Mercurial > hg > Gears > GearsAgda
view hoareBinaryTree1.agda @ 815:e22ebb0f00a3
add replaceRBTNode
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 24 Jan 2024 10:36:44 +0900 |
| parents | 82029d2a8970 |
| children | a16f0b2ce509 |
line wrap: on
line source
module hoareBinaryTree1 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 ) open import Data.Unit hiding ( _≟_ ) -- ; _≤?_ ; _≤_) data treeInvariant {n : Level} {A : Set n} : (tree : bt A) → Set n where t-leaf : treeInvariant leaf t-single : (key : ℕ) → (value : A) → treeInvariant (node key value leaf leaf) t-right : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → key < key₁ → treeInvariant (node key₁ value₁ t₁ t₂) → treeInvariant (node key value leaf (node key₁ value₁ t₁ t₂)) t-left : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → key < key₁ → treeInvariant (node key value t₁ t₂) → treeInvariant (node key₁ value₁ (node key value t₁ t₂) leaf ) t-node : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} → key < key₁ → key₁ < key₂ → treeInvariant (node key value t₁ t₂) → treeInvariant (node key₂ value₂ t₃ t₄) → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) -- -- stack always contains original top at end (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) data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (before after : bt A ) → Set n where r-leaf : replacedTree key value leaf (node key value leaf leaf) r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁) r-right : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} → k < key → replacedTree key value t2 t → replacedTree key value (node k v1 t1 t2) (node k v1 t1 t) r-left : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} → key < k → replacedTree key value t1 t → replacedTree key value (node k v1 t1 t2) (node k v1 t t2) add< : { i : ℕ } (j : ℕ ) → i < suc i + j add< {i} j = begin suc i ≤⟨ m≤m+n (suc i) j ⟩ suc i + j ∎ where open ≤-Reasoning treeTest1 : bt ℕ treeTest1 = node 0 0 leaf (node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf)) treeTest2 : bt ℕ treeTest2 = node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf) treeInvariantTest1 : treeInvariant treeTest1 treeInvariantTest1 = t-right (m≤m+n _ 2) (t-node (add< 0) (add< 1) (t-left (add< 0) (t-single 1 7)) (t-single 5 5) ) stack-top : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) stack-top [] = nothing stack-top (x ∷ s) = just x stack-last : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) stack-last [] = nothing stack-last (x ∷ []) = just x stack-last (x ∷ s) = stack-last s stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] ) stackInvariantTest1 = s-right _ _ _ (add< 3) (s-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-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 rt-property1 : {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf ) rt-property1 {n} {A} key value .leaf .(node key value leaf leaf) r-leaf () rt-property1 {n} {A} key value .(node key _ _ _) .(node key value _ _) r-node () rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-right x rt) = λ () rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-left x rt) = λ () rt-property-leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {repl : bt A} → replacedTree key value leaf repl → repl ≡ node key value leaf leaf rt-property-leaf r-leaf = refl rt-property-¬leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {tree : bt A} → ¬ replacedTree key value tree leaf rt-property-¬leaf () rt-property-key : {n : Level} {A : Set n} {key key₂ key₃ : ℕ} {value value₂ value₃ : A} {left left₁ right₂ right₃ : bt A} → replacedTree key value (node key₂ value₂ left right₂) (node key₃ value₃ left₁ right₃) → key₂ ≡ key₃ rt-property-key r-node = refl rt-property-key (r-right x ri) = refl rt-property-key (r-left x ri) = refl nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x nat-<> : { x y : ℕ } → x < y → y < x → ⊥ nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x open _∧_ depth-1< : {i j : ℕ} → suc i ≤ suc (i Data.Nat.⊔ j ) depth-1< {i} {j} = s≤s (m≤m⊔n _ j) depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i ) depth-2< {i} {j} = s≤s (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 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A)) → treeInvariant tree ∧ stackInvariant key tree tree0 stack → (next : (tree1 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → bt-depth tree1 < bt-depth tree → t ) → (exit : (tree1 : bt A) → (stack : List (bt A)) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t findP key leaf tree0 st Pre _ exit = exit leaf st Pre (case1 refl) findP key (node key₁ v1 tree tree₁) tree0 st Pre next exit with <-cmp key key₁ findP key n tree0 st Pre _ exit | tri≈ ¬a refl ¬c = exit n st