Mercurial > hg > Gears > GearsAgda
annotate hoareBinaryTree.agda @ 672:3676e845d46f
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Tue, 23 Nov 2021 11:32:35 +0900 |
| parents | b5fde9727830 |
| children | a8e2bb44b843 |
| rev | line source |
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1 module hoareBinaryTree where |
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2 |
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3 open import Level renaming (zero to Z ; suc to succ) |
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4 |
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5 open import Data.Nat hiding (compare) |
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6 open import Data.Nat.Properties as NatProp |
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7 open import Data.Maybe |
| 588 | 8 -- open import Data.Maybe.Properties |
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9 open import Data.Empty |
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10 open import Data.List |
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11 open import Data.Product |
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12 |
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13 open import Function as F hiding (const) |
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14 |
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15 open import Relation.Binary |
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16 open import Relation.Binary.PropositionalEquality |
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17 open import Relation.Nullary |
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18 open import logic |
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19 |
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20 |
| 588 | 21 _iso_ : {n : Level} {a : Set n} → ℕ → ℕ → Set |
| 22 d iso d' = (¬ (suc d ≤ d')) ∧ (¬ (suc d' ≤ d)) | |
| 23 | |
| 24 iso-intro : {n : Level} {a : Set n} {x y : ℕ} → ¬ (suc x ≤ y) → ¬ (suc y ≤ x) → _iso_ {n} {a} x y | |
| 25 iso-intro = λ z z₁ → record { proj1 = z ; proj2 = z₁ } | |
| 26 | |
| 590 | 27 -- |
| 28 -- | |
| 29 -- no children , having left node , having right node , having both | |
| 30 -- | |
| 597 | 31 data bt {n : Level} (A : Set n) : Set n where |
| 604 | 32 leaf : bt A |
| 33 node : (key : ℕ) → (value : A) → | |
| 610 | 34 (left : bt A ) → (right : bt A ) → bt A |
| 600 | 35 |
| 620 | 36 node-key : {n : Level} {A : Set n} → bt A → Maybe ℕ |
| 37 node-key (node key _ _ _) = just key | |
| 38 node-key _ = nothing | |
| 39 | |
| 40 node-value : {n : Level} {A : Set n} → bt A → Maybe A | |
| 41 node-value (node _ value _ _) = just value | |
| 42 node-value _ = nothing | |
| 43 | |
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44 bt-depth : {n : Level} {A : Set n} → (tree : bt A ) → ℕ |
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45 bt-depth leaf = 0 |
| 618 | 46 bt-depth (node key value t t₁) = suc (Data.Nat._⊔_ (bt-depth t ) (bt-depth t₁ )) |
| 606 | 47 |
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48 find : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → List (bt A) |
| 604 | 49 → (next : bt A → List (bt A) → t ) → (exit : bt A → List (bt A) → t ) → t |
| 50 find key leaf st _ exit = exit leaf st | |
| 632 | 51 find key (node key₁ v1 tree tree₁) st next exit with <-cmp key key₁ |
| 604 | 52 find key n st _ exit | tri≈ ¬a b ¬c = exit n st |
| 632 | 53 find key n@(node key₁ v1 tree tree₁) st next _ | tri< a ¬b ¬c = next tree (n ∷ st) |
| 54 find key n@(node key₁ v1 tree tree₁) st next _ | tri> ¬a ¬b c = next tree₁ (n ∷ st) | |
| 597 | 55 |
| 604 | 56 {-# TERMINATING #-} |
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57 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 |
| 611 | 58 find-loop {n} {m} {A} {t} key tree st exit = find-loop1 tree st where |
| 604 | 59 find-loop1 : bt A → List (bt A) → t |
| 60 find-loop1 tree st = find key tree st find-loop1 exit | |
| 600 | 61 |
| 611 | 62 replaceNode : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → (bt A → t) → t |
| 632 | 63 replaceNode k v1 leaf next = next (node k v1 leaf leaf) |
| 64 replaceNode k v1 (node key value t t₁) next = next (node k v1 t t₁) | |
| 611 | 65 |
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66 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 |
