Mercurial > hg > Gears > GearsAgda
view RBTree0.agda @ 954:08281092430b
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 06 Oct 2024 17:59:51 +0900 |
| parents | e5288029f850 |
| children |
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module RBTree0 where open import Level hiding (suc ; zero ; _⊔_ ) 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 -- -- -- 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 (bt-depth t ⊔ bt-depth t₁ ) bt-ne : {n : Level} {A : Set n} {key : ℕ} {value : A} {t t₁ : bt A} → ¬ ( leaf ≡ node key value t t₁ ) bt-ne {n} {A} {key} {value} {t} {t₁} () open import Data.Unit hiding ( _≟_ ) -- ; _≤?_ ; _≤_) tr< : {n : Level} {A : Set n} → (key : ℕ) → bt A → Set tr< {_} {A} key leaf = ⊤ tr< {_} {A} key (node key₁ value tr tr₁) = (key₁ < key ) ∧ tr< key tr ∧ tr< key tr₁ tr> : {n : Level} {A : Set n} → (key : ℕ) → bt A → Set tr> {_} {A} key leaf = ⊤ tr> {_} {A} key (node key₁ value tr tr₁) = (key < key₁ ) ∧ tr> key tr ∧ tr> key tr₁ -- -- 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 → tr> key t₁ → tr> key t₂ → 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 → tr< key₁ t₁ → tr< key₁ t₂ → 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 → tr< key₁ t₁ → tr< key₁ t₂ → tr> key₁ t₃ → tr> key₁ t₄ → 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 (path of the tree) -- data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where s-nil : {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) 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 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 nat-<≡ : { x : ℕ } → x < x → ⊥ nat-<≡ (s≤s lt) = nat-<≡ lt nat-≡< : { x y : ℕ } → x ≡ y → x < y → ⊥ nat-≡< refl lt = nat-<≡ lt 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) ⟪ add< _ , ⟪ ⟪ add< _ , _ ⟫ , _ ⟫ ⟫ ⟪ add< _ , ⟪ _ , _ ⟫ ⟫ (t-node _ _ _ (add< 0) (add< 1) ⟪ add< _ , ⟪ _ , _ ⟫ ⟫ _ _ _ (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-nil ) 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-nil ) () 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-nil ) = refl si-property1 (s-right _ _ _ _ si) = refl si-property1 (s-left _ _ _ _ si) = refl si-property2 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 tree1 : bt A} → (stack : List (bt A)) → stackInvariant key tree tree0 (tree1 ∷ stack) → ¬ ( just leaf ≡ stack-last stack ) si-property2 (.leaf ∷ []) (s-right _ _ tree₁ x ()) refl si-property2 (x₁ ∷ x₂ ∷ stack) (s-right _ _ tree₁ x si) eq = si-property2 (x₂ ∷ stack) si eq si-property2 (.leaf ∷ []) (s-left _ _ tree₁ x ()) refl si-property2 (x₁ ∷ x₂ ∷ stack) (s-left _ _ tree₁ x si) eq = si-property2 (x₂ ∷ stack) si eq si-property-< : {n : Level} {A : Set n} {key kp : ℕ} {value₂ : A} {tree orig tree₃ : bt A} → {stack : List (bt A)} → ¬ ( tree ≡ leaf ) → treeInvariant (node kp value₂ tree tree₃ ) → stackInvariant key tree orig (tree ∷ node kp value₂ tree tree₃ ∷ stack) → key < kp si-property-< ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right .(node _ _ _ _) _ .(node _ _ _ _) x s-nil) = ⊥-elim (nat-<> x₁ x₂) si-property-< ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right .(node _ _ _ _) _ .(node _ _ _ _) x (s-right .(node _ _ (node _ _ _ _) (node _ _ _ _)) _ tree₁ x₇ si)) = ⊥-elim (nat-<> x₁ x₂) si-property-< ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right .(node _ _ _ _) _ .(node _ _ _ _) x (s-left .(node _ _ (node _ _ _ _) (node _ _ _ _)) _ tree x₇ si)) = ⊥-elim (nat-<> x₁ x₂) si-property-< ne ti (s-left _ _ _ x (s-right .(node _ _ _ _) _ tree₁ x₁ si)) = x si-property-< ne ti (s-left _ _ _ x (s-left .(node _ _ _ _) _ tree₁ x₁ si)) = x si-property-< ne ti (s-left _ _ _ x s-nil) = x si-property-< ne (t-single _ _) (s-right _ _ tree₁ x si) = ⊥-elim ( ne refl ) si-property-> : {n : Level} {A : Set n} {key kp : ℕ} {value₂ : A} {tree orig tree₃ : bt A} → {stack : List (bt A)} → ¬ ( tree ≡ leaf ) → treeInvariant (node kp value₂ tree₃ tree ) → stackInvariant key tree orig (tree ∷ node kp value₂ tree₃ tree ∷ stack) → kp < key si-property-> ne ti (s-right _ _ tree₁ x s-nil) = x si-property-> ne ti (s-right _ _ tree₁ x (s-right .(node _ _ tree₁ _) _ tree₂ x₁ si)) = x si-property-> ne ti (s-right _ _ tree₁ x (s-left .(node _ _ tree₁ _) _ tree x₁ si)) = x si-property-> ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-left _ _ _ x s-nil) = ⊥-elim (nat-<> x₁ x₂) si-property-> ne (t-node _ _ _ x₂ x₃ x₄ x₅ x₆ x₇ ti ti₁) (s-left _ _ _ x (s-right .(node _ _ _ _) _ tree₁ x₁ si)) = ⊥-elim (nat-<> x₂ x₃) si-property-> ne (t-node _ _ _ x₂ x₃ x₄ x₅ x₆ x₇ ti ti₁) (s-left _ _ _ x (s-left .(node _ _ _ _) _ tree x₁ si)) = ⊥-elim (nat-<> x₂ x₃) si-property-> ne (t-single _ _) (s-left _ _ _ x s-nil) = ⊥-elim ( ne refl ) si-property-> ne (t-single _ _) (s-left _ _ _ x (s-right .(node _ _ leaf leaf) _ tree₁ x₁ si)) = ⊥-elim ( ne refl ) si-property-> ne (t-single _ _) (s-left _ _ _ x (s-left .(node _ _ leaf leaf) _ tree x₁ si)) = ⊥-elim ( ne 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-nil ) = 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 si-property-pne : {n : Level} {A : Set n} {key key₁ : ℕ} {value₁ : A} (tree orig : bt A) → {left right x : bt A} → (stack : List (bt A)) {rest : List (bt A)} → stack ≡ x ∷ node key₁ value₁ left right ∷ rest → stackInvariant key tree orig stack → ¬ ( key ≡ key₁ ) si-property-pne tree orig .(tree ∷ node _ _ _ _ ∷ _) refl (s-right .tree .orig tree₁ x si) eq with si-property1 si ... | refl = ⊥-elim ( nat-≡< (sym eq) x) si-property-pne tree orig .(tree ∷ node _ _ _ _ ∷ _) refl (s-left .tree .orig tree₁ x si) eq with si-property1 si ... | refl = ⊥-elim ( nat-≡< eq x) si-property-parent-node : {n : Level} {A : Set n} {key : ℕ} (tree orig : bt A) {x : bt A} → (stack : List (bt A)) {rest : List (bt A)} → stackInvariant key tree orig stack → ¬ ( stack ≡ x ∷ leaf ∷ rest ) si-property-parent-node {n} {A} tree orig .(tree ∷ leaf ∷ _) (s-right .tree .orig tree₁ x si) refl with si-property1 si ... | () si-property-parent-node {n} {A} tree orig .(tree ∷ leaf ∷ _) (s-left .tree .orig tree₁ x si) refl with si-property1 si ... | () 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 (m≤n⊔m j i) 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 ti-property1 : {n : Level} {A : Set n} {key₁ : ℕ} {value₂ : A} {left right : bt A} → treeInvariant (node key₁ value₂ left right ) → tr< key₁ left ∧ tr> key₁ right ti-property1 {n} {A} {key₁} {value₂} {.leaf} {.leaf} (t-single .key₁ .value₂) = ⟪ tt , tt ⟫ ti-property1 {n} {A} {key₁} {value₂} {.leaf} {.(node key₂ _ _ _)} (t-right .key₁ key₂ x x₁ x₂ t) = ⟪ tt , ⟪ x , ⟪ x₁ , x₂ ⟫ ⟫ ⟫ ti-property1 {n} {A} {key₁} {value₂} {.(node key _ _ _)} {.leaf} (t-left key .key₁ x x₁ x₂ t) = ⟪ ⟪ x , ⟪ x₁ , x₂ ⟫ ⟫ , tt ⟫ ti-property1 {n} {A} {key₁} {value₂} {.(node key _ _ _)} {.(node key₂ _ _ _)} (t-node key .key₁ key₂ x x₁ x₂ x₃ x₄ x₅ t t₁) = ⟪ ⟪ x , ⟪ x₂ , x₃ ⟫ ⟫ , ⟪ x₁ , ⟪ x₄ , x₅ ⟫ ⟫ ⟫ 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 a b t) = t-right _ _ x a b t replaceTree1 k v1 value (t-left _ _ x a b t) = t-left _ _ x a b t replaceTree1 k v1 value (t-node _ _ _ x x₁ a b c d t t₁) = t-node _ _ _ x x₁ a b c d 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 child-replaced-left : {n : Level} {A : Set n} {key key₁ : ℕ} {value : A} {left right : bt A} → key < key₁ → child-replaced key (node key₁ value left right) ≡ left child-replaced-left {n} {A} {key} {key₁} {value} {left} {right} lt = ch00 (node key₁ value left right) refl lt where ch00 : (tree : bt A) → tree ≡ node key₁ value left right → key < key₁ → child-replaced key tree ≡ left ch00 (node key₁ value tree tree₁) refl lt1 with <-cmp key key₁ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a lt1) ... | tri> ¬a ¬b c = ⊥-elim (¬a lt1) child-replaced-right : {n : Level} {A : Set n} {key key₁ : ℕ} {value : A} {left right : bt A} → key₁ < key → child-replaced key (node key₁ value left right) ≡ right child-replaced-right {n} {A} {key} {key₁} {value} {left} {right} lt = ch00 (node key₁ value left right) refl lt where ch00 : (tree : bt A) → tree ≡ node key₁ value left right → key₁ < key → child-replaced key tree ≡ right ch00 (node key₁ value tree tree₁) refl lt1 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim (¬c lt1) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c lt1) ... | tri> ¬a ¬b c = refl child-replaced-eq : {n : Level} {A : Set n} {key key₁ : ℕ} {value : A} {left right : bt A} → key₁ ≡ key → child-replaced key (node key₁ value left right) ≡ node key₁ value left right child-replaced-eq {n} {A} {key} {key₁} {value} {left} {right} lt = ch00 (node key₁ value left right) refl lt where ch00 : (tree : bt A) → tree ≡ node key₁ value left right → key₁ ≡ key → child-replaced key tree ≡ node key₁ value left right ch00 (node key₁ value tree tree₁) refl lt1 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim (¬b (sym lt1)) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim (¬b (sym lt1)) 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 ci : C tree repl stack -- data continuation record replacePR' {n : Level} {A : Set n} (key : ℕ) (value : A) (orig : bt A ) (stack : List (bt A)) : Set n where field tree repl : bt A ti : treeInvariant orig si : stackInvariant key tree orig stack -- treeInvariant of tree and repl is inferred from ti, si and ri. 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 → t) → t replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P) 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) → 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) 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 ) ... | tri≈ ¬a b ¬c = exit (replacePR.tree0 Pre) repl ... | tri> ¬a ¬b c = exit (replacePR.tree0 Pre) (node key₁ value₁ left repl ) 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 ; 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 ... | s-left _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; 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) 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 ; 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 ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; 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 ... | 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 ; 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 ... | s-left _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; 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) ... | 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 ; 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 ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; 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 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 (m≤n⇒m≤1+n 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 ) open _∧_ 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 → t ) → t insertTreeP {n} {m} {A} {t} tree key value ti 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 ∷ [] ⟫ ⟪ ? , s-nil ⟫ -- $ 