Mercurial > hg > Gears > GearsAgda

comparison hoareBinaryTree.agda @ 621:6861bcb9c54d

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 08 Nov 2021 16:36:26 +0900
parents fe8c2d82c05c
children a1849f24fa66
comparison
equal deleted inserted replaced
620:fe8c2d82c05c 621:6861bcb9c54d
96 → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) 96 → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄))
97 97
98 treeInvariantTest1 : treeInvariant (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf))) 98 treeInvariantTest1 : treeInvariant (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf)))
99 treeInvariantTest1 = {!!} 99 treeInvariantTest1 = {!!}
100 100
101 data stackInvariant {n : Level} {A : Set n} : (tree0 : bt A) → (stack : List (bt A)) → Set n where 101 data stackInvariant {n : Level} {A : Set n} : (tree tree0 : bt A) → (stack : List (bt A)) → Set n where
102 s-nil : stackInvariant leaf [] 102 s-nil : stackInvariant leaf leaf []
103 s-single : (tree : bt A) → stackInvariant tree (tree ∷ [] ) 103 s-single : (tree : bt A) → stackInvariant tree tree (tree ∷ [] )
104 s-right : (tree : bt A) → {key : ℕ } → {value : A } { left : bt A} → stackInvariant (node key value left tree ) (tree ∷ node key value left tree ∷ [])
105 s-left : (tree : bt A) → {key : ℕ } → {value : A } { right : bt A} → stackInvariant (node key value tree right) (tree ∷ node key value tree right ∷ [])
106 s-< : (tree0 tree : bt A) → {key : ℕ } → {value : A } { left : bt A} → {st : List (bt A)} 104 s-< : (tree0 tree : bt A) → {key : ℕ } → {value : A } { left : bt A} → {st : List (bt A)}
107 → stackInvariant tree0 (tree ∷ st ) → stackInvariant tree0 ((node key value left tree ) ∷ tree ∷ st ) 105 → stackInvariant (node key value left tree ) tree0 (node key value left tree ∷ st ) → stackInvariant tree tree0 (tree ∷ node key value left tree ∷ st )
108 s-> : (tree0 tree : bt A) → {key : ℕ } → {value : A } { right : bt A} → {st : List (bt A)} 106 s-> : (tree0 tree : bt A) → {key : ℕ } → {value : A } { right : bt A} → {st : List (bt A)}
109 → stackInvariant tree0 (tree ∷ st ) → stackInvariant tree0 ((node key value tree right ) ∷ tree ∷ st ) 107 → stackInvariant (node key value tree right ) tree0 (node key value tree right ∷ st ) → stackInvariant tree tree0 (tree ∷ node key value tree right ∷ st )
110 108
111 data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (tree tree1 : bt A ) → Set n where 109 data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (tree tree1 : bt A ) → Set n where
112 r-leaf : replacedTree key value leaf (node key value leaf leaf) 110 r-leaf : replacedTree key value leaf (node key value leaf leaf)
113 r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁) 111 r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁)
114 r-right : {k : ℕ } {v : A} → {t t1 t2 : bt A} 112 r-right : {k : ℕ } {v : A} → {t t1 t2 : bt A}
115 → k > key → ( replacedTree key value t1 t2 → replacedTree key value (node k v t t1) (node k v t t2) ) 113 → k > key → ( replacedTree key value t1 t2 → replacedTree key value (node k v t t1) (node k v t t2) )
116 r-left : {k : ℕ } {v : A} → {t t1 t2 : bt A} 114 r-left : {k : ℕ } {v : A} → {t t1 t2 : bt A}
117 → k < key → ( replacedTree key value t1 t2 → replacedTree key value (node k v t1 t) (node k v t2 t) ) 115 → k < key → ( replacedTree key value t1 t2 → replacedTree key value (node k v t1 t) (node k v t2 t) )
118 116
119 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A)) 117 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A))
120 → treeInvariant tree ∧ stackInvariant tree0 stack 118 → treeInvariant tree ∧ stackInvariant tree tree0 stack
121 → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree0 stack → bt-depth tree1 < bt-depth tree → t ) 119 → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t )
122 → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree0 stack → t ) → t 120 → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 tree0 stack → t ) → t
123 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st {!!} 121 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st {!!}
124 findP key (node key₁ v tree tree₁) tree0 st Pre next exit with <-cmp key key₁ 122 findP key (node key₁ v tree tree₁) tree0 st Pre next exit with <-cmp key key₁
125 findP key n tree0 st Pre _ exit | tri≈ ¬a b ¬c = exit n tree0 st {!!} 123 findP key n tree0 st Pre _ exit | tri≈ ¬a b ¬c = exit n tree0 st {!!}
126 findP key n@(node key₁ v tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) {!!} {!!} 124 findP key n@(node key₁ v tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) {!!} {!!}
