Mercurial > hg > Members > Moririn
view hoareBinaryTree1.agda @ 742:0e0a30f942ca
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 24 Apr 2023 13:44:44 +0900 |
| parents | c44edea35126 |
| children | 80f027be2c68 |
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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 -- 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 ti-right : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 tree₁ repl) → treeInvariant repl ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-right x ti) = ti ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-left x ti) = t-leaf ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti₁ ti-left : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 repl tree₁ ) → treeInvariant repl ti-left {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf ti-left {_} {_} {_} {_} {key₁} {v1} (t-right x ti) = t-leaf ti-left {_} {_} {_} {_} {key₁} {v1} (t-left x ti) = ti ti-left {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti stackTreeInvariant : {n : Level} {A : Set n} (key : ℕ) (sub tree : bt A) → (stack : List (bt A)) → treeInvariant tree → stackInvariant key sub tree stack → treeInvariant sub stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-nil ) = ti stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si ) = ti-right (si1 si) where si1 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 tree₁ sub ) tree st → treeInvariant (node key₁ v1 tree₁ sub ) si1 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 tree₁ sub ) tree st ti si stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si ) = ti-left ( si2 si) where si2 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 sub tree₁ ) tree st → treeInvariant (node key₁ v1 sub tree₁ ) si2 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 sub tree₁ ) tree st ti si rt-property1 : {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf ) rt-property1 {n} {A} key value .leaf .(node key value leaf leaf) r-leaf () rt-property1 {n} {A} key value .(node key _ _ _) .(node key value _ _) r-node () rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-right x rt) = λ () rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-left x rt) = λ () rt-property-leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {repl : bt A} → replacedTree key value leaf repl → repl ≡ node key value leaf leaf rt-property-leaf r-leaf = refl rt-property-¬leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {tree : bt A} → ¬ replacedTree key value tree leaf rt-property-¬leaf () rt-property-key : {n : Level} {A : Set n} {key key₂ key₃ : ℕ} {value value₂ value₃ : A} {left left₁ right₂ right₃ : bt A} → replacedTree key value (node key₂ value₂ left right₂) (node key₃ value₃ left₁ right₃) → key₂ ≡ key₃ rt-property-key r-node = refl rt-property-key (r-right x ri) = refl rt-property-key (r-left x ri) = refl nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x nat-<> : { x y : ℕ } → x < y → y < x → ⊥ nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x open _∧_ depth-1< : {i j : ℕ} → suc i ≤ suc (i Data.Nat.⊔ j ) depth-1< {i} {j} = s≤s (m≤m⊔n _ j) depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i ) depth-2< {i} {j} = s≤s (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 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) → (treeInvariant tree ) → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value (child-replaced key tree) tree1 → t) → t replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P) (subst (λ j → replacedTree k v1 j (node k v1 t t₁) ) repl00 r-node) where repl00 : node k value t t₁ ≡ child-replaced k (node k value t t₁) repl00 with <-cmp k k ... | tri< a ¬b ¬c = ⊥-elim (¬b refl) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim (¬b refl) replaceP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → {tree : bt A} ( repl : bt A) → (stack : List (bt A)) → replacePR key value tree repl stack (λ _ _ _ → Lift n ⊤) → (next : ℕ → A → {tree1 : bt A } (repl : bt A) → (stack1 : List (bt A)) → replacePR key value tree1 repl stack1 (λ _ _ _ → Lift n ⊤) → length stack1 < length stack → t) → (exit : (tree1 repl : bt A) → treeInvariant tree1 ∧ replacedTree key value tree1 repl → t) → t replaceP key value {tree} repl [] Pre next exit = ⊥-elim ( si-property0 (replacePR.si Pre) refl ) -- can't happen replaceP key value {tree} repl (leaf ∷ []) Pre next exit with si-property-last _ _ _ _ (replacePR.si Pre)-- tree0 ≡ leaf ... | refl = exit (replacePR.tree0 Pre) (node key value leaf leaf) ⟪ replacePR.ti Pre , r-leaf ⟫ replaceP key value {tree} repl (node key₁ value₁ left right ∷ []) Pre next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = exit (replacePR.tree0 Pre) (node key₁ value₁ repl right ) ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ repl right ) repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) repl01 | refl | refl = subst (λ k → replacedTree key value (node key₁ value₁ k right ) (node key₁ value₁ repl right )) repl02 (r-left a repl03) where repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl repl03 = replacePR.ri Pre repl02 : child-replaced key (node key₁ value₁ left right) ≡ left repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a a) ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a) ... | tri≈ ¬a b ¬c = exit (replacePR.tree0 Pre) repl ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) repl repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) repl01 | refl | refl = subst (λ k → replacedTree key value k repl) repl02 (replacePR.ri Pre) where repl02 : child-replaced key (node key₁ value₁ left right) ≡ node key₁ value₁ left right repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b) ... | tri> ¬a ¬b c = exit (replacePR.tree0 Pre) (node key₁ value₁ left repl ) ⟪ replacePR.ti Pre , repl01 ⟫ where repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ left repl ) repl01 with si-property1 (replacePR.si Pre) | si-property-last _ _ _ _ (replacePR.si Pre) repl01 | refl | refl = subst (λ k → replacedTree key value (node key₁ value₁ left k ) (node key₁ value₁ left repl )) repl02 (r-right c repl03) where repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl repl03 = replacePR.ri Pre repl02 : child-replaced key (node key₁ value₁ left right) ≡ right repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim ( ¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬c c) ... | tri> ¬a ¬b c = refl replaceP {n} {_} {A} key value {tree} repl (leaf ∷ st@(tree1 ∷ st1)) Pre next exit = next key value repl st Post ≤-refl where Post : replacePR key value tree1 repl (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ leaf repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ leaf) ≡ leaf repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = ⊥-elim (¬c x) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) ... | tri> ¬a ¬b c = refl repl12 : replacedTree key value (child-replaced key tree1 ) repl repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf ... | s-left {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 leaf tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 leaf tree₁ ) ≡ leaf repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) ... | tri> ¬a ¬b c = ⊥-elim (¬a x) repl12 : replacedTree key value (child-replaced key tree1 ) repl repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf replaceP {n} {_} {A} key value {tree} repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre next exit with <-cmp key key₁ ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value₁ repl right ) st (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ left repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) ... | refl | tri> ¬a ¬b c = refl repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) ... | s-left {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ left repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = refl ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ repl right) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04 (r-left a (replacePR.ri Pre)) ... | tri≈ ¬a b ¬c = next key value (node key₁ value left right ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value left right ) (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 b ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ tree -- (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ tree) ≡ tree repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = ⊥-elim (¬c x) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x) ... | tri> ¬a ¬b c = refl repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) repl12 refl with repl09 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node ... | s-left {_} {_} {tree₁} {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 b ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree tree₁ ) ≡ tree repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) ... | tri> ¬a ¬b c = ⊥-elim (¬a x) repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key tree1 ) (node key₁ value left right ) repl12 refl with repl09 ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st Post ≤-refl where Post : replacePR key value tree1 (node key₁ value₁ left repl ) st (λ _ _ _ → Lift n ⊤) Post with replacePR.si Pre ... | s-right {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right) repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ right repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) ... | tri> ¬a ¬b c = refl repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt) ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt) ... | refl | tri> ¬a ¬b c = refl repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) ... | s-left {_} {_} {tree₁} {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ; ci = lift tt } where repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁ repl09 = si-property1 si repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1) repl10 with si-property1 si ... | refl = si repl03 : child-replaced key (node key₁ value₁ left right) ≡ right repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) ... | tri> ¬a ¬b c = refl repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right repl02 with repl09 | <-cmp key key₂ ... | refl | tri< a ¬b ¬c = refl ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt) ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt) repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1 repl04 = begin node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩ node key₁ value₁ left right ≡⟨ sym repl02 ⟩ child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩ child-replaced key tree1 ∎ where open ≡-Reasoning repl12 : replacedTree key value (child-replaced key tree1 ) (node key₁ value₁ left repl) repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre)) TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ) → (r : Index) → (p : Invraiant r) → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r) ... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) ... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (≤-step lt1) p1 lt1 ) where TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → Invraiant r1 → reduce r1 < reduce r → t TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1)) TerminatingLoop1 (suc j) r r1 n≤j p1 lt with <-cmp (reduce r1) (suc j) ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt ... | tri≈ ¬a b ¬c = loop r1 p1 (λ r2 p2 lt1 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b lt1 ) p2 lt1 ) ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) 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 data RBTree {n : Level} (A : Set n) : (key : ℕ) → Color → (b-depth : ℕ) → Set n where rb-leaf : (key : ℕ) → RBTree A key Black 0 rb-single : (key : ℕ) → (value : A) → (c : Color) → RBTree A key c 1 t-right-red : (key : ℕ) {key₁ : ℕ} → (value : A) → key < key₁ → {d : ℕ } → RBTree A key₁ Black d → RBTree A key Red d t-right-black : (key : ℕ) {key₁ : ℕ} → (value : A) → key < key₁ → {c : Color} → {d : ℕ }→ RBTree A key₁ c d → RBTree A key Black (suc d) t-left-red : (key₁ : ℕ) { key : ℕ} → (value : A) → key < key₁ → {d : ℕ} → RBTree A key Black d → RBTree A key₁ Red d t-left-black : (key₁ : ℕ) {key : ℕ} → (value : A) → key < key₁ → {c : Color} → {d : ℕ} → RBTree A key c d → RBTree A key₁ Black (suc d) t-node-red : (key₁ : ℕ) { key key₂ : ℕ} → (value : A) → key < key₁ → key₁ < key₂ → {d : ℕ} → RBTree A key Black d → RBTree A key₂ Black d → RBTree A key₁ Red d t-node-black : (key₁ : ℕ) {key key₂ : ℕ} → (value : A) → key < key₁ → key₁ < key₂ → {c c1 : Color} {d : ℕ} → RBTree A key c d → RBTree A key₂ c1 d → RBTree A key₁ Black (suc d) color : {n : Level} (A : Set n) {key d : ℕ} {c : Color } → (rb : RBTree A key c d ) → Color color {n} A {k} {d} {c} rb = c rb-key : {n : Level} (A : Set n) {key d : ℕ} {c : Color } → (rb : RBTree A key c d ) → ℕ rb-key {n} A {k} {d} {c} rb = k RB→bt : {n : Level} (A : Set n) {key d : ℕ} {c : Color } → (rb : RBTree A key c d ) → bt A RB→bt {n} A (rb-leaf _) = leaf RB→bt {n} A (rb-single key value _) = node key value leaf leaf RB→bt {n} A (t-right-red key value x rb) = node key value leaf (RB→bt A rb) RB→bt {n} A (t-right-black key value x rb) = node key value leaf (RB→bt A rb) RB→bt {n} A (t-left-red key value x rb) = node key value (RB→bt A rb) leaf RB→bt {n} A (t-left-black key value x rb) = node key value (RB→bt A rb) leaf RB→bt {n} A (t-node-red key value x x₁ rb rb₁) = node key value (RB→bt A rb) (RB→bt A rb₁) RB→bt {n} A (t-node-black key value x x₁ rb rb₁) = node key value (RB→bt A rb) (RB→bt A rb₁) rbt-depth : {n : Level} (A : Set n) {key : ℕ} {c : Color} {d : ℕ} → RBTree A key c d → ℕ rbt-depth A (rb-leaf _) = zero rbt-depth A (rb-single _ value _) = 1 rbt-depth A (t-right-red _ value x ab) = suc ( rbt-depth A ab) rbt-depth A (t-right-black _ value x ab) = suc ( rbt-depth A ab) rbt-depth A (t-left-red _ value x ab) = suc ( rbt-depth A ab) rbt-depth A (t-left-black _ value x ab) = suc ( rbt-depth A ab) rbt-depth A (t-node-red _ value x x₁ ab ab₁) = suc (Data.Nat._⊔_ (rbt-depth A ab ) (rbt-depth A ab₁ )) rbt-depth A (t-node-black _ value x x₁ ab ab₁) = suc (Data.Nat._⊔_ (rbt-depth A ab ) (rbt-depth A ab₁ )) rbt-key : {n : Level} (A : Set n) {key : ℕ} {c : Color} {d : ℕ} → RBTree A key c d → Maybe ℕ rbt-key {n} A (rb-leaf _) = nothing rbt-key {n} A (rb-single key value _) = just