module bt where

open import Data.Nat
open import Level renaming (zero to Z ; suc to succ)
open import Data.List


open import Data.Nat.Properties as NatProp -- <-cmp
open import Relation.Binary

open import Function as F hiding (const)

data bt {n : Level} (A : Set n) : Set n where
  leaf : bt A
  node : (key-t : ℕ) → (value : A) →
    (ltree : bt A ) → (rtree : bt A ) → bt A

record Env {n : Level} (A : Set n) : Set n where
  field
    varn : ℕ
    varv : A
    vart : bt A
    varl : List (bt A)
open Env
{-
bt-depth : {n : Level} {A : Set n} → (env : Env A ) → ℕ
bt-depth env  with vart env
... | leaf = zero
... | node key value ltree rtree = Data.Nat._⊔_ (bt-depth-c ltree ) (bt-depth-c rtree ) where
    bt-depth-c : {n : Level} {A : Set n} → (tree : bt A ) → ℕ
    bt-depth-c leaf = suc zero
    bt-depth-c (node key value lt rt) = suc (Data.Nat._⊔_ (bt-depth-c lt ) ( bt-depth-c rt))
-}

bt-depth : {n : Level} {A : Set n} → (tree : bt A ) → ℕ
bt-depth leaf = 0
bt-depth (node key value t t₁) = Data.Nat._⊔_ (bt-depth t ) (bt-depth t₁ )

bt-depth1 : {n : Level} {A : Set n} → (env : Env A ) → ℕ
bt-depth1 env  with vart env
... | tree = bt-depth-c tree where
    bt-depth-c : {n : Level} {A : Set n} → (tree : bt A ) → ℕ
    bt-depth-c leaf = zero
    bt-depth-c (node key value lt rt) = suc (Data.Nat._⊔_ (bt-depth-c lt ) ( bt-depth-c rt))

test-depth :  ℕ
test-depth = bt-depth (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf)))

test-env : {n : Level} {A : Set n} →  Env ℕ
test-env = record {varn = 0; varv = 4 ; vart = (node 3 2 leaf (node 1 1 leaf (node 3 5 leaf leaf))); varl = []}

test-depth1 :  ℕ
test-depth1 = bt-depth1 record {varn = 0; varv = 4;  vart = (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf))); varl = []}

find : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → List (bt A)
           → (next : bt A → List (bt A) → t )
           → (exit : bt A → List (bt A) → t ) → t
find key leaf st _ exit = exit leaf st
find key (node key₁ v tree tree₁) st next exit with <-cmp key key₁
find key n st _ exit | tri≈ ¬a b ¬c = exit n st
find key n@(node key₁ v tree tree₁) st next _ | tri< a ¬b ¬c = next tree (n ∷ st)
find key n@(node key₁ v tree tree₁) st next _ | tri> ¬a ¬b c = next tree₁ (n ∷ st)

find1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
           → (next : (env : Env A ) → t )
           → (exit : (env : Env A ) → t ) → t
find1 env next exit = find1-c (varn env) (vart env) env next exit where
    find1-c : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A) → (env : Env A )
           → (next : (env : Env A ) → t )
           → (exit : (env : Env A ) → t ) → t
    find1-c key leaf env next exit = exit env
    find1-c key n@(node key-t value ltree rtree) env next exit with <-cmp key key-t
    ... | tri< a ¬b ¬c = find1-c key ltree record env {vart = ltree ; varl = (n ∷ (varl env))} next exit
    ... | tri≈ ¬a b ¬c = exit record env {vart = n}
    ... | tri> ¬a ¬b c = find1-c key rtree record env {vart = rtree ; varl = (n ∷ (varl env))} next exit

{-
... | leaf = exit env
... | n@(node key-t value ltree rtree) with <-cmp (varn env) key-t
... | tri≈ ¬a b ¬c = exit record env {vart = n}
... | tri< a ¬b ¬c = find1 record env {vart = ltree ; varl = (n ∷ (varl env))} next exit
... | tri> ¬a ¬b c = find1 record env {vart = rtree ; varl = (n ∷ (varl env))} next exit
-}

