-node (record { key = record { coler = red ; number = x }; ltree = l; rtree = rl }) )
    ; rtree = rr }) ) n stack
-}

-- skew exitがsplitへ　nextが左回転
skew-bt : {le : Level} {t : Set le} → Env → (next : Env → t) → (exit : Env → t) → t
skew-bt node@record { vart = bt-empty ; varn = varn ; varl = varl } next exit =  exit node
skew-bt node@record { vart = (bt-node record { key = key ; ltree = ltree ; rtree = bt-empty }) ; varn = varn ; varl = varl } next exit = exit node
skew-bt node@record { vart = (bt-node record { key = key₁
  ; ltree = ltree₁
  ; rtree = (bt-node record { key = record { coler = black ; number = number }
  ; ltree = ltree ; rtree = rtree }) }) ; varn = varn ; varl = varl } next exit
    = exit node
skew-bt node@record { vart = (bt-node record { key = key₁
  ; ltree = ltree₁
  ; rtree = (bt-node record { key = record { coler = red ; number = number } ; ltree = ltree ; rtree = rtree }) })
  ; varn = varn ; varl = varl } next exit
    = next node


{-
skew bt-empty n stack = bt-empty
skew node@(bt-node (record { key = _; ltree = _ ; rtree = bt-empty }) ) n stack = node
skew node@(bt-node (record { key = record { coler = y ; number = x }
  ; ltree = l
  ; rtree = r@(bt-node (record { key = record { coler = red ; number = lx }; ltree = ll; rtree = lr }) )}) ) n stack
    = merge-rotate_left node n stack

skew node@(bt-node (record { key = record { coler = y ; number = x }
  ; ltree = l
  ; rtree = (bt-node (record { key = record { coler = black ; number = lx }; ltree = ll; rtree = lr }) ) }) ) n stack
    = split_branch node n stack
-}


-- test : {l : Level} {t : Set l} → (Code : Env → t) → t
-- test next = next (record {vart = bt-node (record { key = record { coler = red ; number = 0 }; ltree = bt-empty; rtree = bt-empty }); varn = 3; varl = []} )


init_node_coler : Env → Env
init_node_coler node@record { vart = bt-empty ; varn = varn ; varl = varl } = node
init_node_coler record { vart = (bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = rtree }) ; varn = varn ; varl = varl }
  = record { vart = (bt-node record { key = record { coler = black ; number = number } ; ltree = ltree ; rtree = rtree }) ; varn = varn ; varl = varl }



merge-tree node@record { vart = vart ; varn = varn ; varl = [] } next exit = exit (init_node_coler node)
merge-tree node@record { vart = vart ; varn = varn ; varl = (bt-empty ∷ varl₁) } next exit = exit node
merge-tree record { vart = vart ; varn = varn ; varl = (bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = rtree } ∷ varl₁) } next exit with <-cmp varn number
merge-tree record { vart = vart ; varn = varn ; varl = (bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = rtree } ∷ varl₁) } next exit | tri≈ ¬a b ¬c
  = next record { vart = (bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = rtree }) ; varn = number ; varl = varl₁ }

merge-tree record { vart = vart ; varn = varn ; varl = (bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = rtree } ∷ varl₁) } next exit | tri> ¬a ¬b c
  = next record { vart = (bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = vart }) ; varn = number ; varl = varl₁ }

merge-tree record { vart = vart ; varn = varn ; varl = (bt-node record { key = record { coler = coler ; number = number } ; ltree = ltree ; rtree = rtree } ∷ varl₁) } next exit | tri< a ¬b ¬c
  = next record { vart = (bt-node record { key = record { coler = coler ; number = number } ; ltree = vart ; rtree = rtree }) ; varn = number ; varl = varl₁ }

{-
merge-tree {le} {t} node n []  = node
merge-tree node n (bt-empty ∷ xs) =  bt-empty
merge-tree target n ( (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) ∷ xs) with <-cmp n x
merge-tree {le} {t} target n ( (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) ∷ xs) | tri≈ ¬a b ¬c
  = merge-tree {le} {t} (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r } ) ) x xs

merge-tree {le} {t} target n ( (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) ∷ xs) | tri> ¬a ¬b c
  = merge-tree {le} {t} (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = target } ) ) x xs

merge-tree {le} {t} target n ( (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) ∷ xs) | tri< a ¬b ¬c
  = merge-tree {le} {t} (bt-node (record { key = record { coler = y; number = x }; ltree = target; rtree = r } ) ) x xs
-}



-- insert
bt-insert : {le : Level} {t : Set le} → Env → (next : Env → t) → (exit : Env → t) → t
-- mergeへ遷移する
bt-insert (record { vart = bt-empty ; varn = varn ; varl = varl }) next exit
  = exit (record { vart = (bt-node (record { key = record { coler = red ; number = varn }; ltree = bt-empty; rtree = bt-empty })) ; varn = varn ; varl = varl })

