--- a/hoareBinaryTree1.agda	Mon May 27 08:46:10 2024 +0900
+++ b/hoareBinaryTree1.agda	Mon May 27 17:32:41 2024 +0900
@@ -884,594 +884,598 @@
 RB-repl→eq  : {n : Level} {A : Set n}  → (tree repl : bt (Color ∧ A)) → {key : ℕ} → {value : A}
      → RBtreeInvariant tree
      → replacedRBTree key value tree repl → black-depth tree ≡ black-depth repl
-RB-repl→eq {n} {A} .leaf .(node _ ⟪ Red , _ ⟫ leaf leaf) rbt rbr-leaf = refl
-RB-repl→eq {n} {A} (node _ ⟪ Red , _ ⟫ _ _) .(node _ ⟪ Red , _ ⟫ _ _) rbt rbr-node = refl
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) rbt rbr-node = refl
-RB-repl→eq {n} {A} (node _ ⟪ Red , _ ⟫ left _) .(node _ ⟪ Red , _ ⟫ left _) (rb-red _ _ x₂ x₃ x₄ rbt rbt₁) (rbr-right x x₁ t) =
-   cong (λ k → black-depth left ⊔ k ) (RB-repl→eq _ _ rbt₁ t)
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ left _) .(node _ ⟪ Black , _ ⟫ left _) (rb-black _ _ x₂ rbt rbt₁) (rbr-right x x₁ t) =
-   cong (λ k → suc (black-depth left ⊔ k)) (RB-repl→eq _ _ rbt₁ t)
-RB-repl→eq {n} {A} (node _ ⟪ Red , _ ⟫ _ right) .(node _ ⟪ Red , _ ⟫ _ right) (rb-red _ _ x₂ x₃ x₄ rbt rbt₁) (rbr-left x x₁ t) = cong (λ k → k ⊔ black-depth right) (RB-repl→eq _ _ rbt t)
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ right) .(node _ ⟪ Black , _ ⟫ _ right) (rb-black _ _ x₂ rbt rbt₁) (rbr-left x x₁ t) = cong (λ k → suc (k ⊔ black-depth right)) (RB-repl→eq _ _ rbt t)
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ t₁ _) .(node _ ⟪ Black , _ ⟫ t₁ _) (rb-black _ _ x₂ rbt rbt₁) (rbr-black-right x x₁ t) = cong (λ k → suc (black-depth t₁ ⊔ k)) (RB-repl→eq _ _ rbt₁ t)
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ t₁) .(node _ ⟪ Black , _ ⟫ _ t₁) (rb-black _ _ x₂ rbt rbt₁) (rbr-black-left x x₁ t) = cong (λ k → suc (k ⊔ black-depth t₁)) (RB-repl→eq _ _ rbt t)
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ t₁ t₂) t₃) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ t₄ t₂) (to-black t₃)) (rb-black _ _ x₃ (rb-red _ _ x₄ x₅ x₆ rbt rbt₂) rbt₁) (rbr-flip-ll {_} {_} {t₄} x x₁ x₂ t) = begin
-   suc (black-depth t₁ ⊔ black-depth t₂ ⊔ black-depth t₃)  ≡⟨ cong (λ k → suc (k ⊔ black-depth t₂ ⊔ black-depth t₃)) (RB-repl→eq _ _ rbt t) ⟩
-   suc (black-depth t₄ ⊔ black-depth t₂) ⊔ suc (black-depth t₃)  ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ black-depth t₂) ⊔ k )  (  to-black-eq t₃ x₁ ) ⟩
-   suc (black-depth t₄ ⊔ black-depth t₂) ⊔ black-depth (to-black t₃) ∎
-     where open ≡-Reasoning
