module hoareBinaryTree where

open import Level renaming (zero to Z ; suc to succ)

open import Data.Nat hiding (compare)
open import Data.Nat.Properties as NatProp
open import Data.Maybe
-- open import Data.Maybe.Properties
open import Data.Empty
open import Data.List
open import Data.Product

open import Function as F hiding (const)

open import Relation.Binary
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary
open import logic


_iso_ : {n : Level} {a : Set n} → ℕ → ℕ → Set
d iso d' = (¬ (suc d ≤ d')) ∧ (¬ (suc d' ≤ d))

iso-intro : {n : Level} {a : Set n} {x y : ℕ} → ¬ (suc x ≤ y) → ¬ (suc y ≤ x) → _iso_ {n} {a} x y
iso-intro = λ z z₁ → record { proj1 = z ; proj2 = z₁ }

--
--
--  no children , having left node , having right node , having both
--
data bt {n : Level} (A : Set n) : Set n where
  leaf : bt A
  node :  (key : ℕ) → (value : A) →
    (left : bt A ) → (right : bt A ) → bt A

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 )

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)

{-# 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

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)

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

{-# 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 $ λ t1 → replace-loop key value t1 st exit 

insertTest1 = insertTree leaf 1 1 (λ x → x )
insertTest2 = insertTree insertTest1 2 1 (λ x → x )

open import Data.Unit hiding ( _≟_ ;  _≤?_ ; _≤_)

treeInvariant : {n : Level} {A : Set n} → (tree : bt A) → Set
treeInvariant leaf = ⊤
treeInvariant (node key value leaf leaf) = ⊤
treeInvariant (node key value leaf n@(node key₁ value₁ t₁ t₂)) =  (key < key₁) ∧ treeInvariant n 
treeInvariant (node key value n@(node key₁ value₁ t t₁) leaf) =  treeInvariant n ∧ (key < key₁) 
treeInvariant (node key value n@(node key₁ value₁ t t₁) m@(node key₂ value₂ t₂ t₃)) = treeInvariant n ∧  (key < key₁) ∧ (key₁ < key₂) ∧ treeInvariant m

treeInvariantTest1  = treeInvariant (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf)))

stackInvariant : {n : Level} {A : Set n} → (tree : bt A) → (stack  : List (bt A)) → Set n
stackInvariant {_} {A} _ [] = Lift _ ⊤
stackInvariant {_} {A} tree (tree1 ∷ [] ) = tree1 ≡ tree
stackInvariant {_} {A} tree (x ∷ tail @ (node key value leaf right ∷ _) ) = (right ≡ x) ∧ stackInvariant tree tail
stackInvariant {_} {A} tree (x ∷ tail @ (node key value left leaf ∷ _) ) = (left ≡ x) ∧ stackInvariant tree tail
stackInvariant {_} {A} tree (x ∷ tail @ (node key value left right ∷ _  )) = ( (left ≡ x) ∧ stackInvariant tree tail) ∨ ( (right ≡ x) ∧ stackInvariant tree tail)
stackInvariant {_} {A} tree s = Lift _ ⊥

rstackInvariant : {n : Level} {A : Set n} → (tree : bt A) → (stack  : List (bt A)) → Set n
rstackInvariant {_} {A} _ [] = Lift _ ⊤
rstackInvariant {_} {A} tree (tree1 ∷ [] ) = tree1 ≡ tree
rstackInvariant {_} {A} tree (node key value leaf right ∷ x ∷ y )  = (right ≡ x) ∧ rstackInvariant tree (x ∷ y)
rstackInvariant {_} {A} tree (node key value left leaf ∷ x ∷ y )  = (left ≡ x) ∧ rstackInvariant tree (x ∷ y)
rstackInvariant {_} {A} tree (node key value left right ∷ x  ∷ y  ) = ( (left ≡ x) ∧ rstackInvariant tree (x ∷ y)) ∨ ( (right ≡ x) ∧ rstackInvariant tree (x ∷ y))
rstackInvariant {_} {A} tree s = Lift _ ⊥

