Mercurial > hg > Members > Moririn
annotate hoareBinaryTree.agda @ 701:690da797cf40
hoareBinaryTree done
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 05 Dec 2021 10:40:44 +0900 |
| parents | adb7c2101f67 |
| children | 5959f7aa0960 |
| 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 |
| 690 | 68 replace key value repl (leaf ∷ []) next exit = exit repl |
| 669 | 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 ) |
| 690 | 73 replace key value repl (leaf ∷ st) next exit = next key value repl st |
| 669 | 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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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 ) |
| 696 | 91 insertTest4 = insertTree insertTest3 2 2 (λ x → x ) -- this is wrong |
| 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)) | |
| 692 | 100 t-left : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key < key₁) → treeInvariant (node key value t₁ t₂) |
| 632 | 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 | |
| 675 | 111 s-single : {tree0 : bt A} → 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 |
| 677 | 117 data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (before after : bt A ) → Set n where |
| 639 | 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} | |
| 677 | 121 → k < key → replacedTree key value t2 t → replacedTree key value (node k v1 t1 t2) (node k v1 t1 t) |
| 639 | 122 r-left : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} |
| 687 | 123 → key < k → replacedTree key value t1 t → replacedTree key value (node k v1 t1 t2) (node k v1 t t2) |
| 652 | 124 |
| 632 | 125 add< : { i : ℕ } (j : ℕ ) → i < suc i + j |
| 126 add< {i} j = begin | |
| 127 suc i ≤⟨ m≤m+n (suc i) j ⟩ | |
| 128 suc i + j ∎ where open ≤-Reasoning | |
| 129 | |
| 130 treeTest1 : bt ℕ | |
| 692 | 131 treeTest1 = node 0 0 leaf (node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf)) |
| 632 | 132 treeTest2 : bt ℕ |
| 692 | 133 treeTest2 = node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf) |
| 632 | 134 |
| 135 treeInvariantTest1 : treeInvariant treeTest1 | |
| 692 | 136 treeInvariantTest1 = t-right (m≤m+n _ 2) (t-node (add< 0) (add< 1) (t-left (add< 0) (t-single 1 7)) (t-single 5 5) ) |
| 605 | 137 |
| 639 | 138 stack-top : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) |
| 139 stack-top [] = nothing | |
| 140 stack-top (x ∷ s) = just x | |
| 606 | 141 |
| 639 | 142 stack-last : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) |
| 143 stack-last [] = nothing | |
| 144 stack-last (x ∷ []) = just x | |
| 145 stack-last (x ∷ s) = stack-last s | |
| 632 | 146 |
| 662 | 147 stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] ) |
| 692 | 148 stackInvariantTest1 = s-right (add< 3) (s-single ) |
| 662 | 149 |
| 666 | 150 si-property0 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 stack → ¬ ( stack ≡ [] ) |
| 675 | 151 si-property0 (s-single ) () |
| 666 | 152 si-property0 (s-right x si) () |
| 153 si-property0 (s-left x si) () | |
| 665 | 154 |
| 666 | 155 si-property1 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 tree1 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 (tree1 ∷ stack) |
| 156 → tree1 ≡ tree | |
| 675 | 157 si-property1 (s-single ) = refl |
| 666 | 158 si-property1 (s-right _ si) = refl |
| 159 si-property1 (s-left _ si) = refl | |
| 662 | 160 |
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161 si-property-last : {n : Level} {A : Set n} (key : ℕ) (tree tree0 : bt A) → (stack : List (bt A)) → stackInvariant key tree tree0 stack |
| 662 | 162 → stack-last stack ≡ just tree0 |
| 675 | 163 si-property-last key t t0 (t ∷ []) (s-single ) = refl |
| 666 | 164 si-property-last key t t0 (.t ∷ x ∷ st) (s-right _ si ) with si-property1 si |
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165 ... | refl = si-property-last key x t0 (x ∷ st) si |
| 666 | 166 si-property-last key t t0 (.t ∷ x ∷ st) (s-left _ si ) with si-property1 si |
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167 ... | refl = si-property-last key x t0 (x ∷ st) si |
| 656 | 168 |
| 642 | 169 ti-right : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 tree₁ repl) → treeInvariant repl |
| 170 ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf | |
| 171 ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-right x ti) = ti | |
| 172 ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-left x ti) = t-leaf | |
| 173 ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti₁ | |
| 174 | |
| 175 ti-left : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 repl tree₁ ) → treeInvariant repl | |
| 176 ti-left {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf | |
| 177 ti-left {_} {_} {_} {_} {key₁} {v1} (t-right x ti) = t-leaf | |
| 178 ti-left {_} {_} {_} {_} {key₁} {v1} (t-left x ti) = ti | |
| 179 ti-left {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti | |
| 180 | |
| 662 | 181 stackTreeInvariant : {n : Level} {A : Set n} (key : ℕ) (sub tree : bt A) → (stack : List (bt A)) |
| 182 → treeInvariant tree → stackInvariant key sub tree stack → treeInvariant sub | |
| 675 | 183 stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-single ) = ti |
| 662 | 184 stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si ) = ti-right (si1 si) where |