127 findP key n@(node key₁ v tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} {!!} 125 findP key n@(node key₁ v tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} {!!}
130 → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value tree tree1 → t) → t 128 → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value tree tree1 → t) → t
131 replaceNodeP k v leaf P next = next (node k v leaf leaf) {!!} {!!} 129 replaceNodeP k v leaf P next = next (node k v leaf leaf) {!!} {!!}
132 replaceNodeP k v (node key value t t₁) P next = next (node k v t t₁) {!!} {!!} 130 replaceNodeP k v (node key value t t₁) P next = next (node k v t t₁) {!!} {!!}
133 131
134 replaceP : {n m : Level} {A : Set n} {t : Set m} 132 replaceP : {n m : Level} {A : Set n} {t : Set m}
135 → (key : ℕ) → (value : A) → (tree repl : bt A) → (stack : List (bt A)) → treeInvariant tree ∧ stackInvariant tree stack ∧ replacedTree key value tree repl 133 → (key : ℕ) → (value : A) → (tree repl : bt A) → (stack : List (bt A)) → treeInvariant tree ∧ stackInvariant repl tree stack ∧ replacedTree key value tree repl
136 → (next : ℕ → A → (tree1 repl : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 stack ∧ replacedTree key value tree1 repl → bt-depth tree1 < bt-depth tree → t ) 134 → (next : ℕ → A → (tree1 repl : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant repl tree1 stack ∧ replacedTree key value tree1 repl → bt-depth tree1 < bt-depth tree → t )
137 → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t 135 → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t
138 replaceP key value tree repl [] Pre next exit = exit tree repl {!!} 136 replaceP key value tree repl [] Pre next exit = exit tree repl {!!}
139 replaceP key value tree repl (leaf ∷ st) Pre next exit = next key value tree {!!} st {!!} {!!} 137 replaceP key value tree repl (leaf ∷ st) Pre next exit = next key value tree {!!} st {!!} {!!}
140 replaceP key value tree repl (node key₁ value₁ left right ∷ st) Pre next exit with <-cmp key key₁ 138 replaceP key value tree repl (node key₁ value₁ left right ∷ st) Pre next exit with <-cmp key key₁
141 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) {!!} st {!!} {!!} 139 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) {!!} st {!!} {!!}
175 RTtoTI1 = {!!} 173 RTtoTI1 = {!!}
176 174
177 insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree 175 insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree
178 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t 176 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t
179 insertTreeP {n} {m} {A} {t} tree key value P exit = 177 insertTreeP {n} {m} {A} {t} tree key value P exit =
180 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ ⟪ P , {!!} ⟫ 178 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant (proj1 p) tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ ⟪ P , {!!} ⟫
181 $ λ p P loop → findP key (proj1 p) tree (proj2 p) {!!} (λ t _ s P1 lt → loop ⟪ t , s ⟫ {!!} lt ) 179 $ λ p P loop → findP key (proj1 p) tree (proj2 p) {!!} (λ t _ s P1 lt → loop ⟪ t , s ⟫ {!!} lt )
182 $ λ t _ s P → replaceNodeP key value t (proj1 P) 180 $ λ t _ s P → replaceNodeP key value t (proj1 P)
183 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A )) 181 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A ))
184 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } 182 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
185 (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ proj1 P , ⟪ {!!} , R ⟫ ⟫ 183 (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ proj1 P , ⟪ {!!} , R ⟫ ⟫
186 $ λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) P1 184 $ λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) {!!}
187 (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) exit 185 (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ {!!} lt ) exit
188 186
189 top-value : {n : Level} {A : Set n} → (tree : bt A) → Maybe A 187 top-value : {n : Level} {A : Set n} → (tree : bt A) → Maybe A
190 top-value leaf = nothing 188 top-value leaf = nothing
191 top-value (node key value tree tree₁) = just value 189 top-value (node key value tree tree₁) = just value
192 190
195 193
196 record findPR {n : Level} {A : Set n} (tree : bt A ) (stack : List (bt A)) : Set n where 194 record findPR {n : Level} {A : Set n} (tree : bt A ) (stack : List (bt A)) : Set n where
197 field 195 field
198 tree0 : bt A 196 tree0 : bt A
199 ti : treeInvariant tree 197 ti : treeInvariant tree
200 si : stackInvariant tree0 stack 198 si : stackInvariant tree tree0 stack
201 199
