Mercurial > hg > Gears > GearsAgda
changeset 790:3bf3433a32d1
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 19 Oct 2023 12:01:08 +0900 |
| parents | b85b2a8e40c1 |
| children | 846663e9858b |
| files | hoareBinaryTree1.agda |
| diffstat | 1 files changed, 186 insertions(+), 195 deletions(-) [+] |
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--- a/hoareBinaryTree1.agda Thu Oct 19 11:49:58 2023 +0900 +++ b/hoareBinaryTree1.agda Thu Oct 19 12:01:08 2023 +0900 @@ -42,34 +42,34 @@ 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-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)) + → 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 ) + → 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 key value t₁ t₂) → treeInvariant (node key₂ value₂ t₃ t₄) - → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) + → treeInvariant (node key₁ value₁ (node key value t₁ t₂) (node key₂ value₂ t₃ t₄)) -- -- stack always contains original top at end (path of the tree) -- data stackInvariant {n : Level} {A : Set n} (key : ℕ) : (top orig : bt A) → (stack : List (bt A)) → Set n where s-nil : {tree0 : bt A} → stackInvariant key tree0 tree0 (tree0 ∷ []) - s-right : (tree tree0 tree₁ : bt A) → {key₁ : ℕ } → {v1 : A } → {st : List (bt A)} + 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)} + 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-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) + → 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) + → 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 @@ -124,7 +124,7 @@ 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 : {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} @@ -154,22 +154,22 @@ treeLeftDown : {n : Level} {A : Set n} {k : ℕ} {v1 : A} → (tree tree₁ : bt A ) → treeInvariant (node k v1 tree tree₁) - → treeInvariant tree + → 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 _ _ _ _) .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 + → 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 + → 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 @@ -211,7 +211,7 @@ si : stackInvariant key tree tree0 stack ri : replacedTree key value (child-replaced key tree ) repl ci : C tree repl stack -- data continuation - + record replacePR' {n : Level} {A : Set n} (key : ℕ) (value : A) (orig : bt A ) (stack : List (bt A)) : Set n where field tree repl : bt A @@ -219,13 +219,13 @@ si : stackInvariant key tree orig stack ri : replacedTree key value (child-replaced key tree) repl -- treeInvariant of tree and repl is inferred from ti, si and ri. - + replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A) → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) → (treeInvariant tree ) → ((tree1 : bt A) → treeInvariant tree1 → replacedTree key value (child-replaced key tree) tree1 → t) → t replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P) - (subst (λ j → replacedTree k v1 j (node k v1 t t₁) ) repl00 r-node) where + (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) @@ -234,7 +234,7 @@ 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 ⊤) + → (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 @@ -244,18 +244,18 @@ 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 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 = 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 : 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 @@ -264,7 +264,7 @@ ... | 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 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 @@ -272,10 +272,10 @@ repl02 with <-cmp key key₁ ... | tri< a ¬b ¬c = ⊥-elim ( ¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬c c) - ... | tri> ¬a ¬b c = refl + ... | 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 + 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 @@ -283,38 +283,38 @@ repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ leaf) ≡ leaf - repl07 with <-cmp key key₂ + 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 : 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 + repl07 with <-cmp key key₂ ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a x) ... | tri> ¬a ¬b c = ⊥-elim (¬a x) repl12 : replacedTree key value (child-replaced key tree1 ) repl repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07) ) (sym (rt-property-leaf (replacePR.ri Pre ))) r-leaf -- repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf -replaceP {n} {_} {A} key value {tree} repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre next exit with <-cmp key key₁ +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 + 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 : 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 + repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) @@ -330,15 +330,15 @@ 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)) + 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 : 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 + repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a) ... | tri> ¬a ¬b c = ⊥-elim (¬a a) @@ -354,10 +354,10 @@ 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 + 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 + 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 @@ -365,41 +365,41 @@ repl10 with si-property1 si ... | refl = si repl07 : child-replaced key (node key₂ v1 tree₁ tree) ≡ tree - repl07 with <-cmp key key₂ + 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 + 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 : 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 + 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 + 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 + 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 : 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 + repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) - ... | tri> ¬a ¬b c = refl + ... | 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) @@ -412,18 +412,18 @@ 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)) + 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 : 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 + repl03 with <-cmp key key₁ ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c) - ... | tri> ¬a ¬b c = refl + ... | 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 @@ -436,20 +436,20 @@ 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)) + 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) + → (r : Index) → (p : Invraiant r) → (loop : (r : Index) → Invraiant r → (next : (r1 : Index) → Invraiant r1 → reduce r1 < reduce r → t ) → t) → t TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r) -... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) -... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (m≤n⇒m≤1+n lt1) p1 lt1 ) where +... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) ) +... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (m≤n⇒m≤1+n lt1) p1 lt1 ) where TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → Invraiant r1 → reduce r1 < reduce r → t - TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1)) + TerminatingLoop1 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 = 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 ) + ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) open _∧_ @@ -457,17 +457,17 @@ → 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 _ 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) = +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) + 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 @@ -493,31 +493,31 @@ 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) = +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) = +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 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 → 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 ⟫ + $ 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 _ 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 : {n : Level} {A : Set n} → (tree : bt A) → Maybe A top-value leaf = nothing top-value (node key value tree tree₁) = just value @@ -531,7 +531,7 @@ Red : Color Black : Color -RB→bt : {n : Level} (A : Set n) → (bt (Color ∧ A)) → bt A +RB→bt : {n : Level} (A : Set n) → (bt (Color ∧ A)) → bt A RB→bt {n} A leaf = leaf RB→bt {n} A (node key ⟪ C , value ⟫ tr t1) = (node key value (RB→bt A tr) (RB→bt A t1)) @@ -541,28 +541,28 @@ 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 ⟪ 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)} → key < key₁ - → black-depth t ≡ black-depth t₁ - → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) + → black-depth t ≡ black-depth t₁ + → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) → RBtreeInvariant (node key ⟪ Red , value ⟫ leaf (node key₁ ⟪ Black , value₁ ⟫ t t₁)) rb-right-black : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} → key < key₁ → {c : Color} → black-depth t ≡ black-depth t₁ - → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) - → RBtreeInvariant (node key ⟪ Black , value ⟫ leaf (node key₁ ⟪ c , value₁ ⟫ t t₁)) + → 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)} → key < key₁ → black-depth t ≡ black-depth t₁ - → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) - → RBtreeInvariant (node key ⟪ Red , value ⟫ (node key₁ ⟪ Black , value₁ ⟫ t t₁) leaf ) + → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ t t₁) + → RBtreeInvariant (node key ⟪ Red , value ⟫ (node key₁ ⟪ Black , value₁ ⟫ t t₁) leaf ) rb-left-black : {key key₁ : ℕ} → {value value₁ : A} → {t t₁ : bt (Color ∧ A)} → key < key₁ → {c : Color} → black-depth t ≡ black-depth t₁ - → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) - → RBtreeInvariant (node key ⟪ Black , value ⟫ (node key₁ ⟪ c , value₁ ⟫ t t₁) leaf) + → RBtreeInvariant (node key₁ ⟪ c , value₁ ⟫ t t₁) + → RBtreeInvariant (node key ⟪ Black , value ⟫ (node key₁ ⟪ c , value₁ ⟫ t t₁) leaf) rb-node-red : {key key₁ key₂ : ℕ} → {value value₁ value₂ : A} → {t₁ t₂ t₃ t₄ : bt (Color ∧ A)} → key < key₁ → key₁ < key₂ → black-depth t₁ ≡ black-depth t₂ → RBtreeInvariant (node key ⟪ Black , value ⟫ t₁ t₂) @@ -574,7 +574,7 @@ → 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 key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄) → RBtreeInvariant (node key₁ ⟪ Black , value₁ ⟫ (node key ⟪ c , value ⟫ t₁ t₂) (node key₂ ⟪ c₁ , value₂ ⟫ t₃ t₄)) @@ -583,7 +583,7 @@ -- a b -- b c d a -- d e e c - -- + -- rtt-right : {ka kb kc kd ke : ℕ} {va vb vc vd ve : A} → {c d e c1 d1 e1 : bt A} → {ctl ctr dtl dtr etl etr : bt A} --kd < kb < ke < ka< kc → {ctl1 ctr1 dtl1 dtr1 etl1 etr1 : bt A} @@ -596,7 +596,7 @@ rtt-left : {ka kb kc kd ke : ℕ} {va vb vc vd ve : A} → {c d e c1 d1 e1 : bt A} → {ctl ctr dtl dtr etl etr : bt A} --kd < kb < ke < ka< kc - → {ctl1 ctr1 dtl1 dtr1 etl1 etr1 : bt A} -- after child + → {ctl1 ctr1 dtl1 dtr1 etl1 etr1 : bt A} -- after child → kd < kb → kb < ke → ke < ka → ka < kc → rotatedTree (node ke ve etl etr) (node ke ve etl1 etr1) → rotatedTree (node kd vd dtl dtr) (node kd vd dtl1 dtr1) @@ -606,7 +606,7 @@ RBtreeLeftDown : {n : Level} {A : Set n} {key : ℕ} {value : A} {c : Color} → (tleft tright : bt (Color ∧ A)) - → RBtreeInvariant (node key ⟪ c , value ⟫ tleft tright) + → RBtreeInvariant (node key ⟪ c , value ⟫ tleft tright) → RBtreeInvariant tleft RBtreeLeftDown leaf leaf (rb-single k1 v) = rb-leaf RBtreeLeftDown leaf (node key ⟪ Black , value ⟫ t1 t2 ) (rb-right-red x bde rbti) = rb-leaf @@ -623,8 +623,8 @@ RBtreeRightDown : {n : Level} {A : Set n} { key : ℕ} {value : A} {c : Color} → (tleft tright : bt (Color ∧ A)) - → RBtreeInvariant (node key ⟪ c , value ⟫ tleft tright) - → RBtreeInvariant tright + → RBtreeInvariant (node key ⟪ c , value ⟫ tleft tright) + → RBtreeInvariant tright RBtreeRightDown leaf leaf (rb-single k1 v1 ) = rb-leaf RBtreeRightDown leaf (node key ⟪ Black , value ⟫ t1 t2 ) (rb-right-red x bde rbti) = rbti RBtreeRightDown leaf (node key ⟪ Black , value ⟫ t1 t2 ) (rb-right-black x bde rbti) = rbti @@ -638,17 +638,17 @@ RBtreeRightDown (node key ⟪ Black , value ⟫ t t₁) (node key₁ ⟪ Red , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til bde2 tir) = tir RBtreeRightDown (node key ⟪ Red , value ⟫ t t₁) (node key₁ ⟪ Red , value1 ⟫ t1 t2) (rb-node-black x x1 bde1 til bde2 tir) = tir -findRBT : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt (Color ∧ A) ) +findRBT : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt (Color ∧ A) ) → (stack : List (bt (Color ∧ A))) → treeInvariant tree ∧ stackInvariant key tree tree0 stack → RBtreeInvariant tree → (next : (tree1 : bt (Color ∧ A) ) → (stack : List (bt (Color ∧ A))) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack - → RBtreeInvariant tree1 + → RBtreeInvariant tree1 → bt-depth tree1 < bt-depth tree → t ) - → (exit : (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) + → (exit : (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) → treeInvariant tree1 ∧ stackInvariant key tree1 tree0 stack - → RBtreeInvariant tree1 + → RBtreeInvariant tree1 → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key ) → t ) → t findRBT key leaf tree0 stack ti rb0 next exit = exit leaf stack ti rb0 (case1 refl) findRBT key n@(node key₁ value left right) tree0 stack ti rb0 next exit with <-cmp key key₁ @@ -661,21 +661,21 @@ data replacedRBTree {n : Level} {A : Set n} (key : ℕ) (value : A) : (before after : bt (Color ∧ A) ) → Set n where rbr-leaf : {ca cb : Color} → replacedRBTree key value leaf (node key ⟪ cb , value ⟫ leaf leaf) - rbr-node : {value₁ : A} → {ca cb : Color } → {t t₁ : bt (Color ∧ A)} → replacedRBTree key value (node key ⟪ ca , value₁ ⟫ t t₁) (node key ⟪ cb , value ⟫ t t₁) + rbr-node : {value₁ : A} → {ca cb : Color } → {t t₁ : bt (Color ∧ A)} → replacedRBTree key value (node key ⟪ ca , value₁ ⟫ t t₁) (node key ⟪ cb , value ⟫ t t₁) rbr-right : {k : ℕ } {v1 : A} → {ca cb : Color} → {t t1 t2 : bt (Color ∧ A)} - → k < key → replacedRBTree key value t2 t → replacedRBTree key value (node k ⟪ ca , v1 ⟫ t1 t2) (node k ⟪ cb , v1 ⟫ t1 t) + → k < key → replacedRBTree key value t2 t → replacedRBTree key value (node k ⟪ ca , v1 ⟫ t1 t2) (node k ⟪ cb , v1 ⟫ t1 t) rbr-left : {k : ℕ } {v1 : A} → {ca cb : Color} → {t t1 t2 : bt (Color ∧ A)} - → k < key → replacedRBTree key value t1 t → replacedRBTree key value (node k ⟪ ca , v1 ⟫ t1 t2) (node k ⟪ cb , v1 ⟫ t t2) + → k < key → replacedRBTree key value t1 t → replacedRBTree key value (node k ⟪ ca , v1 ⟫ t1 t2) (node k ⟪ cb , v1 ⟫ t t2) data ParentGrand {n : Level} {A : Set n} (self : bt A) : (parent uncle grand : bt