--- a/hoareBinaryTree.agda	Mon Nov 08 22:45:19 2021 +0900
+++ b/hoareBinaryTree.agda	Mon Nov 08 23:17:35 2021 +0900
@@ -98,26 +98,26 @@
 treeInvariantTest1  : treeInvariant (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf)))
 treeInvariantTest1  = {!!}
 
-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 ∷ [] ) 
+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-<      : (tree0 tree : bt A) → {key : ℕ } → {value : A } { left  : bt A} → {st : List (bt A)}
-         → stackInvariant (node key value left tree ) tree0 (node key value left tree ∷ st )  → stackInvariant tree tree0 (tree  ∷ node key value left tree ∷ st ) 
+         → 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->      : (tree0 tree : bt A) → {key : ℕ } → {value : A } { right  : bt A} → {st : List (bt A)}
-         → stackInvariant (node key value tree right ) tree0 (node key value tree right ∷ st )  → stackInvariant tree tree0 (tree  ∷ node key value tree right ∷ st ) 
+         → 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 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)
     r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁) 
     r-right : {k : ℕ } {v : A} → {t t1 t2 : bt A}
-          → k > key → ( replacedTree key value t1 t2 →  replacedTree key value (node k v t t1) (node k v t t2) )
+          → k > key →  replacedTree key value t1 t2 →  replacedTree key value (node k v t t1) (node k v t t2) 
     r-left : {k : ℕ } {v : A} → {t t1 t2 : bt A}
-          → k < key → ( replacedTree key value t1 t2 →  replacedTree key value (node k v t1 t) (node k v t2 t) )
+          → k < key →  replacedTree key value t1 t2 →  replacedTree key value (node k v t1 t) (node k v t2 t) 
 
 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A))
-           →  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
+           →  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
 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st {!!}
 findP key (node key₁ v 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 {!!}
@@ -130,8 +130,8 @@
 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 : 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 )
+     → (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 )
      → (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 {!!} {!!}
@@ -175,11 +175,11 @@
 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 (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 key (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 (proj1 (proj2 p)) tree (proj1 p)  ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
+            {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 → 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 
@@ -191,25 +191,25 @@
 insertTreeSpec0 : {n : Level} {A : Set n} → (tree : bt A) → (value : A) → top-value tree ≡ just value → ⊤
 insertTreeSpec0 _ _ _ = tt
 
-record findPR {n : Level} {A : Set n} (tree : bt A ) (stack : List (bt A)) (C : bt A → List (bt A) → Set n) : Set n where
+record findPR {n : Level} {A : Set n} (key : ℕ) (tree : bt A ) (stack : List (bt A)) (C : bt A → List (bt A) → Set n) : Set n where
    field
      tree0 : bt A
      ti : treeInvariant tree0
-     si : stackInvariant tree tree0 stack
+     si : stackInvariant key tree tree0 stack
      ci : C tree stack
    