{-# TERMINATING #-}
find-loop : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → bt A → List (bt A)  → (exit : bt A → List (bt A) → t) → t
find-loop {n} {m} {A} {t} key tree st exit = find-loop1 tree st where
    find-loop1 : bt A → List (bt A) → t
    find-loop1 tree st = find key tree st find-loop1 exit

find-loop-1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
    → (exit : (env : Env A ) → t) → t
find-loop-1 env exit = find1 env exit exit

fin-cg : {n m : Level} {A : Set n} {t : Set m} → t → (env : Env A ) → Env A
fin-cg t = (λ env → env )

test-cg = find-loop-1 record {varn = 5; varv = 4; vart = (node 2 2 (node 1 1 leaf leaf) (node 4 4 leaf (node 5 5 leaf leaf))); varl = []} (λ env → env )

replaceNode : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → (bt A → t) → t
replaceNode k v leaf next = next (node k v leaf leaf)
replaceNode k v (node key value t t₁) next = next (node k v t t₁)

replaceNode1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
    → (next : Env A → t) → t
replaceNode1 env next with vart env
... | leaf = next record env {vart = (node (varn env) (varv env) leaf leaf) }
... | node key-t value ltree rtree = next record env {vart = (node (varn env) (varv env) ltree rtree) }

replace : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → List (bt A)
    → (next : ℕ → A → bt A → List (bt A) → t )
    → (exit : bt A → t) → t
replace key value tree [] next exit = exit tree
replace key value tree (leaf ∷ st) next exit = next key value tree st
replace key value tree (node key₁ value₁ left right ∷ st) next exit with <-cmp key key₁
... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) st
... | tri≈ ¬a b ¬c = next key value (node key₁ value  left right ) st
... | tri> ¬a ¬b c = next key value (node key₁ value₁ left tree ) st

replace1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
    → (next : Env A → t )
    → (exit : Env A → t) → t
replace1 env next exit = replace1-c (varl env) env next exit where
    replace1-c : {n m : Level} {A : Set n} {t : Set m} → List (bt A) → (env : Env A )
        → (next : Env A → t )
        → (exit : Env A → t) → t
    replace1-c st env next exit with st
    ... | [] = exit env
    ... | leaf ∷ st1 = replace1-c st1 env next exit
    ... | n@(node key-t value ltree rtree) ∷ st1 with <-cmp (varn env) (key-t)
    ... | tri< a ¬b ¬c = replace1-c st1 record env{vart = (node key-t value (vart env) rtree); varl = st1} next exit
    ... | tri≈ ¬a b ¬c = replace1-c st1 record env{vart = (node key-t (varv env) ltree rtree); varl = st1} next exit
    ... | tri> ¬a ¬b c = replace1-c st1 record env{vart = (node key-t value ltree (vart env)); varl = st1} next exit

{-
replace1 env next exit = next env
replace1 env tree (node key₁ value₁ left right ∷ st) next exit with <-cmp (varn env) key₁
... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) st
... | tri≈ ¬a b ¬c = next key value (node key₁ value  left right ) st
... | tri> ¬a ¬b c = next key value (node key₁ value₁ left tree ) st
-}

{-# TERMINATING #-}
replace-loop : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → List (bt A)  → (exit : bt A → t) → t
replace-loop {_} {_} {A} {t} key value tree st exit = replace-loop1 key value tree st where
    replace-loop1 : (key : ℕ) → (value : A) → bt A → List (bt A) → t
    replace-loop1 key value tree st = replace key value tree st replace-loop1  exit

insertTree : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : bt A → t ) → t
insertTree tree key value exit = find-loop key tree []
  $ λ t st → replaceNode key value t -- find-loop next
  $ λ t1 → replace-loop key value t1 st exit -- replaceNode exit

replace-loop1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
    → (exit : (env : Env A ) → t) → t
replace-loop1 env exit = replace1 env exit exit

insertTree1 : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : Env A  → t ) → t
insertTree1 tree key value exit = find-loop-1 record{varn = key; varv = value; vart = tree ; varl = []}
  $ λ env → replaceNode1 env
    $ λ env → replace-loop1 env exit

insertTree11 : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : Env A  → t ) → t
insertTree11 tree key value exit = find-loop-1 record{varn = key; varv = value; vart = tree ; varl = []}
  $ λ env → replaceNode1 env exit

test-insert = insertTree1 (node 2 2 (node 1 1 leaf leaf) (node 4 4 leaf (node 5 5 leaf leaf))) 0 5 (λ env → env )

test-insert1 = insertTree11 (node 2 2 (node 1 1 leaf leaf) (node 4 4 leaf (node 5 5 leaf leaf))) 0 5 (λ env → env )