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ t₁ t₂) t₃) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ t₁ t₄) (to-black t₃)) (rb-black _ _ x₄ (rb-red _ _ x₅ x₆ x₇ rbt rbt₂) rbt₁) (rbr-flip-lr {_} {_} {t₄} x x₁ x₂ x₃ t) = begin
-   suc (black-depth t₁ ⊔ black-depth t₂) ⊔ suc (black-depth t₃)  ≡⟨ cong (λ k → suc (black-depth t₁ ⊔ black-depth t₂) ⊔ k )  (  to-black-eq t₃ x₁ ) ⟩
-   suc (black-depth t₁ ⊔ black-depth t₂) ⊔ black-depth (to-black t₃) ≡⟨ cong (λ k → suc (black-depth t₁ ⊔ k ) ⊔ black-depth (to-black t₃)) (RB-repl→eq _ _ rbt₂ t) ⟩
-   suc (black-depth t₁ ⊔ black-depth t₄) ⊔ black-depth (to-black t₃) ∎
-     where open ≡-Reasoning
-RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black t₄) (node _ ⟪ Black , _ ⟫ t₃ t₁)) (rb-black _ _ x₄ rbt (rb-red _ _ x₅ x₆ x₇ rbt₁ rbt₂)) (rbr-flip-rl {t₁} {t₂} {t₃} {t₄} x x₁ x₂ x₃ t) = begin
-   suc (black-depth t₄ ⊔ (black-depth t₂ ⊔ black-depth t₁)) ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ ( k  ⊔ black-depth t₁)) ) (RB-repl→eq _ _ rbt₁ t) ⟩
-   suc (black-depth t₄ ⊔ (black-depth t₃ ⊔ black-depth t₁)) ≡⟨ cong (λ k → k ⊔ suc (black-depth t₃ ⊔ black-depth t₁)) ( to-black-eq t₄ x₁ ) ⟩
-   black-depth (to-black t₄) ⊔ suc (black-depth t₃ ⊔ black-depth t₁) ∎
-     where open ≡-Reasoning
-RB-repl→eq {n} {A} .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) (rb-black _ _ x₃ rbt (rb-red _ _ x₄ x₅ x₆ rbt₁ rbt₂)) (rbr-flip-rr {t₁} {t₂} {t₃} {t₄} x x₁ x₂ t) = begin
-   suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₂)) ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ k ))) ( RB-repl→eq _ _ rbt₂ t) ⟩
-   suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₃)) ≡⟨ cong (λ k → k ⊔ suc (black-depth t₁ ⊔ black-depth t₃)) ( to-black-eq t₄ x₁ ) ⟩
-   black-depth (to-black t₄) ⊔ suc (black-depth t₁ ⊔ black-depth t₃) ∎
-     where open ≡-Reasoning
-RB-repl→eq {n} {A} .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) (rb-black _ _ x₂ (rb-red _ _ x₃ x₄ x₅ rbt rbt₂) rbt₁) (rbr-rotate-ll {t₁} {t₂} {t₃} {t₄} x x₁ t) = begin
-   suc (black-depth t₂ ⊔ black-depth t₁ ⊔ black-depth t₄) ≡⟨ cong suc ( ⊔-assoc (black-depth t₂) (black-depth t₁) (black-depth t₄)) ⟩
-   suc (black-depth t₂ ⊔ (black-depth t₁ ⊔ black-depth t₄)) ≡⟨ cong (λ k → suc (k ⊔ (black-depth t₁ ⊔ black-depth t₄)) ) (RB-repl→eq _ _ rbt t) ⟩
-   suc (black-depth t₃ ⊔ (black-depth t₁ ⊔ black-depth t₄)) ∎
-     where open ≡-Reasoning
-RB-repl→eq {n} {A} .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) (rb-black _ _ x₂ rbt (rb-red _ _ x₃ x₄ x₅ rbt₁ rbt₂)) (rbr-rotate-rr {t₁} {t₂} {t₃} {t₄} x x₁ t) = begin