key rbt-key {n} A (t-right-red key value x rb) = just key rbt-key {n} A (t-right-black key value x rb) = just key rbt-key {n} A (t-left-red key value x rb) = just key rbt-key {n} A (t-left-black key value x rb) = just key rbt-key {n} A (t-node-red key value x x₁ rb rb₁) = just key rbt-key {n} A (t-node-black key value x x₁ rb rb₁) = just key data ParentGrand {n : Level} {A : Set n} (self : bt A) : (parent 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 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 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 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 grand data rotatedTree {n : Level} {A : Set n} : (before after : bt A ) → Set n where rr-node : {t : bt A} → rotatedTree t t rr-right : {ka kb : ℕ } {va vb : A} → {ta tb tc ta1 tb1 tc1 : bt A} → ka < kb → rotatedTree ta ta1 → rotatedTree tb tb1 → rotatedTree tc tc1 → rotatedTree (node ka va (node kb vb ta tb) tc) (node kb vb ta1 (node ka va tb1 tc1) ) rr-left : {ka kb : ℕ } {va vb : A} → {ta tb tc ta1 tb1 tc1 : bt A} → ka < kb → rotatedTree ta ta1 → rotatedTree tb tb1 → rotatedTree tc tc1 → rotatedTree (node kb vb ta (node ka va tb tc) ) (node ka va (node kb vb ta1 tb1) tc1) record PG {n : Level } (A : Set n) (stack : List (bt A)) : Set n where field self parent grand : bt A pg : ParentGrand self parent grand rest : List (bt A) stack=gp : stack ≡ ( self ∷ parent ∷ grand ∷ rest ) 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 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; 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 { self = tree ; parent = node k1 v1 tree t1 ; grand = _ ; pg = s2-s1p2 refl refl ; rest = _ ; stack=gp = refl } ) rbsi-len : {n : Level} {A : Set n} {orig parent grand : bt A} → ParentGrand orig parent grand → ℕ rbsi-len {n} {A} {s} {p} {g} st = ? findRBP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) {key1 d d1 : ℕ} → {c c1 : Color} → (tree : RBTree A key c d ) (orig : RBTree A key1 c1 d1 ) → (stack : List (bt A)) → stackInvariant key (RB→bt A tree) (RB→bt A orig) stack → (next : {key0 d0 : ℕ} {c0 : Color} → (tree1 : RBTree A key0 c0 d0 ) → (stack : List (bt A)) → stackInvariant key (RB→bt A tree1) (RB→bt A orig) stack → rbt-depth A tree1 < rbt-depth A tree → t ) → (exit : {key0 d0 : ℕ} {c0 : Color} → (tree1 : RBTree A key0 c0 d0 ) → (stack : List (bt A)) → stackInvariant key (RB→bt A tree1) (RB→bt A orig) stack → (rbt-depth A tree1 ≡ 0 ) ∨ ( rbt-key A tree1 ≡ just key ) → t ) → t findRBP {n} {m} {A} {t} key {key1} tree orig st si next exit = ? rotateRight : ? rotateRight = ? rotateLeft : ? rotateLeft = ? insertCase5 : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → {key0 key1 key2 d0 d1 d2 : ℕ} {c0 c1 c2 : Color} → (orig : RBTree A key1 c1 d1 ) → (tree : RBTree A key1 c1 d1 ) ( repl : RBTree A key2 c2 d2 ) → (si : ParentGrand ? ? ?) → (ri : rotatedTree (RB→bt A tree) (RB→bt A repl)) → (next : ℕ → A → {k1 k2 d1 d2 : ℕ} {c1 c2 : Color} → (tree1 : RBTree A k1 c1 d1 ) (repl1 : RBTree A k2 c2 d2 ) → (si1 : ParentGrand ? ? ?) → (ri : rotatedTree (RB→bt A tree1) (RB→bt A repl1)) → rbsi-len si1 < rbsi-len si → t ) → (exit : {k1 k2 d1 d2 : ℕ} {c1 c2 : Color} (tree1 : RBTree A k1 c1 d1 ) → (repl1 : RBTree A k2 c2 d2 ) → (ri : rotatedTree (RB→bt A orig) (RB→bt A repl1)) → t ) → t insertCase5 {n} {m} {A} {t} key value orig tree repl si ri next exit = ? where insertCase51 : (key1 : ℕ) (si : ParentGrand ? ? ? ) → t insertCase51 = ? replaceRBP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → {key0 key1 d0 d1 : ℕ} {c0 c1 : Color} → (orig : RBTree A key0 c0 d0 ) → (tree : RBTree A key1 c1 d1 ) → (stack : List (bt A)) → (si : stackInvariant key (RB→bt A tree) (RB→bt A orig) stack ) → (next : {key2 d2 : ℕ} {c2 : Color} → (tree2 : RBTree A key2 c2 d2 ) → {tr to : bt A} → RB→bt A tree2 ≡ tr → RB→bt A orig ≡ to → (stack1 : List (bt A)) → stackInvariant key tr to stack1 → length stack1 < length stack → t ) → (exit : {k1 d1 : ℕ} {c1 : Color} → (repl1 : RBTree A k1 c1 d1 ) → (rot : bt A ) → (ri : rotatedTree (RB→bt A orig) rot ) → replacedTree key value rot (RB→bt A repl1) → t ) → t replaceRBP {n} {m} {A} {t} key value {_} {key1} orig tree stack si next exit = insertCase1 stack _ _ refl refl si where insertCase2 : {k0 k1 d0 d1 d2 : ℕ} {c0 c1 c2 : Color} → (tree : RBTree A k0 c0 d0) → (parent : RBTree A k1 c1 d1) → (grand : RBTree A key1 c2 d2) → (stack : List (bt A)) → (tr to : bt A) → RB→bt A grand ≡ tr → RB→bt A orig ≡ to → (si : stackInvariant key tr to stack ) → (pg : ParentGrand (RB→bt A tree) (RB→bt A parent) tr ) → t insertCase2 tree parent grand stack tr to treq toeq si pg = ? insertCase1 : (stack : List (bt A)) → (tr to : bt A) → RB→bt A tree ≡ tr → RB→bt A orig ≡ to → (si : stackInvariant key tr to stack ) → t insertCase1 = ?