| 669 | 67 replace key value repl [] next exit = exit repl -- can't happen |
| 68 replace key value repl (leaf ∷ []) next exit = exit repl -- can't happen | |
| 69 replace key value repl (node key₁ value₁ left right ∷ []) next exit with <-cmp key key₁ | |
| 70 ... | tri< a ¬b ¬c = exit (node key₁ value₁ repl right ) | |
| 664 | 71 ... | tri≈ ¬a b ¬c = exit (node key₁ value left right ) |
| 669 | 72 ... | tri> ¬a ¬b c = exit (node key₁ value₁ left repl ) |
| 73 replace key value repl (leaf ∷ st) next exit = next key value repl st -- can't happen | |
| 74 replace key value repl (node key₁ value₁ left right ∷ st) next exit with <-cmp key key₁ | |
| 75 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st | |
| 604 | 76 ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st |
| 669 | 77 ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st |
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78 |
| 604 | 79 {-# TERMINATING #-} |
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parents:
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80 replace-loop : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → List (bt A) → (exit : bt A → t) → t |
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81 replace-loop {_} {_} {A} {t} key value tree st exit = replace-loop1 key value tree st where |
| 604 | 82 replace-loop1 : (key : ℕ) → (value : A) → bt A → List (bt A) → t |
| 83 replace-loop1 key value tree st = replace key value tree st replace-loop1 exit | |
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84 |
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85 insertTree : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : bt A → t ) → t |
| 662 | 86 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 |
| 587 | 87 |
| 604 | 88 insertTest1 = insertTree leaf 1 1 (λ x → x ) |
| 611 | 89 insertTest2 = insertTree insertTest1 2 1 (λ x → x ) |
| 669 | 90 insertTest3 = insertTree insertTest2 3 2 (λ x → x ) |
| 91 insertTest4 = insertTree insertTest3 2 2 (λ x → x ) | |
| 587 | 92 |
| 605 | 93 open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_) |
| 94 | |
| 620 | 95 data treeInvariant {n : Level} {A : Set n} : (tree : bt A) → Set n where |
| 96 t-leaf : treeInvariant leaf | |
| 632 | 97 t-single : (key : ℕ) → (value : A) → treeInvariant (node key value leaf leaf) |
| 98 t-right : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key < key₁) → treeInvariant (node key₁ value₁ t₁ t₂) | |
| 99 → treeInvariant (node key value leaf (node key₁ value₁ t₁ t₂)) | |
| 100 t-left : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key₁ < key) → treeInvariant (node key value t₁ t₂) | |
| 101 → treeInvariant (node key₁ value₁ (node key value t₁ t₂) leaf ) | |
| 620 | 102 t-node : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} → (key < key₁) → (key₁ < key₂) |
| 103 → treeInvariant (node key value t₁ t₂) | |
| 104 → treeInvariant (node key₂ value₂ t₃ t₄) | |
| 105 → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) | |
| 605 | 106 |
| 662 | 107 -- |
| 108 -- stack always contains original top at end | |
| 109 -- | |
| 110 data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where | |
| 670 | 111 s-single : {tree0 : bt A} → ¬ ( tree0 ≡ leaf ) → stackInvariant key tree0 tree0 (tree0 ∷ []) |
| 653 | 112 s-right : {tree tree0 tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} |
| 662 | 113 → key₁ < key → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree tree0 (tree ∷ st) |
| 653 | 114 s-left : {tree₁ tree0 tree : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} |
| 662 | 115 → key < key₁ → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree₁ tree0 (tree₁ ∷ st) |
| 639 | 116 |
| 117 data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (tree tree1 : bt A ) → Set n where | |
| 118 r-leaf : replacedTree key value leaf (node key value leaf leaf) | |
| 119 r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁) | |
| 120 r-right : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} | |
| 650 | 121 → k < key → replacedTree key value t1 t2 → replacedTree key value (node k v1 t t1) (node k v1 t t2) |
| 639 | 122 r-left : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} |
| 650 | 123 → k > key → replacedTree key value t1 t2 → replacedTree key value (node k v1 t1 t) (node k v1 t2 t) |
| 639 | 124 |