| 185 si1 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 tree₁ sub ) tree st → treeInvariant (node key₁ v1 tree₁ sub ) | |
| 186 si1 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 tree₁ sub ) tree st ti si | |
| 187 stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si ) = ti-left ( si2 si) where | |
| 188 si2 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 sub tree₁ ) tree st → treeInvariant (node key₁ v1 sub tree₁ ) | |
| 189 si2 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 sub tree₁ ) tree st ti si | |
| 190 | |
| 639 | 191 rt-property1 : {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf ) |
| 192 rt-property1 {n} {A} key value .leaf .(node key value leaf leaf) r-leaf () | |
| 193 rt-property1 {n} {A} key value .(node key _ _ _) .(node key value _ _) r-node () | |
| 677 | 194 rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-right x rt) = λ () |
| 195 rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-left x rt) = λ () | |
| 639 | 196 |
| 690 | 197 rt-property-leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {repl : bt A} → replacedTree key value leaf repl → repl ≡ node key value leaf leaf |
| 198 rt-property-leaf r-leaf = refl | |
| 199 | |
| 698 | 200 rt-property-¬leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {tree : bt A} → ¬ replacedTree key value tree leaf |
| 201 rt-property-¬leaf () | |
| 202 | |
| 692 | 203 rt-property-key : {n : Level} {A : Set n} {key key₂ key₃ : ℕ} {value value₂ value₃ : A} {left left₁ right₂ right₃ : bt A} |
| 204 → replacedTree key value (node key₂ value₂ left right₂) (node key₃ value₃ left₁ right₃) → key₂ ≡ key₃ | |
| 205 rt-property-key r-node = refl | |
| 206 rt-property-key (r-right x ri) = refl | |
| 207 rt-property-key (r-left x ri) = refl | |
| 208 | |
| 698 | 209 nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ |
| 210 nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x | |
| 211 nat-<> : { x y : ℕ } → x < y → y < x → ⊥ | |
| 212 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x | |
| 213 | |
| 214 open _∧_ | |
| 215 | |
| 216 | |
| 632 | 217 depth-1< : {i j : ℕ} → suc i ≤ suc (i Data.Nat.⊔ j ) |
| 218 depth-1< {i} {j} = s≤s (m≤m⊔n _ j) | |
| 219 | |
| 220 depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i ) | |
| 650 | 221 depth-2< {i} {j} = s≤s (m≤n⊔m j i) |
| 611 | 222 |
| 649 | 223 depth-3< : {i : ℕ } → suc i ≤ suc (suc i) |
| 224 depth-3< {zero} = s≤s ( z≤n ) | |
| 225 depth-3< {suc i} = s≤s (depth-3< {i} ) | |
| 226 | |
| 227 | |
| 634 | 228 treeLeftDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A ) |
| 229 → treeInvariant (node k v1 tree tree₁) | |
| 230 → treeInvariant tree | |
| 231 treeLeftDown {n} {A} {_} {v1} leaf leaf (t-single k1 v1) = t-leaf | |
| 232 treeLeftDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = t-leaf | |
| 233 treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = ti | |
| 234 treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti | |
| 235 | |
| 236 treeRightDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A ) | |
| 237 → treeInvariant (node k v1 tree tree₁) | |
| 238 → treeInvariant tree₁ | |
| 239 treeRightDown {n} {A} {_} {v1} .leaf .leaf (t-single _ .v1) = t-leaf | |
| 240 treeRightDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = ti | |
| 241 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = t-leaf | |
| 242 treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti₁ | |
| 243 | |
| 615 | 244 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A)) |
| 662 | 245 → treeInvariant tree ∧ stackInvariant key tree tree0 stack |
| 693 | 246 → (next : (tree1 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t ) |
| 247 → (exit : (tree1 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack | |
| 638 | 248 → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t |
| 693 | 249 findP key leaf tree0 st Pre _ exit = exit leaf st Pre (case1 refl) |
| 632 | 250 findP key (node key₁ v1 tree tree₁) tree0 st Pre next exit with <-cmp key key₁ |
| 693 | 251 findP key n tree0 st Pre _ exit | tri≈ ¬a refl ¬c = exit n st Pre (case2 refl) |
| 252 findP {n} {_} {A} key (node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree (tree ∷ st) | |
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253 ⟪ treeLeftDown tree tree₁ (proj1 Pre) , findP1 a st (proj2 Pre) ⟫ depth-1< where |
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254 findP1 : key < key₁ → (st : List (bt A)) → stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (tree ∷ st) |
| 664 | 255 findP1 a (x ∷ st) si = s-left a si |
| 693 | 256 findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) ⟪ treeRightDown tree tree₁ (proj1 Pre) , s-right c (proj2 Pre) ⟫ depth-2< |
| 606 | 257 |
| 638 | 258 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) |
| 259 replaceTree1 k v1 value (t-single .k .v1) = t-single k value | |
| 260 replaceTree1 k v1 value (t-right x t) = t-right x t | |
| 261 replaceTree1 k v1 value (t-left x t) = t-left x t | |
| 262 replaceTree1 k v1 value (t-node x x₁ t t₁) = t-node x x₁ t t₁ | |
| 263 | |
| 649 | 264 open import Relation.Binary.Definitions |
| 265 | |
| 266 lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥ | |
| 267 lemma3 refl () | |
| 268 lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥ | |
| 269 lemma5 (s≤s z≤n) () | |
| 700 | 270 ¬x<x : {x : ℕ} → ¬ (x < x) |
