Mercurial > hg > Members > Moririn
changeset 722:b088fa199d3d
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 09 Apr 2023 17:15:42 +0900 |
| parents | 2abfce56523a |
| children | 43180a01bfbe |
| files | ModelChecking.agda hoareBinaryTree1.agda redBlackTreeHoare.agda |
| diffstat | 3 files changed, 574 insertions(+), 2 deletions(-) [+] |
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--- a/ModelChecking.agda Sun May 22 19:07:20 2022 +0900 +++ b/ModelChecking.agda Sun Apr 09 17:15:42 2023 +0900 @@ -261,7 +261,7 @@ -- state db ( binary tree of processes ) --- depth first ececution +-- depth first execution -- verify temporal logic poroerries
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/hoareBinaryTree1.agda Sun Apr 09 17:15:42 2023 +0900 @@ -0,0 +1,572 @@ +module hoareBinaryTree1 where + +open import Level renaming (zero to Z ; suc to succ) + +open import Data.Nat hiding (compare) +open import Data.Nat.Properties as NatProp +open import Data.Maybe +-- open import Data.Maybe.Properties +open import Data.Empty +open import Data.List +open import Data.Product + +open import Function as F hiding (const) + +open import Relation.Binary +open import Relation.Binary.PropositionalEquality +open import Relation.Nullary +open import logic + + +_iso_ : {n : Level} {a : Set n} → ℕ → ℕ → Set +d iso d' = (¬ (suc d ≤ d')) ∧ (¬ (suc d' ≤ d)) + +iso-intro : {n : Level} {a : Set n} {x y : ℕ} → ¬ (suc x ≤ y) → ¬ (suc y ≤ x) → _iso_ {n} {a} x y +iso-intro = λ z z₁ → record { proj1 = z ; proj2 = z₁ } + +-- +-- +-- no children , having left node , having right node , having both +-- +data bt {n : Level} (A : Set n) : Set n where + leaf : bt A + node : (key : ℕ) → (value : A) → + (left : bt A ) → (right : bt A ) → bt A + +node-key : {n : Level} {A : Set n} → bt A → Maybe ℕ +node-key (node key _ _ _) = just key +node-key _ = nothing + +node-value : {n : Level} {A : Set n} → bt A → Maybe A +node-value (node _ value _ _) = just value +node-value _ = nothing + +bt-depth : {n : Level} {A : Set n} → (tree : bt A ) → ℕ +bt-depth leaf = 0 +bt-depth (node key value t t₁) = suc (Data.Nat._⊔_ (bt-depth t ) (bt-depth t₁ )) + +open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_) + +data treeInvariant {n : Level} {A : Set n} : (tree : bt A) → Set n where + t-leaf : treeInvariant leaf + t-single : (key : ℕ) → (value : A) → treeInvariant (node key value leaf leaf) + t-right : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key < key₁) → treeInvariant (node key₁ value₁ t₁ t₂) + → treeInvariant (node key value leaf (node key₁ value₁ t₁ t₂)) + t-left : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → (key < key₁) → treeInvariant (node key value t₁ t₂) + → treeInvariant (node key₁ value₁ (node key value t₁ t₂) leaf ) + t-node : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} → (key < key₁) → (key₁ < key₂) + → treeInvariant (node key value t₁ t₂) + → treeInvariant (node key₂ value₂ t₃ t₄) + → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) + +-- +-- stack always contains original top at end +-- +data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where + s-single : {tree0 : bt A} → stackInvariant key tree0 tree0 (tree0 ∷ []) + s-right : {tree tree0 tree₁ : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key₁ < key → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree tree0 (tree ∷ st) + s-left : {tree₁ tree0 tree : bt A} → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + → key < key₁ → stackInvariant key (node key₁ v1 tree₁ tree) tree0 st → stackInvariant key tree₁ tree0 (tree₁ ∷ st) + +data replacedTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (before after : bt A ) → Set n where + r-leaf : replacedTree key value leaf (node key value leaf leaf) + r-node : {value₁ : A} → {t t₁ : bt A} → replacedTree key value (node key value₁ t t₁) (node key value t t₁) + r-right : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} + → k < key → replacedTree key value t2 t → replacedTree