Mercurial > hg > Members > Moririn
annotate hoareBinaryTree.agda @ 626:6465673df5bc
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 08 Nov 2021 22:45:19 +0900 |
| parents | 074fb29ebf57 |
| children | 967547859521 |
| 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 | |
| 51 find key (node key₁ v tree tree₁) st next exit with <-cmp key key₁ | |
| 52 find key n st _ exit | tri≈ ¬a b ¬c = exit n st | |
| 53 find key n@(node key₁ v tree tree₁) st next _ | tri< a ¬b ¬c = next tree (n ∷ st) | |
| 54 find key n@(node key₁ v 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 |
| 63 replaceNode k v leaf next = next (node k v leaf leaf) | |
| 64 replaceNode k v (node key value t t₁) next = next (node k v t t₁) | |
| 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 |
| 604 | 67 replace key value tree [] next exit = exit tree |
| 68 replace key value tree (leaf ∷ st) next exit = next key value tree st | |
| 69 replace key value tree (node key₁ value₁ left right ∷ st) next exit with <-cmp key key₁ | |
| 70 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) st | |
| 71 ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st | |
| 72 ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left tree ) st | |
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73 |
| 604 | 74 {-# TERMINATING #-} |
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75 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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76 replace-loop {_} {_} {A} {t} key value tree st exit = replace-loop1 key value tree st where |
| 604 | 77 replace-loop1 : (key : ℕ) → (value : A) → bt A → List (bt A) → t |
| 78 replace-loop1 key value tree st = replace key value tree st replace-loop1 exit | |
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79 |
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80 insertTree : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : bt A → t ) → t |
| 611 | 81 insertTree tree key value exit = find-loop key tree [] $ λ t st → replaceNode key value t $ λ t1 → replace-loop key value t1 st exit |
| 587 | 82 |
| 604 | 83 insertTest1 = insertTree leaf 1 1 (λ x → x ) |
| 611 | 84 insertTest2 = insertTree insertTest1 2 1 (λ x → x ) |
| 587 | 85 |
| 605 | 86 open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_) |
| 87 | |
| 620 | 88 data treeInvariant {n : Level} {A : Set n} : (tree : bt A) → Set n where |
| 89 t-leaf : treeInvariant leaf | |
| 90 t-single : {key : ℕ} → {value : A} → treeInvariant (node key value leaf leaf) | |
| 91 t-right : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key < key₁) → treeInvariant (node key₁ value₁ t₁ t₂) → treeInvariant (node key value leaf (node key₁ value₁ t₁ t₂)) | |
| 92 t-left : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key₁ < key) → treeInvariant (node key value₁ t₁ t₂) → treeInvariant (node key₁ value₁ (node key value₁ t₁ t₂) leaf ) | |
| 93 t-node : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} → (key < key₁) → (key₁ < key₂) | |
| 94 → treeInvariant (node key value t₁ t₂) | |
| 95 → treeInvariant (node key₂ value₂ t₃ t₄) | |
| 96 → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) | |
| 605 | 97 |
| 620 | 98 treeInvariantTest1 : treeInvariant (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf))) |
| 99 treeInvariantTest1 = {!!} | |
| 605 | 100 |
| 621 | 101 data stackInvariant {n : Level} {A : Set n} : (tree tree0 : bt A) → (stack : List (bt A)) → Set n where |
| 102 s-nil : stackInvariant leaf leaf [] | |
| 103 s-single : (tree : bt A) → stackInvariant tree tree (tree ∷ [] ) | |
| 620 | 104 s-< : (tree0 tree : bt A) → {key : ℕ } → {value : A } { left : bt A} → {st : List (bt A)} |
| 621 | 105 → stackInvariant (node key value left tree ) tree0 (node key value left tree ∷ st ) → stackInvariant tree tree0 (tree ∷ node key value left tree ∷ st ) |
| 106 s-> : (tree0 tree : bt A) → {key : ℕ } → {value : A } { right : bt A} → {st : List (bt A)} | |
| 107 → stackInvariant (node key value tree right ) tree0 (node key value tree right ∷ st ) → stackInvariant tree tree0 (tree ∷ node key value tree right ∷ st ) | |
| 606 | 108 |
