Mercurial > hg > Gears > GearsAgda
view hoareBinaryTree.agda @ 793:08e04ed15468
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 21 Oct 2023 18:51:25 +0900 |
parents | 0b791ae19543 |
children | f2a3f5707075 |
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module hoareBinaryTree where open import Level renaming (zero to Z ; suc to succ) open import Data.Nat hiding (compare) open import Data.Nat.Properties as NatProp open import Data.Maybe -- open import Data.Maybe.Properties open import Data.Empty open import Data.List open import Data.Product open import Function as F hiding (const) open import Relation.Binary open import Relation.Binary.PropositionalEquality open import Relation.Nullary open import logic _iso_ : {n : Level} {a : Set n} → ℕ → ℕ → Set d iso d' = (¬ (suc d ≤ d')) ∧ (¬ (suc d' ≤ d)) iso-intro : {n : Level} {a : Set n} {x y : ℕ} → ¬ (suc x ≤ y) → ¬ (suc y ≤ x) → _iso_ {n} {a} x y iso-intro = λ z z₁ → record { proj1 = z ; proj2 = z₁ } -- -- -- no children , having left node , having right node , having both -- data bt {n : Level} (A : Set n) : Set n where leaf : bt A node : (key : ℕ) → (value : A) → (left : bt A ) → (right : bt A ) → bt A node-key : {n : Level} {A : Set n} → bt A → Maybe ℕ node-key (node key _ _ _) = just key node-key _ = nothing node-value : {n : Level} {A : Set n} → bt A → Maybe A node-value (node _ value _ _) = just value node-value _ = nothing bt-depth : {n : Level} {A : Set n} → (tree : bt A ) → ℕ bt-depth leaf = 0 bt-depth (node key value t t₁) = suc (Data.Nat._⊔_ (bt-depth t ) (bt-depth t₁ )) find' : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → List (bt A) → (next : bt A → List (bt A) → t ) → (exit : bt A → List (bt A) → t ) → t find' key leaf st _ exit = exit leaf st find' key (node key₁ v1 tree tree₁) st next exit with <-cmp key key₁ find' key n st _ exit | tri≈ ¬a b ¬c = exit n st find' key n@(node key₁ v1 tree tree₁) st next _ | tri< a ¬b ¬c = next tree (n ∷ st) find' key n@(node key₁ v1 tree tree₁) st next _ | tri> ¬a ¬b c = next tree₁ (n ∷ st) find : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) → (next : (tree1 : bt A) → (stack : List (bt A)) → t) → (exit : (tree1 : bt A) → (stack : List (bt A)) → t) → t find key leaf st _ exit = exit leaf st find key (node key₁ v1 tree tree₁) st next exit with <-cmp key key₁ find key n st _ exit | tri≈ ¬a refl ¬c = exit n st find {n} {_} {A} key (node key₁ v1 tree tree₁) st next _ | tri< a ¬b ¬c = next tree (tree ∷ st) find key n@(node key₁ v1 tree tree₁) st next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) {-# TERMINATING #-} 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 find-loop {n} {m} {A} {t} key tree st exit = find-loop1 tree st where find-loop1 : bt A → List (bt A) → t find-loop1 tree st = find key tree st find-loop1 exit replaceNode : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → (bt A → t) → t replaceNode k v1 leaf next = next (node k v1 leaf leaf) replaceNode k v1 (node key value t t₁) next = next (node k v1 t t₁) 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 replace key value repl [] next exit = exit repl -- can't happen replace key value repl (leaf ∷ []) next exit = exit repl replace key value repl (node key₁ value₁ left right ∷ []) next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = exit (node key₁ value₁ repl right ) ... | tri≈ ¬a b ¬c = exit (node key₁ value left right ) ... | tri> ¬a ¬b c = exit (node key₁ value₁ left repl ) replace key value repl (leaf ∷ st) next exit = next key value repl st replace key value repl (node key₁ value₁ left right ∷ st) next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st {-# TERMINATING #-} 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 replace-loop {_} {_} {A} {t} key value tree st exit = replace-loop1 key value tree st where replace-loop1 : (key : ℕ) → (value : A) → bt A → List (bt A) → t replace-loop1 key value tree st = replace key value tree st replace-loop1 exit insertTree : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : bt A → t ) → t 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 insertTest1 = insertTree leaf 1 1 (λ x → x ) insertTest2 = insertTree insertTest1 2 1 (λ x → x ) insertTest3 = insertTree insertTest2 3 3 (λ x → x ) insertTest4 = insertTree insertTest3 1 4 (λ x → x ) -- this is wrong updateTree : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (empty : bt A → t ) → (next : A → bt A → t ) → t updateTree {_} {_} {A} {t} tree key value empty next = find-loop key tree ( tree ∷ [] ) $ λ t st → replaceNode key value