Mercurial > hg > Gears > GearsAgda
changeset 779:7e5dfe642064
remove black depth on RBTreeInvariant and introduce black-count
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 26 Jun 2023 20:04:38 +0900 |
| parents | 4d71d0894cfa |
| children | f7e1704305df |
| files | hoareBinaryTree1.agda |
| diffstat | 1 files changed, 102 insertions(+), 89 deletions(-) [+] |
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line diff
--- a/hoareBinaryTree1.agda Mon May 22 19:06:08 2023 +0900 +++ b/hoareBinaryTree1.agda Mon Jun 26 20:04:38 2023 +0900 @@ -530,39 +530,52 @@ Red : Color Black : Color -data RBtreeInvariant {n : Level} {A : Set n} : (tree : bt (Color ∧ A)) → (bd : ℕ) → Set n where - rb-leaf : RBtreeInvariant leaf 0 - rb-single : (key : ℕ) → (value : A) → RBtreeInvariant (node key ⟪ Black , value ⟫ leaf leaf) 1 +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 ) + +data RBtreeInvariant {n : Level} {A : Set n} : (tree : bt (Color ∧ A)) → Set n where + rb-leaf : RBtreeInvariant leaf + rb-single : (key : ℕ) → (value : A) → RBtreeInvariant (node key ⟪ Black , value ⟫ leaf leaf) rb-right-red : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} - → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) 1 - → RBtreeInvariant (node key ⟪ Red , value ⟫ leaf (node key₁ ⟪ Black , value₁ ⟫ t t₁)) 1 + → 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)} → {c : Color} - → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) 1 - → RBtreeInvariant (node key ⟪ Black , value ⟫ leaf (node key₁ ⟪ c , value₁ ⟫ t t₁)) 1 + → 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)} - → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) 1 - → RBtreeInvariant (node key₁ ⟪ Red , value ⟫ (node key ⟪ Black , value₁ ⟫ t t₁) leaf ) 1 + → black-depth t ≡ black-depth t₁ + → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) + → RBtreeInvariant (node key₁ ⟪ Red , value ⟫ (node key ⟪ Black , value₁ ⟫ t t₁) leaf ) rb-left-black : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} → {c : Color} - → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) 1 - → RBtreeInvariant (node key₁ ⟪ Black , value ⟫ (node key ⟪ c , value₁ ⟫ t t₁) leaf) 1 - rb-node-red : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt (Color ∧ A)} → (d : ℕ) - → RBtreeInvariant (node key ⟪ Black , value ⟫ t₁ t₂) d - → RBtreeInvariant (node key₂ ⟪ Black , value₂ ⟫ t₃ t₄) d - → RBtreeInvariant (node key₁ ⟪ Red , value₁ ⟫ (node key ⟪ Black , value ⟫ t₁ t₂) (node key₂ ⟪ Black , value₂ ⟫ t₃ t₄)) d - rb-node-black : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt (Color ∧ A)} → (d : ℕ) + → black-depth t ≡ black-depth t₁ + → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) + → RBtreeInvariant (node key₁ ⟪ Black , value ⟫ (node key ⟪ c , value₁ ⟫ t t₁) leaf) + rb-node-red : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt (Color ∧ A)} + → black-depth t₁ ≡ black-depth t₂ + → RBtreeInvariant (node key ⟪ Black , value ⟫ t₁ t₂) + → black-depth t₃ ≡ black-depth t₄ + → 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)} → {c c₁ : Color} - → RBtreeInvariant (node key ⟪ c , value ⟫ t₁ t₂) d - → RBtreeInvariant (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄) d - → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ (node key ⟪ c , value ⟫ t₁ t₂) (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄)) (suc d) + → black-depth t₁ ≡ black-depth t₂ + → RBtreeInvariant (node key ⟪ c , value ⟫ t₁ t₂) + → black-depth t₃ ≡ black-depth t₄ + → RBtreeInvariant (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄) + → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ (node key ⟪ c , value ⟫ t₁ t₂) (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄)) -tesr-rb-0 : RBtreeInvariant {_} {ℕ} leaf 0 +tesr-rb-0 : RBtreeInvariant {_} {ℕ} leaf tesr-rb-0 = rb-leaf -tesr-rb-1 : RBtreeInvariant {_} {ℕ} (node 10 ⟪ Black , 10 ⟫ leaf leaf) 1 +tesr-rb-1 : RBtreeInvariant {_} {ℕ} (node 10 ⟪ Black , 10 ⟫ leaf leaf) tesr-rb-1 = rb-single 10 10 -tesr-rb-2 : RBtreeInvariant {_} {ℕ} (node 10 ⟪ Red , 10 ⟫ (node 5 ⟪ Black , 5 ⟫ leaf leaf) leaf) 1 -tesr-rb-2 = rb-left-red ( rb-single 10 5) +tesr-rb-2 : RBtreeInvariant {_} {ℕ} (node 10 ⟪ Red , 10 ⟫ (node 5 ⟪ Black , 5 ⟫ leaf leaf) leaf) +tesr-rb-2 = rb-left-red refl ( rb-single 10 5) color : {n : Level} {A : Set n} → (rb : bt (Color ∧ A)) → Color color leaf = Red @@ -620,9 +633,9 @@ od d rd : ℕ tree rot : bt (Color ∧ A) origti : treeInvariant orig - origrb : RBtreeInvariant orig od - treerb : RBtreeInvariant tree d - replrb : RBtreeInvariant repl rd + origrb : RBtreeInvariant orig + treerb : RBtreeInvariant tree + replrb : RBtreeInvariant repl d=rd : ( d ≡ rd ) ∨ ( (suc d ≡ rd ) ∧ (color tree ≡ Red)) si : stackInvariant key tree orig stack rotated : rotatedTree tree rot @@ -637,48 +650,48 @@ -- find b in left and move the b to the right (one down or two down) -- rbi-case1 : {n : Level} {A : Set n} → {key d : ℕ} → {value : A} → (parent repl : bt (Color ∧ A) ) - → RBtreeInvariant parent (suc d) - → RBtreeInvariant repl d + → RBtreeInvariant parent + → RBtreeInvariant repl → {left right : bt (Color ∧ A) } → parent ≡ node key ⟪ Black , value ⟫ left right - → (color right ≡ Red → RBtreeInvariant (node key ⟪ Black , value ⟫ left repl ) (suc d)) - ∧ (color left ≡ Red → RBtreeInvariant (node key ⟪ Black , value ⟫ repl right ) (suc d)) -rbi-case1 {n} {A} {key} = ? + → (color right ≡ Red → RBtreeInvariant (node key ⟪ Black , value ⟫ left repl ) ) + ∧ (color left ≡ Red → RBtreeInvariant (node key ⟪ Black , value ⟫ repl right ) ) +rbi-case1 {n} {A} {key} = {!!} rbi-case31 : {n : Level} {A : Set n} → {kg kp d dp : ℕ} → {vg vp : A} → {cg : Color} → (tree grand parent repl : bt (Color ∧ A) ) - → RBtreeInvariant grand dp - → RBtreeInvariant repl d + → RBtreeInvariant grand + → RBtreeInvariant repl → {uncle left right : bt (Color ∧ A) } → grand ≡ node kg ⟪ cg , vg ⟫ uncle parent → parent ≡ node kp ⟪ Red , vp ⟫ left right → color uncle ≡ Red - → (color left ≡ Red → RBtreeInvariant (node kg ⟪ Red , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ repl right )) dp) - ∧ (color right ≡ Red → RBtreeInvariant (node kg ⟪ Red , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ left repl )) dp) -rbi-case31 {n} {A} {key} = ? + → (color left ≡ Red → RBtreeInvariant (node kg ⟪ Red , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ repl right )) ) + ∧ (color right ≡ Red → RBtreeInvariant (node kg ⟪ Red , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ left repl )) ) +rbi-case31 {n} {A} {key} = {!!