key value (node k v1 t1 t2) (node k v1 t1 t) + r-left : {k : ℕ } {v1 : A} → {t t1 t2 : bt A} + → key < k → replacedTree key value t1 t → replacedTree key value (node k v1 t1 t2) (node k v1 t t2) + +add< : { i : ℕ } (j : ℕ ) → i < suc i + j +add< {i} j = begin + suc i ≤⟨ m≤m+n (suc i) j ⟩ + suc i + j ∎ where open ≤-Reasoning + +treeTest1 : bt ℕ +treeTest1 = node 0 0 leaf (node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf)) +treeTest2 : bt ℕ +treeTest2 = node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf) + +treeInvariantTest1 : treeInvariant treeTest1 +treeInvariantTest1 = t-right (m≤m+n _ 2) (t-node (add< 0) (add< 1) (t-left (add< 0) (t-single 1 7)) (t-single 5 5) ) + +stack-top : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) +stack-top [] = nothing +stack-top (x ∷ s) = just x + +stack-last : {n : Level} {A : Set n} (stack : List (bt A)) → Maybe (bt A) +stack-last [] = nothing +stack-last (x ∷ []) = just x +stack-last (x ∷ s) = stack-last s + +stackInvariantTest1 : stackInvariant 4 treeTest2 treeTest1 ( treeTest2 ∷ treeTest1 ∷ [] ) +stackInvariantTest1 = s-right (add< 3) (s-single ) + +si-property0 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 stack → ¬ ( stack ≡ [] ) +si-property0 (s-single ) () +si-property0 (s-right x si) () +si-property0 (s-left x si) () + +si-property1 : {n : Level} {A : Set n} {key : ℕ} {tree tree0 tree1 : bt A} → {stack : List (bt A)} → stackInvariant key tree tree0 (tree1 ∷ stack) + → tree1 ≡ tree +si-property1 (s-single ) = refl +si-property1 (s-right _ si) = refl +si-property1 (s-left _ si) = refl + +si-property-last : {n : Level} {A : Set n} (key : ℕ) (tree tree0 : bt A) → (stack : List (bt A)) → stackInvariant key tree tree0 stack + → stack-last stack ≡ just tree0 +si-property-last key t t0 (t ∷ []) (s-single ) = refl +si-property-last key t t0 (.t ∷ x ∷ st) (s-right _ si ) with si-property1 si +... | refl = si-property-last key x t0 (x ∷ st) si +si-property-last key t t0 (.t ∷ x ∷ st) (s-left _ si ) with si-property1 si +... | refl = si-property-last key x t0 (x ∷ st) si + +ti-right : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 tree₁ repl) → treeInvariant repl +ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf +ti-right {_} {_} {.leaf} {_} {key₁} {v1} (t-right x ti) = ti +ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-left x ti) = t-leaf +ti-right {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti₁ + +ti-left : {n : Level} {A : Set n} {tree₁ repl : bt A} → {key₁ : ℕ} → {v1 : A} → treeInvariant (node key₁ v1 repl tree₁ ) → treeInvariant repl +ti-left {_} {_} {.leaf} {_} {key₁} {v1} (t-single .key₁ .v1) = t-leaf +ti-left {_} {_} {_} {_} {key₁} {v1} (t-right x ti) = t-leaf +ti-left {_} {_} {_} {_} {key₁} {v1} (t-left x ti) = ti +ti-left {_} {_} {.(node _ _ _ _)} {_} {key₁} {v1} (t-node x x₁ ti ti₁) = ti + +stackTreeInvariant : {n : Level} {A : Set n} (key : ℕ) (sub tree : bt A) → (stack : List (bt A)) + → treeInvariant tree → stackInvariant key sub tree stack → treeInvariant sub +stackTreeInvariant {_} {A} key sub tree (sub ∷ []) ti (s-single ) = ti +stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-right _ si ) = ti-right (si1 si) where + si1 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 tree₁ sub ) tree st → treeInvariant (node key₁ v1 tree₁ sub ) + si1 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 tree₁ sub ) tree st ti si +stackTreeInvariant {_} {A} key sub tree (sub ∷ st) ti (s-left _ si ) = ti-left ( si2 si) where + si2 : {tree₁ : bt A} → {key₁ : ℕ} → {v1 : A} → stackInvariant key (node key₁ v1 sub tree₁ ) tree st → treeInvariant (node key₁ v1 sub tree₁ ) + si2 {tree₁ } {key₁ } {v1 } si = stackTreeInvariant key (node key₁ v1 sub tree₁ ) tree st ti si + +rt-property1 : {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf ) +rt-property1 {n} {A} key value .leaf .(node key value leaf leaf) r-leaf () +rt-property1 {n} {A} key value .(node key _ _ _) .