t $ λ t1 → replace-loop key value t1 st (found? st) where found? : List (bt A) → bt A → t found? [] tree = empty tree -- can't happen found? (leaf ∷ st) tree = empty tree found? (node key value x x₁ ∷ st) tree = next value tree open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_) data treeInvariant {n : Level} {A : Set n} : (tree : bt A) → Set n where t-leaf : treeInvariant leaf t-single : (key : ℕ) → (value : A) → treeInvariant (node key value leaf leaf) 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)) t-left : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key < key₁) → treeInvariant (node key value t₁ t₂) → treeInvariant (node key₁ value₁ (node key value t₁ t₂) leaf ) t-node : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} → (key < key₁) → (key₁ < key₂) → treeInvariant (node key value t₁ t₂) → treeInvariant (node key₂ value₂ t₃ t₄) → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) -- -- stack always contains original top at end -- data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where s-single : {tree0 : bt A} → stackInvariant key tree0 tree0 (tree0 ∷ []) s-right : {tree tree0 tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} → key₁ < key → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree tree0 (tree ∷ st) s-left : {tree₁ tree0 tree : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} → key < key₁ → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree₁ tree0 (tree₁ ∷ st) data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (before after : bt A ) → Set n where r-leaf : replacedTree key value leaf (node key value leaf leaf) r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁) r-right : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} → k < key → replacedTree key value t2 t → replacedTree key value (node k v1 t1 t2) (node k v1 t1 t) r-left : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} → key < k → replacedTree key value t1 t → replacedTree key value (node k v1 t1 t2) (node k v1 t t2) add< : { i : ℕ } (j : ℕ ) → i < suc i + j add< {i} j = begin suc i ≤⟨ m≤m+n (suc i) j ⟩ suc i + j ∎ where open ≤-Reasoning treeTest1 : bt ℕ treeTest1 = node 0 0 leaf (node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf)) treeTest2 : bt ℕ treeTest2 = node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf) treeInvariantTest1 : treeInvariant treeTest1 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) ) stack-top : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) stack-top [] = nothing stack-top (x ∷ s) = just x stack-last : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) stack-last [] = nothing stack-last (x ∷ []) = just x stack-last (x ∷ s) = stack-last s stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] ) stackInvariantTest1 = s-right (add< 3) (s-single ) si-property0 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 stack → ¬ ( stack ≡ [] ) si-property0 (s-single ) () si-property0 (s-right x si) () si-property0 (s-left x si) () si-property1 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 tree1 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 (tree1 ∷ stack) → tree1 ≡ tree si-property1 (s-single ) = refl si-property1 (s-right _ si) = refl si-property1 (s-left _ si) = refl si-property-last : {n : Level} {A : Set n} (key : ℕ) (tree tree0 : bt A) → (stack : List (bt A)) → stackInvariant key tree tree0 stack → stack-last stack ≡ just tree0 si-property-last key t t0 (t ∷ []) (s-single ) = refl si-property-last key t t0 (.t ∷ x ∷ st) (s-right _ si ) with si-property1 si ... | refl = si-property-last key x t0 (x ∷ st) si si-property-last key t t0 (.t ∷ x ∷ st) (s-left _ si ) with si-property1 si ... | refl = si-property-last key x t0 (x ∷ st) si ti-right : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 tree₁ repl) → treeInvariant repl ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-right x ti) = ti ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-left x ti) = t-leaf ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti₁ ti-left : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 repl tree₁ ) → treeInvariant repl ti-left {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf ti-left {_} {_} {_} {_} {key₁} {v1} (t-right x ti) = t-leaf ti-left {_} {_} {_} {_} {key₁} {v1} (t-left x ti) = ti ti-left {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti stackTreeInvariant : {n : Level} {A : Set n} (key : ℕ) (sub tree : bt A) → (stack : List (bt A)) → treeInvariant tree → stackInvariant key sub tree stack → treeInvariant sub stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-single ) = ti stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si ) = ti-right (si1 si) where si1 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 tree₁ sub ) tree st → treeInvariant (node key₁ v1 tree₁ sub ) si1 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 tree₁ sub ) tree st ti si stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si ) = ti-left ( si2 si) where si2 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 sub tree₁ ) tree st → treeInvariant (node key₁ v1 sub tree₁ ) si2 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 sub tree₁ ) tree st ti si rt-property1 : {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf ) rt-property1 {n} {A} key value .leaf .(node key value leaf leaf) r-leaf () rt-property1 {n} {A} key value .(node key _ _ _) .(node key value _ _) r-node () rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-right x rt) = λ () rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-left x rt) = λ () 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 rt-property-leaf r-leaf = refl rt-property-¬leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {tree : bt A} → ¬ replacedTree key value tree leaf rt-property-¬leaf () rt-property-key : {n : Level} {A : Set n} {key key₂ key₃ : ℕ} {value value₂ value₃ : A} {left left₁ right₂ right₃ : bt A} → replacedTree key value (node key₂ value₂ left right₂) (node key₃ value₃ left₁ right₃) → key₂ ≡ key₃ rt-property-key r-node = refl rt-property-key (r-right x ri) = refl rt-property-key (r-left x ri) = refl nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x nat-<> : { x y : ℕ } → x < y → y < x → ⊥ nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x open _∧_ depth-1< : {i j : ℕ} → suc i ≤ suc (i Data.Nat.⊔ j ) depth-1< {i} {j} = s≤s (m≤m⊔n _ j) depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i ) depth-2< {i} {j} = s≤s {! !} depth-3< : {i : ℕ } → suc i ≤ suc (suc i) depth-3< {zero} = s≤s ( z≤n ) depth-3< {suc i} = s≤s (depth-3< {i} ) treeLeftDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A ) → treeInvariant (node k v1 tree tree₁) → treeInvariant tree treeLeftDown {n} {A} {_} {v1} leaf leaf (t-single k1 v1) = t-leaf treeLeftDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = t-leaf treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = ti treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti treeRightDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A ) → treeInvariant (node k v1 tree tree₁) → treeInvariant tree₁ treeRightDown {n} {A} {_} {v1} .leaf .leaf (t-single _ .v1) = t-leaf treeRightDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right x ti) = ti treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left x ti) = t-leaf treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node x x₁ ti ti₁) = ti₁ -- record FindCond {n : Level} {A : Set n} (C : ℕ → bt A → Set n) : Set (Level.suc n) where -- field -- c1 : {key key₁ : ℕ} {v1 : A } { tree tree₁ : bt A } → C key (node key₁ v1 tree tree₁) → key < key₁ → C key tree -- c2 : {key key₁ : ℕ} {v1 : A } { tree tree₁ : bt A } → C key (node key₁ v1 tree tree₁) → key > key₁ → C key tree₁ -- -- -- findP0 : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) -- → {C : ℕ → bt A → Set n} → C key tree → FindCond C -- → (next : (tree1 : bt A) → (stack : List (bt A)) → C key tree1 → bt-depth tree1 < bt-depth tree → t ) -- → (exit : (tree1 : bt A) → (stack : List (bt A)) → C key tree1 -- → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t -- findP0 key leaf st Pre _ _ exit = exit leaf st Pre (case1 refl) -- findP0 key (node key₁ v1 tree tree₁) st Pre _ next exit with <-cmp key key₁ -- findP0 key n st Pre e _ exit | tri≈ ¬a refl ¬c = exit n st Pre (case2 refl) -- findP0 {n} {_} {A} key (node key₁ v1 tree tree₁) st Pre e next _ | tri< a ¬b ¬c = next tree (tree ∷ st) (FindCond.c1 e Pre a) depth-1< -- findP0 key n@(node key₁ v1 tree tree₁) st Pre e next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) (FindCond.c2 e Pre c) depth-2< findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A)) → treeInvariant tree ∧ stackInvariant key tree tree0 stack → (next : (tree1 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t ) → (exit : (tree1 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t findP key leaf tree0 st Pre _ exit = exit leaf st Pre (case1 refl) findP key (node key₁ v1 tree tree₁) tree0 st Pre next exit with <-cmp key key₁ findP key n tree0 st Pre _ exit | tri≈ ¬a refl ¬c = exit n st Pre (case2 refl) findP {n} {_} {A} key (node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree (tree ∷ st) ⟪ treeLeftDown tree tree₁ (proj1 Pre) , findP1 a st (proj2 Pre) ⟫ depth-1< where findP1 : key < key₁ → (st : List (bt A)) → stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (tree ∷ st) findP1 a (x ∷ st) si = s-left a si 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< 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) replaceTree1 k v1 value (t-single .k .v1) = t-single k value replaceTree1 k v1 value (t-right x t) = t-right x t replaceTree1 k v1 value (t-left x t) = t-left x t replaceTree1 k v1 value (t-node x x₁ t t₁) = t-node x x₁ t t₁ open import Relation.Binary.Definitions lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥ lemma3 refl () lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥ lemma5 (s≤s z≤n) () ¬x<x : {x : ℕ} → ¬ (x < x) ¬x<x (s≤s lt) = ¬x<x lt child-replaced : {n : Level} {A : Set n} (key : ℕ) (tree : bt A) → bt A child-replaced key leaf = leaf child-replaced key (node key₁ value left right) with <-cmp key key₁ ... | tri< a ¬b ¬c = left ... | tri≈ ¬a b ¬c = node key₁ value left right ... | tri> ¬a ¬b c = right 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 field tree0 : bt A ti : treeInvariant tree0 si : stackInvariant key tree tree0 stack ri : replacedTree key value (child-replaced key tree ) repl ci : C tree repl stack -- data continuation replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) → (treeInvariant tree ) → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value (child-replaced key tree) tree1 → t) → t replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P) (subst (λ j → replacedTree k v1 j (node k v1 t t₁) ) repl00 r-node) where repl00 : node k value t t₁ ≡ child-replaced k (node k value t t₁) repl00 with <-cmp k k ... | tri< a ¬b ¬c = ⊥-elim (¬b refl) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim (¬b refl) replaceP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → {tree : bt A} ( repl : bt A) → (stack : List (bt A)) → replacePR key value tree repl stack (λ _ _ _ → Lift n ⊤) → (next : ℕ → A → {tree1 : bt A } (repl : bt A) → (stack1 : List (bt A)) → replacePR key value tree1 repl stack1 (λ _ _ _ → Lift n ⊤) → length stack1 < length stack → t) → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t replaceP key value {tree} repl [] Pre next exit = ⊥-elim ( si-property0 (replacePR.si Pre) refl ) -- can't happen replaceP key value {tree} repl (leaf ∷ []) Pre next exit with si-property-last _ _ _ _ (replacePR.si Pre)-- tree0 ≡ leaf ... | refl = exit (replacePR.tree0 Pre) (node key value leaf leaf) ⟪ replacePR.ti Pre , r-leaf ⟫ replaceP key value {tree} repl (node key₁ value₁ left right ∷ []) Pre next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = exit (replacePR.tree0 Pre) (node key₁ value₁ repl right ) ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ repl right ) repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) 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 repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl repl03 = replacePR.ri Pre repl02 : child-replaced key (node key₁ value₁ left right) ≡ left repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a a) ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a) ... | tri≈ ¬a b ¬c = exit (replacePR.tree0 Pre) repl ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) repl repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) repl01 | refl | refl = subst (λ k → replacedTree key value k repl) repl02 (replacePR.ri Pre) where repl02 : child-replaced key (node key₁ value₁ left right) ≡ node key₁ value₁ left right repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b) ... | tri> ¬a ¬b c = exit (replacePR.tree0 Pre) (node key₁ value₁ left repl ) ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ left repl ) repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) 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 repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl repl03 = replacePR.ri Pre repl02 : child-replaced key (node key₁ value₁ left right) ≡ right repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim ( ¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬c c) ... | tri> ¬a ¬b c = refl replaceP {n} {_} {A} key value {tree} repl (leaf ∷ st@(tree1 ∷ st1)) Pre next exit = next key value repl st Post ≤-refl where Post : replacePR key value tree1 repl (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ leaf repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ leaf) ≡ leaf repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = ⊥-elim (¬c x) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) ... | tri> ¬a ¬b c = refl repl12 : replacedTree key value (child-replaced key tree1 ) repl 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 ... | s-left {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 leaf tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 leaf tree₁ ) ≡ leaf repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) ... | tri> ¬a ¬b c = ⊥-elim (¬a x) repl12 : replacedTree key value (child-replaced key tree1 ) repl 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 replaceP {n} {_} {A} key value {tree} repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value₁ repl right ) st (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ left repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) ... | refl | tri> ¬a ¬b c = refl repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) ... | s-left {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ left repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = refl ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value left right ) (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | 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 repl09 : tree1 ≡ node key₂ v1 tree₁ tree -- (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ tree) ≡ tree repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = ⊥-elim (¬c x) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) ... | tri> ¬a ¬b c = refl repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) repl12 refl with repl09 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node ... | 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 repl09 : tree1 ≡ node key₂ v1 tree tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree tree₁ ) ≡ tree repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) ... | tri> ¬a ¬b c = ⊥-elim (¬a x) repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) repl12 refl with repl09 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value₁ left repl ) st (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ right repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) ... | tri> ¬a ¬b c = refl repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) ... | refl | tri> ¬a ¬b c = refl repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) ... | s-left {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ right repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) ... | tri> ¬a ¬b c = refl repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = refl ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) → (r : Index) → (p : Invraiant r) → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r) ... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) ... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (≤-step lt1) p1 lt1 ) where TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → Invraiant r1 → reduce r1 < reduce r → t TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1)) TerminatingLoop1 (suc j) r r1 n≤j p1 lt with <-cmp (reduce r1) (suc j) ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt ... | 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 ) ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) record LoopTermination {n : Level} {Index : Set n } { reduce : Index → ℕ } (r : Index ) (C : Set n) : Set (Level.suc n) where field rd : (r1 : Index) → reduce r1 < reduce r ci : C -- data continuation -- TerminatingLoopC : {l n : Level} {t : Set l} (Index : Set n ) → {C : Set n } → ( reduce : Index → ℕ) -- → (r : Index) → (P : LoopTermination r C ) -- → (loop : (r : Index) → LoopTermination {_} {_} {reduce} r C → (next : (r1 : Index) → LoopTermination r1 C → t ) → t) → t -- TerminatingLoopC {_} {_} {t} Index {C} reduce r P loop with <-cmp 0 (reduce r) -- ... | tri≈ ¬a b ¬c = loop r P (λ r1 p1 → ⊥-elim (lemma3 b (LoopTermination.rd P r1))) -- ... | tri< a ¬b ¬c = loop r P (λ r1 p1 → TerminatingLoop1 (reduce r) r r1 (≤-step (LoopTermination.rd P r1)) p1 (LoopTermination.rd P r1)) where -- TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → {!!} → reduce r1 < reduce r → t -- TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 {!!} (λ r2 P1 → ⊥-elim (lemma5 n≤j (LoopTermination.rd P1 r2))) -- TerminatingLoop1 (suc j) r r1 n≤j p1 lt with <-cmp (reduce r1) (suc j) -- ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt -- ... | tri≈ ¬a b ¬c = loop r1 {!!} (λ r2 p2 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b {!!