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 = ? ; 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 ? -- insertTestP1 = insertTreeP leaf 1 1 t-leaf $ λ _ x0 P0 → insertTreeP x0 2 1 ? $ λ _ x1 P1 → insertTreeP x1 3 2 ? $ λ _ x2 P2 → insertTreeP x2 2 2 ? (λ _ x P → 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 -- is realy inserted? -- other element is preserved? -- deletion? data Color : Set where Red : Color Black : Color RB→bt : {n : Level} (A : Set n) → (bt (Color ∧ A)) → bt A RB→bt {n} A leaf = leaf RB→bt {n} A (node key ⟪ C , value ⟫ tr t1) = (node key value (RB→bt A tr) (RB→bt A t1)) color : {n : Level} {A : Set n} → (bt (Color ∧ A)) → Color color leaf = Black color (node key ⟪ C , value ⟫ rb rb₁) = C to-red : {n : Level} {A : Set n} → (tree : bt (Color ∧ A)) → bt (Color ∧ A) to-red leaf = leaf to-red (node key ⟪ _ , value ⟫ t t₁) = (node key ⟪ Red , value ⟫ t t₁) to-black : {n : Level} {A : Set n} → (tree : bt (Color ∧ A)) → bt (Color ∧ A) to-black leaf = leaf to-black (node key ⟪ _ , value ⟫ t t₁) = (node key ⟪ Black , value ⟫ t t₁) black-depth : {n : Level} {A : Set n} → (tree : bt (Color ∧ A) ) → ℕ black-depth leaf = 1 black-depth (node key ⟪ Red , value ⟫ t t₁) = black-depth t ⊔ black-depth t₁ black-depth (node key ⟪ Black , value ⟫ t t₁) = suc (black-depth t ⊔ black-depth t₁ ) zero≢suc : { m : ℕ } → zero ≡ suc m → ⊥ zero≢suc () suc≢zero : {m : ℕ } → suc m ≡ zero → ⊥ suc≢zero () data RBtreeInvariant {n : Level} {A : Set n} : (tree : bt (Color ∧ A)) → Set n where rb-leaf : RBtreeInvariant leaf rb-red : (key : ℕ) → (value : A) → {left right : bt (Color ∧ A)} → color left ≡ Black → color right ≡ Black → black-depth left ≡ black-depth right → RBtreeInvariant left → RBtreeInvariant right → RBtreeInvariant (node key ⟪ Red , value ⟫ left right) rb-black : (key : ℕ) → (value : A) → {left right : bt (Color ∧ A)} → black-depth left ≡ black-depth right → RBtreeInvariant left → RBtreeInvariant right → RBtreeInvariant (node key ⟪ Black , value ⟫ left right) RightDown : {n : Level} {A : Set n} → bt (Color ∧ A) → bt (Color ∧ A) RightDown leaf = leaf RightDown (node key ⟪ c , value ⟫ t1 t2) = t2 LeftDown : {n : Level} {A : Set n} → bt (Color ∧ A) → bt (Color ∧ A) LeftDown leaf = leaf LeftDown (node key ⟪ c , value ⟫ t1 t2 ) = t1 RBtreeLeftDown : {n : Level} {A : Set n} {key : ℕ} {value : A} {c : Color} → (left right : bt (Color ∧ A)) → RBtreeInvariant (node key ⟪ c , value ⟫ left right) → RBtreeInvariant left RBtreeLeftDown left right (rb-red _ _ x x₁ x₂ rb rb₁) = rb RBtreeLeftDown left right (rb-black _ _ x rb rb₁) = rb RBtreeRightDown : {n : Level} {A : Set n} { key : ℕ} {value : A} {c : Color} → (left right : bt (Color ∧ A)) → RBtreeInvariant (node key ⟪ c , value ⟫ left right) → RBtreeInvariant right RBtreeRightDown left right (rb-red _ _ x x₁ x₂ rb rb₁) = rb₁ RBtreeRightDown left right (rb-black _ _ x rb rb₁) = rb₁ RBtreeEQ : {n : Level} {A : Set n} {key : ℕ} {value : A} {c : Color} → {left right : bt (Color ∧ A)} → RBtreeInvariant (node key ⟪ c , value ⟫ left right) → black-depth left ≡ black-depth right RBtreeEQ (rb-red _ _ x x₁ x₂ rb rb₁) = x₂ RBtreeEQ (rb-black _ _ x rb rb₁) = x RBtreeToBlack : {n : Level} {A : Set n} → (tree : bt (Color ∧ A)) → RBtreeInvariant tree → RBtreeInvariant (to-black tree) RBtreeToBlack leaf rb-leaf = rb-leaf RBtreeToBlack (node key ⟪ Red , value ⟫ left right) (rb-red _ _ x x₁ x₂ rb rb₁) = rb-black key value x₂ rb rb₁ RBtreeToBlack (node key ⟪ Black , value ⟫ left right) (rb-black _ _ x rb rb₁) = rb-black key value x rb rb₁ RBtreeToBlackColor : {n : Level} {A : Set n} → (tree : bt (Color ∧ A)) → RBtreeInvariant tree → color (to-black tree) ≡ Black RBtreeToBlackColor leaf rb-leaf = refl RBtreeToBlackColor (node key ⟪ Red , value ⟫ left right) (rb-red _ _ x x₁ x₂ rb rb₁) = refl RBtreeToBlackColor (node key ⟪ Black , value ⟫ left right) (rb-black _ _ x rb rb₁) = refl RBtreeParentColorBlack : {n : Level} {A : Set n} → (left right : bt (Color ∧ A)) { value : A} {key : ℕ} { c : Color} → RBtreeInvariant (node key ⟪ c , value ⟫ left right) → (color left ≡ Red) ∨ (color right ≡ Red) → c ≡ Black RBtreeParentColorBlack leaf leaf (rb-red _ _ x₁ x₂ x₃ x₄ x₅) (case1 ()) RBtreeParentColorBlack leaf leaf (rb-red _ _ x₁ x₂ x₃ x₄ x₅) (case2 ()) RBtreeParentColorBlack (node key ⟪ Red , proj4 ⟫ left left₁) right (rb-red _ _ () x₁ x₂ rb rb₁) (case1 x₃) RBtreeParentColorBlack (node key ⟪ Black , proj4 ⟫ left left₁) right (rb-red _ _ x x₁ x₂ rb rb₁) (case1 ()) RBtreeParentColorBlack left (node key ⟪ Red , proj4 ⟫ right right₁) (rb-red _ _ x () x₂ rb rb₁) (case2 x₃) RBtreeParentColorBlack left (node key ⟪ Black , proj4 ⟫ right right₁) (rb-red _ _ x x₁ x₂ rb rb₁) (case2 ()) RBtreeParentColorBlack left right (rb-black _ _ x rb rb₁) x₃ = refl RBtreeChildrenColorBlack : {n : Level} {A : Set n} → (left right : bt (Color ∧ A)) { value : Color ∧ A} {key : ℕ} → RBtreeInvariant (node key value left right) → proj1 value ≡ Red → (color left ≡ Black) ∧ (color right ≡ Black) RBtreeChildrenColorBlack left right (rb-red _ _ x x₁ x₂ rbi rbi₁) refl = ⟪ x , x₁ ⟫ -- -- findRBT exit with replaced node -- case-eq node value is replaced, just do replacedTree and rebuild rb-invariant -- case-leaf insert new single node -- case1 if parent node is black, just do replacedTree and rebuild rb-invariant -- case2 if parent node is red, increase blackdepth, do rotatation -- findRBT : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt (Color ∧ A) ) → (stack : List (bt (Color ∧ A))) → RBtreeInvariant tree ∧ stackInvariant key tree tree0 stack → (next : (tree1 : bt (Color ∧ A) ) → (stack : List (bt (Color ∧ A))) → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t ) → (exit : (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t findRBT key leaf tree0 stack rb0 next exit = exit leaf stack rb0 (case1 refl) findRBT key (node key₁ value left right) tree0 stack rb0 next exit with <-cmp key key₁ findRBT key (node key₁ value left right) tree0 stack rb0 next exit | tri< a ¬b ¬c = next left (left ∷ stack) ⟪ RBtreeLeftDown left right (_∧_.proj1 rb0) , s-left _ _ _ a (_∧_.proj2 rb0) ⟫ depth-1< findRBT key n tree0 stack rb0 _ exit | tri≈ ¬a refl ¬c = exit n stack rb0 (case2 refl) findRBT key (node key₁ value left right) tree0 stack rb0 next exit | tri> ¬a ¬b c = next right (right ∷ stack) ⟪ RBtreeRightDown left right (_∧_.proj1 rb0), s-right _ _ _ c (_∧_.proj2 rb0) ⟫ depth-2< findTest : {n m : Level} {A : Set n } {t : Set m } → (key : ℕ) → (tree0 : bt (Color ∧ A)) → RBtreeInvariant tree0 → (exit : (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t findTest {n} {m} {A} {t} k tr0 rb0 exit = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant k (proj1 p) tr0 (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tr0 , tr0 ∷ [] ⟫ ⟪ rb0 , s-nil ⟫ $ λ p RBP loop → findRBT k (proj1 p) tr0 (proj2 p) RBP (λ t1 s1 P2 lt1 → loop ⟪ t1 , s1 ⟫ P2 lt1 ) $ λ tr1 st P2 O → exit tr1 st P2 O testRBTree0 : bt (Color ∧ ℕ) testRBTree0 = node 8 ⟪ Black , 800 ⟫ (node 5 ⟪ Red , 500 ⟫ (node 2 ⟪ Black , 200 ⟫ leaf leaf) (node 6 ⟪ Black , 600 ⟫ leaf leaf)) (node 10 ⟪ Red , 1000 ⟫ (leaf) (node 15 ⟪ Black , 1500 ⟫ (node 14 ⟪ Red , 1400 ⟫ leaf leaf) leaf)) -- testRBI0 : RBtreeInvariant testRBTree0 -- testRBI0 = rb-node-black (add< 2) (add< 1) refl (rb-node-red (add< 2) (add< 0) refl (rb-single 2 200) (rb-single 6 600)) (rb-right-red (add< 4) refl (rb-left-black (add< 0) refl (rb-single 14 1400) )) -- findRBTreeTest : result -- findRBTreeTest = findTest 14 testRBTree0 testRBI0 -- $ λ tr s P O → (record {tree = tr ; stack = s ; ti = (proj1 P) ; si = (proj2 P)}) -- create replaceRBTree with rotate -- -- Parent Grand Relation -- should we require stack-invariant? -- data ParentGrand {n : Level} {A : Set n} (key : ℕ) (self : bt A) : (parent uncle grand : bt A) → Set n where s2-s1p2 : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } → key < kp → parent ≡ node kp vp self n1 → grand ≡ node kg vg parent n2 → ParentGrand key self parent n2 grand s2-1sp2 : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } → kp < key → parent ≡ node kp vp n1 self → grand ≡ node kg vg parent n2 → ParentGrand key self parent n2 grand s2-s12p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } → key < kp → parent ≡ node kp vp self n1 → grand ≡ node kg vg n2 parent → ParentGrand key self parent n2 grand s2-1s2p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } → kp < key → parent ≡ node kp vp n1 self → grand ≡ node kg vg n2 parent → ParentGrand key self parent n2 grand record PG {n : Level } (A : Set n) (key : ℕ) (self : bt A) (stack : List (bt A)) : Set n where field parent grand uncle : bt A pg : ParentGrand key self parent uncle grand rest : List (bt A) stack=gp : stack ≡ ( self ∷ parent ∷ grand ∷ rest ) -- -- RBI : Invariant on InsertCase2 -- color repl ≡ Red ∧ black-depth repl ≡ suc (black-depth tree) -- data RBI-state {n : Level} {A : Set n} (key : ℕ) (value : A) : (tree repl : bt (Color ∧ A) ) → (stak : List (bt (Color ∧ A))) → Set n where rebuild : {tree repl : bt (Color ∧ A) } {stack : List (bt (Color ∧ A))} → color tree ≡ color repl → black-depth repl ≡ black-depth tree → ¬ ( tree ≡ leaf) → RBI-state key value tree repl stack -- one stage up rotate : (tree : bt (Color ∧ A)) → {repl : bt (Color ∧ A) } {stack : List (bt (Color ∧ A))} → color repl ≡ Red → RBI-state key value tree repl stack -- two stages up top-black : {tree repl : bt (Color ∧ A) } → {stack : List (bt (Color ∧ A))} → stack ≡ tree ∷ [] → RBI-state key value tree repl stack record RBI {n : Level} {A : Set n} (key : ℕ) (value : A) (orig : bt (Color ∧ A) ) (stack : List (bt (Color ∧ A))) : Set n where field tree repl : bt (Color ∧ A) origti : treeInvariant orig origrb : RBtreeInvariant orig treerb : RBtreeInvariant tree -- tree node te be replaced replrb : RBtreeInvariant repl si : stackInvariant key tree orig stack state : RBI-state key value tree repl stack tr>-to-black : {n : Level} {A : Set n} {key : ℕ} {tree : bt (Color ∧ A)} → tr> key tree → tr> key (to-black tree) tr>-to-black {n} {A} {key} {leaf} tr = tt tr>-to-black {n} {A} {key} {node key₁ value tree tree₁} tr = tr tr<-to-black : {n : Level} {A : Set n} {key : ℕ} {tree : bt (Color ∧ A)} → tr< key tree → tr< key (to-black tree) tr<-to-black {n} {A} {key} {leaf} tr = tt tr<-to-black {n} {A} {key} {node key₁ value tree tree₁} tr = tr to-black-eq : {n : Level} {A : Set n} (tree : bt (Color ∧ A)) → color tree ≡ Red → suc (black-depth tree) ≡ black-depth (to-black tree) to-black-eq {n} {A} (leaf) () to-black-eq {n} {A} (node key₁ ⟪ Red , proj4 ⟫ tree tree₁) eq = refl to-black-eq {n} {A} (node key₁ ⟪ Black , proj4 ⟫ tree tree₁) () red-children-eq : {n : Level} {A : Set n} {tree left right : bt (Color ∧ A)} → {key : ℕ} → {value : A} → {c : Color} → tree ≡ node key ⟪ c , value ⟫ left right → c ≡ Red → RBtreeInvariant tree → (black-depth tree ≡ black-depth left ) ∧ (black-depth tree ≡ black-depth right) red-children-eq {n} {A} {.