Mercurial > hg > Gears > GearsAgda

changeset 632:b58991f8e2e4

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Thu, 11 Nov 2021 15:48:36 +0900
parents 956ee8ae42b9
children 119f340c0b10
files hoareBinaryTree.agda
diffstat 1 files changed, 56 insertions(+), 27 deletions(-) [+]
line wrap: on
line diff
--- a/hoareBinaryTree.agda	Wed Nov 10 10:18:34 2021 +0900
+++ b/hoareBinaryTree.agda	Thu Nov 11 15:48:36 2021 +0900
@@ -48,10 +48,10 @@
 find : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → List (bt A)
            → (next : bt A → List (bt A) → t ) → (exit : bt A → List (bt A) → t ) → t
 find key leaf st _ exit = exit leaf st
-find key (node key₁ v tree tree₁) st next exit with <-cmp key key₁
+find key (node key₁ v1 tree tree₁) st next exit with <-cmp key key₁
 find key n st _ exit | tri≈ ¬a b ¬c = exit n st
-find key n@(node key₁ v tree tree₁) st next _ | tri< a ¬b ¬c = next tree (n ∷ st)
-find key n@(node key₁ v tree tree₁) st next _ | tri> ¬a ¬b c = next tree₁ (n ∷ st)
+find key n@(node key₁ v1 tree tree₁) st next _ | tri< a ¬b ¬c = next tree (n ∷ st)
+find key n@(node key₁ v1 tree tree₁) st next _ | tri> ¬a ¬b c = next tree₁ (n ∷ st)
 
 {-# TERMINATING #-}
 find-loop : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → bt A → List (bt A)  → (exit : bt A → List (bt A) → t) → t
@@ -60,8 +60,8 @@
     find-loop1 tree st = find key tree st find-loop1  exit
 
 replaceNode : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → (bt A → t) → t
-replaceNode k v leaf next = next (node k v leaf leaf)
-replaceNode k v (node key value t t₁) next = next (node k v t t₁)
+replaceNode k v1 leaf next = next (node k v1 leaf leaf)
+replaceNode k v1 (node key value t t₁) next = next (node k v1 t t₁)
 
 replace : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → List (bt A) → (next : ℕ → A → bt A → List (bt A) → t ) → (exit : bt A → t) → t
 replace key value tree [] next exit = exit tree
@@ -87,47 +87,76 @@
 
 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-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 value t₁ t₂)
+       → treeInvariant (node key₁ value₁ (node key value 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  = {!!}
+add< : { i : ℕ } (j : ℕ ) → i < suc i + j
+add<  {i} j = begin
+        suc i ≤⟨ m≤m+n (suc i) j ⟩
+        suc i + j ∎  where open ≤-Reasoning
+
+treeTest1  : bt ℕ
+treeTest1  =  node 1 0 leaf (node 3 1 (node 2 5 (node 4 7 leaf leaf ) leaf) (node 5 5 leaf leaf))
+treeTest2  : bt ℕ
+treeTest2  =  node 3 1 (node 2 5 (node 4 7 leaf leaf ) leaf) (node 5 5 leaf leaf)
+
+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)}
+    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)}
+    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 ) 
 
+stackInvariantTest0 : stackInvariant {_} {ℕ} 1 leaf leaf []
+stackInvariantTest0 = s-nil
+
+stackInvariantTest1 : stackInvariant 3 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] )
+stackInvariantTest1 = s-right (add< 1) (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)
     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) 
-    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) 
+    r-right : {k : ℕ } {v1 : A} → {t t1 t2 : bt A}
+          → 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)
+
+depth-2< : {i j : ℕ} →   suc i ≤ suc (j Data.Nat.⊔ i )
+depth-2< {i} {j} = s≤s (m≤n⊔m _ i)
 
 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
 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st Pre
-findP key (node key₁ v tree tree₁) tree0 st Pre next exit with <-cmp key key₁
+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 v tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) {!!} {!!}
-findP key n@(node key₁ v tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} {!!}
+findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) {!!} depth-1<
+findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} depth-2<
+-- Pre   : treeInvariant (node key₁ v1 tree tree₁)
+--    →      treeInvariant tree ∧
+--        stackInvariant key (node key₁ v1 tree tree₁) tree0 st
+-        → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st)
+
 
 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (treeInvariant tree )
     → ((tree1 : bt A) → treeInvariant tree1 →  replacedTree key value tree tree1 → t) → t
-replaceNodeP k v leaf P next = next (node k v leaf leaf) {!!} {!!} 
-replaceNodeP k v (node key value t t₁) P next = next (node k v t t₁) {!!} {!!}
+replaceNodeP k v1 leaf P next = next (node k v1 leaf leaf) {!!} {!!} 
+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
@@ -204,16 +233,16 @@
            → (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 (node key₁ v1 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 =
+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 v tree tree₁ ∷ st)
+    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 =  {!!}
-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)
+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 = {!!}
 
@@ -240,9 +269,9 @@
            → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR key tree1 stack1 (findPC key value) →  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 (findPC key value) → t) → t
 findPPC key value leaf st Pre next exit = exit leaf st (case1 refl) Pre  
-findPPC key value (node key₁ v tree tree₁) st Pre next exit with <-cmp key key₁
+findPPC key value (node key₁ v1 tree tree₁) st Pre next exit with <-cmp key key₁
 findPPC key value n st P next exit | tri≈ ¬a b ¬c = exit n st (case2 {!!}) P 
-findPPC {_} {_} {A} key value n@(node key₁ v tree tree₁) st Pre next exit | tri< a ¬b ¬c =
+findPPC {_} {_} {A} key value n@(node key₁ v1 tree tree₁) st Pre next exit | tri< a ¬b ¬c =
           next tree (n ∷ st) (record {ti = findPR.ti Pre  ; si = {!!} ; ci =  {!!} } ) {!!} 
 findPPC key value n st P next exit | tri> ¬a ¬b c = {!!}