| 271 ¬x<x (s≤s lt) = ¬x<x lt | |
| 649 | 272 |
| 687 | 273 child-replaced : {n : Level} {A : Set n} (key : ℕ) (tree : bt A) → bt A |
| 274 child-replaced key leaf = leaf | |
| 275 child-replaced key (node key₁ value left right) with <-cmp key key₁ | |
| 276 ... | tri< a ¬b ¬c = left | |
| 277 ... | tri≈ ¬a b ¬c = node key₁ value left right | |
| 278 ... | tri> ¬a ¬b c = right | |
| 677 | 279 |
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280 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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281 field |
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282 tree0 : bt A |
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283 ti : treeInvariant tree0 |
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284 si : stackInvariant key tree tree0 stack |
| 687 | 285 ri : replacedTree key value (child-replaced key tree ) repl |
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286 ci : C tree repl stack -- data continuation |
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287 |
| 638 | 288 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) |
| 289 → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) | |
| 694 | 290 → (treeInvariant tree ) → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value (child-replaced key tree) tree1 → t) → t |
| 291 replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf | |
| 292 replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P) | |
| 695 | 293 (subst (λ j → replacedTree k v1 j (node k v1 t t₁) ) repl00 r-node) where |
| 694 | 294 repl00 : node k value t t₁ ≡ child-replaced k (node k value t t₁) |
| 295 repl00 with <-cmp k k | |
| 296 ... | tri< a ¬b ¬c = ⊥-elim (¬b refl) | |
| 297 ... | tri≈ ¬a b ¬c = refl | |
| 298 ... | tri> ¬a ¬b c = ⊥-elim (¬b refl) | |
| 606 | 299 |
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300 replaceP : {n m : Level} {A : Set n} {t : Set m} |
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301 → (key : ℕ) → (value : A) → {tree : bt A} ( repl : bt A) |
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302 → (stack : List (bt A)) → replacePR key value tree repl stack (λ _ _ _ → Lift n ⊤) |
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303 → (next : ℕ → A → {tree1 : bt A } (repl : bt A) → (stack1 : List (bt A)) |
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304 → replacePR key value tree1 repl stack1 (λ _ _ _ → Lift n ⊤) → length stack1 < length stack → t) |
| 613 | 305 → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t |
| 675 | 306 replaceP key value {tree} repl [] Pre next exit = ⊥-elim ( si-property0 (replacePR.si Pre) refl ) -- can't happen |
| 307 replaceP key value {tree} repl (leaf ∷ []) Pre next exit with si-property-last _ _ _ _ (replacePR.si Pre)-- tree0 ≡ leaf | |
| 677 | 308 ... | refl = exit (replacePR.tree0 Pre) (node key value leaf leaf) ⟪ replacePR.ti Pre , r-leaf ⟫ |
| 689 | 309 replaceP key value {tree} repl (node key₁ value₁ left right ∷ []) Pre next exit with <-cmp key key₁ |
| 310 ... | tri< a ¬b ¬c = exit (replacePR.tree0 Pre) (node key₁ value₁ repl right ) ⟪ replacePR.ti Pre , repl01 ⟫ where | |
| 311 repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ repl right ) | |
| 312 repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) | |
| 313 repl01 | refl | refl = subst (λ k → replacedTree key value (node key₁ value₁ k right ) (node key₁ value₁ repl right )) repl02 (r-left a repl03) where | |
| 314 repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl | |
| 315 repl03 = replacePR.ri Pre | |
| 316 repl02 : child-replaced key (node key₁ value₁ left right) ≡ left | |
| 317 repl02 with <-cmp key key₁ | |
| 318 ... | tri< a ¬b ¬c = refl | |
| 319 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a a) | |
| 320 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a) | |
| 321 ... | tri≈ ¬a b ¬c = exit (replacePR.tree0 Pre) repl ⟪ replacePR.ti Pre , repl01 ⟫ where | |
| 678 | 322 repl01 : replacedTree key value (replacePR.tree0 Pre) repl |
| 323 repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) | |
| 689 | 324 repl01 | refl | refl = subst (λ k → replacedTree key value k repl) repl02 (replacePR.ri Pre) where |
| 325 repl02 : child-replaced key (node key₁ value₁ left right) ≡ node key₁ value₁ left right | |
| 326 repl02 with <-cmp key key₁ | |
| 327 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b) | |
| 328 ... | tri≈ ¬a b ¬c = refl | |
| 329 ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b) | |
| 330 ... | tri> ¬a ¬b c = exit (replacePR.tree0 Pre) (node key₁ value₁ left repl ) ⟪ replacePR.ti Pre , repl01 ⟫ where | |
| 331 repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ left repl ) | |
| 332 repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) | |
| 333 repl01 | refl | refl = subst (λ k → replacedTree key value (node key₁ value₁ left k ) (node key₁ value₁ left repl )) repl02 (r-right c repl03) where | |
| 334 repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl | |
| 335 repl03 = replacePR.ri Pre | |
| 336 repl02 : child-replaced key (node key₁ value₁ left right) ≡ right | |
| 337 repl02 with <-cmp key key₁ | |
| 338 ... | tri< a ¬b ¬c = ⊥-elim ( ¬c c) | |
| 339 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬c c) | |
| 340 ... | tri> ¬a ¬b c = refl | |
