--- a/hoareBinaryTree.agda	Sun Nov 21 10:54:13 2021 +0900
+++ b/hoareBinaryTree.agda	Sun Nov 21 14:36:55 2021 +0900
@@ -104,7 +104,10 @@
 
 data stackInvariant {n : Level} {A : Set n}  (key : ℕ) : (top orig : bt A) → (stack  : List (bt A)) → Set n where
     s-nil :  {tree : bt A} → stackInvariant key tree tree []
-    s-single :  {tree : bt A} → stackInvariant key tree tree [] → stackInvariant key tree tree (tree ∷ [])
+    s-right0 :  {tree tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } 
+        → key₁ < key  →  stackInvariant key (node key₁ v1 tree₁ tree) (node key₁ v1 tree₁ tree) [] →  stackInvariant key tree (node key₁ v1 tree₁ tree) (tree ∷ (node key₁ v1 tree₁ tree) ∷ [])
+    s-left0 :  {tree₁ tree : bt A} → {key₁ : ℕ } → {v1 : A } 
+        → key  < key₁ →  stackInvariant key (node key₁ v1 tree₁ tree) (node key₁ v1 tree₁ tree) [] →  stackInvariant key tree₁ (node key₁ v1 tree₁ tree) (tree₁ ∷ (node key₁ v1 tree₁ tree) ∷ [])
     s-right :  {tree tree0 tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} 
         → key₁ < key  →  stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → ¬ (st ≡ []) →  stackInvariant key tree tree0 (tree ∷ st)
     s-left :  {tree₁ tree0 tree : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} 
@@ -120,7 +123,8 @@
 
 replFromStack : {n : Level} {A : Set n}  {key : ℕ} {top orig : bt A} → {stack  : List (bt A)} →  stackInvariant key top orig stack → bt A
 replFromStack (s-nil {tree} ) = tree
-replFromStack (s-single {tree} _ ) = tree
+replFromStack (s-right0 {tree} _ _ ) = tree
+replFromStack (s-left0 {tree} _ _ ) = tree
 replFromStack (s-right {tree} x _ st) = tree
 replFromStack (s-left {tree} x _ st) = tree
 
@@ -147,25 +151,31 @@
 stack-last (x ∷ s) = stack-last s
 
 stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] )
-stackInvariantTest1 = s-right (add< 2) (s-single s-nil) (λ ())
+stackInvariantTest1 = s-right0 (add< 2) s-nil
+
+si-nil :  {n : Level} {A : Set n} {key : ℕ} {tree tree0 : bt A} → (si : stackInvariant key tree tree0 []) → tree ≡ tree0
+si-nil s-nil = refl
 
 si-property1 :  {n : Level} {A : Set n} (key : ℕ) (tree tree0 : bt A) → (stack  : List (bt A)) → ¬ (stack ≡ [])  → stackInvariant key tree tree0 stack
    → stack-top stack ≡ just tree
 si-property1 key t t0 [] ne (s-nil ) = ⊥-elim ( ne refl )
-si-property1 key t t0 (t ∷ []) ne (s-single _) = refl
+si-property1 key t t0 (t ∷ _ ∷ []) ne (s-right0 _ _) = refl
+si-property1 key t t0 (t ∷ _ ∷ []) ne (s-left0 _ _) = refl
 si-property1 key t t0 (t ∷ st) _ (s-right _ _ si) = refl
 si-property1 key t t0 (t ∷ st) _ (s-left _ _ si) = refl
 
 si-property-last :  {n : Level} {A : Set n}  (key : ℕ) (tree tree0 : bt A) → (stack  : List (bt A)) → ¬ (stack ≡ [])   → stackInvariant key tree tree0 stack
    → stack-last stack ≡ just tree0
 si-property-last key t t0 [] ne s-nil  = ⊥-elim ( ne refl )
-si-property-last key t t0 (t ∷ [])  _ (s-single _)  = refl
+si-property-last key t t0 (t ∷ _ ∷ [])  _ (s-left0 _ _)  = {!!}
+si-property-last key t t0 (t ∷ _ ∷ [])  _ (s-right0 _ _)  = {!!}
 si-property-last key t t0 (t ∷ [])  _ (s-right _ _ ne)  = ⊥-elim ( ne refl )
 si-property-last key t t0 (t ∷ [])  _ (s-left _ _ ne)  = ⊥-elim ( ne refl )
 si-property-last key t t0 (.t ∷ x ∷ st) ne (s-right _ si _) with  si-property1 key _ _ (x ∷ st) (λ ()) si
 ... | refl = si-property-last key x t0 (x ∷ st)  (λ ()) si
 si-property-last key t t0 (.t ∷ x ∷ st) ne (s-left _ si _) with  si-property1 key _ _ (x ∷ st)  (λ ()) si
 ... | refl = si-property-last key x t0 (x ∷ st)  (λ ()) si
+
 ti-right : {n  : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} →  treeInvariant  (node key₁ v1 tree₁ repl) → treeInvariant repl
 ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf
 ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-right x ti) = ti
@@ -181,7 +191,8 @@
 stackTreeInvariant : {n  : Level} {A : Set n} (key : ℕ) (sub tree : bt A) → (stack : List (bt A))
    →  treeInvariant tree → stackInvariant key sub tree stack  → treeInvariant sub
 stackTreeInvariant {_} {A} key sub tree [] ti s-nil = ti
-stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-single _) = ti
+stackTreeInvariant {_} {A} key sub tree (sub ∷ _ ∷ []) ti (s-left0 _ _) = {!!}
+stackTreeInvariant {_} {A} key sub tree (sub ∷ _ ∷ []) ti (s-right0 _ _) = {!!}
 stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si _) = ti-right (si1 si) where
    si1 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} →  stackInvariant key (node key₁ v1 tree₁ sub ) tree st → treeInvariant  (node key₁ v1 tree₁ sub )
    si1 {tree₁ }  {key₁ }  {v1 }  si = stackTreeInvariant  key (node key₁ v1 tree₁ sub ) tree st ti si
@@ -233,10 +244,11 @@
 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st Pre (case1 refl)
 findP key (node key₁ v1 tree tree₁) tree0 st Pre next exit with <-cmp key key₁
 findP key n tree0 st Pre _ exit | tri≈ ¬a refl ¬c = exit n tree0 st Pre (case2 refl)
-findP {_} {_} {A} key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (tree ∷ st) ⟪ treeLeftDown tree tree₁ (proj1 Pre)  , findP1 a st (proj2 Pre) ⟫ depth-1< where
+findP {_} {_} {A} key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (tree ∷ st)
+       ⟪ treeLeftDown tree tree₁ (proj1 Pre)  , findP1 a st (proj2 Pre) ⟫ depth-1< where
    findP1 : key < key₁ → (st : List (bt A)) →  stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (tree ∷ st)
+   findP1 a (x ∷ st) si = s-left a si (λ ())
    findP1 a [] s-nil = {!!}
-   findP1 a (x ∷ st) si = s-left a si (λ ())
 findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (tree₁ ∷ st) ⟪ treeRightDown tree tree₁ (proj1 Pre) , s-right c (proj2 Pre) {!!} ⟫ depth-2<
 
 replaceTree1 : {n  : Level} {A : Set n} {t t₁ : bt A } → ( k : ℕ ) → (v1 value : A ) →  treeInvariant (node k v1 t t₁) → treeInvariant (node k value t t₁)