Pre (case2 refl) findP {n} {_} {A} key (node key₁ v1 tree tree₁) tree0 st Pre next _ | tri< a ¬b ¬c = next tree (tree ∷ st) ⟪ treeLeftDown tree tree₁ (proj1 Pre) , findP1 a st (proj2 Pre) ⟫ depth-1< where findP1 : key < key₁ → (st : List (bt A)) → stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (tree ∷ st) findP1 a (x ∷ st) si = s-left _ _ _ a si findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) ⟪ treeRightDown tree tree₁ (proj1 Pre) , s-right _ _ _ c (proj2 Pre) ⟫ depth-2< replaceTree1 : {n : Level} {A : Set n} {t t₁ : bt A } → ( k : ℕ ) → (v1 value : A ) → treeInvariant (node k v1 t t₁) → treeInvariant (node k value t t₁) replaceTree1 k v1 value (t-single .k .v1) = t-single k value replaceTree1 k v1 value (t-right x t) = t-right x t replaceTree1 k v1 value (t-left x t) = t-left x t replaceTree1 k v1 value (t-node x x₁ t t₁) = t-node x x₁ t t₁ open import Relation.Binary.Definitions lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥ lemma3 refl () lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥ lemma5 (s≤s z≤n) () ¬x<x : {x : ℕ} → ¬ (x < x) ¬x<x (s≤s lt) = ¬x<x lt child-replaced : {n : Level} {A : Set n} (key : ℕ) (tree : bt A) → bt A child-replaced key leaf = leaf child-replaced key (node key₁ value left right) with <-cmp key key₁ ... | tri< a ¬b ¬c = left ... | tri≈ ¬a b ¬c = node key₁ value left right ... | tri> ¬a ¬b c = right record replacePR {n : Level} {A : Set n} (key : ℕ) (value : A) (tree repl : bt A ) (stack : List (bt A)) (C : bt A → bt A → List (bt A) → Set n) : Set n where field tree0 : bt A ti : treeInvariant tree0 si : stackInvariant key tree tree0 stack ri : replacedTree key value (child-replaced key tree ) repl ci : C tree repl stack -- data continuation 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 ri : replacedTree key value (child-replaced key tree) repl -- 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 → replacedTree key value (child-replaced key tree) tree1 → t) → t replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P) (subst (λ j → replacedTree k v1 j (node k v1 t t₁) ) repl00 r-node) where repl00 : node k value t t₁ ≡ child-replaced k (node k value t t₁) repl00 with <-cmp k k ... | tri< a ¬b ¬c = ⊥-elim (¬b refl) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim (¬b refl) replaceP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → {tree : bt A} ( repl : bt A) → (stack : List (bt A)) → replacePR key value tree repl stack (λ _ _ _ → Lift n ⊤) → (next : ℕ → A → {tree1 : bt A } (repl : bt A) → (stack1 : List (bt A)) → replacePR key value tree1 repl stack1 (λ _ _ _ → Lift n ⊤) → length stack1 < length stack → t) → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t replaceP key value {tree} repl [] Pre next exit = ⊥-elim ( si-property0 (replacePR.si Pre) refl ) -- can't happen replaceP key value {tree} repl (leaf ∷ []) Pre next exit with si-property-last _ _ _ _ (replacePR.si Pre)-- tree0 ≡ leaf ... | refl = exit (replacePR.tree0 Pre) (node key value leaf leaf) ⟪ replacePR.ti Pre , r-leaf ⟫ replaceP key value {tree} repl (node key₁ value₁ left right ∷ []) Pre next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = exit (replacePR.tree0 Pre) (node key₁ value₁ repl right ) ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ repl right ) repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) repl01 | refl | refl = subst (λ k → replacedTree key value (node key₁ value₁ k right ) (node key₁ value₁ repl right )) repl02 (r-left a repl03) where repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl repl03 = replacePR.ri Pre repl02 : child-replaced key (node key₁ value₁ left right) ≡ left repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a a) ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a) ... | tri≈ ¬a b ¬c = exit (replacePR.tree0 Pre) repl ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) repl repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) repl01 | refl | refl = subst (λ k → replacedTree key value k repl) repl02 (replacePR.ri Pre) where repl02 : child-replaced key (node key₁ value₁ left right) ≡ node key₁ value₁ left right repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b) ... | tri> ¬a ¬b c = exit (replacePR.tree0 Pre) (node key₁ value₁ left repl ) ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ left repl ) repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) repl01 | refl | refl = subst (λ k → replacedTree key value (node key₁ value₁ left k ) (node key₁ value₁ left repl )) repl02 (r-right c repl03) where repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl repl03 = replacePR.ri Pre repl02 : child-replaced key (node key₁ value₁ left right) ≡ right repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim ( ¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬c c) ... | tri> ¬a ¬b c = refl replaceP {n} {_} {A} key value {tree} repl (leaf ∷ st@(tree1 ∷ st1)) Pre next exit = next key value repl st Post ≤-refl where Post : replacePR key value tree1 repl (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ leaf repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ leaf) ≡ leaf repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = ⊥-elim (¬c x) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) ... | tri> ¬a ¬b c = refl repl12 : replacedTree key value (child-replaced key tree1 ) repl repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf ... | s-left _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 leaf tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 leaf tree₁ ) ≡ leaf repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) ... | tri> ¬a ¬b c = ⊥-elim (¬a x) repl12 : replacedTree key value (child-replaced key tree1 ) repl repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07) ) (sym (rt-property-leaf (replacePR.ri Pre ))) r-leaf -- repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf replaceP {n} {_} {A} key value {tree} repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value₁ repl right ) st (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ left repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) ... | refl | tri> ¬a ¬b c = refl repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ left repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = refl ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value left right ) (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 b ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ tree -- (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ tree) ≡ tree repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = ⊥-elim (¬c x) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) ... | tri> ¬a ¬b c = refl repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) repl12 refl with repl09 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node ... | s-left _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 b ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree tree₁ ) ≡ tree repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) ... | tri> ¬a ¬b c = ⊥-elim (¬a x) repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) repl12 refl with repl09 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value₁ left repl ) st (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ right repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) ... | tri> ¬a ¬b c = refl repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) ... | refl | tri> ¬a ¬b c = refl repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ right repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) ... | tri> ¬a ¬b c = refl repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = refl ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) → (r : Index) → (p : Invraiant r) → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r) ... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) ... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (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 _∧_ RTtoTI0 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → replacedTree key value tree repl → treeInvariant repl RTtoTI0 .leaf .(node key value leaf leaf) key value ti r-leaf = t-single key value RTtoTI0 .(node key _ leaf leaf) .(node key value leaf leaf) key value (t-single .key _) r-node = t-single key value RTtoTI0 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right x ti) r-node = t-right x ti RTtoTI0 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left x ti) r-node = t-left x ti RTtoTI0 .(node key _ (node _ _ _ _) (node _ _ _ _)) .(node key value (node _ _ _ _) (node _ _ _ _)) key value (t-node x x₁ ti ti₁) r-node = t-node x x₁ ti ti₁ -- r-right case RTtoTI0 (node _ _ leaf leaf) (node _ _ leaf .(node key value leaf leaf)) key value (t-single _ _) (r-right x r-leaf) = t-right x (t-single key value) RTtoTI0 (node _ _ leaf right@(node _ _ _ _)) (node key₁ value₁ leaf leaf) key value (t-right x₁ ti) (r-right x ri) = t-single key₁ value₁ RTtoTI0 (node key₁ _ leaf right@(node key₂ _ _ _)) (node key₁ value₁ leaf right₁@(node key₃ _ _ _)) key value (t-right x₁ ti) (r-right x ri) = t-right (subst (λ k → key₁ < k ) (rt-property-key ri) x₁) (RTtoTI0 _ _ key value ti ri) RTtoTI0 (node key₁ _ (node _ _ _ _) leaf) (node key₁ _ (node key₃ value left right) leaf) key value₁ (t-left x₁ ti) (r-right x ()) RTtoTI0 (node key₁ _ (node key₃ _ _ _) leaf) (node key₁ _ (node key₃ value₃ _ _) (node key value leaf leaf)) key value (t-left x₁ ti) (r-right x r-leaf) = t-node x₁ x ti (t-single key value) RTtoTI0 (node key₁ _ (node _ _ _ _) (node key₂ _ _ _)) (node key₁ _ (node _ _ _ _) (node key₃ _ _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x ri) = t-node x₁ (subst (λ k → key₁ < k) (rt-property-key ri) x₂) ti (RTtoTI0 _ _ key value ti₁ ri) -- r-left case RTtoTI0 .