(node key₁ ⟪ Red , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Red} refl eq₁ rb2@(rb-red key₁ value₁ x x₁ x₂ rb rb₁) = ⟪ ( begin black-depth left ⊔ black-depth right ≡⟨ cong (λ k → black-depth left ⊔ k) (sym (RBtreeEQ rb2)) ⟩ black-depth left ⊔ black-depth left ≡⟨ ⊔-idem _ ⟩ black-depth left ∎ ) , ( begin black-depth left ⊔ black-depth right ≡⟨ cong (λ k → k ⊔ black-depth right ) (RBtreeEQ rb2) ⟩ black-depth right ⊔ black-depth right ≡⟨ ⊔-idem _ ⟩ black-depth right ∎ ) ⟫ where open ≡-Reasoning red-children-eq {n} {A} {tree} {left} {right} {key} {value} {Black} eq () rb black-children-eq : {n : Level} {A : Set n} {tree left right : bt (Color ∧ A)} → {key : ℕ} → {value : A} → {c : Color} → tree ≡ node key ⟪ c , value ⟫ left right → c ≡ Black → RBtreeInvariant tree → (black-depth tree ≡ suc (black-depth left) ) ∧ (black-depth tree ≡ suc (black-depth right)) black-children-eq {n} {A} {.(node key₁ ⟪ Black , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Black} refl eq₁ rb2@(rb-black key₁ value₁ x rb rb₁) = ⟪ ( begin suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (black-depth left ⊔ k)) (sym (RBtreeEQ rb2)) ⟩ suc (black-depth left ⊔ black-depth left) ≡⟨ ⊔-idem _ ⟩ suc (black-depth left) ∎ ) , ( begin suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (k ⊔ black-depth right)) (RBtreeEQ rb2) ⟩ suc (black-depth right ⊔ black-depth right) ≡⟨ ⊔-idem _ ⟩ suc (black-depth right) ∎ ) ⟫ where open ≡-Reasoning black-children-eq {n} {A} {tree} {left} {right} {key} {value} {Red} eq () rb black-depth-cong : {n : Level} (A : Set n) {tree tree₁ : bt (Color ∧ A)} → tree ≡ tree₁ → black-depth tree ≡ black-depth tree₁ black-depth-cong {n} A {leaf} {leaf} refl = refl black-depth-cong {n} A {node key ⟪ Red , value ⟫ left right} {node .key ⟪ Red , .value ⟫ .left .right} refl = cong₂ (λ j k → j ⊔ k ) (black-depth-cong A {left} {left} refl) (black-depth-cong A {right} {right} refl) black-depth-cong {n} A {node key ⟪ Black , value ⟫ left right} {node .key ⟪ Black , .value ⟫ .left .right} refl = cong₂ (λ j k → suc (j ⊔ k) ) (black-depth-cong A {left} {left} refl) (black-depth-cong A {right} {right} refl) black-depth-resp : {n : Level} (A : Set n) (tree tree₁ : bt (Color ∧ A)) → {c1 c2 : Color} { l1 l2 r1 r2 : bt (Color ∧ A)} {key1 key2 : ℕ} {value1 value2 : A} → tree ≡ node key1 ⟪ c1 , value1 ⟫ l1 r1 → tree₁ ≡ node key2 ⟪ c2 , value2 ⟫ l2 r2 → color tree ≡ color tree₁ → black-depth l1 ≡ black-depth l2 → black-depth r1 ≡ black-depth r2 → black-depth tree ≡ black-depth tree₁ black-depth-resp A _ _ {Black} {Black} refl refl refl eql eqr = cong₂ (λ j k → suc (j ⊔ k) ) eql eqr black-depth-resp A _ _ {Red} {Red} refl refl refl eql eqr = cong₂ (λ j k → j ⊔ k ) eql eqr red-children-eq1 : {n : Level} {A : Set n} {tree left right : bt (Color ∧ A)} → {key : ℕ} → {value : A} → {c : Color} → tree ≡ node key ⟪ c , value ⟫ left right → color tree ≡ Red → RBtreeInvariant tree → (black-depth tree ≡ black-depth left ) ∧ (black-depth tree ≡ black-depth right) red-children-eq1 {n} {A} {.(node key₁ ⟪ Red , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Red} refl eq₁ rb2@(rb-red key₁ value₁ x x₁ x₂ rb rb₁) = ⟪ ( begin black-depth left ⊔ black-depth right ≡⟨ cong (λ k → black-depth left ⊔ k) (sym (RBtreeEQ rb2)) ⟩ black-depth left ⊔ black-depth left ≡⟨ ⊔-idem _ ⟩ black-depth left ∎ ) , ( begin black-depth left ⊔ black-depth right ≡⟨ cong (λ k → k ⊔ black-depth right ) (RBtreeEQ rb2) ⟩ black-depth right ⊔ black-depth right ≡⟨ ⊔-idem _ ⟩ black-depth right ∎ ) ⟫ where open ≡-Reasoning red-children-eq1 {n} {A} {.(node key ⟪ Black , value ⟫ left right)} {left} {right} {key} {value} {Black} refl () rb black-children-eq1 : {n : Level} {A : Set n} {tree left right : bt (Color ∧ A)} → {key : ℕ} → {value : A} → {c : Color} → tree ≡ node key ⟪ c , value ⟫ left right → color tree ≡ Black → RBtreeInvariant tree → (black-depth tree ≡ suc (black-depth left) ) ∧ (black-depth tree ≡ suc (black-depth right)) black-children-eq1 {n} {A} {.(node key₁ ⟪ Black , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Black} refl eq₁ rb2@(rb-black key₁ value₁ x rb rb₁) = ⟪ ( begin suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (black-depth left ⊔ k)) (sym (RBtreeEQ rb2)) ⟩ suc (black-depth left ⊔ black-depth left) ≡⟨ ⊔-idem _ ⟩ suc (black-depth left) ∎ ) , ( begin suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (k ⊔ black-depth right)) (RBtreeEQ rb2) ⟩ suc (black-depth right ⊔ black-depth right) ≡⟨ ⊔-idem _ ⟩ suc (black-depth right) ∎ ) ⟫ where open ≡-Reasoning black-children-eq1 {n} {A} {.(node key ⟪ Red , value ⟫ left right)} {left} {right} {key} {value} {Red} refl () rb rbi-from-red-black : {n : Level } {A : Set n} → (n1 rp-left : bt (Color ∧ A)) → (kp : ℕ) → (vp : Color ∧ A) → RBtreeInvariant n1 → RBtreeInvariant rp-left → black-depth n1 ≡ black-depth rp-left → color n1 ≡ Black → color rp-left ≡ Black → ⟪ Red , proj2 vp ⟫ ≡ vp → RBtreeInvariant (node kp vp n1 rp-left) rbi-from-red-black leaf leaf kp vp rb1 rbp deq ceq1 ceq2 ceq3 = subst (λ k → RBtreeInvariant (node kp k leaf leaf)) ceq3 (rb-red kp (proj2 vp) refl refl refl rb-leaf rb-leaf) rbi-from-red-black leaf (node key ⟪ .Black , proj3 ⟫ trpl trpl₁) kp vp rb1 rbp deq ceq1 refl ceq3 = subst (λ k → RBtreeInvariant (node kp k _ _)) ceq3 (rb-red kp (proj2 vp) refl refl deq rb1 rbp) rbi-from-red-black (node key ⟪ .Black , proj3 ⟫ tn1 tn2) leaf kp vp rb1 rbp deq refl ceq2 ceq3 = subst (λ k → RBtreeInvariant (node kp k _ _)) ceq3 (rb-red kp (proj2 vp) refl refl deq rb1 rbp) rbi-from-red-black (node key ⟪ .Black , proj3 ⟫ tn1 tn2) (node key₁ ⟪ .Black , proj4 ⟫ trpl trpl₁) kp vp rb1 rbp deq refl refl ceq3 = subst (λ k → RBtreeInvariant (node kp k _ _)) ceq3 (rb-red kp (proj2 vp) refl refl deq rb1 rbp ) ⊔-succ : {m n : ℕ} → suc (m ⊔ n) ≡ suc m ⊔ suc n ⊔-succ {m} {n} = refl -- -- if we consider tree invariant, this may be much simpler and faster -- stackToPG : {n : Level} {A : Set n} → {key : ℕ } → (tree orig : bt A ) → (stack : List (bt A)) → stackInvariant key tree orig stack → ( stack ≡ orig ∷ [] ) ∨ ( stack ≡ tree ∷ orig ∷ [] ) ∨ PG A key tree stack stackToPG {n} {A} {key} tree .tree .(tree ∷ []) s-nil = case1 refl stackToPG {n} {A} {key} tree .(node _ _ _ tree) .(tree ∷ node _ _ _ tree ∷ []) (s-right _ _ _ x s-nil) = case2 (case1 refl) stackToPG {n} {A} {key} tree .(node k2 v2 t5 (node k1 v1 t2 tree)) (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ [])) (s-right tree (node k2 v2 t5 (node k1 v1 t2 tree)) t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) (node k2 v2 t5 (node k1 v1 t2 tree)) t5 {k2} {v2} x₁ s-nil)) = case2 (case2 record { parent = node k1 v1 t2 tree ; grand = _ ; pg = s2-1s2p x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ _)) (s-right tree orig t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) orig t5 {k2} {v2} x₁ (s-right _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 t2 tree ; grand = _ ; pg = s2-1s2p x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ _)) (s-right tree orig t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) orig t5 {k2} {v2} x₁ (s-left _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 t2 tree ; grand = _ ; pg = s2-1s2p x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree .(node k2 v2 (node k1 v1 t1 tree) t2) .(tree ∷ node k1 v1 t1 tree ∷ node k2 v2 (node k1 v1 t1 tree) t2 ∷ []) (s-right _ _ t1 {k1} {v1} x (s-left _ _ t2 {k2} {v2} x₁ s-nil)) = case2 (case2 record { parent = node k1 v1 t1 tree ; grand = _ ; pg = s2-1sp2 x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 t1 tree ∷ node k2 v2 (node k1 v1 t1 tree) t2 ∷ _) (s-right _ _ t1 {k1} {v1} x (s-left _ _ t2 {k2} {v2} x₁ (s-right _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 t1 tree ; grand = _ ; pg = s2-1sp2 x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 t1 tree ∷ node k2 v2 (node k1 v1 t1 tree) t2 ∷ _) (s-right _ _ t1 {k1} {v1} x (s-left _ _ t2 {k2} {v2} x₁ (s-left _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 t1 tree ; grand = _ ; pg = s2-1sp2 x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree .(node _ _ tree _) .(tree ∷ node _ _ tree _ ∷ []) (s-left _ _ t1 {k1} {v1} x s-nil) = case2 (case1 refl) stackToPG {n} {A} {key} tree .(node _ _ _ (node k1 v1 tree t1)) .(tree ∷ node k1 v1 tree t1 ∷ node _ _ _ (node k1 v1 tree t1) ∷ []) (s-left _ _ t1 {k1} {v1} x (s-right _ _ _ x₁ s-nil)) = case2 (case2 record { parent = node k1 v1 tree t1 ; grand = _ ; pg = s2-s12p x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ _ (node k1 v1 tree t1) ∷ _) (s-left _ _ t1 {k1} {v1} x (s-right _ _ _ x₁ (s-right _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 tree t1 ; grand = _ ; pg = s2-s12p x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ _ (node k1 v1 tree t1) ∷ _) (s-left _ _ t1 {k1} {v1} x (s-right _ _ _ x₁ (s-left _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 tree t1 ; grand = _ ; pg = s2-s12p x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree .(node _ _ (node k1 v1 tree t1) _) .(tree ∷ node k1 v1 tree t1 ∷ node _ _ (node k1 v1 tree t1) _ ∷ []) (s-left _ _ t1 {k1} {v1} x (s-left _ _ _ x₁ s-nil)) = case2 (case2 record { parent = node k1 v1 tree t1 ; grand = _ ; pg = s2-s1p2 x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ (node k1 v1 tree t1) _ ∷ _) (s-left _ _ t1 {k1} {v1} x (s-left _ _ _ x₁ (s-right _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 tree t1 ; grand = _ ; pg = s2-s1p2 x refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ (node k1 v1 tree t1) _ ∷ _) (s-left _ _ t1 {k1} {v1} x (s-left _ _ _ x₁ (s-left _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 tree t1 ; grand = _ ; pg = s2-s1p2 x refl refl ; rest = _ ; stack=gp = refl } ) stackCase1 : {n : Level} {A : Set n} → {key : ℕ } → {tree orig : bt A } → {stack : List (bt A)} → stackInvariant key tree orig stack → stack ≡ orig ∷ [] → tree ≡ orig stackCase1 s-nil refl = refl pg-prop-1 : {n : Level} (A : Set n) → (key : ℕ) → (tree orig : bt A ) → (stack : List (bt A)) → (pg : PG A key tree stack) → (¬ PG.grand pg ≡ leaf ) ∧ (¬ PG.parent pg ≡ leaf) pg-prop-1 {_} A _ tree orig stack pg with PG.pg pg ... | s2-s1p2 _ refl refl = ⟪ (λ () ) , ( λ () ) ⟫ ... | s2-1sp2 _ refl refl = ⟪ (λ () ) , ( λ () ) ⟫ ... | s2-s12p _ refl refl = ⟪ (λ () ) , ( λ () ) ⟫ ... | s2-1s2p _ refl refl = ⟪ (λ () ) , ( λ () ) ⟫ popStackInvariant : {n : Level} {A : Set n} → (rest : List ( bt (Color ∧ A))) → ( tree parent orig : bt (Color ∧ A)) → (key : ℕ) → stackInvariant key tree orig ( tree ∷ parent ∷ rest ) → stackInvariant key parent orig (parent ∷ rest ) popStackInvariant rest tree parent orig key (s-right .tree .orig tree₁ x si) = sc00 where sc00 : stackInvariant key parent orig (parent ∷ rest ) sc00 with si-property1 si ... | refl = si popStackInvariant rest tree parent orig key (s-left .tree .orig tree₁ x si) = sc00 where sc00 : stackInvariant key parent orig (parent ∷ rest ) sc00 with si-property1 si ... | refl = si siToTreeinvariant : {n : Level} {A : Set n} → (rest : List ( bt (Color ∧ A))) → ( tree orig : bt (Color ∧ A)) → (key : ℕ) → treeInvariant orig → stackInvariant key tree orig ( tree ∷ rest ) → treeInvariant tree siToTreeinvariant .