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changeset 620:fe8c2d82c05c

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 07 Nov 2021 23:00:57 +0900
parents a3fbc9b57015
children 6861bcb9c54d
files hoareBinaryTree.agda
diffstat 1 files changed, 34 insertions(+), 20 deletions(-) [+]
line wrap: on
line diff
--- a/hoareBinaryTree.agda	Sun Nov 07 19:43:16 2021 +0900
+++ b/hoareBinaryTree.agda	Sun Nov 07 23:00:57 2021 +0900
@@ -33,6 +33,14 @@
   node :  (key : ℕ) → (value : A) →
     (left : bt A ) → (right : bt A ) → bt A
 
+node-key : {n : Level} {A : Set n} → bt A → Maybe ℕ
+node-key (node key _ _ _) = just key
+node-key _ = nothing
+
+node-value : {n : Level} {A : Set n} → bt A → Maybe A
+node-value (node _ value _ _) = just value
+node-value _ = nothing
+
 bt-depth : {n : Level} {A : Set n} → (tree : bt A ) → ℕ
 bt-depth leaf = 0
 bt-depth (node key value t t₁) = suc (Data.Nat._⊔_ (bt-depth t ) (bt-depth t₁ ))
@@ -77,22 +85,28 @@
 
 open import Data.Unit hiding ( _≟_ ;  _≤?_ ; _≤_)
 
-treeInvariant : {n : Level} {A : Set n} → (tree : bt A) → Set
-treeInvariant leaf = ⊤
-treeInvariant (node key value leaf leaf) = ⊤
-treeInvariant (node key value leaf n@(node key₁ value₁ t₁ t₂)) =  (key < key₁) ∧ treeInvariant n 
-treeInvariant (node key value n@(node key₁ value₁ t t₁) leaf) =  treeInvariant n ∧ (key < key₁) 
-treeInvariant (node key value n@(node key₁ value₁ t t₁) m@(node key₂ value₂ t₂ t₃)) = treeInvariant n ∧  (key < key₁) ∧ (key₁ < key₂) ∧ treeInvariant m
+data treeInvariant {n : Level} {A : Set n} : (tree : bt A) → Set n where
+    t-leaf : treeInvariant leaf 
+    t-single : {key : ℕ} → {value : A} →  treeInvariant (node key value leaf leaf) 
+    t-right : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key < key₁) → treeInvariant (node key₁ value₁ t₁ t₂)  → treeInvariant (node key value leaf (node key₁ value₁ t₁ t₂)) 
+    t-left  : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key₁ < key) → treeInvariant (node key value₁ t₁ t₂)  → treeInvariant (node key₁ value₁ (node key value₁ t₁ t₂) leaf ) 
+    t-node  : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} → (key < key₁) → (key₁ < key₂)
+       → treeInvariant (node key value t₁ t₂) 
+       → treeInvariant (node key₂ value₂ t₃ t₄)
+       → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) 
 
-treeInvariantTest1  = treeInvariant (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf)))
+treeInvariantTest1  : treeInvariant (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf)))
+treeInvariantTest1  = {!!}
 
-stackInvariant : {n : Level} {A : Set n} → (tree : bt A) → (stack  : List (bt A)) → Set n
-stackInvariant {_} {A} _ [] = Lift _ ⊤
-stackInvariant {_} {A} tree (tree1 ∷ [] ) = tree1 ≡ tree
-stackInvariant {_} {A} tree (x ∷ tail @ (node key value leaf right ∷ _) ) = (right ≡ x) ∧ stackInvariant tree tail
-stackInvariant {_} {A} tree (x ∷ tail @ (node key value left leaf ∷ _) ) = (left ≡ x) ∧ stackInvariant tree tail
-stackInvariant {_} {A} tree (x ∷ tail @ (node key value left right ∷ _  )) = ( (left ≡ x) ∧ stackInvariant tree tail) ∨ ( (right ≡ x) ∧ stackInvariant tree tail)
-stackInvariant {_} {A} tree s = Lift _ ⊥
+data stackInvariant {n : Level} {A : Set n} : (tree0 : bt A) → (stack  : List (bt A)) → Set n where
+    s-nil : stackInvariant  leaf [] 
+    s-single : (tree : bt A) → stackInvariant tree (tree ∷ [] ) 
+    s-right  : (tree : bt A) → {key : ℕ } → {value : A } { left  : bt A} → stackInvariant (node key value left tree ) (tree ∷ node key value left tree  ∷ []) 
+    s-left   : (tree : bt A) → {key : ℕ } → {value : A } { right : bt A} → stackInvariant (node key value tree right) (tree ∷ node key value tree right  ∷ []) 
+    s-<      : (tree0 tree : bt A) → {key : ℕ } → {value : A } { left  : bt A} → {st : List (bt A)}
+         → stackInvariant tree0 (tree ∷ st )  → stackInvariant tree0 ((node key value left tree  ) ∷ tree ∷ st ) 
+    s->      : (tree0 tree : bt A) → {key : ℕ } → {value : A } { right : bt A} →  {st : List (bt A)}
+         → stackInvariant tree0 (tree ∷ st )  → stackInvariant tree0 ((node key value tree right ) ∷ tree ∷ 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)
@@ -163,7 +177,7 @@
 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 tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫  ⟪ P , lift tt  ⟫
+   TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant 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 ))
@@ -197,16 +211,16 @@
           next tree (n ∷ st) (record {ti = findPP0 tree tree₁ (findPR.ti (Pre n st)) ; si = findPP2 st (findPR.si (Pre n st))} ) findPP1 where 
     tree0 =  findPR.tree0 (Pre n st)
     findPP0 : (tree tree₁ : bt A) → treeInvariant ( node key₁ v tree tree₁ ) → treeInvariant tree
-    findPP0 leaf t x = tt
-    findPP0 (node key value tree tree₁) leaf x = proj1 x
-    findPP0 (node key value tree tree₁) (node key₁ value₁ t t₁) x = proj1 x
+    findPP0 leaf t x = {!!}
+    findPP0 (node key value tree tree₁) leaf x = proj1 {!!}
+    findPP0 (node key value tree tree₁) (node key₁ value₁ t t₁) x = proj1 {!!}
     findPP2 : (st : List (bt A)) → stackInvariant tree0 st →  stackInvariant tree0 (node key₁ v tree tree₁ ∷ st)
-    findPP2 [] (lift tt) = {!!}
+    findPP2 [] = {!!}
     findPP2 (leaf ∷ st) x = {!!}
     findPP2 (node key value leaf leaf ∷ st) x = {!!}
     findPP2 (node key value leaf (node key₁ value₁ x₂ x₃) ∷ st) x = {!!}
     findPP2 (node key value (node key₁ value₁ x₁ x₃) leaf ∷ st) x = {!!}
-    findPP2 (node key value (node key₁ value₁ x₁ x₃) (node key₂ value₂ x₂ x₄) ∷ st) x = case1 ⟪ {!!} , {!!} ⟫
+    findPP2 (node key value (node key₁ value₁ x₁ x₃) (node key₂ value₂ x₂ x₄) ∷ st) x = {!!}
     findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁)
     findPP1 =  {!!}
 findPP key n@(node key₁ v tree tree₁) st Pre next exit | tri> ¬a ¬b c = next tree₁ (n ∷ st) {!!} findPP2 where -- Cond n st → Cond tree₁ (n ∷ st)