(node _ _ leaf leaf) .(node _ _ (node key value leaf leaf) leaf) key value (t-single _ _) (r-left x r-leaf) = t-left x (t-single _ _ ) RTtoTI0 .(node _ _ leaf (node _ _ _ _)) (node key₁ value₁ (node key value leaf leaf) (node _ _ _ _)) key value (t-right x₁ ti) (r-left x r-leaf) = t-node x x₁ (t-single key value) ti RTtoTI0 (node key₃ _ (node key₂ _ _ _) leaf) (node key₃ _ (node key₁ value₁ left left₁) leaf) key value (t-left x₁ ti) (r-left x ri) = t-left (subst (λ k → k < key₃ ) (rt-property-key ri) x₁) (RTtoTI0 _ _ key value ti ri) -- key₁ < key₃ RTtoTI0 (node key₁ _ (node key₂ _ _ _) (node _ _ _ _)) (node key₁ _ (node key₃ _ _ _) (node _ _ _ _)) key value (t-node x₁ x₂ ti ti₁) (r-left x ri) = t-node (subst (λ k → k < key₁ ) (rt-property-key ri) x₁) x₂ (RTtoTI0 _ _ key value ti ri) ti₁ RTtoTI1 : {n : Level} {A : Set n} → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant repl → replacedTree key value tree repl → treeInvariant tree RTtoTI1 .leaf .(node key value leaf leaf) key value ti r-leaf = t-leaf RTtoTI1 (node key value₁ leaf leaf) .(node key value leaf leaf) key value (t-single .key .value) r-node = t-single key value₁ RTtoTI1 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right x ti) r-node = t-right x ti RTtoTI1 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left x ti) r-node = t-left x ti RTtoTI1 .(node key _ (node _ _ _ _) (node _ _ _ _)) .(node key value (node _ _ _ _) (node _ _ _ _)) key value (t-node x x₁ ti ti₁) r-node = t-node x x₁ ti ti₁ -- r-right case RTtoTI1 (node key₁ value₁ leaf leaf) (node key₁ _ leaf (node _ _ _ _)) key value (t-right x₁ ti) (r-right x r-leaf) = t-single key₁ value₁ RTtoTI1 (node key₁ value₁ leaf (node key₂ value₂ t2 t3)) (node key₁ _ leaf (node key₃ _ _ _)) key value (t-right x₁ ti) (r-right x ri) = t-right (subst (λ k → key₁ < k ) (sym (rt-property-key ri)) x₁) (RTtoTI1 _ _ key value ti ri) -- key₁ < key₂ RTtoTI1 (node _ _ (node _ _ _ _) leaf) (node _ _ (node _ _ _ _) (node key value _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x r-leaf) = t-left x₁ ti RTtoTI1 (node key₄ _ (node key₃ _ _ _) (node key₁ value₁ n n₁)) (node key₄ _ (node key₃ _ _ _) (node key₂ _ _ _)) key value (t-node x₁ x₂ ti ti₁) (r-right x ri) = t-node x₁ (subst (λ k → key₄ < k ) (sym (rt-property-key ri)) x₂) ti (RTtoTI1 _ _ key value ti₁ ri) -- key₄ < key₁ -- r-left case RTtoTI1 (node key₁ value₁ leaf leaf) (node key₁ _ _ leaf) key value (t-left x₁ ti) (r-left x ri) = t-single key₁ value₁ RTtoTI1 (node key₁ _ (node key₂ value₁ n n₁) leaf) (node key₁ _ (node key₃ _ _ _) leaf) key value (t-left x₁ ti) (r-left x ri) = t-left (subst (λ k → k < key₁ ) (sym (rt-property-key ri)) x₁) (RTtoTI1 _ _ key value ti ri) -- key₂ < key₁ RTtoTI1 (node key₁ value₁ leaf _) (node key₁ _ _ _) key value (t-node x₁ x₂ ti ti₁) (r-left x r-leaf) = t-right x₂ ti₁ RTtoTI1 (node key₁ value₁ (node key₂ value₂ n n₁) _) (node key₁ _ _ _) key value (t-node x₁ x₂ ti ti₁) (r-left x ri) = t-node (subst (λ k → k < key₁ ) (sym (rt-property-key ri)) x₁) x₂ (RTtoTI1 _ _ key value ti ri) ti₁ -- key₂ < key₁ insertTreeP : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → treeInvariant tree → (exit : (tree repl : bt A) → treeInvariant repl ∧ replacedTree key value tree repl → t ) → t insertTreeP {n} {m} {A} {t} tree key value P0 exit = TerminatingLoopS (bt A ∧ List (bt A) ) {λ p → treeInvariant (proj1 p) ∧ stackInvariant key (proj1 p) tree (proj2 p) } (λ p → bt-depth (proj1 p)) ⟪ tree , tree ∷ [] ⟫ ⟪ P0 , s-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 = P0 ; si = proj2 P ; ri = R ; ci = lift tt } $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) P1 (λ key value {tree1} repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) $ λ tree repl P → {!!