 findPP : {n m : Level} {A : Set n} {t : Set m}
            → (key : ℕ) → (tree : bt A ) → (stack : List (bt A))
-           → (Pre :  findPR tree stack (λ t s → Lift n ⊤))
-           → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 (λ t s → Lift n ⊤) →  bt-depth tree1 < bt-depth tree   → t )
-           → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → ( tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key)  → findPR tree1 stack1 (λ t s → Lift n ⊤) → t) → t
+           → (Pre :  findPR key tree stack (λ t s → Lift n ⊤))
+           → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR key tree1 stack1 (λ t s → Lift n ⊤) →  bt-depth tree1 < bt-depth tree   → t )
+           → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → ( tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key)  → findPR key tree1 stack1 (λ t s → Lift n ⊤) → t) → t
 findPP key leaf st Pre next exit = exit leaf st (case1 refl) Pre  
 findPP key (node key₁ v tree tree₁) st Pre next exit with <-cmp key key₁
 findPP key n st P next exit | tri≈ ¬a b ¬c = exit n st (case2 {!!}) P 
 findPP {_} {_} {A} key n@(node key₁ v 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 {!!} tree0 st →  stackInvariant {!!} tree0 (node key₁ v tree tree₁ ∷ st)
+    findPP2 : (st : List (bt A)) → stackInvariant key {!!} tree0 st →  stackInvariant key {!!} tree0 (node key₁ v tree tree₁ ∷ st)
     findPP2 = {!!}
     findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁)
     findPP1 =  {!!}
@@ -220,11 +220,11 @@
 insertTreePP : {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
 insertTreePP {n} {m} {A} {t} tree key value  P exit =
-   TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR (proj1 p) (proj2 p) (λ t s → Lift n ⊤) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫  {!!}
+   TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR key (proj1 p) (proj2 p) (λ t s → Lift n ⊤) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫  {!!}
        $ λ p P loop → findPP key (proj1 p) (proj2 p) {!!} (λ t s P1 lt → loop ⟪ t ,  s  ⟫ {!!} 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 (proj1 (proj2 p)) tree (proj1 p)  ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
+            {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 → 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 
@@ -240,23 +240,19 @@
    
 findPPC : {n m : Level} {A : Set n} {t : Set m}
            → (key : ℕ) → (tree : bt A ) → (stack : List (bt A))
-           → (Pre :  findPR tree stack findP-contains)
-           → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 findP-contains →  bt-depth tree1 < bt-depth tree   → t )
-           → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → ( tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key)  → findPR tree1 stack1 findP-contains → t) → t
+           → (Pre :  findPR key tree stack findP-contains)
+           → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR key tree1 stack1 findP-contains →  bt-depth tree1 < bt-depth tree   → t )
+           → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → ( tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key)  → findPR key tree1 stack1 findP-contains → t) → t
 findPPC = {!!}
 
 containsTree : {n m : Level} {A : Set n} {t : Set m} → (tree tree1 : bt A) → (key : ℕ) → (value : A) → treeInvariant tree1 → replacedTree key value tree1 tree  → ⊤
 containsTree {n} {m} {A} {t} tree tree1 key value P RT =
    TerminatingLoopS (bt A ∧ List (bt A) )
-     {λ p → findPR (proj1 p) (proj2 p) findP-contains } (λ p → bt-depth (proj1 p))
+     {λ p → findPR key (proj1 p) (proj2 p) findP-contains } (λ p → bt-depth (proj1 p))
               ⟪ tree1 , []  ⟫ {!!}
        $ λ p P loop → findPPC key (proj1 p) (proj2 p) {!!} (λ t s P1 lt → loop ⟪ t , s ⟫ {!!} lt )  
-       $ λ t1 s1 found? P2 → insertTreeSpec0 t1 value {!!} where
+       $ λ t1 s1 found? P2 → insertTreeSpec0 t1 value (lemma7 {!!} {!!} found?  ) where
            lemma7 : {key : ℕ } {value1 : A } {t1 tree : bt A } { s1 : List (bt A) } →
-              replacedTree key value1 tree t1 → stackInvariant t1 tree s1  → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key)  →  node-key t1 ≡ just key
-           lemma7 {key} {value1} {.(node key value1 leaf leaf)} {leaf} r-leaf s (case1 ())
-           lemma7 {key} {value1} {.(node key value1 leaf leaf)} {leaf} r-leaf s (case2 x) = x
-           lemma7 {.key₁} {value1} {.(node key₁ value1 s1 s2)} {node key₁ value s1 s2} r-node s or = {!!}
-           lemma7 {key} {value1} {.(node key₁ value s1 _)} {node key₁ value s1 s2} (r-right x r) s or = {!!}
-           lemma7 {key} {value1} {.(node key₁ value _ s2)} {node key₁ value s1 s2} (r-left x r) s or = {!!}
+              replacedTree key value1 tree t1 → stackInvariant key t1 tree s1  → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key)  →   top-value t1 ≡ just value
+           lemma7 = ?