A) → Set n where - s2-s1p2 : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } - → parent ≡ node kp vp self n1 → grand ≡ node kg vg parent n2 → ParentGrand self parent n2 grand - s2-1sp2 : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } - → parent ≡ node kp vp n1 self → grand ≡ node kg vg parent n2 → ParentGrand self parent n2 grand - s2-s12p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } - → parent ≡ node kp vp self n1 → grand ≡ node kg vg n2 parent → ParentGrand self parent n2 grand - s2-1s2p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } - → parent ≡ node kp vp n1 self → grand ≡ node kg vg n2 parent → ParentGrand self parent n2 grand + s2-s1p2 : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } + → parent ≡ node kp vp self n1 → grand ≡ node kg vg parent n2 → ParentGrand self parent n2 grand + s2-1sp2 : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } + → parent ≡ node kp vp n1 self → grand ≡ node kg vg parent n2 → ParentGrand self parent n2 grand + s2-s12p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } + → parent ≡ node kp vp self n1 → grand ≡ node kg vg n2 parent → ParentGrand self parent n2 grand + s2-1s2p : {kp kg : ℕ} {vp vg : A} → {n1 n2 : bt A} {parent grand : bt A } + → parent ≡ node kp vp n1 self → grand ≡ node kg vg n2 parent → ParentGrand self parent n2 grand record PG {n : Level } (A : Set n) (self : bt A) (stack : List (bt A)) : Set n where @@ -689,10 +689,10 @@ field od d rd : ℕ tree rot : bt (Color ∧ A) - origti : treeInvariant orig - origrb : RBtreeInvariant orig - treerb : RBtreeInvariant tree - replrb : RBtreeInvariant repl + origti : treeInvariant orig + 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 @@ -700,38 +700,38 @@ -- r (b , b) inserting a node into black node does not change the black depth, but color may change -- → b (b , r ) ∨ b (r , b) --- b (b , b) +-- b (b , b) -- → b (b , r ) ∨ b (r , b) --- b (r , b) inserting to right → b (r , r ) +-- b (r , b) inserting to right → b (r , r ) -- b (r , b) inserting to left may increse black depth, need rotation -- 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 - → RBtreeInvariant repl +rbi-case1 : {n : Level} {A : Set n} → {key d : ℕ} → {value : A} → (parent repl : bt (Color ∧ A) ) + → 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 ) ) ∧ (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 - → RBtreeInvariant repl - → {uncle left right : bt (Color ∧ A) } +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 + → 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 )) ) + → (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 - → RBtreeInvariant repl - → {uncle left right : bt (Color ∧ A) } +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 + → 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 @@ -739,10 +739,10 @@ ∧ (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 - → RBtreeInvariant repl - → {uncle left right : bt (Color ∧ A) } +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 + → 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 @@ -751,7 +751,7 @@ rbi-case51 {n} {A} {key} = {!!} --... | Black = record { --- d = ? ; od = RBI.od rbi ; rd = ? +-- d = ? ; od = RBI.od rbi ; rd = ? -- ; tree = ? ; rot = ? ; repl = ? -- ; treerb = ? ; replrb = ? -- ; d=rd = ? ; si = ? ; rotated = ? ; ri = ? @@ -764,9 +764,9 @@ → ( stack ≡ orig ∷ [] ) ∨ ( stack ≡ tree ∷ orig ∷ [] ) ∨ PG A tree stack stackToPG {n} {A} {key} tree .tree .(tree ∷ []) s-nil = case1 refl stackToPG {n} {A} {key} tree .(node _ _ _ tree) .(tree ∷ node _ _ _ tree ∷ []) (s-right _ _ _ x s-nil) = case2 (case1 refl) -stackToPG {n} {A} {key} tree .