-   suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₂)) ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ k ))) ( RB-repl→eq _ _ rbt₂ t) ⟩
-   suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₃)) ≡⟨ cong suc  (sym ( ⊔-assoc (black-depth t₄) (black-depth t₁) (black-depth t₃))) ⟩
-   suc (black-depth t₄ ⊔ black-depth t₁ ⊔ black-depth t₃) ∎
-     where open ≡-Reasoning
-RB-repl→eq {n} {A} .(node kg ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ _) _) .(node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₂) (node kg ⟪ Red , _ ⟫ t₃ _)) (rb-black .kg _ x₃ (rb-red .kp _ x₄ x₅ x₆ rbt rbt₂) rbt₁) (rbr-rotate-lr {t₀} {t₁} {uncle} t₂ t₃ kg kp kn x x₁ x₂ t) = begin
-   suc (black-depth t₀ ⊔ black-depth t₁ ⊔ black-depth uncle) ≡⟨ cong suc ( ⊔-assoc (black-depth t₀) (black-depth t₁) (black-depth uncle)) ⟩
-   suc (black-depth t₀ ⊔ (black-depth t₁ ⊔ black-depth uncle)) ≡⟨ cong (λ k → suc (black-depth t₀ ⊔ (k ⊔ black-depth uncle))) (RB-repl→eq _ _ rbt₂ t) ⟩
-   suc (black-depth t₀ ⊔ ((black-depth t₂ ⊔ black-depth t₃) ⊔ black-depth uncle)) ≡⟨ cong (λ k → suc (black-depth t₀ ⊔ k )) ( ⊔-assoc (black-depth t₂) (black-depth t₃) (black-depth uncle)) ⟩
-   suc (black-depth t₀ ⊔ (black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth uncle)))  ≡⟨ cong suc (sym ( ⊔-assoc (black-depth t₀) (black-depth t₂) (black-depth t₃ ⊔ black-depth uncle))) ⟩
-   suc (black-depth t₀ ⊔ black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth uncle)) ∎
-     where open ≡-Reasoning
-RB-repl→eq {n} {A} .(node kg ⟪ Black , _ ⟫ _ (node kp ⟪ Red , _ ⟫ _ _)) .(node kn ⟪ Black , _ ⟫ (node kg ⟪ Red , _ ⟫ _ t₂) (node kp ⟪ Red , _ ⟫ t₃ _)) (rb-black .kg _ x₃ rbt (rb-red .kp _ x₄ x₅ x₆ rbt₁ rbt₂)) (rbr-rotate-rl {t₀} {t₁} {uncle} t₂ t₃ kg kp kn x x₁ x₂ t) = begin
-   suc (black-depth uncle ⊔ (black-depth t₁ ⊔ black-depth t₀)) ≡⟨ cong (λ k → suc (black-depth uncle ⊔ (k ⊔ black-depth t₀))) (RB-repl→eq _ _ rbt₁ t) ⟩
-   suc (black-depth uncle ⊔ ((black-depth t₂ ⊔ black-depth t₃)  ⊔ black-depth t₀)) ≡⟨ cong (λ k → suc (black-depth uncle ⊔ k)) ( ⊔-assoc (black-depth t₂) (black-depth t₃) (black-depth t₀)) ⟩
-   suc (black-depth uncle ⊔ (black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth t₀))) ≡⟨ cong suc (sym ( ⊔-assoc (black-depth uncle) (black-depth t₂) (black-depth t₃ ⊔ black-depth t₀))) ⟩
-   suc (black-depth uncle ⊔ black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth t₀)) ∎
-     where open ≡-Reasoning
+RB-repl→eq = ?