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

replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (treeInvariant tree )
    → ((tree : bt A) → treeInvariant tree → (rstack : List (bt A))  → rstackInvariant tree rstack → t) → t
replaceNodeP k v leaf P next = next (node k v leaf leaf) {!!} {!!} {!!}
replaceNodeP k v (node key value t t₁) P next = next (node k v t t₁) {!!} {!!} {!!}

replaceP : {n m : Level} {A : Set n} {t : Set m}
     → (key : ℕ) → (value : A) → (tree repl : bt A) → (stack rstack : List (bt A)) → treeInvariant tree ∧ stackInvariant tree stack ∧ rstackInvariant repl rstack
     → (next : ℕ → A → (tree1 : bt A) → (stack rstack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 stack ∧ rstackInvariant repl rstack → bt-depth tree1 < bt-depth tree   → t )
     → (exit : (tree1 repl : bt A) → (rstack : List (bt A))  → treeInvariant tree1 ∧ rstackInvariant repl rstack → t) → t
replaceP key value tree repl [] rs Pre next exit = exit tree repl {!!} {!!}  
replaceP key value tree repl (leaf ∷ st) rs Pre next exit = next key value tree st {!!} {!!} {!!}
replaceP key value tree repl (node key₁ value₁ left right ∷ st) rs Pre 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 {!!} {!!} {!!}

open import Relation.Binary.Definitions

nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥
nat-≤>  (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x
lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥
lemma3 refl ()
lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥
lemma5 (s≤s z≤n) ()

TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ)
   → (r : Index) → (p : Invraiant r)  
   → (loop : (r : Index)  → Invraiant r → (next : (r1 : Index)  → Invraiant r1 → reduce r1 < reduce r  → t ) → t) → t
TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r)
... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) 
... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (≤-step lt1) p1 lt1 ) where 
    TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j  → Invraiant r1 →  reduce r1 < reduce r → t
    TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1)) 
    TerminatingLoop1 (suc j) r r1  n≤j p1 lt with <-cmp (reduce r1) (suc j)
    ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt 
    ... | tri≈ ¬a b ¬c = loop r1 p1 (λ r2 p2 lt1 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b lt1 ) p2 lt1 )
    ... | tri> ¬a ¬b c =  ⊥-elim ( nat-≤> c n≤j )   

open _∧_

insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree
     → (exit : (tree repl : bt A) → (rstack : List (bt A)) → treeInvariant tree ∧ rstackInvariant repl rstack  → t ) → t
insertTreeP {n} {m} {A} {t} tree key value P exit =
   TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant (proj1 p) (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫  ⟪ P , lift tt  ⟫
       $ λ p P loop → findP key (proj1 p)  (proj2 p) P (λ t s P1 lt → loop ⟪ t ,  s  ⟫ P1 lt )
       $ λ t s P → replaceNodeP key value t (proj1 P)
       $ λ t1 P1 r R → TerminatingLoopS (bt A ∧ List (bt A) ∧ List (bt A))
            {λ p → treeInvariant (proj1 p) ∧ stackInvariant (proj1 p) (proj1 (proj2 p))  ∧ rstackInvariant t1 (proj2 (proj2 p))}
               (λ p → bt-depth (proj1 p)) ⟪ t , ⟪ s , r ⟫ ⟫　⟪ proj1 P , ⟪ proj2 P , R ⟫ ⟫
       $  λ p P1 loop → replaceP key value (proj1 p) t1 (proj1 (proj2 p)) (proj2 (proj2 p)) P1 (λ k v t s s1 P2 lt → loop ⟪ t , ⟪  s , s1 ⟫ ⟫ P2 lt ) exit 
top-value : {n : Level} {A : Set n} → (tree : bt A) →  Maybe A 
top-value leaf = nothing
top-value (node key value tree tree₁) = just value

insertTreeSpec0 : {n : Level} {A : Set n} → (tree : bt A) → (value : A) → top-value tree ≡ just value → ⊤
insertTreeSpec0 _ _ _ = tt

insertTreeSpec1 : {n : Level} {A : Set n}  → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → ⊤
insertTreeSpec1 {n} {A}  tree key value P =
         insertTreeP tree key value P (λ  (tree₁ repl : bt A) (rstack : List (bt A)) 
            (P1 : treeInvariant tree₁ ∧ rstackInvariant repl rstack ) → insertTreeSpec0 tree₁ value (lemma1 {!!} ) ) where
                lemma1 : (tree₁ : bt A) → top-value tree₁ ≡ just value
                lemma1 = {!!}