[] tree .tree key ti s-nil = ti siToTreeinvariant .(node _ _ leaf leaf ∷ []) .leaf .(node _ _ leaf leaf) key (t-single _ _) (s-right .leaf .(node _ _ leaf leaf) .leaf x s-nil) = t-leaf siToTreeinvariant .(node _ _ leaf (node key₁ _ _ _) ∷ []) .(node key₁ _ _ _) .(node _ _ leaf (node key₁ _ _ _)) key (t-right _ key₁ x₁ x₂ x₃ ti) (s-right .(node key₁ _ _ _) .(node _ _ leaf (node key₁ _ _ _)) .leaf x s-nil) = ti siToTreeinvariant .(node _ _ (node key₁ _ _ _) leaf ∷ []) .leaf .(node _ _ (node key₁ _ _ _) leaf) key (t-left key₁ _ x₁ x₂ x₃ ti) (s-right .leaf .(node _ _ (node key₁ _ _ _) leaf) .(node key₁ _ _ _) x s-nil) = t-leaf siToTreeinvariant .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _) ∷ []) .(node key₂ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) key (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right .(node key₂ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) .(node key₁ _ _ _) x s-nil) = ti₁ siToTreeinvariant .(node _ _ tree₁ tree ∷ _) tree orig key ti (s-right .tree .orig tree₁ x si2@(s-right .(node _ _ tree₁ tree) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2 ... | t-single _ _ = t-leaf ... | t-right _ key₁ x₂ x₃ x₄ ti₁ = ti₁ ... | t-left key₁ _ x₂ x₃ x₄ ti₁ = t-leaf ... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ ti₁ ti₂ = ti₂ siToTreeinvariant .(node _ _ tree₁ tree ∷ _) tree orig key ti (s-right .tree .orig tree₁ x si2@(s-left .(node _ _ tree₁ tree) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2 ... | t-single _ _ = t-leaf ... | t-right _ key₁ x₂ x₃ x₄ ti₁ = ti₁ ... | t-left key₁ _ x₂ x₃ x₄ ti₁ = t-leaf ... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ ti₁ ti₂ = ti₂ siToTreeinvariant .(node _ _ leaf leaf ∷ []) .leaf .(node _ _ leaf leaf) key (t-single _ _) (s-left .leaf .(node _ _ leaf leaf) .leaf x s-nil) = t-leaf siToTreeinvariant .(node _ _ leaf (node key₁ _ _ _) ∷ []) .leaf .(node _ _ leaf (node key₁ _ _ _)) key (t-right _ key₁ x₁ x₂ x₃ ti) (s-left .leaf .(node _ _ leaf (node key₁ _ _ _)) .(node key₁ _ _ _) x s-nil) = t-leaf siToTreeinvariant .(node _ _ (node key₁ _ _ _) leaf ∷ []) .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) leaf) key (t-left key₁ _ x₁ x₂ x₃ ti) (s-left .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) leaf) .leaf x s-nil) = ti siToTreeinvariant .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _) ∷ []) .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) key (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-left .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) .(node key₂ _ _ _) x s-nil) = ti siToTreeinvariant .(node _ _ tree tree₁ ∷ _) tree orig key ti (s-left .tree .orig tree₁ x si2@(s-right .(node _ _ tree tree₁) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2 ... | t-single _ _ = t-leaf ... | t-right _ key₁ x₂ x₃ x₄ t = t-leaf ... | t-left key₁ _ x₂ x₃ x₄ t = t ... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ t t₁ = t siToTreeinvariant .(node _ _ tree tree₁ ∷ _) tree orig key ti (s-left .tree .orig tree₁ x si2@(s-left .(node _ _ tree tree₁) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2 ... | t-single _ _ = t-leaf ... | t-right _ key₁ x₂ x₃ x₄ t = t-leaf ... | t-left key₁ _ x₂ x₃ x₄ t = t ... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ t t₁ = t PGtoRBinvariant1 : {n : Level} {A : Set n} → (tree orig : bt (Color ∧ A) ) → {key : ℕ } → (rb : RBtreeInvariant orig) → (stack : List (bt (Color ∧ A))) → (si : stackInvariant key tree orig stack ) → RBtreeInvariant tree PGtoRBinvariant1 tree .tree rb .(tree ∷ []) s-nil = rb PGtoRBinvariant1 tree orig rb (tree ∷ rest) (s-right .tree .orig tree₁ x si) with PGtoRBinvariant1 _ orig rb _ si ... | rb-red _ value x₁ x₂ x₃ t t₁ = t₁ ... | rb-black _ value x₁ t t₁ = t₁ PGtoRBinvariant1 tree orig rb (tree ∷ rest) (s-left .tree .orig tree₁ x si) with PGtoRBinvariant1 _ orig rb _ si ... | rb-red _ value x₁ x₂ x₃ t t₁ = t ... | rb-black _ value x₁ t t₁ = t RBI-child-replaced : {n : Level} {A : Set n} (tr : bt (Color ∧ A)) (key : ℕ) → RBtreeInvariant tr → RBtreeInvariant (child-replaced key tr) RBI-child-replaced {n} {A} leaf key rbi = rbi RBI-child-replaced {n} {A} (node key₁ value tr tr₁) key rbi with <-cmp key key₁ ... | tri< a ¬b ¬c = RBtreeLeftDown _ _ rbi ... | tri≈ ¬a b ¬c = rbi ... | tri> ¬a ¬b c = RBtreeRightDown _ _ rbi -- -- create RBT invariant after findRBT, continue to replaceRBT -- createRBT : {n m : Level} {A : Set n } {t : Set m } → (key : ℕ) (value : A) → (tree0 : bt (Color ∧ A)) → RBtreeInvariant tree0 → treeInvariant tree0 → (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → stackInvariant key tree1 tree0 stack → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → (exit : (stack : List ( bt (Color ∧ A))) (r : RBI key value tree0 stack ) → t ) → t createRBT {n} {m} {A} {t} key value orig rbi ti tree (x ∷ []) si P exit = crbt00 orig P refl where crbt00 : (tree1 : bt (Color ∧ A)) → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) → tree1 ≡ orig → t crbt00 leaf P refl = exit (x ∷ []) record { tree = leaf ; repl = node key ⟪ Red , value ⟫ leaf leaf ; origti = ti ; origrb = rbi ; treerb = rb-leaf ; replrb = rb-red key value refl refl refl rb-leaf rb-leaf ; si = subst (λ k → stackInvariant key k leaf _ ) crbt01 si ; state = rotate leaf refl } where crbt01 : tree ≡ leaf crbt01 with si-property-last _ _ _ _ si | si-property1 si ... | refl | refl = refl crbt00 (node key₁ value₁ left right ) (case1 eq) refl with si-property-last _ _ _ _ si | si-property1 si ... | refl | refl = ⊥-elim (bt-ne (sym eq)) crbt00 (node key₁ value₁ left right ) (case2 eq) refl with si-property-last _ _ _ _ si | si-property1 si ... | refl | refl = exit (x ∷ []) record { tree = node key₁ value₁ left right ; repl = node key₁ ⟪ proj1 value₁ , value ⟫ left right ; origti = ti ; origrb = rbi ; treerb = rbi ; replrb = crbt03 value₁ rbi ; si = si ; state = rebuild refl (crbt01 value₁ ) (λ ()) } where crbt01 : (value₁ : Color ∧ A) → black-depth (node key₁ ⟪ proj1 value₁ , value ⟫ left right) ≡ black-depth (node key₁ value₁ left right) crbt01 ⟪ Red , proj4 ⟫ = refl crbt01 ⟪ Black , proj4 ⟫ = refl crbt03 : (value₁ : Color ∧ A) → RBtreeInvariant (node key₁ value₁ left right) → RBtreeInvariant (node key₁ ⟪ proj1 value₁ , value ⟫ left right) crbt03 ⟪ Red , proj4 ⟫ (rb-red .key₁ .proj4 x x₁ x₂ rbi rbi₁) = rb-red key₁ _ x x₁ x₂ rbi rbi₁ crbt03 ⟪ Black , proj4 ⟫ (rb-black .key₁ .proj4 x rbi rbi₁) = rb-black key₁ _ x rbi rbi₁ keq : ( just key₁ ≡ just key ) → key₁ ≡ key keq refl = refl createRBT {n} {m} {A} {t} key value orig rbi ti tree (x ∷ leaf ∷ stack) si P exit = ⊥-elim (si-property-parent-node _ _ _ si refl) createRBT {n} {m} {A} {t} key value orig rbi ti tree sp@(x ∷ node kp vp pleft pright ∷ stack) si P exit = crbt00 tree P refl where crbt00 : (tree1 : bt (Color ∧ A)) → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) → tree1 ≡ tree → t crbt00 leaf P refl = exit sp record { tree = leaf ; repl = node key ⟪ Red , value ⟫ leaf leaf ; origti = ti ; origrb = rbi ; treerb = rb-leaf ; replrb = rb-red key value refl refl refl rb-leaf rb-leaf ; si = si ; state = rotate leaf refl } crbt00 (node key₁ value₁ left right ) (case1 eq) refl = ⊥-elim (bt-ne (sym eq)) crbt00 (node key₁ value₁ left right ) (case2 eq) refl with si-property-last _ _ _ _ si | si-property1 si ... | eq1 | eq2 = exit sp record { tree = node key₁ value₁ left right ; repl = node key₁ ⟪ proj1 value₁ , value ⟫ left right ; origti = ti ; origrb = rbi ; treerb = treerb ; replrb = crbt03 value₁ treerb ; si = si ; state = rebuild refl (crbt01 value₁ ) (λ ()) } where crbt01 : (value₁ : Color ∧ A) → black-depth (node key₁ ⟪ proj1 value₁ , value ⟫ left right) ≡ black-depth (node key₁ value₁ left right) crbt01 ⟪ Red , proj4 ⟫ = refl crbt01 ⟪ Black , proj4 ⟫ = refl crbt03 : (value₁ : Color ∧ A) → RBtreeInvariant (node key₁ value₁ left right) → RBtreeInvariant (node key₁ ⟪ proj1 value₁ , value ⟫ left right) crbt03 ⟪ Red , proj4 ⟫ (rb-red .key₁ .proj4 x x₁ x₂ rbi rbi₁) = rb-red key₁ _ x x₁ x₂ rbi rbi₁ crbt03 ⟪ Black , proj4 ⟫ (rb-black .key₁ .proj4 x rbi rbi₁) = rb-black key₁ _ x rbi rbi₁ keq : ( just key₁ ≡ just key ) → key₁ ≡ key keq refl = refl treerb : RBtreeInvariant (node key₁ value₁ left right) treerb = PGtoRBinvariant1 _ orig rbi _ si -- -- rotate and rebuild replaceTree and rb-invariant -- replaceRBP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (orig : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig stack ) → (next : (stack1 : List (bt (Color ∧ A))) → (r : RBI key value orig stack1 ) → length stack1 < length stack → t ) → (exit : (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) → RBI key value orig stack1 → t ) → t replaceRBP {n} {m} {A} {t} key value orig stack r next exit = replaceRBP1 where -- we have no grand parent -- eq : stack₁ ≡ RBI.tree r ∷ orig ∷ [] -- change parent color ≡ Black and exit -- one level stack, orig is parent of repl repl = RBI.repl r insertCase4 : (tr0 : bt (Color ∧ A)) → tr0 ≡ orig → (eq : stack ≡ RBI.tree r ∷ orig ∷ []) → ( color repl ≡ Red ) ∨ (color (RBI.tree r) ≡ color repl) → stackInvariant key (RBI.tree r) orig stack → t insertCase4 leaf eq1 eq col si = ⊥-elim (rb04 eq eq1 si) where -- can't happen rb04 : {stack : List ( bt ( Color ∧ A))} → stack ≡ RBI.tree r ∷ orig ∷ [] → leaf ≡ orig → stackInvariant key (RBI.tree r) orig stack → ⊥ rb04 refl refl (s-right tree leaf tree₁ x si) = si-property2 _ (s-right tree leaf tree₁ x si) refl rb04 refl refl (s-left tree₁ leaf tree x si) = si-property2 _ (s-left tree₁ leaf tree x si) refl insertCase4 tr0@(node key₁ value₁ left right) refl eq col si with <-cmp key key₁ ... | tri< a ¬b ¬c = rb07 col where rb04 : stackInvariant key (RBI.tree r) orig stack → stack ≡ RBI.tree r ∷ orig ∷ [] → tr0 ≡ orig → left ≡ RBI.tree r rb04 (s-left tree₁ .(node key₁ value₁ left right) tree {key₂} x s-nil) refl refl = refl rb04 (s-right tree .