# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1636700941 -32400
# Node ID 189cf03bda5f51d5bdf02a8b191ac4b951f291f4
# Parent  119f340c0b10464d75d04ab71fc2b93e0f06dce3
...

diff -r 119f340c0b10 -r 189cf03bda5f hoareBinaryTree.agda
--- a/hoareBinaryTree.agda	Thu Nov 11 15:57:19 2021 +0900
+++ b/hoareBinaryTree.agda	Fri Nov 12 16:09:01 2021 +0900
@@ -138,11 +138,27 @@
 depth-2< : {i j : ℕ} →   suc i ≤ suc (j Data.Nat.⊔ i )
 depth-2< {i} {j} = s≤s (m≤n⊔m _ i)
 
-lemma11  : {n : Level} {A : Set n} {v1 : A}  → (key key₁ : ℕ) → (tree tree₁ : bt A )
-      → key < key₁
+treeLeftDown  : {n : Level} {A : Set n} {k : ℕ} {v1 : A}  → (tree tree₁ : bt A )
+      → treeInvariant (node k v1 tree tree₁)
+      →      treeInvariant tree 
+treeLeftDown {n} {A} {_} {v1} leaf leaf (t-single k1 v1) = t-leaf
+treeLeftDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = t-leaf
+treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = ti 
+treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti
+
+treeRightDown  : {n : Level} {A : Set n} {k : ℕ} {v1 : A}  → (tree tree₁ : bt A )
+      → treeInvariant (node k v1 tree tree₁)
+      →      treeInvariant tree₁ 
+treeRightDown {n} {A} {_} {v1} .leaf .leaf (t-single _ .v1) = t-leaf
+treeRightDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = ti
+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)
-      →      treeInvariant tree 
-lemma11  = {!!}
+      → 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)
@@ -156,7 +172,9 @@
 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) ⟪ lemma11 {!!} {!!} {!!} {!!} {!!} (proj1 Pre)  , {!!}  ⟫ depth-1<
+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<
 
 
@@ -248,10 +266,10 @@
     findPP2 : (st : List (bt A)) → stackInvariant key {!!} tree0 st →  stackInvariant key {!!} tree0 (node key₁ v1 tree tree₁ ∷ st)
     findPP2 = {!!}
     findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁)
-    findPP1 =  {!!}
+    findPP1 =  depth-1<
 findPP key n@(node key₁ v1 tree tree₁) st Pre next exit | tri> ¬a ¬b c = next tree₁ (n ∷ st) {!!} findPP2 where -- Cond n st → Cond tree₁ (n ∷ st)
     findPP2 : suc (bt-depth tree₁) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁)
-    findPP2 = {!!}
+    findPP2 = depth-2<
 
 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
@@ -285,8 +303,8 @@
 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 key (proj1 p) (proj2 p) (findPC key value ) } (λ p → bt-depth (proj1 p))
-              ⟪ tree1 , []  ⟫ {!!}
+     {λ p → findPR key (proj1 p) (proj2 p) (findPC key value ) } (λ p → bt-depth (proj1 p)) -- findPR key tree1 [] (findPC key value)
+              ⟪ tree1 , []  ⟫ record { tree0 = tree ; ti = {!!} ; si = {!!} ; ci = record { tree1 = tree ; ci = RT } }
        $ λ p P loop → findPPC key value (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt )  
        $ λ t1 s1 found? P2 → insertTreeSpec0 t1 value (lemma6 t1 s1 found? P2) where
            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