(node k2 v2 t5 (node k1 v1 t2 tree)) (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ [])) (s-right tree (node k2 v2 t5 (node k1 v1 t2 tree)) t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) (node k2 v2 t5 (node k1 v1 t2 tree)) t5 {k2} {v2} x₁ s-nil)) = case2 (case2 +stackToPG {n} {A} {key} tree .(node k2 v2 t5 (node k1 v1 t2 tree)) (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ [])) (s-right tree (node k2 v2 t5 (node k1 v1 t2 tree)) t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) (node k2 v2 t5 (node k1 v1 t2 tree)) t5 {k2} {v2} x₁ s-nil)) = case2 (case2 record { parent = node k1 v1 t2 tree ; grand = _ ; pg = s2-1s2p refl refl ; rest = _ ; stack=gp = refl } ) -stackToPG {n} {A} {key} tree orig (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ _)) (s-right tree orig t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) orig t5 {k2} {v2} x₁ (s-right _ _ _ x₂ si))) = case2 (case2 +stackToPG {n} {A} {key} tree orig (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ _)) (s-right tree orig t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) orig t5 {k2} {v2} x₁ (s-right _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 t2 tree ; grand = _ ; pg = s2-1s2p refl refl ; rest = _ ; stack=gp = refl } ) stackToPG {n} {A} {key} tree orig (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ _)) (s-right tree orig t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) orig t5 {k2} {v2} x₁ (s-left _ _ _ x₂ si))) = case2 (case2 record { parent = node k1 v1 t2 tree ; grand = _ ; pg = s2-1s2p refl refl ; rest = _ ; stack=gp = refl } ) @@ -796,24 +796,24 @@ stackCase1 s-nil refl = refl PGtoRBinvariant : {n : Level} {A : Set n} → {key d0 ds dp dg : ℕ } → (tree orig : bt (Color ∧ A) ) - → RBtreeInvariant orig + → RBtreeInvariant orig → (stack : List (bt (Color ∧ A))) → (pg : PG (Color ∧ A) tree stack ) - → RBtreeInvariant tree ∧ RBtreeInvariant (PG.parent pg) ∧ RBtreeInvariant (PG.grand pg) + → RBtreeInvariant tree ∧ RBtreeInvariant (PG.parent pg) ∧ RBtreeInvariant (PG.grand pg) PGtoRBinvariant = {!!} -RBI-child-replaced : {n : Level} {A : Set n} (tr : bt (Color ∧ A)) (key : ℕ) → RBtreeInvariant tr → RBtreeInvariant (child-replaced key tr) +RBI-child-replaced : {n : Level} {A : Set n} (tr : bt (Color ∧ A)) (key : ℕ) → RBtreeInvariant tr → RBtreeInvariant (child-replaced key tr) RBI-child-replaced {n} {A} leaf key rbi = rbi RBI-child-replaced {n} {A} (node key₁ value tr tr₁) key rbi with <-cmp key key₁ ... | tri< a ¬b ¬c = RBtreeLeftDown _ _ rbi ... | tri≈ ¬a b ¬c = rbi ... | tri> ¬a ¬b c = RBtreeRightDown _ _ rbi --- findRBT : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt (Color ∧ A) ) +-- 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 --- → (exit : (tree1 : bt (Color ∧ A)) → (stack : List (bt (Color ∧ A))) +-- → {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 : 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 (node key₁ value left right) stack ti {d} rb0 exit with <-cmp key key₁ @@ -822,12 +822,12 @@ --... | tri> ¬a ¬b c = {!!} rotateLeft : {n m : Level} {A : Set n} {t : Set m} - → (key : ℕ) → (value : A) - → (orig tree : bt (Color ∧ A)) - → (stack : List (bt (Color ∧ A))) + → (key : ℕ) → (value : A) + → (orig tree : bt (Color ∧ A)) + → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig tree stack ) - → (next : (current : bt (Color ∧ A)) → (stack1 : List (bt (Color ∧ A))) - → (r : RBI key value orig current stack1 ) + → (next : (current : bt (Color ∧ A)) → (stack1 : List (bt (Color ∧ A))) + → (r : RBI key value orig current stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) @@ -841,12 +841,12 @@ ... | case2 (case2 pg) = {!!