# HG changeset patch # User Shinji KONO # Date 1637490704 -32400 # Node ID 323533798054cf44526a33b01b2ca8e800000f51 # Parent 712e2998c76b0c810f0c6bd0423a7d8d4b802f2a ... diff -r 712e2998c76b -r 323533798054 hoareBinaryTree.agda --- a/hoareBinaryTree.agda Sun Nov 21 19:03:22 2021 +0900 +++ b/hoareBinaryTree.agda Sun Nov 21 19:31:44 2021 +0900 @@ -102,13 +102,13 @@ → 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) : (top orig : bt A) → (stack : List (bt A)) → Set n where - s-nil : {tree tree0 : bt A} → stackInvariant key tree tree0 [] - s-single : {tree tree0 : bt A} → stackInvariant key tree tree0 [] → stackInvariant key tree0 tree0 (tree0 ∷ []) +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 ∷ []) 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 → ¬ (st ≡ []) → stackInvariant key tree tree0 (tree ∷ st) + → key₁ < key → stackInvariant key tree0 (node key₁ v1 tree₁ tree ∷ st) → stackInvariant key tree0 (tree ∷ node key₁ v1 tree₁ 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 → ¬ (st ≡ []) → stackInvariant key tree₁ tree0 (tree₁ ∷ st) + → key < key₁ → stackInvariant key tree0 (node key₁ v1 tree₁ tree ∷ st) → stackInvariant key tree0 (tree₁ ∷ node key₁ v1 tree₁ 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) @@ -118,11 +118,11 @@ 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} {top orig : bt A} → {stack : List (bt A)} → stackInvariant key top orig stack → bt A +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 (s-single {tree} _) = tree -replFromStack (s-right {tree} x _ st) = tree -replFromStack (s-left {tree} x _ st) = 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 @@ -146,26 +146,8 @@ stack-last (x ∷ []) = just x stack-last (x ∷ s) = stack-last s -stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] ) -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)) → ¬ (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 s-nil) = 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 +stackInvariantTest1 : stackInvariant 4 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] ) +stackInvariantTest1 = s-right (add< 2) s-single 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 @@ -179,17 +161,6 @@ 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 → ¬ (stack ≡ []) → treeInvariant sub -stackTreeInvariant {_} {A} key sub tree [] ti s-nil ne = ⊥-elim ( ne refl ) -stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-single s-nil ) _ = ti -stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si ne) _ = 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 ne -stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si ne) _ = 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 ne - 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 () @@ -227,19 +198,18 @@ open _∧_ findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A)) - → treeInvariant tree ∧ ((¬ (stack ≡ []) → stackInvariant key tree tree0 stack )) - → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ ((¬ (stack ≡ []) → stackInvariant key tree1 tree0 stack)) → bt-depth tree1 < bt-depth tree → t ) - → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ ((¬ (stack ≡ []) → stackInvariant key tree1 tree0 stack)) + → 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 → (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) , (λ ne → findP1 a st (proj2 Pre )) ⟫ depth-1< where - findP1 : key < key₁ → (st : List (bt A)) → ( ¬ (st ≡ [] ) → stackInvariant key (node key₁ v1 tree tree₁) tree0 st ) → stackInvariant key tree tree0 (tree ∷ st) - findP1 a [] si = {!!} -- s-single s-nil - findP1 a (x ∷ st) si with si {!!} - ... | t = s-left a t {!!} + ⟪ 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< 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) @@ -267,35 +237,30 @@ 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 tree-st tree0 stack ∧ replacedTree key value tree repl + → (stack : List (bt A)) → treeInvariant tree0 ∧ stackInvariant key 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 tree-st tree0 stack1 ∧ replacedTree key value tree1 repl → length stack1 < length stack → t) + → treeInvariant tree0 ∧ stackInvariant key 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 _ = {!!} +... | s-single = {!!} 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 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 : stackInvariant key tree0 (leaf ∷ node key₁ value₁ left right ∷ st) → stackInvariant key tree0 (node key₁ value₁ tree right ∷ st ) + repl5 si = {!!} ... | 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 si1 = {!!} -... | 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-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 -- 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)) = {!!} @@ -331,11 +296,11 @@ 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 (proj1 p) tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ ⟪ P , {!!} ⟫ + TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant key 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 (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } + {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant key 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 @@ -351,7 +316,7 @@ field tree0 : bt A ti : treeInvariant tree0 - si : stackInvariant key tree tree0 stack + si : stackInvariant key tree0 stack ci : C tree stack -- data continuation findPP : {n m : Level} {A : Set n} {t : Set m} @@ -365,7 +330,7 @@ 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< @@ -380,7 +345,7 @@ $ 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 (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } + {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant key 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 @@ -412,6 +377,6 @@ 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 t1 tree0 s1 → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key) → top-value t1 ≡ just value + replacedTree key value t1 tree1 → stackInvariant key tree0 s1 → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key) → top-value t1 ≡ just value lemma7 = {!!}