| 613 | 109 data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (tree tree1 : bt A ) → Set n where |
| 615 | 110 r-leaf : replacedTree key value leaf (node key value leaf leaf) |
| 111 r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁) | |
| 112 r-right : {k : ℕ } {v : A} → {t t1 t2 : bt A} | |
| 113 → k > key → ( replacedTree key value t1 t2 → replacedTree key value (node k v t t1) (node k v t t2) ) | |
| 114 r-left : {k : ℕ } {v : A} → {t t1 t2 : bt A} | |
| 115 → k < key → ( replacedTree key value t1 t2 → replacedTree key value (node k v t1 t) (node k v t2 t) ) | |
| 611 | 116 |
| 615 | 117 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A)) |
| 621 | 118 → treeInvariant tree ∧ stackInvariant tree tree0 stack |
| 119 → (next : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t ) | |
| 120 → (exit : (tree1 tree0 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant tree1 tree0 stack → t ) → t | |
| 615 | 121 findP key leaf tree0 st Pre _ exit = exit leaf tree0 st {!!} |
| 122 findP key (node key₁ v tree tree₁) tree0 st Pre next exit with <-cmp key key₁ | |
| 123 findP key n tree0 st Pre _ exit | tri≈ ¬a b ¬c = exit n tree0 st {!!} | |
| 124 findP key n@(node key₁ v tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree tree0 (n ∷ st) {!!} {!!} | |
| 125 findP key n@(node key₁ v tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ tree0 (n ∷ st) {!!} {!!} | |
| 606 | 126 |
| 611 | 127 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (treeInvariant tree ) |
| 613 | 128 → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value tree tree1 → t) → t |
| 129 replaceNodeP k v leaf P next = next (node k v leaf leaf) {!!} {!!} | |
| 130 replaceNodeP k v (node key value t t₁) P next = next (node k v t t₁) {!!} {!!} | |
| 606 | 131 |
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132 replaceP : {n m : Level} {A : Set n} {t : Set m} |
| 621 | 133 → (key : ℕ) → (value : A) → (tree repl : bt A) → (stack : List (bt A)) → treeInvariant tree ∧ stackInvariant repl tree stack ∧ replacedTree key value tree repl |
| 134 → (next : ℕ → A → (tree1 repl : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant repl tree1 stack ∧ replacedTree key value tree1 repl → bt-depth tree1 < bt-depth tree → t ) | |
| 613 | 135 → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t |
| 136 replaceP key value tree repl [] Pre next exit = exit tree repl {!!} | |
| 614 | 137 replaceP key value tree repl (leaf ∷ st) Pre next exit = next key value tree {!!} st {!!} {!!} |
| 613 | 138 replaceP key value tree repl (node key₁ value₁ left right ∷ st) Pre next exit with <-cmp key key₁ |
| 614 | 139 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) {!!} st {!!} {!!} |
| 140 ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) {!!} st {!!} {!!} | |
| 141 ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left tree ) {!!} st {!!} {!!} | |
| 606 | 142 |
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143 open import Relation.Binary.Definitions |
| 606 | 144 |
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145 nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ |
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146 nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x |
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147 lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥ |
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148 lemma3 refl () |
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149 lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥ |
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150 lemma5 (s≤s z≤n) () |
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151 |
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152 TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) |
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153 → (r : Index) → (p : Invraiant r) |
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154 → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t |
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155 TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r) |
