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changeset 655:d0394c191d84

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 21 Nov 2021 10:54:13 +0900
parents 48c6e6961ea5
children 30690aed1819
files hoareBinaryTree.agda
diffstat 1 files changed, 42 insertions(+), 50 deletions(-) [+]
line wrap: on
line diff
--- a/hoareBinaryTree.agda	Sun Nov 21 10:09:05 2021 +0900
+++ b/hoareBinaryTree.agda	Sun Nov 21 10:54:13 2021 +0900
@@ -103,14 +103,12 @@
        → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) 
 
 data stackInvariant {n : Level} {A : Set n}  (key : ℕ) : (top orig : bt A) → (stack  : List (bt A)) → Set n where
-    s-left0 :  {tree tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } 
-        →  key < key₁  → stackInvariant key (node key₁ v1 tree tree₁) (node key₁ v1 tree tree₁) (node key₁ v1 tree tree₁ ∷ [])
-    s-right0 :  {tree₁ tree : bt A} → {key₁ : ℕ } → {v1 : A } 
-        →  key₁ < key  → stackInvariant key (node key₁ v1 tree tree₁) (node key₁ v1 tree tree₁) (node key₁ v1 tree tree₁ ∷ [])
+    s-nil :  {tree : bt A} → stackInvariant key tree tree []
+    s-single :  {tree : bt A} → stackInvariant key tree tree [] → stackInvariant key tree tree (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 →  stackInvariant key tree tree0 (tree ∷ st)
+        → 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)} 
-        → key  < key₁ →  stackInvariant key (node key₁ v1 tree₁ tree) tree0 st →  stackInvariant key tree₁ tree0 (tree₁ ∷ st)
+        → key  < key₁ →  stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → ¬ (st ≡ []) →  stackInvariant key tree₁ tree0 (tree₁ ∷ 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)
@@ -121,16 +119,10 @@
           → k > key →  replacedTree key value t1 t2 →  replacedTree key value (node k v1 t1 t) (node k v1 t2 t) 
 
 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-right0 {tree} x) = tree
-replFromStack (s-left0 {tree} x) = tree
-replFromStack (s-right {tree} x st) = tree
-replFromStack (s-left {tree} x st) = tree
-
-stackInvariant-leaf : {n : Level} {A : Set n}  {key : ℕ} {top orig : bt A} → {stack  : List (bt A)} →  stackInvariant key top orig stack → ¬ (orig ≡ leaf)
-stackInvariant-leaf {_} {_} {_} {_} {_} (s-right0 x) ()
-stackInvariant-leaf {_} {_} {_} {_} {_} (s-left0 x) ()
-stackInvariant-leaf {_} {_} {_} {_} {_} (s-right x st) eq = stackInvariant-leaf st eq 
-stackInvariant-leaf {_} {_} {_} {_} {_} (s-left x st) eq = stackInvariant-leaf st eq 
+replFromStack (s-nil {tree} ) = tree
+replFromStack (s-single {tree} _ ) = tree
+replFromStack (s-right {tree} x _ st) = tree
+replFromStack (s-left {tree} x _ st) = tree
 
 add< : { i : ℕ } (j : ℕ ) → i < suc i + j
 add<  {i} j = begin
@@ -155,24 +147,25 @@
 stack-last (x ∷ s) = stack-last s
 
 stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] )
-stackInvariantTest1 = s-right (add< 2) (s-right0 (add< 2))
+stackInvariantTest1 = s-right (add< 2) (s-single s-nil) (λ ())
 
-si-property1 :  {n : Level} {A : Set n} (key : ℕ) (tree tree0 : bt A) → (stack  : List (bt A)) → stackInvariant key tree tree0 stack
+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 (t ∷ st) (s-right0 _ ) = refl
-si-property1 key t t0 (t ∷ st) (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-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 ∷ 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)) → stackInvariant key tree tree0 stack
+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 (.t ∷ []) (s-right0 _ ) = refl
-si-property-last key t t0 (.t ∷ []) (s-left0 _ ) = refl
-si-property-last key t t0 (.t ∷ x ∷ st) (s-right _ si) with  si-property1 key _ _ (x ∷ st) si
-... | refl = si-property-last key x t0 (x ∷ st) si
-si-property-last key t t0 (.t ∷ x ∷ st) (s-left _ si) with  si-property1 key _ _ (x ∷ st) si
-... | refl = si-property-last key x t0 (x ∷ st) si
-
+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-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
@@ -187,12 +180,12 @@
 