| 662 | 125 replFromStack : {n : Level} {A : Set n} {key : ℕ} {top orig : bt A} → {stack : List (bt A)} → stackInvariant key top orig stack → bt A |
| 670 | 126 replFromStack (s-single {tree} _ ) = tree |
| 661 | 127 replFromStack (s-right {tree} x st) = tree |
| 128 replFromStack (s-left {tree} x st) = tree | |
| 652 | 129 |
| 632 | 130 add< : { i : ℕ } (j : ℕ ) → i < suc i + j |
| 131 add< {i} j = begin | |
| 132 suc i ≤⟨ m≤m+n (suc i) j ⟩ | |
| 133 suc i + j ∎ where open ≤-Reasoning | |
| 134 | |
| 135 treeTest1 : bt ℕ | |
| 136 treeTest1 = node 1 0 leaf (node 3 1 (node 2 5 (node 4 7 leaf leaf ) leaf) (node 5 5 leaf leaf)) | |
| 137 treeTest2 : bt ℕ | |
| 138 treeTest2 = node 3 1 (node 2 5 (node 4 7 leaf leaf ) leaf) (node 5 5 leaf leaf) | |
| 139 | |
| 140 treeInvariantTest1 : treeInvariant treeTest1 | |
| 141 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) ) | |
| 605 | 142 |
| 639 | 143 stack-top : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) |
| 144 stack-top [] = nothing | |
| 145 stack-top (x ∷ s) = just x | |
| 606 | 146 |
| 639 | 147 stack-last : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) |
| 148 stack-last [] = nothing | |
| 149 stack-last (x ∷ []) = just x | |
| 150 stack-last (x ∷ s) = stack-last s | |
| 632 | 151 |
| 662 | 152 stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] ) |
| 670 | 153 stackInvariantTest1 = s-right (add< 2) (s-single (λ ())) |
| 662 | 154 |
| 666 | 155 si-property0 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 stack → ¬ ( stack ≡ [] ) |
| 670 | 156 si-property0 (s-single _ ) () |
| 666 | 157 si-property0 (s-right x si) () |
| 158 si-property0 (s-left x si) () | |
| 665 | 159 |
| 666 | 160 si-property1 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 tree1 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 (tree1 ∷ stack) |
| 161 → tree1 ≡ tree | |
| 670 | 162 si-property1 (s-single _ ) = refl |
| 666 | 163 si-property1 (s-right _ si) = refl |
| 164 si-property1 (s-left _ si) = refl | |
| 662 | 165 |
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166 si-property-last : {n : Level} {A : Set n} (key : ℕ) (tree tree0 : bt A) → (stack : List (bt A)) → stackInvariant key tree tree0 stack |
| 662 | 167 → stack-last stack ≡ just tree0 |
| 670 | 168 si-property-last key t t0 (t ∷ []) (s-single _) = refl |
| 666 | 169 si-property-last key t t0 (.t ∷ x ∷ st) (s-right _ si ) with si-property1 si |
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170 ... | refl = si-property-last key x t0 (x ∷ st) si |
| 666 | 171 si-property-last key t t0 (.t ∷ x ∷ st) (s-left _ si ) with si-property1 si |
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172 ... | refl = si-property-last key x t0 (x ∷ st) si |
| 656 | 173 |
| 642 | 174 ti-right : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 tree₁ repl) → treeInvariant repl |
| 175 ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf | |
| 176 ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-right x ti) = ti | |
| 177 ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-left x ti) = t-leaf | |
| 178 ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti₁ | |
| 179 | |
| 180 ti-left : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 repl tree₁ ) → treeInvariant repl | |
| 181 ti-left {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf | |
| 182 ti-left {_} {_} {_} {_} {key₁} {v1} (t-right x ti) = t-leaf | |
| 183 ti-left {_} {_} {_} {_} {key₁} {v1} (t-left x ti) = ti | |
| 184 ti-left {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti | |
| 185 | |
| 662 | 186 stackTreeInvariant : {n : Level} {A : Set n} (key : ℕ) (sub tree : bt A) → (stack : List (bt A)) |
| 187 → treeInvariant tree → stackInvariant key sub tree stack → treeInvariant sub | |
| 670 | 188 stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-single _) = ti |
| 662 | 189 stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si ) = ti-right (si1 si) where |
| 190 si1 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 tree₁ sub ) tree st → treeInvariant (node key₁ v1 tree₁ sub ) | |
| 191 si1 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 tree₁ sub ) tree st ti si | |
| 192 stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si ) = ti-left ( si2 si) where | |
| 193 si2 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 sub tree₁ ) tree st → treeInvariant (node key₁ v1 sub tree₁ ) | |
