--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/hoareBinaryTree1.agda	Wed Feb 26 18:27:54 2020 +0900
@@ -0,0 +1,67 @@
+module hoareBinaryTree1 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
+
+
+data bt {n : Level} (A : Set n) : Set n where
+  bt-leaf :  bt A
+  bt-node : (key : ℕ) → A →
+    (ltree : bt A) → (rtree : bt A) → bt A
+
+bt-find : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → List (bt A)
+    → ( bt A → List (bt A) → t ) → t
+bt-find {n} {m} {A} {t}  key leaf@(bt-leaf) stack exit = exit leaf stack
+bt-find {n} {m} {A} {t}  key (bt-node key₁ AA tree tree₁) stack next with <-cmp key key₁
+bt-find {n} {m} {A} {t}  key node@(bt-node key₁ AA tree tree₁) stack exit | tri≈ ¬a b ¬c = exit node stack
+bt-find {n} {m} {A} {t}  key node@(bt-node key₁ AA ltree rtree) stack next | tri< a ¬b ¬c = bt-find key ltree (node ∷ stack) next
+bt-find {n} {m} {A} {t}  key node@(bt-node key₁ AA ltree rtree) stack next | tri> ¬a ¬b c = bt-find key rtree (node ∷ stack) next
+
+bt-replace : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → A → bt A → List (bt A) → (bt A → t ) → t
+bt-replace {n} {m} {A} {t} ikey a otree stack next = bt-replace0 otree where
+    bt-replace1 : bt A → List (bt A) → t
+    bt-replace1 tree [] = next tree
+    bt-replace1 node ((bt-leaf) ∷ stack) = bt-replace1 node stack
+    bt-replace1 node ((bt-node key₁ b x x₁) ∷ stack) = bt-replace1 (bt-node key₁ b node x₁) stack
+    bt-replace0 : (tree : bt A) → t
+    bt-replace0 tree@(bt-node key _ ltr rtr) = bt-replace1 (bt-node ikey a ltr rtr) stack  -- find case
+    bt-replace0 bt-leaf = bt-replace1 (bt-node ikey a bt-leaf bt-leaf) stack
+
+
+
+
+bt-empty : {n : Level} {A : Set n} → bt A
+bt-empty = bt-leaf
+
+bt-insert : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → A → bt A → (bt A → t ) → t
+bt-insert key a tree next = bt-find key tree [] (λ mtree stack → bt-replace key a mtree  stack (λ tree → next tree) )
+
+find-test :  bt ℕ
+find-test = bt-find 5 bt-empty [] (λ x y → x)
+
+
+insert-test :  bt ℕ
+insert-test = bt-insert 5 7 bt-empty (λ x → x)
+
+insert-test1 :  bt ℕ
+insert-test1 = bt-insert 5 7 bt-empty (λ x → bt-insert 15 17 x (λ y → y))
+
+
+tree+stack : {n : Level} {A : Set n} → (tree mtree : bt A) →  (stack : List (bt A))  → Set n
+tree+stack {n} {A} bt-leaf mtree stack = (mtree ≡ bt-leaf) ∧ (stack ≡ [])
+tree+stack {n} {A} (bt-node key x tree tree₁) mtree stack = bt-replace key x mtree stack (λ ntree → ntree ≡ tree) 
+