(node key value _ _) r-node () +rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-right x rt) = λ () +rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-left x rt) = λ () + +rt-property-leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {repl : bt A} → replacedTree key value leaf repl → repl ≡ node key value leaf leaf +rt-property-leaf r-leaf = refl + +rt-property-¬leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {tree : bt A} → ¬ replacedTree key value tree leaf +rt-property-¬leaf () + +rt-property-key : {n : Level} {A : Set n} {key key₂ key₃ : ℕ} {value value₂ value₃ : A} {left left₁ right₂ right₃ : bt A} + → replacedTree key value (node key₂ value₂ left right₂) (node key₃ value₃ left₁ right₃) → key₂ ≡ key₃ +rt-property-key r-node = refl +rt-property-key (r-right x ri) = refl +rt-property-key (r-left x ri) = refl + +nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ +nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x +nat-<> : { x y : ℕ } → x < y → y < x → ⊥ +nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x + +open _∧_ + + +depth-1< : {i j : ℕ} → suc i ≤ suc (i Data.Nat.⊔ j ) +depth-1< {i} {j} = s≤s (m≤m⊔n _ j) + +depth-2< : {i j : ℕ} → suc i ≤ suc (j Data.Nat.⊔ i ) +depth-2< {i} {j} = s≤s (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-single ⟫ + $ λ p P loop → findP key (proj1 p) tree (proj2 p) P (λ t s P1 lt → loop ⟪ t , s ⟫ P1 lt ) + $ λ t s P C → replaceNodeP key value t C (proj1 P) + $ λ t1 P1 R → TerminatingLoopS (List (bt A) ∧ bt A ∧ bt A ) + {λ p → replacePR key value (proj1 (proj2 p)) (proj2 (proj2 p)) (proj1 p) (λ _ _ _ → Lift n ⊤ ) } + (λ p → length (proj1 p)) ⟪ s , ⟪ t , t1 ⟫ ⟫ record { tree0 = tree ; ti = P0 ; si = proj2 P ; ri = R ; ci = lift tt } + $ λ p P1 loop → replaceP key value (proj2 (proj2 p)) (proj1 p) P1 + (λ key value {tree1} repl1 stack P2 lt → loop ⟪ stack , ⟪ tree1 , repl1 ⟫ ⟫ P2 lt ) + $ λ tree repl P → exit tree repl ⟪ RTtoTI0 _ _ _ _ (proj1 P) (proj2 P) , proj2 P ⟫ + +insertTestP1 = insertTreeP leaf 1 1 t-leaf + $ λ _ 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 _ → 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 rbInvariant {n : Level} {A : Set n} : bt A → Color → ℕ → Set n where + rb-leaf : rbInvariant leaf Black 0 + rb-single : (key : ℕ) → (value : A) → (c : Color) → rbInvariant (node key value leaf leaf) c 1 + t-right-red : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → {d : ℕ } → rbInvariant (node key₁ value₁ t₁ t₂) Black d + → rbInvariant (node key value leaf (node key₁ value₁ t₁ t₂)) Red d + t-right-black : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → {c : Color} → {d : ℕ }→ rbInvariant (node key₁ value₁ t₁ t₂) c d + → rbInvariant (node key value leaf (node key₁ value₁ t₁ t₂)) Black (suc d) + t-left-red : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → {d : ℕ} → rbInvariant (node key value t₁ t₂) Black d + → rbInvariant (node key₁ value₁ (node key value t₁ t₂) leaf ) Red d + t-left-black : {key key₁ : ℕ} → {value value₁ : A} → {t₁ t₂ : bt A} → {c : Color} → {d : ℕ} → rbInvariant (node key value t₁ t₂) c d + → rbInvariant (node key₁ value₁ (node key value t₁ t₂) leaf ) Black (suc d) + t-node-red : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} {d : ℕ} + → rbInvariant (node key value t₁ t₂) Black d + → rbInvariant (node key₂ value₂ t₃ t₄) Black d + → rbInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) Red d + t-node-black : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt A} {c c1 : Color} {d : ℕ} + → rbInvariant (node key value t₁ t₂) c d + → rbInvariant (node key₂ value₂ t₃ t₄) c1 d + → rbInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) Black (suc d) +
--- a/redBlackTreeHoare.agda Sun May 22 19:07:20 2022 +0900 +++ b/redBlackTreeHoare.agda Sun Apr 09 17:15:42 2023 +0900 @@ -6,7 +6,7 @@ open import Data.Nat hiding (compare) open import Data.Nat.Properties as NatProp open import Data.Maybe -open import Data.Bool +open import Data.Bool hiding ( _<_ ) open import Data.Empty open import Relation.Binary