| 194 si2 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 sub tree₁ ) tree st ti si | |
| 195 | |
| 639 | 196 rt-property1 : {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf ) |
| 197 rt-property1 {n} {A} key value .leaf .(node key value leaf leaf) r-leaf () | |
| 198 rt-property1 {n} {A} key value .(node key _ _ _) .(node key value _ _) r-node () | |
| 199 rt-property1 {n} {A} key value .(node _ _ _ _) .(node _ _ _ _) (r-right x rt) () | |
| 200 rt-property1 {n} {A} key value .(node _ _ _ _) .(node _ _ _ _) (r-left x rt) () | |
| 201 | |
| 632 | 202 depth-1< : {i j : ℕ} → suc i ≤ suc (i Data.Nat.⊔ j ) |
| 203 depth-1< {i} {j} = s≤s (m≤m⊔n _ j) | |
| 204 | |
| 205 depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i ) | |
| 650 | 206 depth-2< {i} {j} = s≤s (m≤n⊔m j i) |
| 611 | 207 |
| 649 | 208 depth-3< : {i : ℕ } → suc i ≤ suc (suc i) |
| 209 depth-3< {zero} = s≤s ( z≤n ) | |
| 210 depth-3< {suc i} = s≤s (depth-3< {i} ) | |
| 211 | |
| 212 | |
| 634 | 213 treeLeftDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A ) |
| 214 → treeInvariant (node k v1 tree tree₁) | |
| 215 → treeInvariant tree | |
| 216 treeLeftDown {n} {A} {_} {v1} leaf leaf (t-single k1 v1) = t-leaf | |
| 217 treeLeftDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = t-leaf | |
| 218 treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = ti | |
| 219 treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti | |
| 220 | |
| 221 treeRightDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A ) | |
| 222 → treeInvariant (node k v1 tree tree₁) | |
| 223 → treeInvariant tree₁ | |
| 224 treeRightDown {n} {A} {_} {v1} .leaf .leaf (t-single _ .v1) = t-leaf | |
| 225 treeRightDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = ti | |
| 226 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = t-leaf | |
| 227 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti₁ | |
| 228 | |
| 664 | 229 nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ |
| 230 nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x | |
| 231 nat-<> : { x y : ℕ } → x < y → y < x → ⊥ | |
| 232 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x | |
| 633 | 233 |
| 234 open _∧_ | |
| 235 | |
| 615 | 236 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A)) |
| 662 | 237 → treeInvariant tree ∧ stackInvariant key tree tree0 stack |
| 238 → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t ) | |
| 239 → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack | |
| 638 | 240 → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t |
| 241 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st Pre (case1 refl) | |
| 632 | 242 findP key (node key₁ v1 tree tree₁) tree0 st Pre next exit with <-cmp key key₁ |
| 638 | 243 findP key n tree0 st Pre _ exit | tri≈ ¬a refl ¬c = exit n tree0 st Pre (case2 refl) |
| 664 | 244 findP {n} {_} {A} key (node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (tree ∷ st) |
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245 ⟪ treeLeftDown tree tree₁ (proj1 Pre) , findP1 a st (proj2 Pre) ⟫ depth-1< where |
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246 findP1 : key < key₁ → (st : List (bt A)) → stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (tree ∷ st) |
| 664 | 247 findP1 a (x ∷ st) si = s-left a si |
| 662 | 248 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< |
| 606 | 249 |
| 638 | 250 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) |
| 251 replaceTree1 k v1 value (t-single .k .v1) = t-single k value | |
| 252 replaceTree1 k v1 value (t-right x t) = t-right x t | |
| 253 replaceTree1 k v1 value (t-left x t) = t-left x t | |
| 254 replaceTree1 k v1 value (t-node x x₁ t t₁) = t-node x x₁ t t₁ | |
| 255 | |
| 649 | 256 open import Relation.Binary.Definitions |
| 257 | |
| 258 lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥ | |
| 259 lemma3 refl () | |
| 260 lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥ | |
| 261 lemma5 (s≤s z≤n) () | |
| 262 | |
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263 record replacePR {n : Level} {A : Set n} (key : ℕ) (value : A) (tree repl : bt A ) (stack : List (bt A)) (C : bt A → bt A → List (bt A) → Set n) : Set n where |
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264 field |