(node key₁ _ tree₁ tree) tree₁ x s-nil) refl refl with si-property1 si ... | refl = ⊥-elim ( nat-<> x a ) rb06 : black-depth repl ≡ black-depth right rb06 = ? rb08 : (color (RBI.tree r) ≡ color repl) → RBtreeInvariant (node key₁ value₁ repl right) rb08 ceq with proj1 value₁ in coeq ... | Red = subst (λ k → RBtreeInvariant (node key₁ k repl right)) (cong (λ k → ⟪ k , _ ⟫) (sym coeq) ) (rb-red _ (proj2 value₁) rb09 (proj2 (RBtreeChildrenColorBlack _ _ (RBI.origrb r) coeq)) rb06 (RBI.replrb r) (RBtreeRightDown _ _ (RBI.origrb r))) where rb09 : color repl ≡ Black rb09 = trans (trans (sym ceq) (sym (cong color (rb04 si eq refl) ))) (proj1 (RBtreeChildrenColorBlack _ _ (RBI.origrb r) coeq)) ... | Black = subst (λ k → RBtreeInvariant (node key₁ k repl right)) (cong (λ k → ⟪ k , _ ⟫) (sym coeq) ) (rb-black _ (proj2 value₁) rb06 (RBI.replrb r) (RBtreeRightDown _ _ (RBI.origrb r))) rb07 : ( color repl ≡ Red ) ∨ (color (RBI.tree r) ≡ color repl) → t rb07 (case2 cc ) = exit (orig ∷ []) refl record { tree = orig ; repl = node key₁ value₁ repl right ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = RBI.origrb r ; replrb = rb08 cc ; si = s-nil ; state = top-black refl } rb07 (case1 repl-red) = exit (orig ∷ []) refl record { tree = orig ; repl = to-black (node key₁ value₁ repl right) ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = RBI.origrb r ; replrb = rb-black _ _ rb06 (RBI.replrb r) (RBtreeRightDown _ _ (RBI.origrb r)) ; si = s-nil ; state = top-black refl } ... | tri≈ ¬a b ¬c = ⊥-elim ( si-property-pne _ _ stack eq si b ) -- can't happen ... | tri> ¬a ¬b c = rb07 col where rb04 : stackInvariant key (RBI.tree r) orig stack → stack ≡ RBI.tree r ∷ orig ∷ [] → tr0 ≡ orig → right ≡ RBI.tree r rb04 (s-right tree .(node key₁ _ tree₁ tree) tree₁ x s-nil) refl refl = refl rb04 (s-left tree₁ .(node key₁ value₁ left right) tree {key₂} x si) refl refl with si-property1 si ... | refl = ⊥-elim ( nat-<> x c ) rb06 : black-depth repl ≡ black-depth left rb06 = ? rb08 : (color (RBI.tree r) ≡ color repl) → RBtreeInvariant (node key₁ value₁ left repl ) rb08 ceq with proj1 value₁ in coeq ... | Red = subst (λ k → RBtreeInvariant (node key₁ k left repl )) (cong (λ k → ⟪ k , _ ⟫) (sym coeq) ) (rb-red _ (proj2 value₁) (proj1 (RBtreeChildrenColorBlack _ _ (RBI.origrb r) coeq)) rb09 (sym rb06) (RBtreeLeftDown _ _ (RBI.origrb r))(RBI.replrb r)) where rb09 : color repl ≡ Black rb09 = trans (trans (sym ceq) (sym (cong color (rb04 si eq refl) ))) (proj2 (RBtreeChildrenColorBlack _ _ (RBI.origrb r) coeq)) ... | Black = subst (λ k → RBtreeInvariant (node key₁ k left repl )) (cong (λ k → ⟪ k , _ ⟫) (sym coeq) ) (rb-black _ (proj2 value₁) (sym rb06) (RBtreeLeftDown _ _ (RBI.origrb r)) (RBI.replrb r)) rb07 : ( color repl ≡ Red ) ∨ (color (RBI.tree r) ≡ color repl) → t rb07 (case2 cc ) = exit (orig ∷ []) refl record { tree = orig ; repl = node key₁ value₁ left repl ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = RBI.origrb r ; replrb = rb08 cc ; si = s-nil ; state = top-black refl } rb07 (case1 repl-red ) = exit (orig ∷ []) refl record { tree = orig ; repl = to-black (node key₁ value₁ left repl) ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = RBI.origrb r ; replrb = rb-black _ _ (sym rb06) (RBtreeLeftDown _ _ (RBI.origrb r)) (RBI.replrb r) ; si = s-nil ; state = top-black refl } rebuildCase : (ceq : color (RBI.tree r) ≡ color repl) → (bdepth-eq : black-depth repl ≡ black-depth (RBI.tree r)) → (¬ RBI.tree r ≡ leaf) → t rebuildCase ceq bdepth-eq ¬leaf with stackToPG (RBI.tree r) orig stack (RBI.si r) ... | case1 x = exit stack x r where -- single node case rb00 : stack ≡ orig ∷ [] → orig ≡ RBI.tree r rb00 refl = si-property1 (RBI.si r) ... | case2 (case1 x) = insertCase4 orig refl x (case2 ceq) (RBI.si r) -- one level stack, orig is parent of repl ... | case2 (case2 pg) = rebuildCase1 pg where rb00 : (pg : PG (Color ∧ A) key (RBI.tree r) stack) → stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) rb00 pg = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r) treerb : (pg : PG (Color ∧ A) key (RBI.tree r) stack) → RBtreeInvariant (PG.parent pg) treerb pg = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ (rb00 pg)) rb08 : (pg : PG (Color ∧ A) key (RBI.tree r) stack) → treeInvariant (PG.parent pg) rb08 pg = siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ (rb00 pg)) rebuildCase1 : (PG (Color ∧ A) key (RBI.tree r) stack) → t rebuildCase1 pg with PG.pg pg ... | s2-s1p2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = rebuildCase2 where rebuildCase2 : t rebuildCase2 = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb02 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) rb06 (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where rb01 : bt (Color ∧ A) rb01 = node kp vp repl n1 rb03 : black-depth n1 ≡ black-depth repl rb03 = ? rb02 : RBtreeInvariant rb01 rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg) ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value (trans (sym ceq) cx) cx₁ (sym rb03) (RBI.replrb r) t₁ ... | rb-black .kp value ex t t₁ = rb-black kp value (sym rb03) (RBI.replrb r) t₁ rb05 : treeInvariant (node kp vp (RBI.tree r) n1 ) rb05 = subst (λ k → treeInvariant k) x (rb08 pg) rb04 : key < kp rb04 = lt rb06 : black-depth rb01 ≡ black-depth (PG.parent pg) rb06 = trans (rb07 vp) ( cong black-depth (sym x) ) where rb07 : (vp : Color ∧ A) → black-depth (node kp vp repl n1) ≡ black-depth (node kp vp (RBI.tree r) n1 ) rb07 ⟪ Black , proj4 ⟫ = cong (λ k → suc (k ⊔ black-depth n1 )) bdepth-eq rb07 ⟪ Red , proj4 ⟫ = cong (λ k → (k ⊔ black-depth n1 )) bdepth-eq ... | s2-1sp2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = rebuildCase2 where rebuildCase2 : t rebuildCase2 = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb02 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) rb06 (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where rb01 : bt (Color ∧ A) rb01 = node kp vp n1 repl rb03 : black-depth n1 ≡ black-depth repl rb03 = ? rb02 : RBtreeInvariant rb01 rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg) ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value cx (trans (sym ceq) cx₁) rb03 t (RBI.replrb r) ... | rb-black .kp value ex t t₁ = rb-black kp value rb03 t (RBI.replrb r) rb05 : treeInvariant (node kp vp n1 (RBI.tree r) ) rb05 = subst (λ k → treeInvariant k) x (rb08 pg) rb04 : kp < key rb04 = lt rb06 : black-depth rb01 ≡ black-depth (PG.parent pg) rb06 = trans (rb07 vp) ( cong black-depth (sym x) ) where rb07 : (vp : Color ∧ A) → black-depth (node kp vp n1 repl) ≡ black-depth (node kp vp n1 (RBI.tree r)) rb07 ⟪ Black , proj4 ⟫ = cong (λ k → suc (black-depth n1 ⊔ k)) bdepth-eq rb07 ⟪ Red , proj4 ⟫ = cong (λ k → (black-depth n1 ⊔ k)) bdepth-eq ... | s2-s12p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = rebuildCase2 where rebuildCase2 : t rebuildCase2 = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb02 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) rb06 (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where rb01 : bt (Color ∧ A) rb01 = node kp vp repl n1 rb03 : black-depth n1 ≡ black-depth repl rb03 = ? rb02 : RBtreeInvariant rb01 rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg) ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value (trans (sym ceq) cx) cx₁ (sym rb03) (RBI.replrb r) t₁ ... | rb-black .kp value ex t t₁ = rb-black kp value (sym rb03) (RBI.replrb r) t₁ rb05 : treeInvariant (node kp vp (RBI.tree r) n1) rb05 = subst (λ k → treeInvariant k) x (rb08 pg) rb04 : key < kp rb04 = lt rb06 : black-depth rb01 ≡ black-depth (PG.parent pg) rb06 = trans (rb07 vp) ( cong black-depth (sym x) ) where rb07 : (vp : Color ∧ A) → black-depth (node kp vp repl n1) ≡ black-depth (node kp vp (RBI.tree r) n1) rb07 ⟪ Black , proj4 ⟫ = cong (λ k → suc (k ⊔ black-depth n1 )) bdepth-eq rb07 ⟪ Red , proj4 ⟫ = cong (λ k → (k ⊔ black-depth n1 )) bdepth-eq ... | s2-1s2p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = rebuildCase2 where rebuildCase2 : t rebuildCase2 = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb02 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) rb06 (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where rb01 : bt (Color ∧ A) rb01 = node kp vp n1 repl rb03 : black-depth n1 ≡ black-depth repl rb03 = ? rb02 : RBtreeInvariant rb01 rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg) ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value cx (trans (sym ceq) cx₁) rb03 t (RBI.replrb r) ... | rb-black .kp value ex t t₁ = rb-black kp value rb03 t (RBI.replrb r) rb05 : treeInvariant (node kp vp n1 (RBI.tree r)) rb05 = subst (λ k → treeInvariant k) x (rb08 pg) rb04 : kp < key rb04 = si-property-> ¬leaf rb05 (subst (λ k → stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ k ∷ PG.grand pg ∷ PG.rest pg)) x (rb00 pg)) rb06 : black-depth rb01 ≡ black-depth (PG.parent pg) rb06 = trans (rb07 vp) ( cong black-depth (sym x) ) where rb07 : (vp : Color ∧ A) → black-depth (node kp vp n1 repl) ≡ black-depth (node kp vp n1 (RBI.tree r)) rb07 ⟪ Black , proj4 ⟫ = cong (λ k → suc (black-depth n1 ⊔ k)) bdepth-eq rb07 ⟪ Red , proj4 ⟫ = cong (λ k → (black-depth n1 ⊔ k)) bdepth-eq -- -- both parent and uncle are Red, rotate then goto rebuild -- insertCase5 : (repl1 : bt (Color ∧ A)) → repl1 ≡ repl → (pg : PG (Color ∧ A) key (RBI.tree r) stack) → color repl ≡ Red → color (PG.uncle pg) ≡ Black → color (PG.parent pg) ≡ Red → t insertCase5 leaf eq pg repl-red uncle-black pcolor = ⊥-elim ( rb00 repl repl-red (cong color (sym eq))) where rb00 : (repl : bt (Color ∧ A)) → color repl ≡ Red → color repl ≡ Black → ⊥ rb00 (node key ⟪ Black , proj4 ⟫ repl repl₁) () eq1 rb00 (node key ⟪ Red , proj4 ⟫ repl repl₁) eq () insertCase5 (node rkey vr rp-left rp-right) eq pg repl-red uncle-black pcolor = insertCase51 where rb00 : stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) rb00 = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r) rb15 : suc (length (PG.rest pg)) < length stack rb15 = begin suc (suc (length (PG.rest pg))) ≤⟨ <-trans (n<1+n _) (n<1+n _) ⟩ length (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) ≡⟨ cong length (sym (PG.stack=gp pg)) ⟩ length stack ∎ where open ≤-Reasoning rb02 : RBtreeInvariant (PG.grand pg) rb02 = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)) rb09 : RBtreeInvariant (PG.parent pg) rb09 = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ rb00) rb08 : treeInvariant (PG.parent pg) rb08 = siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00) insertCase51 : t insertCase51 with PG.pg pg ... | s2-s1p2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase52 where insertCase52 : t insertCase52 = next (PG.grand pg ∷ PG.rest pg) record { tree = PG.grand pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb02 ; replrb = subst (λ k → RBtreeInvariant k) rb10 (rb-black _ _ rb18 (RBI.replrb r) (rb-red _ _ rb16 uncle-black rb19 rb06 rb05)) ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; state = rebuild (trans (cong color x₁) (cong proj1 (sym rb14))) rb17 (subst (λ k → ¬ (k ≡ leaf)) (sym x₁) (λ ())) } rb15 where rb01 : bt (Color ∧ A) rb01 = to-black (node kp vp (node rkey vr rp-left rp-right) (to-red (node kg vg n1 (PG.uncle pg)))) rb04 : key < kp rb04 = lt rb16 : color n1 ≡ Black rb16 = proj2 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) x rb09) (trans (cong color (sym x)) pcolor)) rb13 : ⟪ Red , proj2 vp ⟫ ≡ vp rb13 with subst (λ k → color k ≡ Red ) x pcolor ... | refl = refl rb14 : ⟪ Black , proj2 vg ⟫ ≡ vg rb14 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) (case1 pcolor) ... | refl = refl rb10 : node kp ⟪ Black , proj2 vp ⟫ repl (node kg ⟪ Red , proj2 vg ⟫ n1 (PG.uncle pg)) ≡ rb01 rb10 = begin to-black (node kp vp repl (to-red (node kg vg n1 (PG.uncle pg)))) ≡⟨ cong (λ k → node _ _ k _) (sym eq) ⟩ rb01 ∎ where open ≡-Reasoning rb12 : node kg ⟪ Black , proj2 vg ⟫ (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) (PG.uncle pg) ≡ PG.grand pg rb12 = begin node kg ⟪ Black , proj2 vg ⟫ (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) (PG.uncle pg) ≡⟨ cong₂ (λ j k → node kg j (node kp k (RBI.tree r) n1) (PG.uncle pg) ) rb14 rb13 ⟩ node kg vg _ (PG.uncle pg) ≡⟨ cong (λ k → node _ _ k _) (sym x) ⟩ node kg vg (PG.parent pg) (PG.uncle pg) ≡⟨ sym x₁ ⟩ PG.grand pg ∎ where open ≡-Reasoning rb05 : RBtreeInvariant (PG.uncle pg) rb05 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) rb06 : RBtreeInvariant n1 rb06 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x rb09) rb19 : black-depth n1 ≡ black-depth (PG.uncle pg) rb19 = trans (sym ( proj2 (red-children-eq x (sym (cong proj1 rb13)) rb09) )) (RBtreeEQ (subst (λ k → RBtreeInvariant k) x₁ rb02)) rb18 : black-depth repl ≡ black-depth n1 ⊔ black-depth (PG.uncle pg) rb18 = ? rb17 : suc (black-depth (node rkey vr rp-left rp-right) ⊔ (black-depth n1 ⊔ black-depth (PG.uncle pg))) ≡ black-depth (PG.grand pg) rb17 = ? ... | s2-1sp2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase52 where insertCase52 : t insertCase52 = next (PG.grand pg ∷ PG.rest pg) record { tree = PG.grand pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb02 ; replrb = rb10 ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; state = rebuild rb33 rb17 (subst (λ k → ¬ (k ≡ leaf)) (sym x₁) (λ ())) } rb15 where -- inner case, repl is decomposed -- lt : kp < key -- x : PG.parent pg ≡ node kp vp n1 (RBI.tree r) -- x₁ : PG.grand pg ≡ node kg vg (PG.parent pg) (PG.uncle pg) -- eq : node rkey vr rp-left rp-right ≡ RBI.repl r rb01 : bt (Color ∧ A) rb01 = to-black (node rkey vr (node kp vp n1 rp-left) (to-red (node kg vg rp-right (PG.uncle pg)))) rb04 : kp < key rb04 = lt rb21 : key < kg -- this can be a part of ParentGand relation rb21 = si-property-< (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) (subst (λ k → treeInvariant k ) x₁ (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)))) (subst (λ k → stackInvariant key _ orig (PG.parent pg ∷ k ∷ PG.rest pg)) x₁ (popStackInvariant _ _ _ _ _ rb00)) rb16 : color n1 ≡ Black rb16 = proj1 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) x rb09) (trans (cong color (sym x)) pcolor)) rb13 : ⟪ Red , proj2 vp ⟫ ≡ vp rb13 with subst (λ k → color k ≡ Red ) x pcolor ... | refl = refl rb14 : ⟪ Black , proj2 vg ⟫ ≡ vg rb14 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) (case1 pcolor) ... | refl = refl rb33 : color (PG.grand pg) ≡ Black rb33 = subst (λ k → color k ≡ Black ) (sym x₁) (sym (cong proj1 rb14)) rb23 : vr ≡ ⟪ Red , proj2 vr ⟫ rb23 with subst (λ k → color k ≡ Red ) (sym eq) repl-red ... | refl = refl rb20 : color rp-right ≡ Black rb20 = proj2 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r) ) (cong proj1 rb23)) rb24 : node kg ⟪ Black , proj2 vg ⟫ (node kp ⟪ Red , proj2 vp ⟫ n1 (RBI.tree r)) (PG.uncle pg) ≡ PG.grand pg rb24 = trans (trans (cong₂ (λ j k → node kg j (node kp k n1 (RBI.tree r)) (PG.uncle pg)) rb14 rb13 ) (cong (λ k → node kg vg k (PG.uncle pg)) (sym x))) (sym x₁) rb25 : node rkey ⟪ Black , proj2 vr ⟫ (node kp ⟪ Red , proj2 vp ⟫ n1 rp-left) (node kg ⟪ Red , proj2 vg ⟫ rp-right (PG.uncle pg)) ≡ rb01 rb25 = begin node rkey ⟪ Black , proj2 vr ⟫ (node kp ⟪ Red , proj2 vp ⟫ n1 rp-left) (node kg ⟪ Red , proj2 vg ⟫ rp-right (PG.uncle pg)) ≡⟨ cong (λ k → node _ _ (node kp k n1 rp-left) _ ) rb13 ⟩ node rkey ⟪ Black , proj2 vr ⟫ (node kp vp n1 rp-left) (node kg ⟪ Red , proj2 vg ⟫ rp-right (PG.uncle pg)) ≡⟨ refl ⟩ rb01 ∎ where open ≡-Reasoning rb05 : RBtreeInvariant (PG.uncle pg) rb05 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) rb06 : RBtreeInvariant n1 rb06 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x rb09) rb26 : RBtreeInvariant rp-left rb26 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r)) rb28 : RBtreeInvariant rp-right rb28 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r)) rb31 : RBtreeInvariant (node rkey vr rp-left rp-right ) rb31 = subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r) rb18 : black-depth rp-right ≡ black-depth (PG.uncle pg) rb18 = ? rb27 : black-depth n1 ≡ black-depth rp-left rb27 = ? rb19 : black-depth (node kp vp n1 rp-left) ≡ black-depth rp-right ⊔ black-depth (PG.uncle pg) rb19 = begin black-depth (node kp vp n1 rp-left) ≡⟨ black-depth-resp A (node kp vp n1 rp-left) (node kp vp rp-left rp-left) refl refl refl rb27 refl ⟩ black-depth (node kp vp rp-left rp-left) ≡⟨ black-depth-resp A (node kp vp rp-left rp-left) (node rkey vr rp-left rp-right) refl refl (trans (sym (cong proj1 rb13)) (sym (cong proj1 rb23))) refl (RBtreeEQ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r))) ⟩ black-depth (node rkey vr rp-left rp-right) ≡⟨ black-depth-cong A eq ⟩ black-depth (RBI.repl r) ≡⟨ proj2 (red-children-eq1 (sym eq) repl-red (RBI.replrb r)) ⟩ black-depth rp-right ≡⟨ sym ( ⊔-idem _ ) ⟩ black-depth rp-right ⊔ black-depth rp-right ≡⟨ cong (λ k → black-depth rp-right ⊔ k) rb18 ⟩ black-depth rp-right ⊔ black-depth (PG.uncle pg) ∎ where open ≡-Reasoning rb29 : color n1 ≡ Black rb29 = proj1 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) x rb09) (sym (cong proj1 rb13)) ) rb30 : color rp-left ≡ Black rb30 = proj1 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r)) (cong proj1 rb23)) rb32 : suc (black-depth (PG.uncle pg)) ≡ black-depth (PG.grand pg) rb32 = sym (proj2 ( black-children-eq1 x₁ rb33 rb02 )) rb10 : RBtreeInvariant (node rkey ⟪ Black , proj2 vr ⟫ (node kp vp n1 rp-left) (node kg ⟪ Red , proj2 vg ⟫ rp-right (PG.uncle pg))) rb10 = rb-black _ _ rb19 (rbi-from-red-black _ _ kp vp rb06 rb26 rb27 rb29 rb30 rb13) (rb-red _ _ rb20 uncle-black rb18 rb28 rb05) rb17 : suc (black-depth (node kp vp n1 rp-left) ⊔ (black-depth rp-right ⊔ black-depth (PG.uncle pg))) ≡ black-depth (PG.grand pg) rb17 = begin suc (black-depth (node kp vp n1 rp-left) ⊔ (black-depth rp-right ⊔ black-depth (PG.uncle pg))) ≡⟨ cong (λ k → suc (k ⊔ _)) rb19 ⟩ suc ((black-depth rp-right ⊔ black-depth (PG.uncle pg)) ⊔ (black-depth rp-right ⊔ black-depth (PG.uncle pg))) ≡⟨ cong suc ( ⊔-idem _) ⟩ suc (black-depth rp-right ⊔ black-depth (PG.uncle pg)) ≡⟨ cong (λ k → suc (k ⊔ _)) rb18 ⟩ suc (black-depth (PG.uncle pg) ⊔ black-depth (PG.uncle pg)) ≡⟨ cong suc (⊔-idem _) ⟩ suc (black-depth (PG.uncle pg)) ≡⟨ rb32 ⟩ black-depth (PG.grand pg) ∎ where open ≡-Reasoning ... | s2-s12p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase52 where insertCase52 : t insertCase52 = next (PG.grand pg ∷ PG.rest pg) record { tree = PG.grand pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb02 ; replrb = rb10 ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; state = rebuild rb33 rb17 (subst (λ k → ¬ (k ≡ leaf)) (sym x₁) (λ ())) } rb15 where -- inner case, repl is decomposed -- lt : key < kp -- x : PG.parent pg ≡ node kp vp (RBI.tree r) n1 -- x₁ : PG.grand pg ≡ node kg vg (PG.uncle pg) (PG.parent pg) -- eq : node rkey vr rp-left rp-right ≡ RBI.repl r rb01 : bt (Color ∧ A) rb01 = to-black (node rkey vr (to-red (node kg vg (PG.uncle pg) rp-left )) (node kp vp rp-right n1)) rb04 : key < kp rb04 = lt rb21 : kg < key -- this can be a part of ParentGand relation rb21 = si-property-> (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) (subst (λ k → treeInvariant k ) x₁ (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)))) (subst (λ k → stackInvariant key _ orig (PG.parent pg ∷ k ∷ PG.rest pg)) x₁ (popStackInvariant _ _ _ _ _ rb00)) rb16 : color n1 ≡ Black rb16 = proj2 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) x rb09) (trans (cong color (sym x)) pcolor)) rb13 : ⟪ Red , proj2 vp ⟫ ≡ vp rb13 with subst (λ k → color k ≡ Red ) x pcolor ... | refl = refl rb14 : ⟪ Black , proj2 vg ⟫ ≡ vg rb14 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) (case2 pcolor) ... | refl = refl rb33 : color (PG.grand pg) ≡ Black rb33 = subst (λ k → color k ≡ Black ) (sym x₁) (sym (cong proj1 rb14)) rb23 : vr ≡ ⟪ Red , proj2 vr ⟫ rb23 with subst (λ k → color k ≡ Red ) (sym eq) repl-red ... | refl = refl rb20 : color rp-right ≡ Black rb20 = proj2 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r) ) (cong proj1 rb23)) rb24 : node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) ≡ PG.grand pg rb24 = begin node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) ≡⟨ cong₂ (λ j k → node kg j (PG.uncle pg) (node kp k (RBI.tree r) n1) ) rb14 rb13 ⟩ node kg vg (PG.uncle pg) (node kp vp (RBI.tree r) n1) ≡⟨ cong (λ k → node kg vg (PG.uncle pg) k ) (sym x) ⟩ node kg vg (PG.uncle pg) (PG.parent pg) ≡⟨ sym x₁ ⟩ PG.grand pg ∎ where open ≡-Reasoning rb25 : node rkey ⟪ Black , proj2 vr ⟫ (node kg ⟪ Red , proj2 vg ⟫ (PG.uncle pg) rp-left) (node kp ⟪ Red , proj2 vp ⟫ rp-right n1) ≡ rb01 rb25 = begin node rkey ⟪ Black , proj2 vr ⟫ (node kg ⟪ Red , proj2 vg ⟫ (PG.uncle pg) rp-left) (node kp ⟪ Red , proj2 vp ⟫ rp-right n1) ≡⟨ cong (λ k → node rkey ⟪ Black , proj2 vr ⟫ (node kg ⟪ Red , proj2 vg ⟫ (PG.uncle pg) rp-left) (node kp k rp-right n1)) rb13 ⟩ node rkey ⟪ Black , proj2 vr ⟫ (node kg ⟪ Red , proj2 vg ⟫ (PG.uncle pg) rp-left) (node kp vp rp-right n1) ≡⟨ refl ⟩ rb01 ∎ where open ≡-Reasoning rb05 : RBtreeInvariant (PG.uncle pg) rb05 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) rb06 : RBtreeInvariant n1 rb06 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x rb09) rb26 : RBtreeInvariant rp-left rb26 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r)) rb28 : RBtreeInvariant rp-right rb28 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r)) rb31 : RBtreeInvariant (node rkey vr rp-left rp-right ) rb31 = subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r) rb18 : black-depth (PG.uncle pg) ≡ black-depth rp-left rb18 = ? rb27 : black-depth rp-right ≡ black-depth n1 rb27 = ? rb19 : black-depth (PG.uncle pg) ⊔ black-depth rp-left ≡ black-depth (node kp vp rp-right n1) rb19 = sym ( begin black-depth (node kp vp rp-right n1) ≡⟨ black-depth-resp A (node kp vp rp-right n1) (node kp vp rp-right rp-right) refl refl refl refl (sym rb27) ⟩ black-depth (node kp vp rp-right rp-right) ≡⟨ black-depth-resp A (node kp vp rp-right rp-right) (node rkey vr rp-left rp-right) refl refl (trans (sym (cong proj1 rb13)) (sym (cong proj1 rb23))) (sym (RBtreeEQ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r)))) refl ⟩ black-depth (node rkey vr rp-left rp-right) ≡⟨ black-depth-cong A eq ⟩ black-depth (RBI.repl r) ≡⟨ proj1 (red-children-eq1 (sym eq) repl-red (RBI.replrb r)) ⟩ black-depth rp-left ≡⟨ sym ( ⊔-idem _ ) ⟩ black-depth rp-left ⊔ black-depth rp-left ≡⟨ cong (λ k → k ⊔ black-depth rp-left ) (sym rb18) ⟩ black-depth (PG.uncle pg) ⊔ black-depth rp-left ∎ ) where open ≡-Reasoning rb29 : color n1 ≡ Black rb29 = proj2 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) x rb09) (sym (cong proj1 rb13)) ) rb30 : color