Mercurial > hg > Gears > GearsAgda

changeset 662:a8959c8340e0

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 21 Nov 2021 21:52:03 +0900
parents 323533798054
children cf5095488bbd
files hoareBinaryTree.agda
diffstat 1 files changed, 66 insertions(+), 37 deletions(-) [+]
line wrap: on
line diff
--- a/hoareBinaryTree.agda	Sun Nov 21 19:31:44 2021 +0900
+++ b/hoareBinaryTree.agda	Sun Nov 21 21:52:03 2021 +0900
@@ -65,12 +65,7 @@
 
 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
-replace key value tree (leaf ∷ []) next exit = exit (node key value leaf leaf)
-replace key value tree (leaf ∷ leaf ∷ st) next exit = exit (node key value leaf leaf)
-replace key value tree (leaf ∷ node key₁ value₁ left right ∷ st) next exit with <-cmp key key₁
-... | tri< a ¬b ¬c = next key value (node key₁ value₁ (node key value leaf leaf) right ) st
-... | tri≈ ¬a b ¬c = next key value (node key₁ value  left right ) st
-... | tri> ¬a ¬b c = next key value (node key₁ value₁ left (node key value leaf leaf) ) st
+replace key value tree (leaf ∷ _) next exit = exit (node key value leaf leaf)
 replace key value tree (node key₁ value₁ left right ∷ st) next exit with <-cmp key key₁
 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) st
 ... | tri≈ ¬a b ¬c = next key value (node key₁ value  left right ) st
@@ -83,7 +78,7 @@
     replace-loop1 key value tree st = replace key value tree st replace-loop1  exit
 
 insertTree : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : bt A → t ) → t
-insertTree tree key value exit = find-loop key tree [] $ λ t st → replaceNode key value t $ λ t1 → replace-loop key value t1 st exit 
+insertTree tree key value exit = find-loop key tree ( tree ∷ [] ) $ λ t st → replaceNode key value t $ λ t1 → replace-loop key value t1 st exit 
 
 insertTest1 = insertTree leaf 1 1 (λ x → x )
 insertTest2 = insertTree insertTest1 2 1 (λ x → x )
@@ -102,13 +97,15 @@
        → treeInvariant (node key₂ value₂ t₃ t₄)
        → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) 
 
-data stackInvariant {n : Level} {A : Set n}  (key : ℕ) : (orig : bt A) → (stack  : List (bt A)) → Set n where
-    s-nil :  {tree0 : bt A} → stackInvariant key tree0 []
-    s-single :  {tree0 : bt A} →  stackInvariant key tree0 (tree0 ∷ [])
+--
+--  stack always contains original top at end
+--
+data stackInvariant {n : Level} {A : Set n}  (key : ℕ) : (top orig : bt A) → (stack  : List (bt A)) → Set n where
+    s-single :  {tree0 : bt A} →  stackInvariant key tree0 tree0 (tree0 ∷ [])
     s-right :  {tree tree0 tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} 
-        → key₁ < key  →  stackInvariant key tree0 (node key₁ v1 tree₁ tree ∷ st) →  stackInvariant key tree0 (tree ∷ node key₁ v1 tree₁ tree ∷ st)
+        → key₁ < key  →  stackInvariant key (node key₁ v1 tree₁ tree) tree0 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 tree0 (node key₁ v1 tree₁ tree ∷ st) →  stackInvariant key tree0 (tree₁ ∷ node key₁ v1 tree₁ tree ∷ st)
+        → key  < key₁ →  stackInvariant key (node key₁ v1 tree₁ tree) tree0 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)
@@ -118,8 +115,7 @@
     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) 
 
-replFromStack : {n : Level} {A : Set n}  {key : ℕ} {orig : bt A} → {stack  : List (bt A)} →  stackInvariant key orig stack → bt A
-replFromStack (s-nil {tree} ) = tree
+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-single {tree} ) = tree
 replFromStack (s-right {tree} x  st) = tree
 replFromStack (s-left {tree} x  st) = tree
@@ -146,8 +142,24 @@
 stack-last (x ∷ []) = just x
 stack-last (x ∷ s) = stack-last s
 
-stackInvariantTest1 : stackInvariant 4 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] )
-stackInvariantTest1 = s-right (add< 2) s-single 
+stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] )
+stackInvariantTest1 = s-right (add< 2) s-single  
+
+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 [] 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)) → ¬ (stack ≡ [])   → stackInvariant key tree tree0 stack
+   → stack-last stack ≡ just tree0
+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 ∷ 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
@@ -161,6 +173,16 @@
 ti-left {_} {_} {_} {_} {key₁} {v1} (t-left x ti) = ti
 ti-left {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti
 
