--- a/hoareBinaryTree.agda	Sun Nov 14 15:50:30 2021 +0900
+++ b/hoareBinaryTree.agda	Mon Nov 15 15:04:06 2021 +0900
@@ -110,19 +110,19 @@
 treeInvariantTest1  : treeInvariant treeTest1
 treeInvariantTest1  = t-right (m≤m+n _ 1) (t-node (add< 0) (add< 1) (t-left (add< 1) (t-single 4 7)) (t-single 5 5) )
 
-data stackInvariant {n : Level} {A : Set n} (key0 : ℕ) : (tree tree0 : bt A) → (stack  : List (bt A)) → Set n where
-    s-nil : stackInvariant  key0 leaf leaf [] 
-    s-single : (tree : bt A) → stackInvariant key0 tree tree (tree ∷ [] ) 
-    s-right      : {tree0 tree : bt A} → {key : ℕ } → {value : A } { left  : bt A} → {st : List (bt A)}
-         → key < key0 → stackInvariant key0(node key value left tree ) tree0 (node key value left tree ∷ st )  → stackInvariant key0 tree tree0 (tree  ∷ node key value left tree ∷ st ) 
-    s-left      : {tree0 tree : bt A} → {key : ℕ } → {value : A } { right  : bt A} → {st : List (bt A)}
-         → key0 < key → stackInvariant key0(node key value tree right ) tree0 (node key value tree right ∷ st )  → stackInvariant key0 tree tree0 (tree  ∷ node key value tree right ∷ st ) 
+data stackInvariant {n : Level} {A : Set n}  : (tree tree0 : bt A) → (stack  : List (bt A)) → Set n where
+    s-nil : stackInvariant  leaf leaf [] 
+    s-single : (tree : bt A) → stackInvariant tree tree (tree ∷ [] ) 
+    s-right :  {tree0 tree tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)}
+        →  stackInvariant (node key₁ v1 tree tree₁) tree0 st → stackInvariant tree₁ tree0 (tree₁ ∷ st)
+    s-left :  {tree0 tree tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)}
+        →  stackInvariant (node key₁ v1 tree tree₁) tree0 st → stackInvariant tree tree0 (tree  ∷ st)
 
-stackInvariantTest0 : stackInvariant {_} {ℕ} 1 leaf leaf []
+stackInvariantTest0 : stackInvariant {_} {ℕ} leaf leaf []
 stackInvariantTest0 = s-nil
 
-stackInvariantTest1 : stackInvariant 3 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] )
-stackInvariantTest1 = s-right (add< 1) (s-single treeTest1 )
+stackInvariantTest1 : stackInvariant treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] )
+stackInvariantTest1 = s-right (s-single treeTest1 )
 
 data replacedTree  {n : Level} {A : Set n} (key : ℕ) (value : A)  : (tree tree1 : bt A ) → Set n where
     r-leaf : replacedTree key value leaf (node key value leaf leaf)
@@ -131,7 +131,6 @@
           → k > key →  replacedTree key value t1 t2 →  replacedTree key value (node k v1 t t1) (node k v1 t t2) 
     r-left : {k : ℕ } {v1 : A} → {t t1 t2 : bt A}
           → k < key →  replacedTree key value t1 t2 →  replacedTree key value (node k v1 t1 t) (node k v1 t2 t) 
-
 depth-1< : {i j : ℕ} →   suc i ≤ suc (i Data.Nat.⊔ j )
 depth-1< {i} {j} = s≤s (m≤m⊔n _ j)
 
@@ -154,28 +153,22 @@
 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = t-leaf
 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti₁
 
-siConsLeft   : {n : Level } {A : Set n} (key  key₁ : ℕ) → { v1 : A } (tree tree₁ tree0 : bt A ) (st : List (bt A))  
-      → key < key₁ →  stackInvariant key (node key₁ v1 tree tree₁) tree0 st
-      → treeInvariant (node key₁ v1 tree tree₁)
-      → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st)
-siConsLeft {n} {A} k k1 {v1} t t1 t0 st k<k1 ti si   = {!!}
-
 --        stackInvariant key (node key₁ v1 tree tree₁) tree0 st
 --        → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st)
 
 open _∧_
 
 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A))
-           →  treeInvariant tree ∧ stackInvariant key tree tree0 stack  
-           → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree   → t )
-           → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → t ) → t
+           →  treeInvariant tree ∧ stackInvariant tree tree0 stack  
+           → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 tree0 stack → bt-depth tree1 < bt-depth tree   → t )
+           → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 tree0 stack → t ) → t
 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st Pre
 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 b ¬c = exit n tree0 st Pre
-findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) ⟪ treeLeftDown tree tree₁ (proj1 Pre)  , findP1 a (proj2 Pre) ⟫ depth-1< where
-   findP1 : key < key₁ →  stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st)
-   findP1 a si = siConsLeft  key  key₁ {v1} tree tree₁ tree0 st  a  si (proj1 Pre) 
-findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} depth-2<
+findP 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 (proj2 Pre) ⟫ depth-1< where
+   findP1 : key < key₁ →  stackInvariant (node key₁ v1 tree tree₁) tree0 st → stackInvariant tree tree0 (tree ∷ st)
+   findP1 a si = s-left si
+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 (proj2 Pre) ⟫ depth-2<
 