} -- -- case4 increase the black depth -- rbi-case41 : {n : Level} {A : Set n} → {kg kp d dp : ℕ} → {vg vp : A} → {cg : Color} → (tree grand parent repl : bt (Color ∧ A) ) - → RBtreeInvariant grand dp - → RBtreeInvariant repl d + → RBtreeInvariant grand + → RBtreeInvariant repl → {uncle left right : bt (Color ∧ A) } → grand ≡ node kg ⟪ cg , vg ⟫ uncle parent → parent ≡ node kp ⟪ Red , vp ⟫ left right → color uncle ≡ Black - → (color left ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ repl right )) (suc dp) ) - ∧ (color right ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ left repl )) (suc dp) ) -rbi-case41 {n} {A} {key} = ? + → (color left ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ repl right )) ) + ∧ (color right ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ left repl )) ) +rbi-case41 {n} {A} {key} = {!!} rbi-case51 : {n : Level} {A : Set n} → {kg kp d dp : ℕ} → {vg vp : A} → {cg : Color} → (tree grand parent repl : bt (Color ∧ A) ) - → RBtreeInvariant grand dp - → RBtreeInvariant repl d + → RBtreeInvariant grand + → RBtreeInvariant repl → {uncle left right : bt (Color ∧ A) } → grand ≡ node kg ⟪ cg , vg ⟫ uncle parent → parent ≡ node kp ⟪ Red , vp ⟫ left right → color uncle ≡ Black - → (color left ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ repl right )) (suc dp) ) - ∧ (color right ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ left repl )) (suc dp) ) -rbi-case51 {n} {A} {key} = ? + → (color left ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ repl right )) ) + ∧ (color right ≡ Red → RBtreeInvariant (node kg ⟪ Black , vg ⟫ uncle (node kp ⟪ Black , vp ⟫ left repl )) ) +rbi-case51 {n} {A} {key} = {!!} --... | Black = record { -- d = ? ; od = RBI.od rbi ; rd = ? @@ -726,24 +739,24 @@ stackCase1 s-nil refl = refl PGtoRBinvariant : {n : Level} {A : Set n} → {key d0 ds dp dg : ℕ } → (tree orig : bt (Color ∧ A) ) - → RBtreeInvariant orig d0 + → RBtreeInvariant orig → (stack : List (bt (Color ∧ A))) → (pg : PG (Color ∧ A) tree stack ) - → RBtreeInvariant tree ds ∧ RBtreeInvariant (PG.parent pg) dp ∧ RBtreeInvariant (PG.grand pg) dg + → RBtreeInvariant tree ∧ RBtreeInvariant (PG.parent pg) ∧ RBtreeInvariant (PG.grand pg) PGtoRBinvariant = {!!} findRBT : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt (Color ∧ A) ) → (stack : List (bt (Color ∧ A))) → treeInvariant tree0 ∧ stackInvariant key tree tree0 stack - → {d : ℕ} → RBtreeInvariant tree0 d + → {d : ℕ} → RBtreeInvariant tree0 → (exit : (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack - → {d1 : ℕ} → RBtreeInvariant tree1 d1 + → {d1 : ℕ} → RBtreeInvariant tree1 → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t -findRBT {_} {_} {A} {t} key tree0 leaf stack ti {d} rb0 exit = ? +findRBT {_} {_} {A} {t} key tree0 leaf stack ti {d} rb0 exit = {!!} findRBT {_} {_} {A} {t} key tree0 (node key₁ value left right) stack ti {d} rb0 exit with <-cmp key key₁ -... | tri< a ¬b ¬c = ? -... | tri≈ ¬a b ¬c = ? -... | tri> ¬a ¬b c = ? +... | tri< a ¬b ¬c = {!!} +... | tri≈ ¬a b ¬c = {!!} +... | tri> ¬a ¬b c = {!!} rotateLeft : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) @@ -757,10 +770,10 @@ → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t -rotateLeft {n} {m} {A} {t} key value = ? where +rotateLeft {n} {m} {A} {t} key value = {!!} where rotateLeft1 : t - rotateLeft1 with stackToPG ? ? ? ? - ... | case1 x = ? -- {!!