} ) p2 {!!} ) -- ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) -- -- record ReplCond {n : Level} {A : Set n} (C : ℕ → bt A → List (bt A) → Set n) : Set (Level.suc n) where -- field -- c1 : (key : ℕ) → (repl : bt A) → (stack : List (bt A)) → C key repl stack -- -- replaceP0 : {n m : Level} {A : Set n} {t : Set m} -- → (key : ℕ) → (value : A) → ( repl : bt A) -- → (stack : List (bt A)) -- → {C : ℕ → (repl : bt A ) → List (bt A) → Set n} → C key repl stack → ReplCond C -- → (next : ℕ → A → (repl : bt A) → (stack1 : List (bt A)) -- → C key repl stack → length stack1 < length stack → t) -- → (exit : (repl : bt A) → C key repl [] → t) → t -- replaceP0 key value repl [] Pre _ next exit = exit repl {!!} -- replaceP0 key value repl (leaf ∷ []) Pre _ next exit = exit (node key value leaf leaf) {!!} -- replaceP0 key value repl (node key₁ value₁ left right ∷ []) Pre e next exit with <-cmp key key₁ -- ... | tri< a ¬b ¬c = exit (node key₁ value₁ repl right ) {!!} -- ... | tri≈ ¬a b ¬c = exit repl {!!} -- ... | tri> ¬a ¬b c = exit (node key₁ value₁ left repl ) {!!} -- replaceP0 {n} {_} {A} key value repl (leaf ∷ st@(tree1 ∷ st1)) Pre e next exit = next key value repl st {!!} ≤-refl -- replaceP0 {n} {_} {A} key value repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre e next exit with <-cmp key key₁ -- ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st {!!} ≤-refl -- ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st {!!} ≤-refl -- ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st {!!} ≤-refl -- -- open _∧_ RTtoTI0 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → replacedTree key value tree repl → treeInvariant repl RTtoTI0 .leaf .(node key value leaf leaf) key value ti r-leaf = t-single key value RTtoTI0 .(node key _ leaf leaf) .(node key value leaf leaf) key value (t-single .key _) r-node = t-single key value RTtoTI0 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right x ti) r-node = t-right x ti RTtoTI0 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left x ti) r-node = t-left x ti 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 -- r-right case 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) RTtoTI0 (node _ _ leaf right@(node _ _ _ _)) (node key₁ value₁ leaf leaf) key value (t-right x₁ ti) (r-right x ri) = t-single key₁ value₁ RTtoTI0 (node key₁ _ leaf right@(node key₂ _ _ _)) (node key₁ value₁ leaf right₁@(node key₃ _ _ _)) key value (t-right x₁ ti) (r-right x ri) = t-right (subst (λ k → key₁ < k ) (rt-property-key ri) x₁) (RTtoTI0 _ _ key value ti ri) RTtoTI0 (node key₁ _ (node _ _ _ _) leaf) (node key₁ _ (node key₃ value left right) leaf) key value₁ (t-left x₁ ti) (r-right x ()) 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) = t-node x₁ x ti (t-single key value) RTtoTI0 (node key₁ _ (node _ _ _ _) (node key₂ _ _ _)) (node key₁ _ (node _ _ _ _) (node key₃ _ _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x ri) = t-node x₁ (subst (λ k → key₁ < k) (rt-property-key ri) x₂) ti (RTtoTI0 _ _ key value ti₁ ri) -- r-left case 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 _ _ ) 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 RTtoTI0 (node key₃ _ (node key₂ _ _ _) leaf) (node key₃ _ (node key₁ value₁ left left₁) leaf) key value (t-left x₁ ti) (r-left x ri) = t-left (subst (λ k → k < key₃ ) (rt-property-key ri) x₁) (RTtoTI0 _ _ key value ti ri) -- key₁ < key₃ 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 RTtoTI1 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant repl → replacedTree key value tree repl → treeInvariant tree RTtoTI1 .leaf .(node key value leaf leaf) key value ti r-leaf = t-leaf RTtoTI1 (node key value₁ leaf leaf) .(node key value leaf leaf) key value (t-single .key .value) r-node = t-single key value₁ RTtoTI1 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right x ti) r-node = t-right x ti RTtoTI1 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left x ti) r-node = t-left x ti 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 -- r-right case 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 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) = t-right (subst (λ k → key₁ < k ) (sym (rt-property-key ri)) x₁) (RTtoTI1 _ _ key value ti ri) -- key₁ < key₂ RTtoTI1 (node _ _ (node _ _ _ _) leaf) (node _ _ (node _ _ _ _) (node key value _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x r-leaf) = t-left x₁ ti 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 -- r-left case RTtoTI1 (node key₁ value₁ leaf leaf) (node key₁ _ _ leaf) key