+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-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
+   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
+
 rt-property1 :  {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf )
 rt-property1 {n} {A} key value .leaf .(node key value leaf leaf) r-leaf ()
 rt-property1 {n} {A} key value .(node key _ _ _) .(node key value _ _) r-node ()
@@ -198,19 +220,21 @@
 open _∧_
 
 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A))
-        →  treeInvariant tree ∧ stackInvariant key tree0 stack 
-        → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree0 stack → bt-depth tree1 < bt-depth tree   → t )
-        → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree0 stack
+           →  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
                  → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key )  → t ) → t
 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 {n} {_} {A} key (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  tree0 st   → stackInvariant key tree0 (tree ∷ st)
-   findP1 a [] si = ?
-   findP1 a (x ∷ st) 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) ,  {!!} ⟫ depth-2<
+findP {n} {_} {A} key nd@(node key₁ v1 tree tree₁) tree0 st  Pre next _ | tri< a ¬b ¬c = next tree tree0 (tree ∷ st)
+       ⟪ treeLeftDown tree tree₁ (proj1 Pre)  , {!!} ⟫ depth-1< where
+   findP1 : key < key₁ → (st : List (bt A)) →  stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (nd ∷ st)
+   findP1 a (x ∷ st) si = {!!} -- s-left a ? ?    stackInvariant key (node key₁ v1 tree tree₁) tree0 (x ∷ st)
+                                            --  → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ x ∷ st)
+   findP1 a [] si = {!!} --  stackInvariant key (node key₁ v1 tree tree₁) tree0 []
+                    --     → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ [])
+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
@@ -237,28 +261,33 @@
 
 replaceP : {n m : Level} {A : Set n} {t : Set m}
      → (key : ℕ) → (value : A) → {tree0 tree tree-st : bt A} ( repl : bt A)
-     → (stack : List (bt A)) → treeInvariant tree0 ∧ stackInvariant key tree0 stack ∧ replacedTree key value tree repl
+     → (stack : List (bt A)) → treeInvariant tree0 ∧ stackInvariant key tree-st tree0 stack ∧ replacedTree key value tree repl
      → (next : ℕ → A → {tree0 tree1 tree-st : bt A } (repl : bt A) → (stack1 : List (bt A))
-         → treeInvariant tree0 ∧ stackInvariant key tree0 stack1 ∧ replacedTree key value tree1 repl → length stack1 < length stack → t)
+         → 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-single   = {!!}
+... | 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 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 tree0 (leaf ∷ node key₁ value₁ left right ∷ st) → stackInvariant key  tree0 (node key₁ value₁ tree right ∷ st )
-    repl5 si = {!!}
+    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 {!!} 
 ... | 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-single  = {!!}
+... | s-single = {!!}
 ... | 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
@@ -296,11 +325,11 @@
 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 key  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 C → replaceNodeP key value t C (proj1 P)
        $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A ))
-            {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant key  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 → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ proj1 P , ⟪ {!!}  , R ⟫ ⟫
        $  λ p P1 loop → replaceP key value  (proj2 (proj2 p)) (proj1 p) {!!}
             (λ key value repl1 stack P2 lt → loop ⟪ stack , ⟪ {!!} , repl1  ⟫ ⟫ {!!} lt )  exit 
@@ -316,7 +345,7 @@
    field
      tree0 : bt A
      ti : treeInvariant tree0
-     si : stackInvariant key tree0 stack
+     si : stackInvariant key tree tree0 stack
      ci : C tree stack     -- data continuation
    
 findPP : {n m : Level} {A : Set n} {t : Set m}
@@ -330,7 +359,7 @@
 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 v1 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 =  depth-1<
@@ -345,7 +374,7 @@
        $ λ p P loop → findPP key (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t ,  s  ⟫ P1 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 key  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 → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ {!!} , ⟪ {!!}  , R ⟫ ⟫
        $  λ p P1 loop → replaceP key value  (proj2 (proj2 p)) (proj1 p) {!!}
             (λ key value repl1 stack P2 lt → loop ⟪ stack , ⟪ {!!} , repl1  ⟫ ⟫ {!!} lt )  exit 
@@ -377,6 +406,6 @@
            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
            lemma6 t1 s1 found? P2 = lemma7 t1 s1 (findPR.tree0 P2) ( findPC.tree1  (findPR.ci P2)) ( findPC.ci  (findPR.ci P2)) (findPR.si P2) found? where
               lemma7 :  (t1 : bt A) ( s1 : List (bt A) ) (tree0 tree1 : bt A) →
-                 replacedTree key value t1 tree1 → stackInvariant key tree0 s1  → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key)  →   top-value t1 ≡ just value
+                 replacedTree key value t1 tree1 → stackInvariant key t1 tree0 s1  → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key)  →   top-value t1 ≡ just value
               lemma7 = {!!}