} {!!} {!!} {!!} rr + rotateLeft1 with stackToPG {!!} {!!} {!!} {!!} + ... | case1 x = {!!} -- {!!} {!!} {!!} {!!} rr ... | case2 (case1 x) = {!!} ... | case2 (case2 pg) = {!!} @@ -776,9 +789,9 @@ → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t -rotateRight {n} {m} {A} {t} key value = ? where +rotateRight {n} {m} {A} {t} key value = {!!} where rotateRight1 : t - rotateRight1 with stackToPG ? ? ? ? + rotateRight1 with stackToPG {!!} {!!} {!!} {!!} ... | case1 x = {!!} ... | case2 (case1 x) = {!!} ... | case2 (case2 pg) = {!!} @@ -795,18 +808,18 @@ → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t -insertCase5 {n} {m} {A} {t} key value = ? where +insertCase5 {n} {m} {A} {t} key value = {!!} where insertCase51 : t - insertCase51 with stackToPG ? ? ? ? + insertCase51 with stackToPG {!!} {!!} {!!} {!!} ... | case1 eq = {!!} ... | case2 (case1 eq ) = {!!} ... | case2 (case2 pg) with PG.pg pg - ... | s2-s1p2 x x₁ = ? -- rotateRight {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} next exit + ... | 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 + ... | 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) @@ -824,7 +837,7 @@ → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t -replaceRBP {n} {m} {A} {t} key value orig repl stack r next exit = ? where +replaceRBP {n} {m} {A} {t} key value orig repl stack r next exit = {!!} 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 @@ -832,27 +845,27 @@ 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 ⟪ 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) = ? + 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) = ? + 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) = ? + 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) = ? + 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 @@ -860,24 +873,24 @@ insertCase1 : t insertCase1 with stackToPG (RBI.tree r) orig stack (RBI.si r) ... | case1 eq = exit repl stack eq record { -- exit rot repl rbir (subst (λ k → rotatedTree k rot) (stackCase1 si eq) roti) ri - od = ? ; d = ? ; rd = ? - ; tree = RBI.tree r ; rot = ? - ; origti = ? - ; origrb = ? - ; treerb = ? - ; replrb = ? - ; d=rd = ? - ; si = ? - ; rotated = ? - ; ri = ? } - ... | case2 (case1 eq ) = ? where - insertCase12 : (to : bt (Color ∧ A)) → {d : ℕ} → RBtreeInvariant orig d → to ≡ orig - → (rr : bt (Color ∧ A)) → {dr : ℕ} → RBtreeInvariant ? dr → rr ≡ ? - → {stack : List (bt (Color ∧ A))} → (si : stackInvariant key ? to stack ) - → stack ≡ ? ∷ to ∷ [] → t - insertCase12 = ? + od = {!!} ; d = {!!} ; rd = {!!} + ; tree = RBI.tree r ; rot = {!!} + ; origti = {!!} + ; origrb = {!!} + ; treerb = {!!} + ; replrb = {!!} + ; d=rd = {!!} + ; si = {!!} + ; rotated = {!!} + ; ri = {!!} } + ... | case2 (case1 eq ) = {!!} where + insertCase12 : (to : bt (Color ∧ A)) → {d : ℕ} → RBtreeInvariant orig → to ≡ orig + → (rr : bt (Color ∧ A)) → {dr : ℕ} → RBtreeInvariant {!!} → rr ≡ {!!} + → {stack : List (bt (Color ∧ A))} → (si : stackInvariant key {!!} to stack ) + → stack ≡ {!!} ∷ to ∷ [] → t + insertCase12 = {!!} -- exit rot repl rbir ? ? - ... | case2 (case2 pg) = ? -- insertCase2 tree (PG.parent pg) (PG.uncle pg) (PG.grand pg) stack si (PG.pg pg) + ... | case2 (case2 pg) = {!!} -- insertCase2 tree (PG.parent pg) (PG.uncle pg) (PG.grand pg) stack si (PG.pg pg)