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265 tree0 : bt A |
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266 ti : treeInvariant tree0 |
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267 si : stackInvariant key tree tree0 stack |
| 672 | 268 ri : replacedTree key value tree repl |
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269 ci : C tree repl stack -- data continuation |
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270 |
| 638 | 271 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) |
| 272 → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) | |
| 273 → (treeInvariant tree ) → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value tree tree1 → t) → t | |
| 274 replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf | |
| 275 replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P) r-node | |
| 606 | 276 |
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277 replaceP : {n m : Level} {A : Set n} {t : Set m} |
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278 → (key : ℕ) → (value : A) → {tree : bt A} ( repl : bt A) |
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279 → (stack : List (bt A)) → replacePR key value tree repl stack (λ _ _ _ → Lift n ⊤) |
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280 → (next : ℕ → A → {tree1 : bt A } (repl : bt A) → (stack1 : List (bt A)) |
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281 → replacePR key value tree1 repl stack1 (λ _ _ _ → Lift n ⊤) → length stack1 < length stack → t) |
| 613 | 282 → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t |
| 672 | 283 replaceP key value {tree} repl [] Pre next exit = ⊥-elim ( si-property0 (replacePR.si Pre) refl ) -- can't happen |
| 284 replaceP key value {tree} repl (leaf ∷ []) Pre next exit with si-property-last _ _ _ _ (replacePR.si Pre) -- tree0 ≡ leaf | |
| 285 ... | refl = exit (replacePR.tree0 Pre) (node key value leaf leaf) ⟪ replacePR.ti Pre , r-leaf ⟫ | |
| 286 replaceP key value {tree} repl (node key₁ value₁ left right ∷ []) Pre next exit = exit (replacePR.tree0 Pre) repl ⟪ replacePR.ti Pre , repl02 ⟫ where | |
| 287 repl01 : tree ≡ replacePR.tree0 Pre | |
| 288 repl01 with replacePR.si Pre | |
| 289 ... | s-single x = refl | |
| 290 repl05 : just (node key₁ value₁ left right) ≡ just (replacePR.tree0 Pre) | |
| 291 repl05 = si-property-last _ _ _ _ (replacePR.si Pre) | |
| 292 repl02 : replacedTree key value (replacePR.tree0 Pre) repl | |
| 293 repl02 = subst (λ k → replacedTree key value k repl ) repl01 (replacePR.ri Pre) | |
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294 replaceP {n} {_} {A} key value {tree} repl (leaf ∷ st@(x ∷ xs)) Pre next exit = {!!} -- can't happen |
| 672 | 295 replaceP {n} {_} {A} key value {tree} repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre next exit with <-cmp key key₁ |
| 296 ... | tri< a ¬b ¬c = next key value {tree1} (node key₁ value₁ repl right ) st Post ≤-refl where | |
| 297 Post : replacePR key value tree1 (node key₁ value₁ repl right ) st (λ _ _ _ → Lift n ⊤) | |
| 298 Post with replacePR.si Pre | |
| 299 ... | s-right lt si1 with si-property1 si1 -- lt : suc key₂ ≤ key -- not allowed | |
| 300 ... | refl = record { tree0 = replacePR.tree0 Pre; ti = replacePR.ti Pre ; si = si1; ri = repl04 } where | |
| 301 repl03 : replacedTree key value tree1 {!!} | |
| 302 repl03 = r-right lt ( replacePR.ri Pre) | |
| 303 repl04 : replacedTree key value tree1 (node key₁ value₁ repl right) | |
| 304 repl04 = subst (λ k → replacedTree key value tree1 k ) {!!} (r-right lt ( replacePR.ri Pre)) | |
| 305 Post | s-left lt si1 with si-property1 si1 | |
| 306 ... | refl = record { tree0 = replacePR.tree0 Pre; ti = replacePR.ti Pre ; si = si1; ri = repl06 } where | |
| 307 repl05 : replacedTree key value tree1 {!!} | |
| 308 repl05 = r-left lt ( replacePR.ri Pre) | |
| 309 repl06 : replacedTree key value tree1 (node key₁ value₁ repl right) | |
| 310 repl06 = subst (λ k → replacedTree key value tree1 k ) {!!} (r-left lt ( replacePR.ri Pre)) | |
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311 ... | tri≈ ¬a b ¬c = next key value {{!!}} (node key₁ value left right ) st {!!} ≤-refl |
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312 ... | tri> ¬a ¬b c = next key value {{!!}} (node key₁ value₁ left tree ) st {!!} ≤-refl |
| 644 | 313 |