} rotateRight : {n m : Level} {A : Set n} {t : Set m} - → (key : ℕ) → (value : A) - → (orig tree : bt (Color ∧ A)) - → (stack : List (bt (Color ∧ A))) + → (key : ℕ) → (value : A) + → (orig tree : bt (Color ∧ A)) + → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig tree stack ) - → (next : (current : bt (Color ∧ A)) → (stack1 : List (bt (Color ∧ A))) - → (r : RBI key value orig current stack1 ) + → (next : (current : bt (Color ∧ A)) → (stack1 : List (bt (Color ∧ A))) + → (r : RBI key value orig current stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) @@ -860,12 +860,12 @@ ... | case2 (case2 pg) = {!!} insertCase5 : {n m : Level} {A : Set n} {t : Set m} - → (key : ℕ) → (value : A) - → (orig tree : bt (Color ∧ A)) - → (stack : List (bt (Color ∧ A))) + → (key : ℕ) → (value : A) + → (orig tree : bt (Color ∧ A)) + → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig tree stack ) - → (next : (tree1 : (bt (Color ∧ A))) → (stack1 : List (bt (Color ∧ A))) - → (r : RBI key value orig tree1 stack1 ) + → (next : (tree1 : (bt (Color ∧ A))) → (stack1 : List (bt (Color ∧ A))) + → (r : RBI key value orig tree1 stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) @@ -883,31 +883,31 @@ ... | 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) + -- = insertCase2 tree (PG.parent pg) (PG.uncle pg) (PG.grand pg) stack si (PG.pg pg) -- if we have replacedNode on RBTree, we have RBtreeInvariant replaceRBP : {n m : Level} {A : Set n} {t : Set m} - → (key : ℕ) → (value : A) - → (orig repl : bt (Color ∧ A)) - → (stack : List (bt (Color ∧ A))) + → (key : ℕ) → (value : A) + → (orig repl : bt (Color ∧ A)) + → (stack : List (bt (Color ∧ A))) → (r : RBI key value orig repl stack ) - → (next : (repl1 : (bt (Color ∧ A))) → (stack1 : List (bt (Color ∧ A))) - → (r : RBI key value orig repl1 stack1 ) + → (next : (repl1 : (bt (Color ∧ A))) → (stack1 : List (bt (Color ∧ A))) + → (r : RBI key value orig repl1 stack1 ) → length stack1 < length stack → t ) → (exit : (repl : bt (Color ∧ A) ) → (stack1 : List (bt (Color ∧ A))) → stack1 ≡ (orig ∷ []) → RBI key value orig repl stack1 → t ) → t replaceRBP {n} {m} {A} {t} key value orig repl stack r next exit = insertCase1 where - insertCase2 : (tree parent uncle grand : bt (Color ∧ A)) - → (stack : List (bt (Color ∧ A))) → (si : stackInvariant key tree orig stack ) + insertCase2 : (tree parent uncle grand : bt (Color ∧ A)) + → (stack : List (bt (Color ∧ A))) → (si : stackInvariant key tree orig stack ) → (pg : ParentGrand tree parent uncle grand ) → t - insertCase2 tree leaf uncle grand stack si (s2-s1p2 () x₁) - insertCase2 tree leaf uncle grand stack si (s2-1sp2 () x₁) - insertCase2 tree leaf uncle grand stack si (s2-s12p () x₁) - insertCase2 tree leaf uncle grand stack si (s2-1s2p () x₁) + insertCase2 tree leaf uncle grand stack si (s2-s1p2 () x₁) + insertCase2 tree leaf uncle grand stack si (s2-1sp2 () x₁) + insertCase2 tree leaf uncle grand stack si (s2-s12p () x₁) + insertCase2 tree leaf uncle grand stack si (s2-1s2p () x₁) insertCase2 tree parent@(node kp ⟪ Red , _ ⟫ _ _) uncle grand stack si pg = {!!} -- next {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) leaf grand stack si pg = {!!} insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Red , _ ⟫ _ _ ) grand stack si pg = {!!} -- next {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} @@ -917,13 +917,13 @@ -- node kp ⟪ Black , proj3 ⟫ left right ≡ node kp₁ vp tree n1 -- grand ≡ node kg vg (node kp ⟪ Black , proj3 ⟫ left right) (node ku ⟪ Black , proj4 ⟫ left₁ right₁) insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Black , _ ⟫ _ _) grand stack si (s2-1sp2 x x₁) = {!!