} --exit tree repl ⟪ RTtoTI0 _ _ _ _ (proj1 P) (proj2 P) , proj2 P ⟫ insertTestP1 = insertTreeP leaf 1 1 t-leaf $ λ _ x0 P0 → insertTreeP x0 2 1 (proj1 P0) $ λ _ x1 P1 → insertTreeP x1 3 2 (proj1 P1) $ λ _ x2 P2 → insertTreeP x2 2 2 (proj1 P2) (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 black-depth : {n : Level} {A : Set n} → (tree : bt (Color ∧ A) ) → ℕ black-depth leaf = 0 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-single : {c : Color} → (key : ℕ) → (value : A) → RBtreeInvariant (node key ⟪ c , value ⟫ leaf leaf) rb-right-red : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} → key < key₁ → black-depth t ≡ black-depth t₁ → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) → RBtreeInvariant (node key ⟪ Red , value ⟫ leaf (node key₁ ⟪ Black , value₁ ⟫ t t₁)) rb-right-black : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} → key < key₁ → {c : Color} → black-depth t ≡ black-depth t₁ → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) → RBtreeInvariant (node key ⟪ Black , value ⟫ leaf (node key₁ ⟪ c , value₁ ⟫ t t₁)) rb-left-red : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} → key₁ < key → black-depth t ≡ black-depth t₁ → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) → RBtreeInvariant (node key ⟪ Red , value ⟫ (node key₁ ⟪ Black , value₁ ⟫ t t₁) leaf ) rb-left-black : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} → {c : Color} → key₁ < key → black-depth t ≡ black-depth t₁ → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) → RBtreeInvariant (node key ⟪ Black , value ⟫ (node key₁ ⟪ c , value₁ ⟫ t t₁) leaf) rb-node-red : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt (Color ∧ A)} → key < key₁ → key₁ < key₂ → black-depth (node key ⟪ Black , value ⟫ t₁ t₂) ≡ black-depth (node key₂ ⟪ Black , value₂ ⟫ t₃ t₄) → RBtreeInvariant (node key ⟪ Black , value ⟫ t₁ t₂) → RBtreeInvariant (node key₂ ⟪ Black , value₂ ⟫ t₃ t₄) → RBtreeInvariant (node key₁ ⟪ Red , value₁ ⟫ (node key ⟪ Black , value ⟫ t₁ t₂) (node key₂ ⟪ Black , value₂ ⟫ t₃ t₄)) rb-node-black : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt (Color ∧ A)} → key < key₁ → key₁ < key₂ → {c c₁ : Color} → black-depth (node key ⟪ c , value ⟫ t₁ t₂) ≡ black-depth (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄) → RBtreeInvariant (node key ⟪ c , value ⟫ t₁ t₂) → RBtreeInvariant (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄) → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ (node key ⟪ c , value ⟫ t₁ t₂) (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄)) -- data rotatedTree {n : Level} {A : Set n} : (before after : bt A) → Set n where -- rtt-unit : {t : bt A} → rotatedTree t t -- rtt-node : {left left' right right' : bt A} → {ke : ℕ} {ve : A} → -- rotatedTree left left' → rotatedTree right right' → rotatedTree (node ke ve left right) (node ke ve left' right') -- -- a b -- -- b c d a -- -- d e e c -- -- -- rtt-right : {ka kb kc kd ke : ℕ} {va vb vc vd ve : A} → {c d e c1 d1 e1 : bt A} → {ctl ctr dtl dtr etl etr : bt A} -- --kd < kb < ke < ka< kc -- → {ctl1 ctr1 dtl1 dtr1 etl1 etr1 : bt A} -- → kd < kb → kb < ke → ke < ka → ka < kc -- → rotatedTree (node ke ve etl etr) (node ke ve etl1 etr1) -- → rotatedTree (node kd vd dtl dtr) (node kd vd dtl1 dtr1) -- → rotatedTree (node kc vc ctl ctr) (node kc vc ctl1 ctr1) -- → rotatedTree (node ka va (node kb vb (node kd vd dtl dtr) (node ke ve etl etr)) (node kc vc ctl ctr)) -- (node kb vb (node kd vd dtl1 dtr1) (node ka va (node ke ve etl1 etr1) (node kc vc ctl1 ctr1))) -- -- rtt-left : {ka kb kc kd ke : ℕ} {va vb vc vd ve : A} → {c d e c1 d1 e1 : bt A} → {ctl ctr dtl dtr etl etr : bt A} -- --kd < kb < ke < ka< kc -- → {ctl1 ctr1 dtl1 dtr1 etl1 etr1 : bt A} -- after child -- → kd < kb → kb < ke → ke < ka → ka < kc -- → rotatedTree (node ke ve etl etr) (node ke ve etl1 etr1) -- → rotatedTree (node kd vd dtl dtr) (node kd vd dtl1 dtr1) -- → rotatedTree (node kc vc ctl ctr) (node kc vc ctl1 ctr1) -- → rotatedTree (node kb vb (node kd vd dtl1 dtr1) (node ka va (node ke ve etl1 etr1) (node kc vc ctl1 ctr1))) -- (node ka va (node kb vb (node kd vd dtl dtr) (node ke ve etl etr)) (node kc vc ctl ctr)) -- 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} → (tleft tright : bt (Color ∧ A)) → RBtreeInvariant (node key ⟪ c , value ⟫ tleft tright) → RBtreeInvariant tleft RBtreeLeftDown leaf leaf (rb-single k1 v) = rb-leaf RBtreeLeftDown leaf (node key ⟪ Black , value ⟫ t1 t2 ) (rb-right-red x bde rbti) = rb-leaf RBtreeLeftDown leaf (node key ⟪ Black , value ⟫ t1 t2 ) (rb-right-black x bde rbti) = rb-leaf RBtreeLeftDown leaf (node key ⟪ Red , value ⟫ t1 t2 ) (rb-right-black x bde rbti)= rb-leaf RBtreeLeftDown (node key ⟪ Black , value ⟫ t t₁) leaf (rb-left-black x bde ti) = ti RBtreeLeftDown (node key ⟪ Black , value ⟫ t t₁) leaf (rb-left-red x bde ti)= ti RBtreeLeftDown (node key ⟪ Red , value ⟫ t t₁) leaf (rb-left-black x bde ti) = ti RBtreeLeftDown (node key ⟪ Black , value ⟫ t t₁) (node key₁ ⟪ Black , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = til RBtreeLeftDown (node key ⟪ Black , value ⟫ t t₁) (node key₁ ⟪ Black , value1 ⟫ t1 t2) (rb-node-red x x1 bde1 til tir) = til RBtreeLeftDown (node key ⟪ Red , value ⟫ t t₁) (node key₁ ⟪ Black , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = til RBtreeLeftDown (node key ⟪ Black , value ⟫ t t₁) (node key₁ ⟪ Red , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = til RBtreeLeftDown (node key ⟪ Red , value ⟫ t t₁) (node key₁ ⟪ Red , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = til RBtreeRightDown : {n : Level} {A : Set n} { key : ℕ} {value : A} {c : Color} → (tleft tright : bt (Color ∧ A)) → RBtreeInvariant (node key ⟪ c , value ⟫ tleft tright) → RBtreeInvariant tright RBtreeRightDown leaf leaf (rb-single k1 v1 ) = rb-leaf RBtreeRightDown leaf (node key ⟪ Black , value ⟫ t1 t2 ) (rb-right-red x bde rbti) = rbti RBtreeRightDown leaf (node key ⟪ Black , value ⟫ t1 t2 ) (rb-right-black x bde rbti) = rbti RBtreeRightDown leaf (node key ⟪ Red , value ⟫ t1 t2 ) (rb-right-black x bde rbti)= rbti RBtreeRightDown (node key ⟪ Black , value ⟫ t t₁) leaf (rb-left-black x bde ti) = rb-leaf RBtreeRightDown (node key ⟪ Black , value ⟫ t t₁) leaf (rb-left-red x bde ti) = rb-leaf RBtreeRightDown (node key ⟪ Red , value ⟫ t t₁) leaf (rb-left-black x bde ti) = rb-leaf RBtreeRightDown (node key ⟪ Black , value ⟫ t t₁) (node key₁ ⟪ Black , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = tir RBtreeRightDown (node key ⟪ Black , value ⟫ t t₁) (node key₁ ⟪ Black , value1 ⟫ t1 t2) (rb-node-red x x1 bde1 til tir) = tir RBtreeRightDown (node key ⟪ Red , value ⟫ t t₁) (node key₁ ⟪ Black , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = tir RBtreeRightDown (node key ⟪ Black , value ⟫ t t₁) (node key₁ ⟪ Red , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = tir RBtreeRightDown (node key ⟪ Red , value ⟫ t t₁) (node key₁ ⟪ Red , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til tir) = tir -- -- 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)) record result {n : Level} {A : Set n} {key : ℕ} {tree0 : bt (Color ∧ A)} : Set n where field tree : bt (Color ∧ A) stack : List (bt (Color ∧ A)) ti : RBtreeInvariant tree si : stackInvariant key tree tree0 stack 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 data replacedRBTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (before after : bt (Color ∧ A) ) → Set n where rbr-leaf : {ca cb : Color} → replacedRBTree key value leaf (node key ⟪ cb , value ⟫ leaf leaf) rbr-node : {value₁ : A} → {ca cb : Color } → {t t₁ : bt (Color ∧ A)} → replacedRBTree key value (node key ⟪ ca , value₁ ⟫ t t₁) (node key ⟪ cb , value ⟫ t t₁) rbr-right : {k : ℕ } {v1 : A} → {ca cb : Color} → {t t1 t2 : bt (Color ∧ A)} → k < key → replacedRBTree key value t2 t → replacedRBTree key value (node k ⟪ ca , v1 ⟫ t1 t2) (node k ⟪ cb , v1 ⟫ t1 t) rbr-left : {k : ℕ } {v1 : A} → {ca cb : Color} → {t t1 t2 : bt (Color ∧ A)} → key < k → replacedRBTree key value t1 t → replacedRBTree key value (node k ⟪ ca , v1 ⟫ t1 t2) (node k ⟪ cb , v1 ⟫ t t2) -- k < key → key < k data ParentGrand {n : Level} {A : Set 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 } → parent ≡ node kp vp self n1 → grand ≡ node kg vg parent n2 → ParentGrand self parent n2 grand s2-1sp2 : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } → parent ≡ node kp vp n1 self → grand ≡ node kg vg parent n2 → ParentGrand self parent n2 grand s2-s12p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } → parent ≡ node kp vp self n1 → grand ≡ node kg vg n2 parent → ParentGrand self parent n2 grand s2-1s2p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } → parent ≡ node kp vp n1 self → grand ≡ node kg vg n2 parent → ParentGrand self parent n2 grand record PG {n : Level } (A : Set n) (self : bt A) (stack : List (bt A)) : Set n where field parent grand uncle : bt A pg : ParentGrand 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) : (tree repl : bt (Color ∧ A) ) → Set n where rebuild : {tree repl : bt (Color ∧ A) } → black-depth repl ≡ black-depth (child-replaced key tree) → RBI-state key tree repl rotate : {tree repl : bt (Color ∧ A) } → color repl ≡ Red → black-depth repl ≡ suc (black-depth (child-replaced key tree)) → RBI-state key tree repl record RBI {n : Level} {A : Set n} (key : ℕ) (value : A) (orig repl : bt (Color ∧ A) ) (stack : List (bt (Color ∧ A))) : Set n where field tree : 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 rotated : replacedRBTree key value tree repl state : RBI-state key tree repl 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 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 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 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 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 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 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 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 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 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 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 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 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 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 PGtoRBinvariant : {n : Level} {A : Set n} → {key d0 ds dp dg : ℕ } → (tree orig : bt (Color ∧ A) ) → RBtreeInvariant orig → (stack : List (bt (Color ∧ A))) → (pg : PG (Color ∧ A) tree stack ) → RBtreeInvariant tree ∧ RBtreeInvariant (PG.parent pg) ∧ RBtreeInvariant (PG.grand pg) PGtoRBinvariant = {!!