202 findPP : {n m : Level} {A : Set n} {t : Set m} 200 findPP : {n m : Level} {A : Set n} {t : Set m}
203 → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) 201 → (key : ℕ) → (tree : bt A ) → (stack : List (bt A))
204 → (Pre : bt A → List (bt A) → findPR tree stack ) 202 → (Pre : findPR tree stack )
205 → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 → bt-depth tree1 < bt-depth tree → t ) 203 → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 → bt-depth tree1 < bt-depth tree → t )
206 → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 → t) → t 204 → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 → t) → t
207 findPP key leaf st Pre next exit = exit leaf st (Pre leaf st ) 205 findPP key leaf st Pre next exit = exit leaf st Pre
208 findPP key (node key₁ v tree tree₁) st Pre next exit with <-cmp key key₁ 206 findPP key (node key₁ v tree tree₁) st Pre next exit with <-cmp key key₁
209 findPP key n st P next exit | tri≈ ¬a b ¬c = exit n st (P n st) 207 findPP key n st P next exit | tri≈ ¬a b ¬c = exit n st P
210 findPP {_} {_} {A} key n@(node key₁ v tree tree₁) st Pre next exit | tri< a ¬b ¬c = 208 findPP {_} {_} {A} key n@(node key₁ v tree tree₁) st Pre next exit | tri< a ¬b ¬c =
211 next tree (n ∷ st) (record {ti = findPP0 tree tree₁ (findPR.ti (Pre n st)) ; si = findPP2 st (findPR.si (Pre n st))} ) findPP1 where 209 next tree (n ∷ st) (record {ti = findPP0 tree tree₁ (findPR.ti Pre ) ; si = findPP2 st (findPR.si Pre)} ) findPP1 where
212 tree0 = findPR.tree0 (Pre n st) 210 tree0 = findPR.tree0 Pre
213 findPP0 : (tree tree₁ : bt A) → treeInvariant ( node key₁ v tree tree₁ ) → treeInvariant tree 211 findPP0 : (tree tree₁ : bt A) → treeInvariant ( node key₁ v tree tree₁ ) → treeInvariant tree
214 findPP0 leaf t x = {!!} 212 findPP0 leaf t x = {!!}
215 findPP0 (node key value tree tree₁) leaf x = proj1 {!!} 213 findPP0 (node key value tree tree₁) leaf x = proj1 {!!}
216 findPP0 (node key value tree tree₁) (node key₁ value₁ t t₁) x = proj1 {!!} 214 findPP0 (node key value tree tree₁) (node key₁ value₁ t t₁) x = proj1 {!!}
217 findPP2 : (st : List (bt A)) → stackInvariant tree0 st → stackInvariant tree0 (node key₁ v tree tree₁ ∷ st) 215 findPP2 : (st : List (bt A)) → stackInvariant {!!} tree0 st → stackInvariant {!!} tree0 (node key₁ v tree tree₁ ∷ st)
218 findPP2 [] = {!!} 216 findPP2 [] = {!!}
219 findPP2 (leaf ∷ st) x = {!!} 217 findPP2 (leaf ∷ st) x = {!!}
220 findPP2 (node key value leaf leaf ∷ st) x = {!!} 218 findPP2 (node key value leaf leaf ∷ st) x = {!!}
221 findPP2 (node key value leaf (node key₁ value₁ x₂ x₃) ∷ st) x = {!!} 219 findPP2 (node key value leaf (node key₁ value₁ x₂ x₃) ∷ st) x = {!!}
222 findPP2 (node key value (node key₁ value₁ x₁ x₃) leaf ∷ st) x = {!!} 220 findPP2 (node key value (node key₁ value₁ x₁ x₃) leaf ∷ st) x = {!!}
232 insertTreePP {n} {m} {A} {t} tree key value P exit = 230 insertTreePP {n} {m} {A} {t} tree key value P exit =
233 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR (proj1 p) (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ {!!} 231 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR (proj1 p) (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ {!!}
234 $ λ p P loop → findPP key (proj1 p) (proj2 p) {!!} (λ t s P1 lt → loop ⟪ t , s ⟫ {!!} lt ) 232 $ λ p P loop → findPP key (proj1 p) (proj2 p) {!!} (λ t s P1 lt → loop ⟪ t , s ⟫ {!!} lt )
235 $ λ t s P → replaceNodeP key value t {!!} 233 $ λ t s P → replaceNodeP key value t {!!}
236 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A )) 234 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A ))
237 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } 235 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) }
238 (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ {!!} , ⟪ {!!} , R ⟫ ⟫ 236 (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫　⟪ {!!} , ⟪ {!!} , R ⟫ ⟫
239 $ λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) P1 237 $ λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) {!!}
240 (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) exit 238 (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ {!!} lt ) exit
241 239
242 -- findP key tree stack = findPP key tree stack {findPR} → record { ti = tree-invariant tree ; si stack-invariant tree stack } → 240 -- findP key tree stack = findPP key tree stack {findPR} → record { ti = tree-invariant tree ; si stack-invariant tree stack } →
243 241
244 record findP-contains {n : Level} {A : Set n} (tree : bt A ) (stack : List (bt A)) : Set n where 242 record findP-contains {n : Level} {A : Set n} (tree : bt A ) (stack : List (bt A)) : Set n where
245 field 243 field