} - -- rotateLeft key value orig tree {!!} repl rbio {!!} {!!} stack si {!!} ri {!!} {!!} + -- rotateLeft key value orig tree {!!} repl rbio {!!} {!!} stack si {!!} ri {!!} {!!} -- (λ a b c d e f h i j k l m → insertCase5 key value a b c d {!!} {!!} h i j k l {!!} {!!} next exit ) exit -- tree is right of parent, parent is left of grand rotateLeft -- node kp ⟪ Black , proj3 ⟫ left right ≡ node kp₁ vp n1 tree -- grand ≡ node kg vg (node kp ⟪ Black , proj3 ⟫ left right) (node ku ⟪ Black , proj4 ⟫ left₁ right₁) insertCase2 tree parent@(node kp ⟪ Black , _ ⟫ _ _) (node ku ⟪ Black , _ ⟫ _ _) grand stack si (s2-s12p x x₁) = {!!} - -- rotateRight key value orig tree {!!} repl rbio {!!} {!!} stack si {!!} ri {!!} {!!} + -- 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 @@ -935,39 +935,30 @@ -- grand ≡ node kg vg (node ku ⟪ Black , proj4 ⟫ left₁ right₁) (node kp ⟪ Black , proj3 ⟫ left right) insertCase1 : t insertCase1 with stackToPG (RBI.tree r) orig stack (RBI.si r) - ... | case1 eq = exit repl stack eq r - ... | case2 (case1 eq) = insertCase12 orig (RBI.origrb r) refl (child-replaced key orig) (RBI-child-replaced orig key (RBI.origrb r) ) refl - ( child-replaced key orig ∷ orig ∷ []) (rb02 orig refl (RBI.si r)) refl where -- insertCase12 + ... | case1 eq = exit repl stack eq r + ... | case2 (case1 eq) = rb02 orig refl (RBI.si r) where rb01 : stackInvariant key (RBI.tree r) orig stack rb01 = RBI.si r rb02 : (tr0 : bt (Color ∧ A)) → tr0 ≡ orig → stackInvariant key (RBI.tree r) orig stack - → stackInvariant key (child-replaced key orig) orig (child-replaced key orig ∷ orig ∷ []) - rb02 leaf eq si = ? - rb02 (node key₁ value tr tr₁) eq si with <-cmp key key₁ - ... | tri< a ¬b ¬c = s-left _ _ _ a ? - ... | tri≈ ¬a b ¬c = ? - ... | tri> ¬a ¬b c = s-right _ _ _ c ? - insertCase12 : (to : bt (Color ∧ A)) → RBtreeInvariant orig → to ≡ orig - → (rr : bt (Color ∧ A)) → RBtreeInvariant rr → rr ≡ (child-replaced key orig) - → (stack1 : List (bt (Color ∧ A))) → (si : stackInvariant key rr to stack1 ) - → stack1 ≡ rr ∷ to ∷ [] → t - insertCase12 to or t≡o rr rbt r≡r stack1 sti stack≡ = next rr stack1 - record { - od = RBI.od r ; d = RBI.rd r ; rd = RBI.rd r ; --r - tree = {!!} ; rot = RBI.rot r - ; origti = RBI.origti r - ; origrb = RBI.origrb r - ; treerb = {!!} --subst (λ k → RBtreeInvariant k) (sym t≡o) (or) - ; replrb = rbt --subst (λ k → RBtreeInvariant k) (sym t≡o) (RBI.origrb r) - ; d=rd = {!!} - ; si = {!!} --subst (λ k → stackInvariant key rr k stack1) (t≡o) (sti) - ; rotated = {!!} - ; ri = {!!} --subst (λ k → replacedRBTree key value (child-replaced key (RBI.rot r)) k) (sym t≡o) ({!ct!}) - } - {!!} - -- exit rot repl rbir ? ? - ... | case2 (case2 pg) = {!!} -- insertCase2 tree (PG.parent pg) (PG.uncle pg) (PG.grand pg) stack si (PG.pg pg) + → t + rb02 leaf eq si = ? -- can't happen + rb02 (node key₁ value left right) eq si with <-cmp key key₁ + ... | tri< a ¬b ¬c = exit ? ? ? record { + od = RBI.od r ; d = RBI.rd r ; rd = RBI.rd r ; --r + tree = {!!} ; rot = RBI.rot r + ; origti = RBI.origti r + ; origrb = RBI.origrb r + ; treerb = {!!} --subst (λ k → RBtreeInvariant k) (sym t≡o) (or) + ; replrb = ? --subst (λ k → RBtreeInvariant k) (sym t≡o) (RBI.origrb r) + ; d=rd = {!!} + ; si = {!!} --subst (λ k → stackInvariant key rr k stack1) (t≡o) (sti) + ; rotated = {!!} + ; ri = {!!} --subst (λ k → replacedRBTree key value (child-replaced key (RBI.rot r)) k) (sym t≡o) ({!ct!}) + } + ... | tri≈ ¬a b ¬c = ? -- can't happen + ... | tri> ¬a ¬b c = ? + ... | case2 (case2 pg) = {!!} -- insertCase2 tree (PG.parent pg) (PG.uncle pg) (PG.grand pg) stack si (PG.pg pg)