value (t-left x₁ ti) (r-left x ri) = t-single key₁ value₁ RTtoTI1 (node key₁ _ (node key₂ value₁ n n₁) leaf) (node key₁ _ (node key₃ _ _ _) leaf) key value (t-left x₁ ti) (r-left x ri) = t-left (subst (λ k → k < key₁ ) (sym (rt-property-key ri)) x₁) (RTtoTI1 _ _ key value ti ri) -- key₂ < key₁ RTtoTI1 (node key₁ value₁ leaf _) (node key₁ _ _ _) key value (t-node x₁ x₂ ti ti₁) (r-left x r-leaf) = t-right x₂ ti₁ RTtoTI1 (node key₁ value₁ (node key₂ value₂ n n₁) _) (node key₁ _ _ _) key value (t-node x₁ x₂ ti ti₁) (r-left x ri) = t-node (subst (λ k → k < key₁ ) (sym (rt-property-key ri)) x₁) x₂ (RTtoTI1 _ _ key value ti ri) ti₁ -- key₂ < key₁ insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → (exit : (tree repl : bt A) → treeInvariant repl ∧ replacedTree key value tree repl → t ) → t insertTreeP {n} {m} {A} {t} tree key value P0 exit = 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 ⟫ $ λ p P loop → findP key (proj1 p) tree (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) $ λ t s P C → replaceNodeP key value t C (proj1 P) $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ bt A ∧ bt A ) {λ p → replacePR key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) (λ _ _ _ → Lift n ⊤ ) } (λ p → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫ record { tree0 = tree ; ti = P0 ; si = proj2 P ; ri = R ; ci = lift tt } $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) P1 (λ key value {tree1} repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) $ λ tree repl P → exit tree repl ⟪ RTtoTI0 _ _ _ _ (proj1 P) (proj2 P) , proj2 P ⟫ insertTestP1 = insertTreeP leaf 1 1 t-leaf $ λ _ x P → insertTreeP x 2 1 (proj1 P) $ λ _ x P → insertTreeP x 3 2 (proj1 P) $ λ _ x P → insertTreeP x 2 2 (proj1 P) (λ _ x _ → x ) top-value : {n : Level} {A : Set n} → (tree : bt A) → Maybe A top-value leaf = nothing top-value (node key value tree tree₁) = just value record findPR {n : Level} {A : Set n} (key : ℕ) (tree : bt A ) (stack : List (bt A)) (C : ℕ → bt A → List (bt A) → Set n) : Set n where field tree0 : bt A ti0 : treeInvariant tree0 ti : treeInvariant tree si : stackInvariant key tree tree0 stack ci : C key tree stack -- data continuation record findExt {n : Level} {A : Set n} (key : ℕ) (C : ℕ → bt A → List (bt A) → Set n) : Set (Level.suc n) where field c1 : {key₁ : ℕ} {tree tree₁ : bt A } {st : List (bt A)} {v1 : A} → findPR key (node key₁ v1 tree tree₁) st C → key < key₁ → C key tree (tree ∷ st) c2 : {key₁ : ℕ} {tree tree₁ : bt A } {st : List (bt A)} {v1 : A} → findPR key (node key₁ v1 tree tree₁) st C → key > key₁ → C key tree₁ (tree₁ ∷ st) findPP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → (stack : List (bt A)) → {C : ℕ → bt A → List (bt A) → Set n } → findPR key tree stack C → findExt key C → (next : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack C → bt-depth tree1 < bt-depth tree → t ) → (exit : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack C → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t findPP key leaf st Pre _ _ exit = exit leaf st Pre (case1 refl) findPP key (node key₁ v1 tree tree₁) st Pre _ next exit with <-cmp key key₁ findPP key n st Pre _ _ exit | tri≈ ¬a refl ¬c = exit n st Pre (case2 refl) findPP {n} {_} {A} key (node key₁ v1 tree tree₁) st Pre e next _ | tri< a ¬b ¬c = next tree (tree ∷ st) record { tree0 = findPR.tree0 Pre ; ti0 = findPR.ti0 Pre ; ti = treeLeftDown tree tree₁ (findPR.ti Pre) ; si = findP1 a st (findPR.si Pre) ; ci = findExt.c1 e Pre a } depth-1< where findP1 : key < key₁ → (st : List (bt A)) → stackInvariant key (node key₁ v1 tree tree₁) (findPR.tree0 Pre) st → stackInvariant key tree (findPR.tree0 Pre) (tree ∷ st) findP1 a (x ∷ st) si = s-left a si findPP key n@(node key₁ v1 tree tree₁) st Pre e next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) record { tree0 = findPR.tree0 Pre ; ti0 = findPR.ti0 Pre ; ti = treeRightDown tree tree₁ (findPR.ti Pre) ; si = s-right c (findPR.si Pre) ; ci = findExt.c2 e Pre c } depth-2< insertTreePP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → (exit : (tree repl : bt A) → treeInvariant tree ∧ replacedTree key value tree repl → t ) → t insertTreePP {n} {m} {A} {t} tree key value P0 exit = 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 ∷ [] ⟫ record { tree0 = tree ; ti = P0 ; ti0 = P0 ;si = s-single ; ci = lift tt } $ λ p P loop → findPP key (proj1 p) (proj2 p) P record { c1 = λ _ _ → lift tt ; c2 = λ _ _ → lift tt } (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) $ λ t s P C → replaceNodeP key value t C (findPR.ti P) $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ bt A ∧ bt A ) {λ p → replacePR key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) (λ _ _ _ → Lift n ⊤ ) } (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 } $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) P1 (λ key value {tree1} repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) exit record findPC {n : Level} {A : Set n} (value : A) (key1 : ℕ) (tree : bt A ) (stack : List (bt A)) : Set n where field tree1 : bt A ci : replacedTree key1 value tree1 tree findPPC1 : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A ) → (stack : List (bt A)) → findPR key tree stack (findPC value ) → (next : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack (findPC value ) → bt-depth tree1 < bt-depth tree → t ) → (exit : (tree1 : bt A) → (stack : List (bt A)) → findPR key tree1 stack (findPC value ) → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t findPPC1 {n} {_} {A} key value tree stack Pr next exit = findPP key tree stack Pr findext next exit where findext01 : {key₂ : ℕ} {tree₁ : bt A} {tree₂ : bt A} {st : List (bt A)} {v1 : A} → (Pre : findPR key (node key₂ v1 tree₁ tree₂) st (findPC value) ) → key < key₂ → findPC value key tree₁ (tree₁ ∷ st) findext01 Pre a with findPC.ci (findPR.ci Pre) | findPC.tree1 (findPR.ci Pre) | inspect findPC.tree1 (findPR.ci Pre) ... | r-leaf | leaf | record { eq = refl } = ⊥-elim ( nat-≤> a ≤-refl) ... | r-node | node key value t1 t3 | record { eq = refl } = ⊥-elim ( nat-≤> a ≤-refl ) ... | r-right x t | t1 | t2 = ⊥-elim (nat-<> x a) ... | r-left x ri | node key value t1 t3 | record { eq = refl } = record { tree1 = t1 ; ci = ri } findext02 : {key₂ : ℕ} {tree₁ : bt A} {tree₂ : bt A} {st : List (bt A)} {v1 : A} → (Pre : findPR key (node key₂ v1 tree₁ tree₂) st (findPC value) ) → key > key₂ → findPC value key tree₂ (tree₂ ∷ st) findext02 Pre c with findPC.ci (findPR.ci Pre) | findPC.tree1 (findPR.ci Pre) | inspect findPC.tree1 (findPR.ci Pre) ... | r-leaf | leaf | record { eq = refl } = ⊥-elim ( nat-≤> c ≤-refl) ... | r-node | node key value t1 t3 | record { eq = refl } = ⊥-elim ( nat-≤> c ≤-refl ) ... | r-left x t | t1 | t2 = ⊥-elim (nat-<> x c) ... | r-right x ri | node key value t1 t3 | record { eq = refl } = record { tree1 = t3 ; ci = ri } findext : findExt key (findPC value ) findext = record { c1 = findext01 ; c2 = findext02 } insertTreeSpec0 : {n : Level} {A : Set n} → (tree : bt A) → (value : A) → top-value tree ≡ just value → ⊤ insertTreeSpec0 _ _ _ = tt containsTree : {n : Level} {A : Set n} → (tree tree1 : bt A) → (key : ℕ) → (value : A) → treeInvariant tree1 → replacedTree key value tree1 tree → ⊤ containsTree {n} {A} tree tree1 key value P RT = TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → findPR key (proj1 p) (proj2 p) (findPC value ) } (λ p → bt-depth (proj1 p)) ⟪ tree , tree ∷ [] ⟫ record { tree0 = tree ; ti0 = RTtoTI0 _ _ _ _ P RT ; ti = RTtoTI0 _ _ _ _ P RT ; si = s-single ; ci = record { tree1 = tree1 ; ci = RT } } $ λ p P loop → findPPC1 key value (proj1 p) (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) $ λ t1 s1 P2 found? → insertTreeSpec0 t1 value (lemma6 t1 s1 found? P2) where lemma6 : (t1 : bt A) (s1 : List (bt A)) (found? : (t1 ≡ leaf) ∨ (node-key t1 ≡ just key)) (P2 : findPR key t1 s1 (findPC value )) → top-value t1 ≡ just value 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 lemma8 : {tree1 t1 : bt A } → replacedTree key value tree1 t1 → node-key t1 ≡ just key → top-value t1 ≡ just value lemma8 {.leaf} {node key value .leaf .leaf} r-leaf refl = refl lemma8 {.(node key _ t1 t2)} {node key value t1 t2} r-node refl = refl lemma8 {.(node key value t1 _)} {node key value t1 t2} (r-right x ri) refl = ⊥-elim (¬x<x x) lemma8 {.(node key value _ t2)} {node key value t1 t2} (r-left x ri) refl = ⊥-elim (¬x<x x) lemma7 : (t1 : bt A) ( s1 : List (bt A) ) (tree0 tree1 : bt A) → replacedTree key value tree1 t1 → stackInvariant key t1 tree0 s1 → ( t1 ≡ leaf ) ∨ ( node-key t1 ≡ just key) → top-value t1 ≡ just value lemma7 .leaf (.leaf ∷ []) .leaf tree1 () s-single (case1 refl) 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 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 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 containsTree1 : {n : Level} {A : Set n} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → ⊤ containsTree1 {n} {A} tree key value ti = insertTreeP tree key value ti $ λ tree0 tree1 P → containsTree tree1 tree0 key value (RTtoTI1 _ _ _ _ (proj1 P) (proj2 P) ) (proj2 P)