| 690 | 341 replaceP {n} {_} {A} key value {tree} repl (leaf ∷ st@(tree1 ∷ st1)) Pre next exit = next key value repl st Post ≤-refl where |
| 342 Post : replacePR key value tree1 repl (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) | |
| 343 Post with replacePR.si Pre | |
| 344 ... | s-right {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where | |
| 345 repl09 : tree1 ≡ node key₂ v1 tree₁ leaf | |
| 346 repl09 = si-property1 si | |
| 347 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 348 repl10 with si-property1 si | |
| 349 ... | refl = si | |
| 350 repl07 : child-replaced key (node key₂ v1 tree₁ leaf) ≡ leaf | |
| 351 repl07 with <-cmp key key₂ | |
| 352 ... | tri< a ¬b ¬c = ⊥-elim (¬c x) | |
| 353 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) | |
| 354 ... | tri> ¬a ¬b c = refl | |
| 355 repl12 : replacedTree key value (child-replaced key tree1 ) repl | |
| 356 repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf | |
| 357 ... | s-left {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where | |
| 358 repl09 : tree1 ≡ node key₂ v1 leaf tree₁ | |
| 359 repl09 = si-property1 si | |
| 360 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 361 repl10 with si-property1 si | |
| 362 ... | refl = si | |
| 363 repl07 : child-replaced key (node key₂ v1 leaf tree₁ ) ≡ leaf | |
| 364 repl07 with <-cmp key key₂ | |
| 365 ... | tri< a ¬b ¬c = refl | |
| 366 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) | |
| 367 ... | tri> ¬a ¬b c = ⊥-elim (¬a x) | |
| 368 repl12 : replacedTree key value (child-replaced key tree1 ) repl | |
| 369 repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf | |
| 683 | 370 replaceP {n} {_} {A} key value {tree} repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre next exit with <-cmp key key₁ |
| 371 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st Post ≤-refl where | |
| 675 | 372 Post : replacePR key value tree1 (node key₁ value₁ repl right ) st (λ _ _ _ → Lift n ⊤) |
| 687 | 373 Post with replacePR.si Pre |
| 688 | 374 ... | s-right {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where |
| 375 repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) | |
| 376 repl09 = si-property1 si | |
| 377 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 378 repl10 with si-property1 si | |
| 379 ... | refl = si | |
| 380 repl03 : child-replaced key (node key₁ value₁ left right) ≡ left | |
| 381 repl03 with <-cmp key key₁ | |
| 382 ... | tri< a1 ¬b ¬c = refl | |
| 383 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) | |
| 384 ... | tri> ¬a ¬b c = ⊥-elim (¬a a) | |
| 385 repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right | |
| 386 repl02 with repl09 | <-cmp key key₂ | |
| 387 ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) | |
| 689 | 388 ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) |
| 688 | 389 ... | refl | tri> ¬a ¬b c = refl |
| 390 repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 | |
| 391 repl04 = begin | |
| 392 node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ | |
| 393 node key₁ value₁ left right ≡⟨ sym repl02 ⟩ | |
| 394 child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ | |
| 395 child-replaced key tree1 ∎ where open ≡-Reasoning | |
| 396 repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) | |
| 397 repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) | |
| 687 | 398 ... | s-left {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where |
| 688 | 399 repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ |
| 683 | 400 repl09 = si-property1 si |
| 401 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 402 repl10 with si-property1 si | |
| 403 ... | refl = si | |
| 687 | 404 repl03 : child-replaced key (node key₁ value₁ left right) ≡ left |
| 405 repl03 with <-cmp key key₁ | |
| 406 ... | tri< a1 ¬b ¬c = refl | |
| 407 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) | |
| 408 ... | tri> ¬a ¬b c = ⊥-elim (¬a a) | |
| 409 repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right | |
| 410 repl02 with repl09 | <-cmp key key₂ | |
| 411 ... | refl | tri< a ¬b ¬c = refl | |
| 412 ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) | |
| 413 ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) | |
| 414 repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 | |
| 415 repl04 = begin | |
| 416 node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ | |
| 417 node key₁ value₁ left right ≡⟨ sym repl02 ⟩ | |
| 418 child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ | |
| 419 child-replaced key tree1 ∎ where open ≡-Reasoning | |
| 420 repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) | |
| 421 repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) | |
| 690 | 422 ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st Post ≤-refl where -- can't happen |
| 423 Post : replacePR key value tree1 (node key₁ value left right ) (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) | |
| 424 Post with replacePR.si Pre | |
| 691 | 425 ... | s-right {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 b ; ci = lift tt } where |
| 690 | 426 repl09 : tree1 ≡ node key₂ v1 tree₁ tree -- (node key₁ value₁ left right) |
| 427 repl09 = si-property1 si | |
| 428 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 429 repl10 with si-property1 si | |
| 430 ... | refl = si | |
