Mercurial > hg > Gears > GearsAgda

comparison hoareBinaryTree.agda @ 634:189cf03bda5f

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 12 Nov 2021 16:09:01 +0900
parents 119f340c0b10
children e30dcd03c07f
comparison
equal deleted inserted replaced
633:119f340c0b10 634:189cf03bda5f
136 depth-1< {i} {j} = s≤s (m≤m⊔n _ j) 136 depth-1< {i} {j} = s≤s (m≤m⊔n _ j)
137 137
138 depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i ) 138 depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i )
139 depth-2< {i} {j} = s≤s (m≤n⊔m _ i) 139 depth-2< {i} {j} = s≤s (m≤n⊔m _ i)
140 140
141 lemma11 : {n : Level} {A : Set n} {v1 : A} → (key key₁ : ℕ) → (tree tree₁ : bt A ) 141 treeLeftDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A )
142 → key < key₁ 142 → treeInvariant (node k v1 tree tree₁)
143 → treeInvariant tree
144 treeLeftDown {n} {A} {_} {v1} leaf leaf (t-single k1 v1) = t-leaf
145 treeLeftDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = t-leaf
146 treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = ti
147 treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti
148
149 treeRightDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A )
150 → treeInvariant (node k v1 tree tree₁)
151 → treeInvariant tree₁
152 treeRightDown {n} {A} {_} {v1} .leaf .leaf (t-single _ .v1) = t-leaf
153 treeRightDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = ti
154 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = t-leaf
155 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti₁
156
157 siConsLeft : {n : Level } {A : Set n} (key key₁ : ℕ) → { v1 : A } (tree tree₁ tree0 : bt A ) (st : List (bt A))
158 → key < key₁ → stackInvariant key (node key₁ v1 tree tree₁) tree0 st
143 → treeInvariant (node key₁ v1 tree tree₁) 159 → treeInvariant (node key₁ v1 tree tree₁)
144 → treeInvariant tree 160 → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st)
145 lemma11 = {!!} 161 siConsLeft {n} {A} k k1 {v1} t t1 t0 st k<k1 ti si = {!!}
146 162
147 -- stackInvariant key (node key₁ v1 tree tree₁) tree0 st 163 -- stackInvariant key (node key₁ v1 tree tree₁) tree0 st
148 -- → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st) 164 -- → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st)
149 165
150 open _∧_ 166 open _∧_
154 → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t ) 170 → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t )
155 → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → t ) → t 171 → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → t ) → t
156 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st Pre 172 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st Pre
157 findP key (node key₁ v1 tree tree₁) tree0 st Pre next exit with <-cmp key key₁ 173 findP key (node key₁ v1 tree tree₁) tree0 st Pre next exit with <-cmp key key₁
158 findP key n tree0 st Pre _ exit | tri≈ ¬a b ¬c = exit n tree0 st Pre 174 findP key n tree0 st Pre _ exit | tri≈ ¬a b ¬c = exit n tree0 st Pre
159 findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) ⟪ lemma11 {!!} {!!} {!!} {!!} {!!} (proj1 Pre) , {!!} ⟫ depth-1< 175 findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) ⟪ treeLeftDown tree tree₁ (proj1 Pre) , findP1 a (proj2 Pre) ⟫ depth-1< where
176 findP1 : key < key₁ → stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (node key₁ v1 tree tree₁ ∷ st)
177 findP1 a si = siConsLeft key key₁ {v1} tree tree₁ tree0 st a si (proj1 Pre)
160 findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} depth-2< 178 findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} depth-2<
161 179
162 180
163 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (treeInvariant tree ) 181 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (treeInvariant tree )
164 → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value tree tree1 → t) → t 182 → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value tree tree1 → t) → t
246 next tree (n ∷ st) (record {ti = findPR.ti Pre ; si = findPP2 st (findPR.si Pre) ; ci = lift tt} ) findPP1 where 264 next tree (n ∷ st) (record {ti = findPR.ti Pre ; si = findPP2 st (findPR.si Pre) ; ci = lift tt} ) findPP1 where
247 tree0 = findPR.tree0 Pre 265 tree0 = findPR.tree0 Pre
248 findPP2 : (st : List (bt A)) → stackInvariant key {!!} tree0 st → stackInvariant key {!!} tree0 (node key₁ v1 tree tree₁ ∷ st) 266 findPP2 : (st : List (bt A)) → stackInvariant key {!!} tree0 st → stackInvariant key {!!} tree0 (node key₁ v1 tree tree₁ ∷ st)
249 findPP2 = {!!} 267 findPP2 = {!!}
250 findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁) 268 findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁)
251 findPP1 = {!!} 269 findPP1 = depth-1<
252 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) 270 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)
253 findPP2 : suc (bt-depth tree₁) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁) 271 findPP2 : suc (bt-depth tree₁) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁)
254 findPP2 = {!!} 272 findPP2 = depth-2<
255 273
256 insertTreePP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree 274 insertTreePP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree
257 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t 275 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t
258 insertTreePP {n} {m} {A} {t} tree key value P exit = 276 insertTreePP {n} {m} {A} {t} tree key value P exit =
259 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR key (proj1 p) (proj2 p) (λ t s → Lift n ⊤) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ {!!} 277 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR key (proj1 p) (proj2 p) (λ t s → Lift n ⊤) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ {!!}
283 findPPC key value n st P next exit | tri> ¬a ¬b c = {!!} 301 findPPC key value n st P next exit | tri> ¬a ¬b c = {!!}
284 302
285 containsTree : {n m : Level} {A : Set n} {t : Set m} → (tree tree1 : bt A) → (key : ℕ) → (value : A) → treeInvariant tree1 → replacedTree key value tree1 tree → ⊤ 303 containsTree : {n m : Level} {A : Set n} {t : Set m} → (tree tree1 : bt A) → (key : ℕ) → (value : A) → treeInvariant tree1 → replacedTree key value tree1 tree → ⊤
286 containsTree {n} {m} {A} {t} tree tree1 key value P RT = 304 containsTree {n} {m} {A} {t} tree tree1 key value P RT =
287 TerminatingLoopS (bt A ∧ List (bt A) ) 305 TerminatingLoopS (bt A ∧ List (bt A) )
288 {λ p → findPR key (proj1 p) (proj2 p) (findPC key value ) } (λ p → bt-depth (proj1 p)) 306 {λ p → findPR key (proj1 p) (proj2 p) (findPC key value ) } (λ p → bt-depth (proj1 p)) -- findPR key tree1 [] (findPC key value)
289 ⟪ tree1 , [] ⟫ {!!} 307 ⟪ tree1 , [] ⟫ record { tree0 = tree ; ti = {!!} ; si = {!!} ; ci = record { tree1 = tree ; ci = RT } }
290 $ λ p P loop → findPPC key value (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) 308 $ λ p P loop → findPPC key value (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt )
291 $ λ t1 s1 found? P2 → insertTreeSpec0 t1 value (lemma6 t1 s1 found? P2) where 309 $ λ t1 s1 found? P2 → insertTreeSpec0 t1 value (lemma6 t1 s1 found? P2) where
292 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 310 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
293 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 311 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
294 lemma7 : (t1 : bt A) ( s1 : List (bt A) ) (tree0 tree1 : bt A) → 312 lemma7 : (t1 : bt A) ( s1 : List (bt A) ) (tree0 tree1 : bt A) →