 
 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (treeInvariant tree )
@@ -184,8 +177,8 @@
 replaceNodeP k v1 (node key value t t₁) P next = next (node k v1 t t₁) {!!} {!!}
 
 replaceP : {n m : Level} {A : Set n} {t : Set m}
-     → (key : ℕ) → (value : A) → (tree repl : bt A) → (stack : List (bt A)) → treeInvariant tree ∧ stackInvariant key repl tree stack ∧ replacedTree key value tree repl
-     → (next : ℕ → A → (tree1 repl : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key repl tree1 stack ∧ replacedTree key value tree1 repl → bt-depth tree1 < bt-depth tree   → t )
+     → (key : ℕ) → (value : A) → (tree repl : bt A) → (stack : List (bt A)) → treeInvariant tree ∧ stackInvariant repl tree stack ∧ replacedTree key value tree repl
+     → (next : ℕ → A → (tree1 repl : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant repl tree1 stack ∧ replacedTree key value tree1 repl → bt-depth tree1 < bt-depth tree   → t )
      → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t
 replaceP key value tree repl [] Pre next exit = exit tree repl {!!} 
 replaceP key value tree repl (leaf ∷ st) Pre next exit = next key value tree {!!} st {!!} {!!}
@@ -229,11 +222,11 @@
 insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree
      → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t
 insertTreeP {n} {m} {A} {t} tree key value P exit =
-   TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant key (proj1 p) tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫  ⟪ P , {!!}  ⟫
+   TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant (proj1 p) tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫  ⟪ P , {!!}  ⟫
        $ λ p P loop → findP key (proj1 p)  tree (proj2 p) {!!} (λ t _ s P1 lt → loop ⟪ t ,  s  ⟫ {!!} lt )
        $ λ t _ s P → replaceNodeP key value t (proj1 P)
        $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A ))
-            {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant key (proj1 (proj2 p)) tree (proj1 p)  ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
+            {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) tree (proj1 p)  ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
                (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ proj1 P , ⟪ {!!}  , R ⟫ ⟫
        $  λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) {!!}
             (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1  , repl1  ⟫ ⟫ {!!} lt )  exit 
@@ -249,7 +242,7 @@
    field
      tree0 : bt A
      ti : treeInvariant tree0
-     si : stackInvariant key tree tree0 stack
+     si : stackInvariant tree tree0 stack
      ci : C tree stack     -- data continuation
    
 findPP : {n m : Level} {A : Set n} {t : Set m}
@@ -263,7 +256,7 @@
 findPP {_} {_} {A} key n@(node key₁ v1 tree tree₁) st Pre next exit | tri< a ¬b ¬c =
           next tree (n ∷ st) (record {ti = findPR.ti Pre  ; si = findPP2 st (findPR.si Pre) ; ci = lift tt} ) findPP1 where 
     tree0 =  findPR.tree0 Pre 
-    findPP2 : (st : List (bt A)) → stackInvariant key {!!} tree0 st →  stackInvariant key {!!} tree0 (node key₁ v1 tree tree₁ ∷ st)
+    findPP2 : (st : List (bt A)) → stackInvariant {!!} tree0 st →  stackInvariant {!!} tree0 (node key₁ v1 tree tree₁ ∷ st)
     findPP2 = {!!}
     findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁)
     findPP1 =  depth-1<
@@ -278,7 +271,7 @@
        $ λ p P loop → findPP key (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t ,  s  ⟫ P1 lt )
        $ λ t s _ P → replaceNodeP key value t {!!}
        $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A ))
-            {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant key (proj1 (proj2 p)) tree (proj1 p)  ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
+            {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) tree (proj1 p)  ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
                (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ {!!} , ⟪ {!!}  , R ⟫ ⟫
        $  λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) {!!}
             (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1  , repl1  ⟫ ⟫ {!!} lt )  exit 
@@ -310,6 +303,6 @@
            lemma6 : (t1 : bt A) (s1 : List (bt A)) (found? : (t1 ≡ leaf) ∨ (node-key t1 ≡ just key)) (P2 : findPR key t1 s1 (findPC key value)) → top-value t1 ≡ just value
            lemma6 t1 s1 found? P2 = lemma7 t1 s1 (findPR.tree0 P2) ( findPC.tree1  (findPR.ci P2)) ( findPC.ci  (findPR.ci P2)) (findPR.si P2) found? where
               lemma7 :  (t1 : bt A) ( s1 : List (bt A) ) (tree0 tree1 : bt A) →
-                 replacedTree key value t1 tree1 → stackInvariant key t1 tree0 s1  → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key)  →   top-value t1 ≡ just value
+                 replacedTree key value t1 tree1 → stackInvariant t1 tree0 s1  → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key)  →   top-value t1 ≡ just value
               lemma7 = {!!}