| 431 repl07 : child-replaced key (node key₂ v1 tree₁ tree) ≡ tree | |
| 432 repl07 with <-cmp key key₂ | |
| 433 ... | tri< a ¬b ¬c = ⊥-elim (¬c x) | |
| 434 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) | |
| 435 ... | tri> ¬a ¬b c = refl | |
| 691 | 436 repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) |
| 437 repl12 refl with repl09 | |
| 438 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node | |
| 439 ... | s-left {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 b ; ci = lift tt } where | |
| 690 | 440 repl09 : tree1 ≡ node key₂ v1 tree tree₁ |
| 441 repl09 = si-property1 si | |
| 442 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 443 repl10 with si-property1 si | |
| 444 ... | refl = si | |
| 445 repl07 : child-replaced key (node key₂ v1 tree tree₁ ) ≡ tree | |
| 446 repl07 with <-cmp key key₂ | |
| 447 ... | tri< a ¬b ¬c = refl | |
| 448 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) | |
| 449 ... | tri> ¬a ¬b c = ⊥-elim (¬a x) | |
| 691 | 450 repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) |
| 451 repl12 refl with repl09 | |
| 452 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node | |
| 690 | 453 ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st Post ≤-refl where |
| 454 Post : replacePR key value tree1 (node key₁ value₁ left repl ) st (λ _ _ _ → Lift n ⊤) | |
| 455 Post with replacePR.si Pre | |
| 456 ... | s-right {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where | |
| 457 repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) | |
| 458 repl09 = si-property1 si | |
| 459 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 460 repl10 with si-property1 si | |
| 461 ... | refl = si | |
| 462 repl03 : child-replaced key (node key₁ value₁ left right) ≡ right | |
| 463 repl03 with <-cmp key key₁ | |
| 464 ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) | |
| 465 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) | |
| 466 ... | tri> ¬a ¬b c = refl | |
| 467 repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right | |
| 468 repl02 with repl09 | <-cmp key key₂ | |
| 469 ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) | |
| 470 ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) | |
| 471 ... | refl | tri> ¬a ¬b c = refl | |
| 472 repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 | |
| 473 repl04 = begin | |
| 474 node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ | |
| 475 node key₁ value₁ left right ≡⟨ sym repl02 ⟩ | |
| 476 child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ | |
| 477 child-replaced key tree1 ∎ where open ≡-Reasoning | |
| 478 repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) | |
| 479 repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) | |
| 480 ... | s-left {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where | |
| 481 repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ | |
| 482 repl09 = si-property1 si | |
| 483 repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) | |
| 484 repl10 with si-property1 si | |
| 485 ... | refl = si | |
| 486 repl03 : child-replaced key (node key₁ value₁ left right) ≡ right | |
| 487 repl03 with <-cmp key key₁ | |
| 488 ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) | |
| 489 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) | |
| 490 ... | tri> ¬a ¬b c = refl | |
| 491 repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right | |
| 492 repl02 with repl09 | <-cmp key key₂ | |
| 493 ... | refl | tri< a ¬b ¬c = refl | |
| 494 ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) | |
| 495 ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) | |
| 496 repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 | |
| 497 repl04 = begin | |
| 498 node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ | |
| 499 node key₁ value₁ left right ≡⟨ sym repl02 ⟩ | |
| 500 child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ | |
| 501 child-replaced key tree1 ∎ where open ≡-Reasoning | |
| 502 repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) | |
| 503 repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) | |
| 644 | 504 |
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505 TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) |
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506 → (r : Index) → (p : Invraiant r) |
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507 → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t |
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508 TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r) |
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509 ... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) |
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510 ... | 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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511 TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → Invraiant r1 → reduce r1 < reduce r → t |
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512 TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1)) |
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513 TerminatingLoop1 (suc j) r r1 n≤j p1 lt with <-cmp (reduce r1) (suc j) |
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514 ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt |
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515 ... | 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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516 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) |
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517 |
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518 open _∧_ |
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519 |
| 615 | 520 RTtoTI0 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 521 → replacedTree key value tree repl → treeInvariant repl | |
| 692 | 522 RTtoTI0 .leaf .(node key value leaf leaf) key value ti r-leaf = t-single key value |
| 523 RTtoTI0 .(node key _ leaf leaf) .(node key value leaf leaf) key value (t-single .key _) r-node = t-single key value | |
| 524 RTtoTI0 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right x ti) r-node = t-right x ti | |
| 525 RTtoTI0 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left x ti) r-node = t-left x ti | |
| 526 RTtoTI0 .(node key _ (node _ _ _ _) (node _ _ _ _)) .(node key value (node _ _ _ _) (node _ _ _ _)) key value (t-node x x₁ ti ti₁) r-node = t-node x x₁ ti ti₁ | |
| 701 | 527 -- r-right case |
| 692 | 528 RTtoTI0 (node _ _ leaf leaf) (node _ _ leaf .(node key value leaf leaf)) key value (t-single _ _) (r-right x r-leaf) = t-right x (t-single key value) |
| 529 RTtoTI0 (node _ _ leaf right@(node _ _ _ _)) (node key₁ value₁ leaf leaf) key value (t-right x₁ ti) (r-right x ri) = t-single key₁ value₁ | |
| 693 | 530 RTtoTI0 (node key₁ _ leaf right@(node key₂ _ _ _)) (node key₁ value₁ leaf right₁@(node key₃ _ _ _)) key value (t-right x₁ ti) (r-right x ri) = |
| 531 t-right (subst (λ k → key₁ < k ) (rt-property-key ri) x₁) (RTtoTI0 _ _ key value ti ri) | |
| 692 | 532 RTtoTI0 (node key₁ _ (node _ _ _ _) leaf) (node key₁ _ (node key₃ value left right) leaf) key value₁ (t-left x₁ ti) (r-right x ()) |
| 533 RTtoTI0 (node key₁ _ (node key₃ _ _ _) leaf) (node key₁ _ (node key₃ value₃ _ _) (node key value leaf leaf)) key value (t-left x₁ ti) (r-right x r-leaf) = | |
| 534 t-node x₁ x ti (t-single key value) | |
| 693 | 535 RTtoTI0 (node key₁ _ (node _ _ _ _) (node key₂ _ _ _)) (node key₁ _ (node _ _ _ _) (node key₃ _ _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x ri) = |
| 536 t-node x₁ (subst (λ k → key₁ < k) (rt-property-key ri) x₂) ti (RTtoTI0 _ _ key value ti₁ ri) | |
| 701 | 537 -- r-left case |
| 700 | 538 RTtoTI0 .(node _ _ leaf leaf) .(node _ _ (node key value leaf leaf) leaf) key value (t-single _ _) (r-left x r-leaf) = t-left x (t-single _ _ ) |
| 701 | 539 RTtoTI0 .(node _ _ leaf (node _ _ _ _)) (node key₁ value₁ (node key value leaf leaf) (node _ _ _ _)) key value (t-right x₁ ti) (r-left x r-leaf) = t-node x x₁ (t-single key value) ti |
| 540 RTtoTI0 (node key₃ _ (node key₂ _ _ _) leaf) (node key₃ _ (node key₁ value₁ left left₁) leaf) key value (t-left x₁ ti) (r-left x ri) = | |
| 541 t-left (subst (λ k → k < key₃ ) (rt-property-key ri) x₁) (RTtoTI0 _ _ key value ti ri) -- key₁ < key₃ | |
| 542 RTtoTI0 (node key₁ _ (node key₂ _ _ _) (node _ _ _ _)) (node key₁ _ (node key₃ _ _ _) (node _ _ _ _)) key value (t-node x₁ x₂ ti ti₁) (r-left x ri) = t-node (subst (λ k → k < key₁ ) (rt-property-key ri) x₁) x₂ (RTtoTI0 _ _ key value ti ri) ti₁ | |
| 615 | 543 |
| 544 RTtoTI1 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant repl | |
| 545 → replacedTree key value tree repl → treeInvariant tree | |
| 701 | 546 RTtoTI1 .leaf .(node key value leaf leaf) key value ti r-leaf = t-leaf |
| 547 RTtoTI1 (node key value₁ leaf leaf) .(node key value leaf leaf) key value (t-single .key .value) r-node = t-single key value₁ | |
| 548 RTtoTI1 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right x ti) r-node = t-right x ti | |
| 549 RTtoTI1 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left x ti) r-node = t-left x ti | |
| 550 RTtoTI1 .(node key _ (node _ _ _ _) (node _ _ _ _)) .(node key value (node _ _ _ _) (node _ _ _ _)) key value (t-node x x₁ ti ti₁) r-node = t-node x x₁ ti ti₁ | |
| 551 -- r-right case | |
| 552 RTtoTI1 (node key₁ value₁ leaf leaf) (node key₁ _ leaf (node _ _ _ _)) key value (t-right x₁ ti) (r-right x r-leaf) = t-single key₁ value₁ | |
| 553 RTtoTI1 (node key₁ value₁ leaf (node key₂ value₂ t2 t3)) (node key₁ _ leaf (node key₃ _ _ _)) key value (t-right x₁ ti) (r-right x ri) = | |
| 554 t-right (subst (λ k → key₁ < k ) (sym (rt-property-key ri)) x₁) (RTtoTI1 _ _ key value ti ri) -- key₁ < key₂ | |
| 555 RTtoTI1 (node _ _ (node _ _ _ _) leaf) (node _ _ (node _ _ _ _) (node key value _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x r-leaf) = | |
| 556 t-left x₁ ti | |
| 557 RTtoTI1 (node key₄ _ (node key₃ _ _ _) (node key₁ value₁ n n₁)) (node key₄ _ (node key₃ _ _ _) (node key₂ _ _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x ri) = t-node x₁ (subst (λ k → key₄ < k ) (sym (rt-property-key ri)) x₂) ti (RTtoTI1 _ _ key value ti₁ ri) -- key₄ < key₁ | |
| 558 -- r-left case | |
| 559 RTtoTI1 (node key₁ value₁ leaf leaf) (node key₁ _ _ leaf) key value (t-left x₁ ti) (r-left x ri) = t-single key₁ value₁ | |
| 560 RTtoTI1 (node key₁ _ (node key₂ value₁ n n₁) leaf) (node key₁ _ (node key₃ _ _ _) leaf) key value (t-left x₁ ti) (r-left x ri) = | |
| 561 t-left (subst (λ k → k < key₁ ) (sym (rt-property-key ri)) x₁) (RTtoTI1 _ _ key value ti ri) -- key₂ < key₁ | |