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156 ... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) |
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157 ... | 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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158 TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → Invraiant r1 → reduce r1 < reduce r → t |
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159 TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1)) |
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160 TerminatingLoop1 (suc j) r r1 n≤j p1 lt with <-cmp (reduce r1) (suc j) |
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161 ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt |
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162 ... | 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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163 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) |
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164 |
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165 open _∧_ |
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166 |
| 615 | 167 RTtoTI0 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 168 → replacedTree key value tree repl → treeInvariant repl | |
| 169 RTtoTI0 = {!!} | |
| 170 | |
| 171 RTtoTI1 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant repl | |
| 172 → replacedTree key value tree repl → treeInvariant tree | |
| 173 RTtoTI1 = {!!} | |
| 614 | 174 |
| 611 | 175 insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 613 | 176 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t |
| 610 | 177 insertTreeP {n} {m} {A} {t} tree key value P exit = |
| 621 | 178 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant (proj1 p) tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ ⟪ P , {!!} ⟫ |
| 615 | 179 $ λ p P loop → findP key (proj1 p) tree (proj2 p) {!!} (λ t _ s P1 lt → loop ⟪ t , s ⟫ {!!} lt ) |
| 180 $ λ t _ s P → replaceNodeP key value t (proj1 P) | |
| 614 | 181 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A )) |
| 621 | 182 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } |
| 615 | 183 (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫ ⟪ proj1 P , ⟪ {!!} , R ⟫ ⟫ |
| 621 | 184 $ λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) {!!} |
| 185 (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ {!!} lt ) exit | |
| 614 | 186 |
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187 top-value : {n : Level} {A : Set n} → (tree : bt A) → Maybe A |
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188 top-value leaf = nothing |
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189 top-value (node key value tree tree₁) = just value |
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190 |
| 612 | 191 insertTreeSpec0 : {n : Level} {A : Set n} → (tree : bt A) → (value : A) → top-value tree ≡ just value → ⊤ |
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192 insertTreeSpec0 _ _ _ = tt |
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193 |
| 623 | 194 record findPR {n : Level} {A : Set n} (tree : bt A ) (stack : List (bt A)) (C : bt A → List (bt A) → Set n) : Set n where |
| 618 | 195 field |
| 619 | 196 tree0 : bt A |
| 622 | 197 ti : treeInvariant tree0 |
| 621 | 198 si : stackInvariant tree tree0 stack |
| 623 | 199 ci : C tree stack |
| 618 | 200 |
| 616 | 201 findPP : {n m : Level} {A : Set n} {t : Set m} |
| 202 → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) | |
| 624 | 203 → (Pre : findPR tree stack (λ t s → Lift n ⊤)) |
| 204 → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 (λ t s → Lift n ⊤) → bt-depth tree1 < bt-depth tree → t ) | |
| 625 | 205 → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → ( tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key) → findPR tree1 stack1 (λ t s → Lift n ⊤) → t) → t |
| 206 findPP key leaf st Pre next exit = exit leaf st (case1 refl) Pre | |
| 624 | 207 findPP key (node key₁ v tree tree₁) st Pre next exit with <-cmp key key₁ |
| 625 | 208 findPP key n st P next exit | tri≈ ¬a b ¬c = exit n st (case2 {!!}) P |