} 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 -- replaceRBTNode : {n m : Level} {A : Set n } {t : Set m } → (key : ℕ) (value : A) → (tree0 : bt (Color ∧ A)) → RBtreeInvariant tree0 → (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → RBtreeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack → (exit : (r : RBI key value tree0 tree1 stack ) → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key )) → t replaceRBTNode = ? -- -- RBT is blanced with the stack, simply rebuild tree without totation -- rebuildRBT : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (orig repl : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig repl stack ) → black-depth repl ≡ black-depth (child-replaced key (RBI.tree r)) → (next : (repl1 : (bt (Color ∧ A))) → (stack1 : List (bt (Color ∧ A))) → (r : RBI key value orig repl1 stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t rebuildRBT = ? rotateLeft : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (orig tree : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig tree stack ) → (next : (current : bt (Color ∧ A)) → (stack1 : List (bt (Color ∧ A))) → (r : RBI key value orig current stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t rotateLeft {n} {m} {A} {t} key value = {!!} where rotateLeft1 : t rotateLeft1 with stackToPG {!!} {!!} {!!} {!!} ... | case1 x = {!!} -- {!!} {!!} {!!} {!!} rr ... | case2 (case1 x) = {!!} ... | case2 (case2 pg) = {!!} rotateRight : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (orig tree : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig tree stack ) → (next : (current : bt (Color ∧ A)) → (stack1 : List (bt (Color ∧ A))) → (r : RBI key value orig current stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t rotateRight {n} {m} {A} {t} key value = {!!} where rotateRight1 : t rotateRight1 with stackToPG {!!} {!!} {!!} {!!} ... | case1 x = {!!} ... | case2 (case1 x) = {!!} ... | case2 (case2 pg) = {!!} insertCase5 : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (orig tree : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig tree stack ) → (next : (tree1 : (bt (Color ∧ A))) → (stack1 : List (bt (Color ∧ A))) → (r : RBI key value orig tree1 stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t insertCase5 {n} {m} {A} {t} key value = {!!} where insertCase51 : t insertCase51 with stackToPG {!!} {!!} {!!} {!!} ... | case1 eq = {!!} ... | case2 (case1 eq ) = {!!} ... | case2 (case2 pg) with PG.pg pg ... | s2-s1p2 x x₁ = {!!} -- rotateRight {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} next exit -- x : PG.parent pg ≡ node kp vp tree n1 -- x₁ : PG.grand pg ≡ node kg vg (PG.parent pg) (PG.uncle pg) ... | s2-1sp2 x x₁ = {!!} -- rotateLeft {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} next exit ... | s2-s12p x x₁ = {!!} -- rotateLeft {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} next exit ... | s2-1s2p x x₁ = {!!} -- rotateLeft {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} next exit -- = insertCase2 tree (PG.parent pg) (PG.uncle pg) (PG.grand pg) stack si (PG.pg pg) -- -- replaced node increase blackdepth, so we need tree rotate -- -- case2 tree is Red -- -- go upward until -- -- if root -- insert top -- if unkle is leaf or Black -- go insertCase5/6 -- -- make color tree ≡ Black , color unkle ≡ Black, color grand ≡ Red -- loop with grand as repl -- -- case5/case6 rotation -- -- rotate and rebuild replaceTree and rb-invariant replaceRBP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (orig repl : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig repl stack ) → (next : (repl1 : (bt (Color ∧ A))) → (stack1 : List (bt (Color ∧ A))) → (r : RBI key value orig repl1 stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t replaceRBP {n} {m} {A} {t} key value orig repl stack r next exit with RBI.state r ... | rebuild bdepth-eq = rebuildRBT key value orig repl stack r bdepth-eq next exit ... | rotate repl-red pbdeth< with stackToPG (RBI.tree r) orig stack (RBI.si r) ... | case1 eq = exit repl stack eq r -- no stack, replace top node ... | case2 (case1 eq) = insertCase12 orig refl (RBI.si r) where insertCase2 : (tree parent uncle grand : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → (si : stackInvariant key tree orig stack ) → (pg : ParentGrand tree parent uncle grand ) → t insertCase2 tree leaf uncle grand stack si (s2-s1p2 () x₁) insertCase2 tree leaf uncle grand stack si (s2-1sp2 () x₁) insertCase2 tree leaf uncle grand stack si (s2-s12p () x₁) insertCase2 tree leaf uncle grand stack si (s2-1s2p () x₁) insertCase2 tree parent@(node kp ⟪ Red , _ ⟫ _ _) uncle grand stack si pg = {!!