| 562 RTtoTI1 (node key₁ value₁ leaf _) (node key₁ _ _ _) key value (t-node x₁ x₂ ti ti₁) (r-left x r-leaf) = t-right x₂ ti₁ | |
| 563 RTtoTI1 (node key₁ value₁ (node key₂ value₂ n n₁) _) (node key₁ _ _ _) key value (t-node x₁ x₂ ti ti₁) (r-left x ri) = | |
| 564 t-node (subst (λ k → k < key₁ ) (sym (rt-property-key ri)) x₁) x₂ (RTtoTI1 _ _ key value ti ri) ti₁ -- key₂ < key₁ | |
| 614 | 565 |
| 611 | 566 insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 696 | 567 → (exit : (tree repl : bt A) → treeInvariant repl ∧ replacedTree key value tree repl → t ) → t |
| 693 | 568 insertTreeP {n} {m} {A} {t} tree key value P0 exit = |
| 569 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 , tree ∷ [] ⟫ ⟪ P0 , s-single ⟫ | |
| 696 | 570 $ λ p P loop → findP key (proj1 p) tree (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) |
| 693 | 571 $ λ t s P C → replaceNodeP key value t C (proj1 P) |
| 572 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ bt A ∧ bt A ) | |
| 573 {λ p → replacePR key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) (λ _ _ _ → Lift n ⊤ ) } | |
| 696 | 574 (λ p → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫ record { tree0 = tree ; ti = P0 ; si = proj2 P ; ri = R ; ci = lift tt } |
| 693 | 575 $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) P1 |
| 696 | 576 (λ key value {tree1} repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) |
| 577 $ λ tree repl P → exit tree repl ⟪ RTtoTI0 _ _ _ _ (proj1 P) (proj2 P) , proj2 P ⟫ | |
| 614 | 578 |
| 696 | 579 insertTestP1 = insertTreeP leaf 1 1 t-leaf |
| 580 $ λ _ x P → insertTreeP x 2 1 (proj1 P) | |
| 581 $ λ _ x P → insertTreeP x 3 2 (proj1 P) | |
| 582 $ λ _ x P → insertTreeP x 2 2 (proj1 P) (λ _ x _ → x ) | |
| 694 | 583 |
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584 top-value : {n : Level} {A : Set n} → (tree : bt A) → Maybe A |
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585 top-value leaf = nothing |
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586 top-value (node key value tree tree₁) = just value |
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587 |
| 612 | 588 insertTreeSpec0 : {n : Level} {A : Set n} → (tree : bt A) → (value : A) → top-value tree ≡ just value → ⊤ |
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589 insertTreeSpec0 _ _ _ = tt |
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590 |
| 627 | 591 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 | 592 field |
| 619 | 593 tree0 : bt A |
| 696 | 594 ti0 : treeInvariant tree0 |
| 695 | 595 ti : treeInvariant tree |
| 662 | 596 si : stackInvariant key tree tree0 stack |
| 631 | 597 ci : C tree stack -- data continuation |
| 618 | 598 |
| 695 | 599 findPP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) |
| 600 → findPR key tree stack (λ _ _ → Lift n ⊤) | |
| 601 → (next : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack (λ _ _ → Lift n ⊤) → bt-depth tree1 < bt-depth tree → t ) | |
| 602 → (exit : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack (λ _ _ → Lift n ⊤) | |
| 603 → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t | |
| 604 findPP key leaf st Pre _ exit = exit leaf st Pre (case1 refl) | |
| 632 | 605 findPP key (node key₁ v1 tree tree₁) st Pre next exit with <-cmp key key₁ |
| 695 | 606 findPP key n st Pre _ exit | tri≈ ¬a refl ¬c = exit n st Pre (case2 refl) |
| 607 findPP {n} {_} {A} key (node key₁ v1 tree tree₁) st Pre next _ | tri< a ¬b ¬c = next tree (tree ∷ st) | |
| 696 | 608 record { tree0 = findPR.tree0 Pre ; ti0 = findPR.ti0 Pre ; ti = treeLeftDown tree tree₁ (findPR.ti Pre) ; si = findP1 a st (findPR.si Pre) ; ci = lift tt } depth-1< where |
| 695 | 609 findP1 : key < key₁ → (st : List (bt A)) → stackInvariant key (node key₁ v1 tree tree₁) |
| 610 (findPR.tree0 Pre) st → stackInvariant key tree (findPR.tree0 Pre) (tree ∷ st) | |
| 611 findP1 a (x ∷ st) si = s-left a si | |
| 612 findPP key n@(node key₁ v1 tree tree₁) st Pre next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) | |
| 696 | 613 record { tree0 = findPR.tree0 Pre ; ti0 = findPR.ti0 Pre ; ti = treeRightDown tree tree₁ (findPR.ti Pre) ; si = s-right c (findPR.si Pre) ; ci = lift tt } depth-2< |
| 616 | 614 |
| 618 | 615 insertTreePP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 616 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t | |
| 695 | 617 insertTreePP {n} {m} {A} {t} tree key value P0 exit = |
| 618 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR key (proj1 p) (proj2 p) (λ _ _ → Lift n ⊤) } (λ p → bt-depth (proj1 p)) ⟪ tree , tree ∷ [] ⟫ | |
| 696 | 619 record { tree0 = tree ; ti = P0 ; ti0 = P0 ;si = s-single ; ci = lift tt } |
| 695 | 620 $ λ p P loop → findPP key (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) |
| 621 $ λ t s P C → replaceNodeP key value t C (findPR.ti P) | |
| 622 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ bt A ∧ bt A ) | |
| 623 {λ p → replacePR key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) (λ _ _ _ → Lift n ⊤ ) } | |
| 696 | 624 (λ p → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫ record { tree0 = findPR.tree0 P ; ti = findPR.ti0 P ; si = findPR.si P ; ri = R ; ci = lift tt } |
| 695 | 625 $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) P1 |