 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 (sub ∷ []) ti (s-right0 _ ) = ti
-stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-left0 _ ) = ti
-stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si) = ti-right (si1 si) where
+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 ∷ 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
-stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si) = ti-left ( si2 si) where
+stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si _) = ti-left ( si2 si) where
    si2 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} →  stackInvariant key (node key₁ v1 sub tree₁ ) tree st → treeInvariant  (node key₁ v1 sub tree₁ )
    si2 {tree₁ }  {key₁ }  {v1 }  si = stackTreeInvariant  key (node key₁ v1 sub tree₁ ) tree st ti si
 
@@ -240,10 +233,11 @@
 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 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 key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (tree ∷ st)
-   findP1 a si = s-left a 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 c (proj2 Pre) ⟫ depth-2<
+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 [] 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)
 replaceTree1 k v1 value (t-single .k .v1) = t-single k value
@@ -275,30 +269,28 @@
          → treeInvariant tree0 ∧ stackInvariant key tree-st tree0 stack1 ∧ replacedTree key value tree1 repl → length stack1 < length stack → t)
      → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t
 replaceP key value {tree0} {tree} {tree-st} repl [] Pre next exit with proj1 (proj2 Pre)
-... | ()
+... | t = {!!} 
 replaceP {_} {_} {A} key value {tree0} {tree} {tree-st} repl (leaf ∷ []) Pre next exit with proj1 (proj2 Pre)
-... | s-right x ()
-... | s-left x ()
+... | s-right x t _ = {!!}
+... | s-left x t _ = {!!}
 replaceP key value {tree0} {tree} {tree-st} repl (leaf ∷ leaf ∷ st) Pre next exit with proj1 (proj2 Pre)
-... | s-right x ()
-... | s-left x ()
+... | s-right x t _ = {!!}
+... | s-left x t _ = {!!}
 replaceP {_} {_} {A} key value {tree0} {tree} {tree-st} repl (leaf ∷ node key₁ value₁ left right ∷ st) Pre next exit with <-cmp key key₁
 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) (node key₁ value₁ tree right ∷ st)
                   ⟪ proj1 Pre , ⟪ repl5 (proj1 (proj2 Pre)) , r-left a (proj2 (proj2 Pre))  ⟫ ⟫ ≤-refl where
     repl5 :  stackInvariant key tree-st tree0 (leaf ∷ node key₁ value₁ left right ∷ st) → stackInvariant key (node key₁ value₁ tree right) tree0 (node key₁ value₁ tree right ∷ st )
-    repl5 si with si-property1 _ _ _ _ si
-    repl5 (s-right x si) | refl = s-left a {!!}
-    repl5 (s-left x si) | refl = s-left a {!!}
+    repl5 si with si-property1 _ _ _ _ {!!} si
+    repl5 (s-right x si _) | refl = s-left a {!!} {!!}
+    repl5 (s-left x si _) | refl = s-left a {!!} {!!}
 ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right) st {!!} depth-3<
 ... | tri> ¬a ¬b c = next key value (node key₁ value₁ repl right) st {!!} depth-3<
 replaceP key value {tree0} {tree} {tree-st} repl (node key₁ value₁ left right ∷ st) Pre next exit with <-cmp key key₁
 ... | tri> ¬a ¬b c = next key value  (node key₁ value₁ repl right ) st {!!}  ≤-refl
 ... | tri≈ ¬a b ¬c = next key value  (node key value left right ) st {!!}  ≤-refl where -- this case won't happen
 ... | tri< a ¬b ¬c with proj1 (proj2 Pre)
-... | s-right0 x = {!!}
-... | s-left0 x = {!!}
-... | s-right x si1 = {!!}
-... | s-left x si1 = next key value (node key₁ value₁ repl right )  st ⟪ proj1 Pre , ⟪ si1 ,  r-left a (proj2 (proj2 Pre)) ⟫ ⟫ ≤-refl   
+... | s-right x si1 _ = {!!}
+... | s-left x si1 _ = next key value (node key₁ value₁ repl right )  st ⟪ proj1 Pre , ⟪ si1 ,  r-left a (proj2 (proj2 Pre)) ⟫ ⟫ ≤-refl   
 -- = next key value (node key₁ value₁ repl right )  st ⟪ proj1 Pre , ⟪ repl2 (proj1 (proj2 Pre)) ,  r-left a {!!}  ⟫ ⟫ ≤-refl   where
 --    repl2 : stackInvariant key tree tree0 (node key₁ value₁ left right ∷ st) → stackInvariant key (node key₁ value₁ left right) tree0 st
 --    repl2 (s-single .(node key₁ value₁ left right)) = {!!}