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314 TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) |
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315 → (r : Index) → (p : Invraiant r) |
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316 → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t |
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317 TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r) |
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318 ... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) |
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319 ... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (≤-step lt1) p1 lt1 ) where |
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320 TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → Invraiant r1 → reduce r1 < reduce r → t |
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321 TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1)) |
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322 TerminatingLoop1 (suc j) r r1 n≤j p1 lt with <-cmp (reduce r1) (suc j) |
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323 ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt |
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324 ... | tri≈ ¬a b ¬c = loop r1 p1 (λ r2 p2 lt1 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b lt1 ) p2 lt1 ) |
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325 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) |
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326 |
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327 open _∧_ |
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328 |
| 615 | 329 RTtoTI0 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 330 → replacedTree key value tree repl → treeInvariant repl | |
| 331 RTtoTI0 = {!!} | |
| 332 | |
| 333 RTtoTI1 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant repl | |
| 334 → replacedTree key value tree repl → treeInvariant tree | |
| 335 RTtoTI1 = {!!} | |
| 614 | 336 |
| 611 | 337 insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 613 | 338 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t |
| 610 | 339 insertTreeP {n} {m} {A} {t} tree key value P exit = |
| 662 | 340 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 , {!!} ⟫ |
| 615 | 341 $ λ p P loop → findP key (proj1 p) tree (proj2 p) {!!} (λ t _ s P1 lt → loop ⟪ t , s ⟫ {!!} lt ) |
| 638 | 342 $ λ t _ s P C → replaceNodeP key value t C (proj1 P) |
| 614 | 343 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A )) |
| 662 | 344 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant key (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } |
| 639 | 345 (λ p → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫ ⟪ proj1 P , ⟪ {!!} , R ⟫ ⟫ |
| 644 | 346 $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) {!!} |
| 347 (λ key value repl1 stack P2 lt → loop ⟪ stack , ⟪ {!!} , repl1 ⟫ ⟫ {!!} lt ) exit | |
| 614 | 348 |
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349 top-value : {n : Level} {A : Set n} → (tree : bt A) → Maybe A |
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350 top-value leaf = nothing |
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351 top-value (node key value tree tree₁) = just value |
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352 |
| 612 | 353 insertTreeSpec0 : {n : Level} {A : Set n} → (tree : bt A) → (value : A) → top-value tree ≡ just value → ⊤ |
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354 insertTreeSpec0 _ _ _ = tt |
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355 |
| 627 | 356 record findPR {n : Level} {A : Set n} (key : ℕ) (tree : bt A ) (stack : List (bt A)) (C : bt A → List (bt A) → Set n) : Set n where |
| 618 | 357 field |
| 619 | 358 tree0 : bt A |
| 622 | 359 ti : treeInvariant tree0 |
| 662 | 360 si : stackInvariant key tree tree0 stack |
| 631 | 361 ci : C tree stack -- data continuation |
| 618 | 362 |
| 616 | 363 findPP : {n m : Level} {A : Set n} {t : Set m} |
| 364 → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) | |
| 627 | 365 → (Pre : findPR key tree stack (λ t s → Lift n ⊤)) |
| 366 → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR key tree1 stack1 (λ t s → Lift n ⊤) → bt-depth tree1 < bt-depth tree → t ) | |
| 367 → (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 | |