| 626 (λ key value {tree1} repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) exit | |
| 618 | 627 |
| 698 | 628 record findPC {n : Level} {A : Set n} (key1 : ℕ) (value : A) (tree : bt A ) (stack : List (bt A)) : Set n where |
| 616 | 629 field |
| 630 tree1 : bt A | |
| 698 | 631 ci : replacedTree key1 value tree1 tree |
| 616 | 632 |
| 698 | 633 findPPC : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A ) → (stack : List (bt A)) |
| 634 → findPR key tree stack (findPC key value ) | |
| 635 → (next : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack (findPC key value ) → bt-depth tree1 < bt-depth tree → t ) | |
| 636 → (exit : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack (findPC key value ) | |
| 695 | 637 → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t |
| 698 | 638 findPPC key value leaf st Pre _ exit = exit leaf st Pre (case1 refl) |
| 639 findPPC key value (node key₁ v1 tree tree₁) st Pre next exit with <-cmp key key₁ | |
| 640 findPPC key value n st Pre _ exit | tri≈ ¬a refl ¬c = exit n st Pre (case2 refl) | |
| 641 findPPC {n} {_} {A} key value (node key₁ v1 tree tree₁) st Pre next _ | tri< a ¬b ¬c = next tree (tree ∷ st) | |
| 697 | 642 record { tree0 = findPR.tree0 Pre ; ti0 = findPR.ti0 Pre ; ti = treeLeftDown tree tree₁ (findPR.ti Pre) ; si = s-left a (findPR.si Pre) |
| 698 | 643 ; ci = findP2 } depth-1< where |
| 644 findP2 : findPC key value tree (tree ∷ st) | |
| 645 findP2 with findPC.ci (findPR.ci Pre) | findPC.tree1 (findPR.ci Pre) | inspect findPC.tree1 (findPR.ci Pre) | |
| 646 findP2 | r-node | leaf | _ = ⊥-elim ( nat-≤> a ≤-refl ) | |
| 647 findP2 | r-node | node key value t t₁ | _ = ⊥-elim ( nat-≤> a ≤-refl ) | |
| 648 findP2 | (r-right x ri) | t | _ = ⊥-elim (nat-<> x a) | |
| 649 findP2 | (r-left x ri) | node key value t t₁ | record { eq = refl } = record { tree1 = t ; ci = ri } | |
| 650 findP2 | r-left x ri | leaf | record { eq = () } | |
| 651 findP2 | r-leaf | leaf | record { eq = eq } = ⊥-elim ( nat-≤> a ≤-refl ) | |
| 652 findPPC key value n@(node key₁ v1 tree tree₁) st Pre next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) | |
| 696 | 653 record { tree0 = findPR.tree0 Pre ; ti0 = findPR.ti0 Pre ; ti = treeRightDown tree tree₁ (findPR.ti Pre) ; si = s-right c (findPR.si Pre) |
| 698 | 654 ; ci = findP2 } depth-2< where |
| 655 findP2 : findPC key value tree₁ (tree₁ ∷ st) | |
| 656 findP2 with findPC.ci (findPR.ci Pre) | findPC.tree1 (findPR.ci Pre) | inspect findPC.tree1 (findPR.ci Pre) | |
| 657 findP2 | r-node | node key value ti ti₁ | eq = ⊥-elim ( nat-≤> c ≤-refl ) | |
| 658 findP2 | r-left x ri | ti | eq = ⊥-elim ( nat-<> x c ) | |
| 659 findP2 | r-right x ri | node key value t t₁ | record { eq = refl } = record { tree1 = t₁ ; ci = ri } | |
| 660 findP2 | r-leaf | leaf | record { eq = eq } = ⊥-elim ( nat-≤> c ≤-refl ) | |
| 624 | 661 |
| 700 | 662 containsTree : {n : Level} {A : Set n} → (tree tree1 : bt A) → (key : ℕ) → (value : A) → treeInvariant tree1 → replacedTree key value tree1 tree → ⊤ |
| 663 containsTree {n} {A} tree tree1 key value P RT = | |
| 617 | 664 TerminatingLoopS (bt A ∧ List (bt A) ) |
| 698 | 665 {λ p → findPR key (proj1 p) (proj2 p) (findPC key value ) } (λ p → bt-depth (proj1 p)) |
| 666 ⟪ tree , tree ∷ [] ⟫ record { tree0 = tree ; ti0 = RTtoTI0 _ _ _ _ P RT ; ti = RTtoTI0 _ _ _ _ P RT ; si = s-single | |
| 667 ; ci = record { tree1 = tree1 ; ci = RT } } | |
| 668 $ λ p P loop → findPPC key value (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) | |
| 695 | 669 $ λ t1 s1 P2 found? → insertTreeSpec0 t1 value (lemma6 t1 s1 found? P2) where |
| 698 | 670 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 |
| 671 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 | |
| 700 | 672 lemma8 : {tree1 t1 : bt A } → replacedTree key value tree1 t1 → node-key t1 ≡ just key → top-value t1 ≡ just value |
| 673 lemma8 {.leaf} {node key value .leaf .leaf} r-leaf refl = refl | |
| 674 lemma8 {.(node key _ t1 t2)} {node key value t1 t2} r-node refl = refl | |
| 675 lemma8 {.(node key value t1 _)} {node key value t1 t2} (r-right x ri) refl = ⊥-elim (¬x<x x) | |
| 676 lemma8 {.(node key value _ t2)} {node key value t1 t2} (r-left x ri) refl = ⊥-elim (¬x<x x) | |
| 698 | 677 lemma7 : (t1 : bt A) ( s1 : List (bt A) ) (tree0 tree1 : bt A) |
| 678 → replacedTree key value tree1 t1 → stackInvariant key t1 tree0 s1 | |
| 679 → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key) → top-value t1 ≡ just value | |
| 680 lemma7 .leaf (.leaf ∷ []) .leaf tree1 () s-single (case1 refl) | |
| 700 | 681 lemma7 (node key value t1 t2) (.(node key value t1 t2) ∷ []) .(node key value t1 t2) tree1 ri s-single (case2 x) = lemma8 ri x |
| 682 lemma7 (node key value t1 t2) (.(node key value t1 t2) ∷ x₁ ∷ s1) tree0 tree1 ri (s-right x si) (case2 x₂) = lemma8 ri x₂ | |
| 683 lemma7 (node key value t1 t2) (.(node key value t1 t2) ∷ x₁ ∷ s1) tree0 tree1 ri (s-left x si) (case2 x₂) = lemma8 ri x₂ | |
| 684 | |
| 615 | 685 |
| 700 | 686 containsTree1 : {n : Level} {A : Set n} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → ⊤ |
| 687 containsTree1 {n} {A} tree key value ti = | |
| 688 insertTreeP tree key value ti | |
| 689 $ λ tree0 tree1 P → containsTree tree1 tree0 key value (RTtoTI1 _ _ _ _ (proj1 P) (proj2 P) ) (proj2 P) -- (proj1 P) (proj2 P) | |
| 690 | |
| 691 |