rp-left ≡ Black rb30 = proj1 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) (sym eq) (RBI.replrb r)) (cong proj1 rb23)) rb32 : suc (black-depth (PG.uncle pg)) ≡ black-depth (PG.grand pg) rb32 = sym (proj1 ( black-children-eq1 x₁ rb33 rb02 )) rb10 : RBtreeInvariant (node rkey ⟪ Black , proj2 vr ⟫ (node kg ⟪ Red , proj2 vg ⟫ (PG.uncle pg) rp-left ) (node kp vp rp-right n1) ) rb10 = rb-black _ _ rb19 (rb-red _ _ uncle-black rb30 rb18 rb05 rb26 ) (rbi-from-red-black _ _ _ _ rb28 rb06 rb27 rb20 rb16 rb13 ) rb17 : suc (black-depth (PG.uncle pg) ⊔ black-depth rp-left ⊔ black-depth (node kp vp rp-right n1)) ≡ black-depth (PG.grand pg) rb17 = begin suc (black-depth (PG.uncle pg) ⊔ black-depth rp-left ⊔ black-depth (node kp vp rp-right n1)) ≡⟨ cong (λ k → suc (_ ⊔ k)) (sym rb19) ⟩ suc (black-depth (PG.uncle pg) ⊔ black-depth rp-left ⊔ (black-depth (PG.uncle pg) ⊔ black-depth rp-left)) ≡⟨ cong suc ( ⊔-idem _) ⟩ suc (black-depth (PG.uncle pg) ⊔ black-depth rp-left ) ≡⟨ cong (λ k → suc (_ ⊔ k)) (sym rb18) ⟩ suc (black-depth (PG.uncle pg) ⊔ black-depth (PG.uncle pg)) ≡⟨ cong suc (⊔-idem _) ⟩ suc (black-depth (PG.uncle pg)) ≡⟨ rb32 ⟩ black-depth (PG.grand pg) ∎ where open ≡-Reasoning ... | s2-1s2p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase52 where insertCase52 : t insertCase52 = next (PG.grand pg ∷ PG.rest pg) record { tree = PG.grand pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb02 ; replrb = subst (λ k → RBtreeInvariant k) rb10 (rb-black _ _ rb18 (rb-red _ _ uncle-black rb16 rb19 rb05 rb06) (RBI.replrb r) ) ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; state = rebuild rb33 rb17 (subst (λ k → ¬ (k ≡ leaf)) (sym x₁) (λ ())) } rb15 where -- outer case, repl is not decomposed -- lt : kp < key -- x : PG.parent pg ≡ node kp vp n1 (RBI.tree r) -- x₁ : PG.grand pg ≡ node kg vg (PG.uncle pg) (PG.parent pg) rb01 : bt (Color ∧ A) rb01 = to-black (node kp vp (to-red (node kg vg (PG.uncle pg) n1) )(node rkey vr rp-left rp-right)) rb04 : kp < key rb04 = lt rb16 : color n1 ≡ Black rb16 = proj1 (RBtreeChildrenColorBlack _ _ (subst (λ k → RBtreeInvariant k) x rb09) (trans (cong color (sym x)) pcolor)) rb13 : ⟪ Red , proj2 vp ⟫ ≡ vp rb13 with subst (λ k → color k ≡ Red ) x pcolor ... | refl = refl rb14 : ⟪ Black , proj2 vg ⟫ ≡ vg rb14 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) (case2 pcolor) ... | refl = refl rb33 : color (PG.grand pg) ≡ Black rb33 = subst (λ k → color k ≡ Black ) (sym x₁) (sym (cong proj1 rb14)) rb10 : node kp ⟪ Black , proj2 vp ⟫ (node kg ⟪ Red , proj2 vg ⟫ (PG.uncle pg) n1 ) repl ≡ rb01 rb10 = cong (λ k → node _ _ _ k ) (sym eq) rb12 : node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ n1 (RBI.tree r)) ≡ PG.grand pg rb12 = begin node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ n1 (RBI.tree r)) ≡⟨ cong₂ (λ j k → node kg j (PG.uncle pg) (node kp k n1 (RBI.tree r) ) ) rb14 rb13 ⟩ node kg vg (PG.uncle pg) _ ≡⟨ cong (λ k → node _ _ _ k) (sym x) ⟩ node kg vg (PG.uncle pg) (PG.parent pg) ≡⟨ sym x₁ ⟩ PG.grand pg ∎ where open ≡-Reasoning rb05 : RBtreeInvariant (PG.uncle pg) rb05 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb02) rb06 : RBtreeInvariant n1 rb06 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x rb09) rb19 : black-depth (PG.uncle pg) ≡ black-depth n1 rb19 = sym (trans (sym ( proj1 (red-children-eq x (sym (cong proj1 rb13)) rb09) )) (sym (RBtreeEQ (subst (λ k → RBtreeInvariant k) x₁ rb02)))) rb18 : black-depth (PG.uncle pg) ⊔ black-depth n1 ≡ black-depth repl rb18 = ? rb17 : suc (black-depth (PG.uncle pg) ⊔ black-depth n1 ⊔ black-depth (node rkey vr rp-left rp-right)) ≡ black-depth (PG.grand pg) rb17 = ? replaceRBP1 : t replaceRBP1 with RBI.state r ... | rebuild ceq bdepth-eq ¬leaf = rebuildCase ceq bdepth-eq ¬leaf ... | top-black eq = exit stack (trans eq (cong (λ k → k ∷ []) rb00)) r where rb00 : RBI.tree r ≡ orig rb00 with si-property-last _ _ _ _ (subst (λ k → stackInvariant key (RBI.tree r) orig k) (eq) (RBI.si r)) ... | refl = refl ... | rotate _ repl-red with stackToPG (RBI.tree r) orig stack (RBI.si r) ... | case1 eq = exit stack eq r -- no stack, replace top node ... | case2 (case1 eq) = insertCase4 orig refl eq (case1 repl-red) (RBI.si r) -- one level stack, orig is parent of repl ... | case2 (case2 pg) with color (PG.parent pg) in pcolor ... | Black = insertCase1 where -- Parent is Black, make color repl ≡ color tree then goto rebuildCase rb00 : (pg : PG (Color ∧ A) key (RBI.tree r) stack) → stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) rb00 pg = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r) treerb : (pg : PG (Color ∧ A) key (RBI.tree r) stack) → RBtreeInvariant (PG.parent pg) treerb pg = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ (rb00 pg)) rb08 : (pg : PG (Color ∧ A) key (RBI.tree r) stack) → treeInvariant (PG.parent pg) rb08 pg = siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ (rb00 pg)) insertCase1 : t insertCase1 with PG.pg pg ... | s2-s1p2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb06 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) (rb05 (trans (sym (cong color x)) pcolor)) (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where rb01 : bt (Color ∧ A) rb01 = node kp vp repl n1 rb03 : key < kp rb03 = lt rb04 : ⟪ Black , proj2 vp ⟫ ≡ vp rb04 with subst (λ k → color k ≡ Black) x pcolor ... | refl = refl rb07 : black-depth repl ≡ black-depth n1 rb07 = ? rb05 : proj1 vp ≡ Black → black-depth rb01 ≡ black-depth (PG.parent pg) rb05 refl = ? rb06 : RBtreeInvariant rb01 rb06 = subst (λ k → RBtreeInvariant (node kp k repl n1)) rb04 ( rb-black _ _ rb07 (RBI.replrb r) (RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x (treerb pg)))) ... | s2-1sp2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb06 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) (rb05 (trans (sym (cong color x)) pcolor)) (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where --- x : PG.parent pg ≡ node kp vp n1 (RBI.tree r) --- x₁ : PG.grand pg ≡ node kg vg (PG.parent pg) (PG.uncle pg) rb01 : bt (Color ∧ A) rb01 = node kp vp n1 repl rb03 : kp < key rb03 = lt rb04 : ⟪ Black , proj2 vp ⟫ ≡ vp rb04 with subst (λ k → color k ≡ Black) x pcolor ... | refl = refl rb07 : black-depth repl ≡ black-depth n1 rb07 = ? rb05 : proj1 vp ≡ Black → black-depth rb01 ≡ black-depth (PG.parent pg) rb05 refl = ? rb06 : RBtreeInvariant rb01 rb06 = subst (λ k → RBtreeInvariant (node kp k n1 repl )) rb04 ( rb-black _ _ (sym rb07) (RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x (treerb pg))) (RBI.replrb r) ) insertCase1 | s2-s12p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb06 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) (rb05 (trans (sym (cong color x)) pcolor)) (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where rb01 : bt (Color ∧ A) rb01 = node kp vp repl n1 rb03 : key < kp rb03 = lt rb04 : ⟪ Black , proj2 vp ⟫ ≡ vp rb04 with subst (λ k → color k ≡ Black) x pcolor ... | refl = refl rb07 : black-depth repl ≡ black-depth n1 rb07 = ? rb05 : proj1 vp ≡ Black → black-depth rb01 ≡ black-depth (PG.parent pg) rb05 refl = ? rb06 : RBtreeInvariant rb01 rb06 = subst (λ k → RBtreeInvariant (node kp k repl n1)) rb04 ( rb-black _ _ rb07 (RBI.replrb r) (RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x (treerb pg)))) insertCase1 | s2-1s2p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = next (PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) record { tree = PG.parent pg ; repl = rb01 ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = treerb pg ; replrb = rb06 ; si = popStackInvariant _ _ _ _ _ (rb00 pg) ; state = rebuild (cong color x) (rb05 (trans (sym (cong color x)) pcolor)) (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) } (subst₂ (λ j k → j < length k ) refl (sym (PG.stack=gp pg)) ≤-refl) where -- x : PG.parent pg ≡ node kp vp (RBI.tree r) n1 -- x₁ : PG.grand pg ≡ node kg vg (PG.uncle pg) (PG.parent pg) rb01 : bt (Color ∧ A) rb01 = node kp vp n1 repl rb03 : kp < key rb03 = lt rb04 : ⟪ Black , proj2 vp ⟫ ≡ vp rb04 with subst (λ k → color k ≡ Black) x pcolor ... | refl = refl rb07 : black-depth repl ≡ black-depth n1 rb07 = ? rb05 : proj1 vp ≡ Black → black-depth rb01 ≡ black-depth (PG.parent pg) rb05 refl = ? rb06 : RBtreeInvariant rb01 rb06 = subst (λ k → RBtreeInvariant (node kp k n1 repl )) rb04 ( rb-black _ _ (sym rb07) (RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x (treerb pg))) (RBI.replrb r) ) ... | Red with PG.uncle pg in uneq ... | leaf = insertCase5 repl refl pg repl-red (cong color uneq) pcolor ... | node key₁ ⟪ Black , value₁ ⟫ t₁ t₂ = insertCase5 repl refl pg repl-red (cong color uneq) pcolor ... | node key₁ ⟪ Red , value₁ ⟫ t₁ t₂ with PG.pg pg -- insertCase2 : uncle and parent are Red, flip color and go up by two level ... | s2-s1p2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase2 where insertCase2 : t insertCase2 = next (PG.grand pg ∷ (PG.rest pg)) record { tree = PG.grand pg ; repl = to-red (node kg vg (to-black (node kp vp repl n1)) (to-black (PG.uncle pg))) ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb01 ; replrb = rb-red _ _ refl (RBtreeToBlackColor _ rb02) rb12 (rb-black _ _ rb13 (RBI.replrb r) rb04) (RBtreeToBlack _ rb02) ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; state = rotate _ refl } rb15 where rb00 : stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) rb00 = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r) rb01 : RBtreeInvariant (PG.grand pg) rb01 = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)) rb02 : RBtreeInvariant (PG.uncle pg) rb02 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb03 : RBtreeInvariant (PG.parent pg) rb03 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb04 : RBtreeInvariant n1 rb04 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x rb03) rb05 : { tree : bt (Color ∧ A) } → tree ≡ PG.uncle pg → tree ≡ node key₁ ⟪ Red , value₁ ⟫ t₁ t₂ → color (PG.uncle pg) ≡ Red rb05 refl refl = refl rb08 : treeInvariant (PG.parent pg) rb08 = siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00) rb07 : treeInvariant (node kp vp (RBI.tree r) n1) rb07 = subst (λ k → treeInvariant k) x (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00)) rb06 : key < kp rb06 = lt rb10 : vg ≡ ⟪ Black , proj2 vg ⟫ rb10 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) (case1 pcolor) ... | refl = refl rb11 : vp ≡ ⟪ Red , proj2 vp ⟫ rb11 with subst (λ k → color k ≡ Red) x pcolor ... | refl = refl rb09 : PG.grand pg ≡ node kg ⟪ Black , proj2 vg ⟫ (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) (PG.uncle pg) rb09 = begin PG.grand pg ≡⟨ x₁ ⟩ node kg vg (PG.parent pg) (PG.uncle pg) ≡⟨ cong (λ k → node kg vg k (PG.uncle pg)) x ⟩ node kg vg (node kp vp (RBI.tree r) n1) (PG.uncle pg) ≡⟨ cong₂ (λ j k → node kg j (node kp k (RBI.tree r) n1) (PG.uncle pg)) rb10 rb11 ⟩ node kg ⟪ Black , proj2 vg ⟫ (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) (PG.uncle