| 625 | 368 findPP key leaf st Pre next exit = exit leaf st (case1 refl) Pre |
| 632 | 369 findPP key (node key₁ v1 tree tree₁) st Pre next exit with <-cmp key key₁ |
| 625 | 370 findPP key n st P next exit | tri≈ ¬a b ¬c = exit n st (case2 {!!}) P |
| 632 | 371 findPP {_} {_} {A} key n@(node key₁ v1 tree tree₁) st Pre next exit | tri< a ¬b ¬c = |
| 624 | 372 next tree (n ∷ st) (record {ti = findPR.ti Pre ; si = findPP2 st (findPR.si Pre) ; ci = lift tt} ) findPP1 where |
| 621 | 373 tree0 = findPR.tree0 Pre |
| 662 | 374 findPP2 : (st : List (bt A)) → stackInvariant key {!!} tree0 st → stackInvariant key {!!} tree0 (node key₁ v1 tree tree₁ ∷ st) |
| 623 | 375 findPP2 = {!!} |
| 618 | 376 findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁) |
| 634 | 377 findPP1 = depth-1< |
| 632 | 378 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) |
| 618 | 379 findPP2 : suc (bt-depth tree₁) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁) |
| 634 | 380 findPP2 = depth-2< |
| 616 | 381 |
| 618 | 382 insertTreePP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 383 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t | |
| 624 | 384 insertTreePP {n} {m} {A} {t} tree key value P exit = |
| 627 | 385 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 , [] ⟫ {!!} |
| 630 | 386 $ λ p P loop → findPP key (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) |
| 638 | 387 $ λ t s _ P → replaceNodeP key value t {!!} {!!} |
| 618 | 388 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A )) |
| 662 | 389 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant key (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } |
| 639 | 390 (λ p → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫ ⟪ {!!} , ⟪ {!!} , R ⟫ ⟫ |
| 644 | 391 $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) {!!} |
| 392 (λ key value repl1 stack P2 lt → loop ⟪ stack , ⟪ {!!} , repl1 ⟫ ⟫ {!!} lt ) exit | |
| 618 | 393 |
| 629 | 394 record findPC {n : Level} {A : Set n} (key1 : ℕ) (value1 : A) (tree : bt A ) (stack : List (bt A)) : Set n where |
| 616 | 395 field |
| 396 tree1 : bt A | |
| 617 | 397 ci : replacedTree key1 value1 tree tree1 |
| 616 | 398 |
| 624 | 399 findPPC : {n m : Level} {A : Set n} {t : Set m} |
| 628 | 400 → (key : ℕ) → (value : A) → (tree : bt A ) → (stack : List (bt A)) |
| 629 | 401 → (Pre : findPR key tree stack (findPC key value)) |
| 402 → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR key tree1 stack1 (findPC key value) → bt-depth tree1 < bt-depth tree → t ) | |
| 403 → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → ( tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key) → findPR key tree1 stack1 (findPC key value) → t) → t | |
| 404 findPPC key value leaf st Pre next exit = exit leaf st (case1 refl) Pre | |
| 632 | 405 findPPC key value (node key₁ v1 tree tree₁) st Pre next exit with <-cmp key key₁ |
| 629 | 406 findPPC key value n st P next exit | tri≈ ¬a b ¬c = exit n st (case2 {!!}) P |
| 632 | 407 findPPC {_} {_} {A} key value n@(node key₁ v1 tree tree₁) st Pre next exit | tri< a ¬b ¬c = |
| 629 | 408 next tree (n ∷ st) (record {ti = findPR.ti Pre ; si = {!!} ; ci = {!!} } ) {!!} |
| 409 findPPC key value n st P next exit | tri> ¬a ¬b c = {!!} | |
| 624 | 410 |
| 618 | 411 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 → ⊤ |
| 615 | 412 containsTree {n} {m} {A} {t} tree tree1 key value P RT = |
| 617 | 413 TerminatingLoopS (bt A ∧ List (bt A) ) |
| 634 | 414 {λ p → findPR key (proj1 p) (proj2 p) (findPC key value ) } (λ p → bt-depth (proj1 p)) -- findPR key tree1 [] (findPC key value) |
| 415 ⟪ tree1 , [] ⟫ record { tree0 = tree ; ti = {!!} ; si = {!!} ; ci = record { tree1 = tree ; ci = RT } } | |
| 630 | 416 $ λ p P loop → findPPC key value (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) |
| 629 | 417 $ λ t1 s1 found? P2 → insertTreeSpec0 t1 value (lemma6 t1 s1 found? P2) where |
| 418 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 | |
| 419 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 | |
| 420 lemma7 : (t1 : bt A) ( s1 : List (bt A) ) (tree0 tree1 : bt A) → | |
| 662 | 421 replacedTree key value t1 tree1 → stackInvariant key t1 tree0 s1 → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key) → top-value t1 ≡ just value |
| 629 | 422 lemma7 = {!!} |
| 615 | 423 |