-- 場合分けを行う
bt-insert record {vart = (bt-node record { key = record { coler = y; number = x } ; ltree = ltree ; rtree = rtree }) ; varn = n ; varl = varl } next exit with <-cmp x n
bt-insert node@(record { vart = (bt-node record { key = key ; ltree = ltree ; rtree = rtree }) ; varn = varn ; varl = varl } ) next exit | tri≈ ¬a b ¬c
  = exit node

bt-insert record {vart = (bt-node record { key = key ; ltree = ltree ; rtree = rtree }) ; varn = varn ; varl = varl } next exit | tri> ¬a ¬b c
  = next (record {vart = ltree ; varn = varn ; varl = (bt-node record { key = key ; ltree = bt-empty ; rtree = rtree }) ∷ varl })

bt-insert record {vart = (bt-node record { key = key ; ltree = ltree ; rtree = rtree }) ; varn = varn ; varl = varl } next exit | tri< a ¬b ¬c
  = next (record {vart = rtree ; varn = varn ; varl = (bt-node record { key = key ; ltree = ltree ; rtree = bt-empty }) ∷ varl })

-- normal loop without termination
{-# TERMINATING #-}
insert : {l : Level} {t : Set l} → Env → (exit : Env → t) → t
insert env exit = bt-insert env (λ env → insert env exit ) exit

{-# TERMINATING #-}
merge : {l : Level} {t : Set l} → Env → (exit : Env → t) → t

{-# TERMINATING #-}
split-p : {l : Level} {t : Set l} → Env → (exit : Env → t) → t
split-p env exit = split env (λ env → merge env (λ env → merge env exit ) ) exit

{-# TERMINATING #-}
rotate_right : {l : Level} {t : Set l} → Env → (exit : Env → t) → t
rotate_right env exit = merge-rotate_right env (λ env → split-p env (λ env → merge env exit ) ) exit

{-# TERMINATING #-}
split-b : {l : Level} {t : Set l} → Env → (exit : Env → t) → t
split-b env exit = split_branch env (λ env → rotate_right env exit ) λ env → merge env exit

{-# TERMINATING #-}
rotate_left : {l : Level} {t : Set l} → Env → (exit : Env → t) → t
rotate_left env exit = merge-rotate_left env (λ env → split-b env exit ) exit

{-# TERMINATING #-}
skew : {l : Level} {t : Set l} → Env → (exit : Env → t) → t
skew env exit = skew-bt env (λ env → rotate_left env (λ env → split-b env exit ) ) exit

merge env exit = merge-tree env (λ env → skew env exit ) exit


-- equivalent of whileLoopP but it looks like an induction on varn

--whileTestP : {l : Level} {t : Set l} → (c10 : ℕ) → (c11 : ℕ) → (Code : Envc → t) → t
--whileTestP c10 n next = next (record {varn = c10 ; vari = 0 ; c10 = c10 } )

whileTestP : {l : Level} {t : Set l} → (tree : rbt) → (n : ℕ) → (Code : Env → t) → t
whileTestP tree n next = next (record {vart = tree ; varn = n ; varl = [] } )

whileTestPCall : (tree : rbt) → (n : ℕ)  → Env
whileTestPCall tree n = whileTestP {_} {_} tree n (λ env → insert env (λ env → merge env (λ env →  env)) )

test1 = whileTestPCall bt-empty 0
test2 = whileTestPCall (vart test1) 1
test3 = whileTestPCall (vart test2) 2
test4 = whileTestPCall (vart test3) 3
test5 = whileTestPCall (vart test4) 4
test6 = whileTestPCall (vart test5) 5
test7 = whileTestPCall (vart test6) 6

--whileTestPCall tree n = whileTestP {_} {_} tree n (λ env → merge env (λ env2 →  env2))
{-
bt-insert bt-empty n stack = merge-tree {le} {t} (bt-node (record { key = record { coler = red ; number = n }; ltree = bt-empty; rtree = bt-empty }) ) n stack
bt-insert (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) n stack with <-cmp x n
bt-insert (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) n stack | tri≈ ¬a b ¬c
  = (bt-node (record { key = record { coler = y ; number = x }; ltree = l; rtree = r }) )
bt-insert {le} {t} (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) n stack | tri> ¬a ¬b c
  = bt-insert {le} {t} l n ( (bt-node (record { key = record { coler = y ; number = x }; ltree = bt-empty; rtree = r }) ) ∷ stack)
bt-insert  {le} {t} (bt-node (record { key = record { coler = y; number = x }; ltree = l; rtree = r }) ) n stack | tri< a ¬b ¬c
  = bt-insert {le} {t} r n ( (bt-node (record { key = record { coler = y ; number = x }; ltree = l; rtree = bt-empty }) ) ∷ stack )
-}

--insert : bst → ℕ → bst
--insert node x = init_node_coler (bt-insert node x)

{-
-- 以下test部分
test-rbt1 : {le : Level} {t : Set le} → bst
test-rbt1 {le} {t} = bt-insert {le} {t} bt-empty 0 []

test-rbt2 : {le : Level} {t : Set le} → bst
test-rbt2 {le} {t} = bt-insert {le} {t} test-rbt1 1 []

--test-rbt3 = bt-insert test-rbt2 2 []
--test-rbt4 = bt-insert test-rbt3 3 []
--test-rbt5 = bt-insert test-rbt4 4 []
--test-rbt6 = bt-insert test-rbt5 5 []
-}