| 624 | 209 findPP {_} {_} {A} key n@(node key₁ v tree tree₁) st Pre next exit | tri< a ¬b ¬c = |
| 210 next tree (n ∷ st) (record {ti = findPR.ti Pre ; si = findPP2 st (findPR.si Pre) ; ci = lift tt} ) findPP1 where | |
| 621 | 211 tree0 = findPR.tree0 Pre |
| 212 findPP2 : (st : List (bt A)) → stackInvariant {!!} tree0 st → stackInvariant {!!} tree0 (node key₁ v tree tree₁ ∷ st) | |
| 623 | 213 findPP2 = {!!} |
| 618 | 214 findPP1 : suc ( bt-depth tree ) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁) |
| 215 findPP1 = {!!} | |
| 624 | 216 findPP key n@(node key₁ v tree tree₁) st Pre next exit | tri> ¬a ¬b c = next tree₁ (n ∷ st) {!!} findPP2 where -- Cond n st → Cond tree₁ (n ∷ st) |
| 618 | 217 findPP2 : suc (bt-depth tree₁) ≤ suc (bt-depth tree Data.Nat.⊔ bt-depth tree₁) |
| 218 findPP2 = {!!} | |
| 616 | 219 |
| 618 | 220 insertTreePP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree |
| 221 → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t | |
| 624 | 222 insertTreePP {n} {m} {A} {t} tree key value P exit = |
| 223 TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR (proj1 p) (proj2 p) (λ t s → Lift n ⊤) } (λ p → bt-depth (proj1 p)) ⟪ tree , [] ⟫ {!!} | |
| 618 | 224 $ λ p P loop → findPP key (proj1 p) (proj2 p) {!!} (λ t s P1 lt → loop ⟪ t , s ⟫ {!!} lt ) |
| 625 | 225 $ λ t s _ P → replaceNodeP key value t {!!} |
| 618 | 226 $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ (bt A ∧ bt A )) |
| 621 | 227 {λ p → treeInvariant (proj1 (proj2 p)) ∧ stackInvariant (proj1 (proj2 p)) tree (proj1 p) ∧ replacedTree key value (proj1 (proj2 p)) (proj2 (proj2 p)) } |
| 618 | 228 (λ p → bt-depth (proj1 (proj2 p))) ⟪ s , ⟪ t , t1 ⟫ ⟫ ⟪ {!!} , ⟪ {!!} , R ⟫ ⟫ |
| 621 | 229 $ λ p P1 loop → replaceP key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) {!!} |
| 230 (λ key value tree1 repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ {!!} lt ) exit | |
| 618 | 231 |
| 616 | 232 -- findP key tree stack = findPP key tree stack {findPR} → record { ti = tree-invariant tree ; si stack-invariant tree stack } → |
| 233 | |
| 617 | 234 record findP-contains {n : Level} {A : Set n} (tree : bt A ) (stack : List (bt A)) : Set n where |
| 616 | 235 field |
| 617 | 236 key1 : ℕ |
| 237 value1 : A | |
| 616 | 238 tree1 : bt A |
| 617 | 239 ci : replacedTree key1 value1 tree tree1 |
| 616 | 240 |
| 624 | 241 findPPC : {n m : Level} {A : Set n} {t : Set m} |
| 242 → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) | |
| 243 → (Pre : findPR tree stack findP-contains) | |
| 244 → (next : (tree1 : bt A) → (stack1 : List (bt A)) → findPR tree1 stack1 findP-contains → bt-depth tree1 < bt-depth tree → t ) | |
| 625 | 245 → (exit : (tree1 : bt A) → (stack1 : List (bt A)) → ( tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key) → findPR tree1 stack1 findP-contains → t) → t |
| 624 | 246 findPPC = {!!} |
| 247 | |
| 618 | 248 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 | 249 containsTree {n} {m} {A} {t} tree tree1 key value P RT = |
| 617 | 250 TerminatingLoopS (bt A ∧ List (bt A) ) |
| 624 | 251 {λ p → findPR (proj1 p) (proj2 p) findP-contains } (λ p → bt-depth (proj1 p)) |
| 252 ⟪ tree1 , [] ⟫ {!!} | |
| 253 $ λ p P loop → findPPC key (proj1 p) (proj2 p) {!!} (λ t s P1 lt → loop ⟪ t , s ⟫ {!!} lt ) | |
| 625 | 254 $ λ t1 s1 found? P2 → insertTreeSpec0 t1 value {!!} where |
| 626 | 255 lemma7 : {key : ℕ } {value1 : A } {t1 tree : bt A } { s1 : List (bt A) } → |
| 256 replacedTree key value1 tree t1 → stackInvariant t1 tree s1 → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key) → node-key t1 ≡ just key | |
| 257 lemma7 {key} {value1} {.(node key value1 leaf leaf)} {leaf} r-leaf s (case1 ()) | |
| 258 lemma7 {key} {value1} {.(node key value1 leaf leaf)} {leaf} r-leaf s (case2 x) = x | |
| 259 lemma7 {.key₁} {value1} {.(node key₁ value1 s1 s2)} {node key₁ value s1 s2} r-node s or = {!!} | |
| 260 lemma7 {key} {value1} {.(node key₁ value s1 _)} {node key₁ value s1 s2} (r-right x r) s or = {!!} | |
| 261 lemma7 {key} {value1} {.(node key₁ value _ s2)} {node key₁ value s1 s2} (r-left x r) s or = {!!} | |
| 615 | 262 |