pg) ∎ where open ≡-Reasoning rb13 : black-depth repl ≡ black-depth n1 rb13 = ? rb12 : suc (black-depth repl ⊔ black-depth n1) ≡ black-depth (to-black (PG.uncle pg)) rb12 = ? rb15 : suc (length (PG.rest pg)) < length stack rb15 = begin suc (suc (length (PG.rest pg))) ≤⟨ <-trans (n<1+n _) (n<1+n _) ⟩ length (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) ≡⟨ cong length (sym (PG.stack=gp pg)) ⟩ length stack ∎ where open ≤-Reasoning ... | s2-1sp2 {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase2 where insertCase2 : t insertCase2 = next (PG.grand pg ∷ (PG.rest pg)) record { tree = PG.grand pg ; repl = to-red (node kg vg (to-black (node kp vp n1 repl)) (to-black (PG.uncle pg)) ) ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb01 ; replrb = rb-red _ _ refl (RBtreeToBlackColor _ rb02) rb12 (rb-black _ _ (sym rb13) rb04 (RBI.replrb r)) (RBtreeToBlack _ rb02) ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; state = rotate _ refl } rb15 where -- -- lt : kp < key -- x : PG.parent pg ≡ node kp vp n1 (RBI.tree r) --- x₁ : PG.grand pg ≡ node kg vg (PG.parent pg) (PG.uncle pg) -- rb00 : stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) rb00 = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r) rb01 : RBtreeInvariant (PG.grand pg) rb01 = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)) rb02 : RBtreeInvariant (PG.uncle pg) rb02 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb03 : RBtreeInvariant (PG.parent pg) rb03 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb04 : RBtreeInvariant n1 rb04 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x rb03) rb05 : { tree : bt (Color ∧ A) } → tree ≡ PG.uncle pg → tree ≡ node key₁ ⟪ Red , value₁ ⟫ t₁ t₂ → color (PG.uncle pg) ≡ Red rb05 refl refl = refl rb08 : treeInvariant (PG.parent pg) rb08 = siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00) rb07 : treeInvariant (node kp vp n1 (RBI.tree r) ) rb07 = subst (λ k → treeInvariant k) x (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00)) rb06 : kp < key rb06 = lt rb21 : key < kg -- this can be a part of ParentGand relation rb21 = si-property-< (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) (subst (λ k → treeInvariant k ) x₁ (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)))) (subst (λ k → stackInvariant key _ orig (PG.parent pg ∷ k ∷ PG.rest pg)) x₁ (popStackInvariant _ _ _ _ _ rb00)) rb10 : vg ≡ ⟪ Black , proj2 vg ⟫ rb10 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) (case1 pcolor) ... | refl = refl rb11 : vp ≡ ⟪ Red , proj2 vp ⟫ rb11 with subst (λ k → color k ≡ Red) x pcolor ... | refl = refl rb09 : node kg ⟪ Black , proj2 vg ⟫ (node kp ⟪ Red , proj2 vp ⟫ n1 (RBI.tree r) ) (PG.uncle pg) ≡ PG.grand pg rb09 = sym ( begin PG.grand pg ≡⟨ x₁ ⟩ node kg vg (PG.parent pg) (PG.uncle pg) ≡⟨ cong (λ k → node kg vg k (PG.uncle pg)) x ⟩ node kg vg (node kp vp n1 (RBI.tree r) ) (PG.uncle pg) ≡⟨ cong₂ (λ j k → node kg j (node kp k n1 (RBI.tree r) ) (PG.uncle pg) ) rb10 rb11 ⟩ node kg ⟪ Black , proj2 vg ⟫ (node kp ⟪ Red , proj2 vp ⟫ n1 (RBI.tree r) ) (PG.uncle pg) ∎ ) where open ≡-Reasoning rb13 : black-depth repl ≡ black-depth n1 rb13 = ? rb12 : suc (black-depth n1 ⊔ black-depth repl) ≡ black-depth (to-black (PG.uncle pg)) rb12 = ? rb15 : suc (length (PG.rest pg)) < length stack rb15 = begin suc (suc (length (PG.rest pg))) ≤⟨ <-trans (n<1+n _) (n<1+n _) ⟩ length (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) ≡⟨ cong length (sym (PG.stack=gp pg)) ⟩ length stack ∎ where open ≤-Reasoning ... | s2-s12p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase2 where insertCase2 : t insertCase2 = next (PG.grand pg ∷ (PG.rest pg)) record { tree = PG.grand pg ; repl = to-red (node kg vg (to-black (PG.uncle pg)) (to-black (node kp vp repl n1)) ) ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb01 ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; replrb = rb-red _ _ (RBtreeToBlackColor _ rb02) refl rb12 (RBtreeToBlack _ rb02) (rb-black _ _ rb13 (RBI.replrb r) rb04) ; state = rotate _ refl } rb15 where -- x : PG.parent pg ≡ node kp vp (RBI.tree r) n1 -- x₁ : PG.grand pg ≡ node kg vg (PG.uncle pg) (PG.parent pg) rb00 : stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) rb00 = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r) rb01 : RBtreeInvariant (PG.grand pg) rb01 = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)) rb02 : RBtreeInvariant (PG.uncle pg) rb02 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb03 : RBtreeInvariant (PG.parent pg) rb03 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb04 : RBtreeInvariant n1 rb04 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x rb03) rb05 : { tree : bt (Color ∧ A) } → tree ≡ PG.uncle pg → tree ≡ node key₁ ⟪ Red , value₁ ⟫ t₁ t₂ → color (PG.uncle pg) ≡ Red rb05 refl refl = refl rb08 : treeInvariant (PG.parent pg) rb08 = siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00) rb07 : treeInvariant (node kp vp (RBI.tree r) n1) rb07 = subst (λ k → treeInvariant k) x (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00)) rb06 : key < kp rb06 = lt rb21 : kg < key -- this can be a part of ParentGand relation rb21 = si-property-> (subst (λ k → ¬ (k ≡ leaf)) (sym x) (λ ())) (subst (λ k → treeInvariant k ) x₁ (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)))) (subst (λ k → stackInvariant key _ orig (PG.parent pg ∷ k ∷ PG.rest pg)) x₁ (popStackInvariant _ _ _ _ _ rb00)) rb10 : vg ≡ ⟪ Black , proj2 vg ⟫ rb10 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) (case2 pcolor) ... | refl = refl rb11 : vp ≡ ⟪ Red , proj2 vp ⟫ rb11 with subst (λ k → color k ≡ Red) x pcolor ... | refl = refl rb09 : node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) ≡ PG.grand pg rb09 = sym ( begin PG.grand pg ≡⟨ x₁ ⟩ node kg vg (PG.uncle pg) (PG.parent pg) ≡⟨ cong (λ k → node kg vg (PG.uncle pg) k) x ⟩ node kg vg (PG.uncle pg) (node kp vp (RBI.tree r) n1) ≡⟨ cong₂ (λ j k → node kg j (PG.uncle pg) (node kp k (RBI.tree r) n1) ) rb10 rb11 ⟩ node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ (RBI.tree r) n1) ∎ ) where open ≡-Reasoning rb13 : black-depth repl ≡ black-depth n1 rb13 = ? rb12 : black-depth (to-black (PG.uncle pg)) ≡ suc (black-depth repl ⊔ black-depth n1) rb12 = ? rb17 : suc (black-depth repl ⊔ black-depth n1) ⊔ black-depth (to-black (PG.uncle pg)) ≡ black-depth (PG.grand pg) rb17 = begin suc (black-depth repl ⊔ black-depth n1) ⊔ black-depth (to-black (PG.uncle pg)) ≡⟨ cong (λ k → k ⊔ black-depth (to-black (PG.uncle pg))) (sym rb12) ⟩ black-depth (to-black (PG.uncle pg)) ⊔ black-depth (to-black (PG.uncle pg)) ≡⟨ ⊔-idem _ ⟩ black-depth (to-black (PG.uncle pg)) ≡⟨ sym (to-black-eq (PG.uncle pg) (cong color uneq )) ⟩ suc (black-depth (PG.uncle pg)) ≡⟨ sym ( proj1 (black-children-eq x₁ (cong proj1 rb10) rb01 )) ⟩ black-depth (PG.grand pg) ∎ where open ≡-Reasoning rb15 : suc (length (PG.rest pg)) < length stack rb15 = begin suc (suc (length (PG.rest pg))) ≤⟨ <-trans (n<1+n _) (n<1+n _) ⟩ length (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) ≡⟨ cong length (sym (PG.stack=gp pg)) ⟩ length stack ∎ where open ≤-Reasoning ... | s2-1s2p {kp} {kg} {vp} {vg} {n1} {n2} lt x x₁ = insertCase2 where --- lt : kp < key --- x : PG.parent pg ≡ node kp vp n1 (RBI.tree r) --- x₁ : PG.grand pg ≡ node kg vg (PG.uncle pg) (PG.parent pg) insertCase2 : t insertCase2 = next (PG.grand pg ∷ (PG.rest pg)) record { tree = PG.grand pg ; repl = to-red (node kg vg (to-black (PG.uncle pg)) (to-black (node kp vp n1 repl )) ) ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = rb01 ; replrb = rb-red _ _ (RBtreeToBlackColor _ rb02) refl rb12 (RBtreeToBlack _ rb02) (rb-black _ _ (sym rb13) rb04 (RBI.replrb r) ) ; si = popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00) ; state = rotate _ refl } rb15 where rb00 : stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) rb00 = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r) rb01 : RBtreeInvariant (PG.grand pg) rb01 = PGtoRBinvariant1 _ orig (RBI.origrb r) _ (popStackInvariant _ _ _ _ _ (popStackInvariant _ _ _ _ _ rb00)) rb02 : RBtreeInvariant (PG.uncle pg) rb02 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb03 : RBtreeInvariant (PG.parent pg) rb03 = RBtreeRightDown _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) rb04 : RBtreeInvariant n1 rb04 = RBtreeLeftDown _ _ (subst (λ k → RBtreeInvariant k) x rb03) rb05 : { tree : bt (Color ∧ A) } → tree ≡ PG.uncle pg → tree ≡ node key₁ ⟪ Red , value₁ ⟫ t₁ t₂ → color (PG.uncle pg) ≡ Red rb05 refl refl = refl rb08 : treeInvariant (PG.parent pg) rb08 = siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00) rb07 : treeInvariant (node kp vp n1 (RBI.tree r) ) rb07 = subst (λ k → treeInvariant k) x (siToTreeinvariant _ _ _ _ (RBI.origti r) (popStackInvariant _ _ _ _ _ rb00)) rb06 : kp < key rb06 = lt rb10 : vg ≡ ⟪ Black , proj2 vg ⟫ rb10 with RBtreeParentColorBlack _ _ (subst (λ k → RBtreeInvariant k) x₁ rb01) (case2 pcolor) ... | refl = refl rb11 : vp ≡ ⟪ Red , proj2 vp ⟫ rb11 with subst (λ k → color k ≡ Red) x pcolor ... | refl = refl rb09 : PG.grand pg ≡ node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ n1 (RBI.tree r) ) rb09 = begin PG.grand pg ≡⟨ x₁ ⟩ node kg vg (PG.uncle pg) (PG.parent pg) ≡⟨ cong (λ k → node kg vg (PG.uncle pg) k) x ⟩ node kg vg (PG.uncle pg) (node kp vp n1 (RBI.tree r) ) ≡⟨ cong₂ (λ j k → node kg j (PG.uncle pg) (node kp k n1 (RBI.tree r) ) ) rb10 rb11 ⟩ node kg ⟪ Black , proj2 vg ⟫ (PG.uncle pg) (node kp ⟪ Red , proj2 vp ⟫ n1 (RBI.tree r) ) ∎ where open ≡-Reasoning rb13 : black-depth repl ≡ black-depth n1 rb13 = ? rb12 : black-depth (to-black (PG.uncle pg)) ≡ suc (black-depth n1 ⊔ black-depth repl) rb12 = ? rb15 : suc (length (PG.rest pg)) < length stack rb15 = begin suc (suc (length (PG.rest pg))) ≤⟨ <-trans (n<1+n _) (n<1+n _) ⟩ length (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg) ≡⟨ cong length (sym (PG.stack=gp pg)) ⟩ length stack ∎ where open ≤-Reasoning insertRBTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt (Color ∧ A)) → (key : ℕ) → (value : A) → treeInvariant tree → RBtreeInvariant tree → (exit : (stack : List (bt (Color ∧ A))) → stack ≡ (tree ∷ []) → RBI key value tree stack → t ) → t insertRBTreeP {n} {m} {A} {t} orig key value ti rbi exit = TerminatingLoopS (bt (Color ∧ A) ∧ List (bt (Color ∧ A))) {λ p → RBtreeInvariant (proj1 p) ∧ stackInvariant key (proj1 p) orig (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ orig , orig ∷ [] ⟫ ⟪ rbi , s-nil ⟫ $ λ p RBP findLoop → findRBT key (proj1 p) orig (proj2 p) RBP (λ t1 s1 P2 lt1 → findLoop ⟪ t1 , s1 ⟫ P2 lt1 ) $ λ tr1 st P2 O → createRBT key value orig rbi ti tr1 st (proj2 P2) O $ λ st rbi → TerminatingLoopS (List (bt (Color ∧ A))) {λ stack → RBI key value orig stack } (λ stack → length stack ) st rbi $ λ stack rbi replaceLoop → replaceRBP key value orig stack rbi (λ stack1 rbi1 lt → replaceLoop stack1 rbi1 lt ) $ λ stack eq r → exit stack eq r