} -- next {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) leaf grand stack si pg = {!!} insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Red , _ ⟫ _ _ ) grand stack si pg = {!!} -- next {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Black , _ ⟫ _ _) grand stack si (s2-s1p2 x x₁) = {!!} -- insertCase5 key value orig tree {!!} rbio {!!} {!!} stack si {!!} ri {!!} {!!} next exit -- tree is left of parent, parent is left of grand -- node kp ⟪ Black , proj3 ⟫ left right ≡ node kp₁ vp tree n1 -- grand ≡ node kg vg (node kp ⟪ Black , proj3 ⟫ left right) (node ku ⟪ Black , proj4 ⟫ left₁ right₁) insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Black , _ ⟫ _ _) grand stack si (s2-1sp2 x x₁) = {!!} -- rotateLeft key value orig tree {!!} repl rbio {!!} {!!} stack si {!!} ri {!!} {!!} -- (λ a b c d e f h i j k l m → insertCase5 key value a b c d {!!} {!!} h i j k l {!!} {!!} next exit ) exit -- tree is right of parent, parent is left of grand rotateLeft -- node kp ⟪ Black , proj3 ⟫ left right ≡ node kp₁ vp n1 tree -- grand ≡ node kg vg (node kp ⟪ Black , proj3 ⟫ left right) (node ku ⟪ Black , proj4 ⟫ left₁ right₁) insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Black , _ ⟫ _ _) grand stack si (s2-s12p x x₁) = {!!} -- rotateRight key value orig tree {!!} repl rbio {!!} {!!} stack si {!!} ri {!!} {!!} -- (λ a b c d e f h i j k l m → insertCase5 key value a b c d {!!} {!!} h i j k l {!!} {!!} next exit ) exit -- tree is left of parent, parent is right of grand, rotateRight -- node kp ⟪ Black , proj3 ⟫ left right ≡ node kp₁ vp tree n1 -- grand ≡ node kg vg (node ku ⟪ Black , proj4 ⟫ left₁ right₁) (node kp ⟪ Black , proj3 ⟫ left right) insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Black , _ ⟫ _ _) grand stack si (s2-1s2p x x₁) = {!!} -- insertCase5 key value orig tree {!!} repl rbio {!!} {!!} stack si {!!} ri {!!} {!!} next exit -- tree is right of parent, parent is right of grand -- node kp ⟪ Black , proj3 ⟫ left right ≡ node kp₁ vp n1 tree -- grand ≡ node kg vg (node ku ⟪ Black , proj4 ⟫ left₁ right₁) (node kp ⟪ Black , proj3 ⟫ left right) -- one level stack, orig is parent of repl rb01 : stackInvariant key (RBI.tree r) orig stack rb01 = RBI.si r insertCase12 : (tr0 : bt (Color ∧ A)) → tr0 ≡ orig → stackInvariant key (RBI.tree r) orig stack → t insertCase12 leaf eq1 si = {!!} -- can't happen insertCase12 tr0@(node key₁ value₁ left right) refl si with <-cmp key key₁ ... | tri< a ¬b ¬c = {!!} 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 ) ... | tri≈ ¬a b ¬c = {!!} -- can't happen ... | tri> ¬a ¬b c = insertCase13 value₁ refl where -- -- orig B -- / \ -- left tree → rot → repl R -- -- B => B B => B -- / \ / \ / \ rotate L / \ -- L L1 L R3 L R -- bad B B -- / \ / \ / \ 1 : child-replace --- L L2 L B L L 2: child-replace ( unbalanced ) -- / \ 3: child-replace ( rotated / balanced ) -- L L -- 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 ) -- -- RBI key value (node key₁ ⟪ Black , value₄ ⟫ left right) repl stack -- insertCase13 : (v : Color ∧ A ) → v ≡ value₁ → t insertCase13 ⟪ Black , value₄ ⟫ refl = exit (node key₁ ⟪ Black , value₄ ⟫ left repl) (orig ∷ []) refl record { tree = orig ; origti = RBI.origti r ; origrb = RBI.origrb r ; treerb = RBI.origrb r ; replrb = ? ; si = s-nil ; rotated = ? ; ri = ? ; state = ? } where rb09 : {n : Level} {A : Set n} → {key key1 key2 : ℕ} {value value1 : A} {t1 t2 : bt (Color ∧ A)} → RBtreeInvariant (node key ⟪ Red , value ⟫ leaf (node key1 ⟪ Black , value1 ⟫ t1 t2)) → key < key1 rb09 (rb-right-red x x0 x2) = x -- rb05 should more general tkey : {n : Level} {A : Set n } → (rbt : bt (Color ∧ A)) → ℕ tkey (node key value t t2) = key tkey leaf = {!!} -- key is none rb051 : {n : Level} {A : Set n} {key key1 : ℕ } {value : A} {t t1 t2 : bt (Color ∧ A)} {c : Color} → replacedTree key ⟪ ? , value ⟫ (node key1 ⟪ c , value ⟫ t1 t2) (node key1 ⟪ c , value ⟫ t1 t) → key1 < key rb051 = {!!} rb052 : {key key₁ : ℕ} → stackInvariant key (RBI.tree r) orig stack → key < key₁ rb052 = {!!} insertCase13 ⟪ Red , value₄ ⟫ eq with color (RBI.tree r) ... | Black = exit {!!} {!!} {!!} {!!} ... | Red = exit {!!} {!!} {!!} {!!} -- r = orig RBI.tree b -- / \ => / \ -- b b → r RBI.tree r r = orig o (r/b) ... | case2 (case2 pg) = {!!} -- insertCase2 tree (PG.parent pg) (PG.uncle pg) (PG.grand pg) stack si (PG.pg pg)