+--   RB-repl→eq {n} {A} .leaf .(node _ ⟪ Red , _ ⟫ leaf leaf) rbt rbr-leaf = refl
+--   RB-repl→eq {n} {A} (node _ ⟪ Red , _ ⟫ _ _) .(node _ ⟪ Red , _ ⟫ _ _) rbt rbr-node = refl
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) rbt rbr-node = refl
+--   RB-repl→eq {n} {A} (node _ ⟪ Red , _ ⟫ left _) .(node _ ⟪ Red , _ ⟫ left _) (rb-red _ _ x₂ x₃ x₄ rbt rbt₁) (rbr-right x x₁ t) =
+--      cong (λ k → black-depth left ⊔ k ) (RB-repl→eq _ _ rbt₁ t)
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ left _) .(node _ ⟪ Black , _ ⟫ left _) (rb-black _ _ x₂ rbt rbt₁) (rbr-right x x₁ t) =
+--      cong (λ k → suc (black-depth left ⊔ k)) (RB-repl→eq _ _ rbt₁ t)
+--   RB-repl→eq {n} {A} (node _ ⟪ Red , _ ⟫ _ right) .(node _ ⟪ Red , _ ⟫ _ right) (rb-red _ _ x₂ x₃ x₄ rbt rbt₁) (rbr-left x x₁ t) = cong (λ k → k ⊔ black-depth right) (RB-repl→eq _ _ rbt t)
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ right) .(node _ ⟪ Black , _ ⟫ _ right) (rb-black _ _ x₂ rbt rbt₁) (rbr-left x x₁ t) = cong (λ k → suc (k ⊔ black-depth right)) (RB-repl→eq _ _ rbt t)
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ t₁ _) .(node _ ⟪ Black , _ ⟫ t₁ _) (rb-black _ _ x₂ rbt rbt₁) (rbr-black-right x x₁ t) = cong (λ k → suc (black-depth t₁ ⊔ k)) (RB-repl→eq _ _ rbt₁ t)
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ t₁) .(node _ ⟪ Black , _ ⟫ _ t₁) (rb-black _ _ x₂ rbt rbt₁) (rbr-black-left x x₁ t) = cong (λ k → suc (k ⊔ black-depth t₁)) (RB-repl→eq _ _ rbt t)
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ t₁ t₂) t₃) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ t₄ t₂) (to-black t₃)) (rb-black _ _ x₃ (rb-red _ _ x₄ x₅ x₆ rbt rbt₂) rbt₁) (rbr-flip-ll {_} {_} {t₄} x x₁ x₂ t) = begin
+--      suc (black-depth t₁ ⊔ black-depth t₂ ⊔ black-depth t₃)  ≡⟨ cong (λ k → suc (k ⊔ black-depth t₂ ⊔ black-depth t₃)) (RB-repl→eq _ _ rbt t) ⟩
+--      suc (black-depth t₄ ⊔ black-depth t₂) ⊔ suc (black-depth t₃)  ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ black-depth t₂) ⊔ k )  (  to-black-eq t₃ x₁ ) ⟩
+--      suc (black-depth t₄ ⊔ black-depth t₂) ⊔ black-depth (to-black t₃) ∎
+--        where open ≡-Reasoning
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ t₁ t₂) t₃) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ t₁ t₄) (to-black t₃)) (rb-black _ _ x₄ (rb-red _ _ x₅ x₆ x₇ rbt rbt₂) rbt₁) (rbr-flip-lr {_} {_} {t₄} x x₁ x₂ x₃ t) = begin
+--      suc (black-depth t₁ ⊔ black-depth t₂) ⊔ suc (black-depth t₃)  ≡⟨ cong (λ k → suc (black-depth t₁ ⊔ black-depth t₂) ⊔ k )  (  to-black-eq t₃ x₁ ) ⟩
+--      suc (black-depth t₁ ⊔ black-depth t₂) ⊔ black-depth (to-black t₃) ≡⟨ cong (λ k → suc (black-depth t₁ ⊔ k ) ⊔ black-depth (to-black t₃)) (RB-repl→eq _ _ rbt₂ t) ⟩
+--      suc (black-depth t₁ ⊔ black-depth t₄) ⊔ black-depth (to-black t₃) ∎
+--        where open ≡-Reasoning
+--   RB-repl→eq {n} {A} (node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black t₄) (node _ ⟪ Black , _ ⟫ t₃ t₁)) (rb-black _ _ x₄ rbt (rb-red _ _ x₅ x₆ x₇ rbt₁ rbt₂)) (rbr-flip-rl {t₁} {t₂} {t₃} {t₄} x x₁ x₂ x₃ t) = begin
+--      suc (black-depth t₄ ⊔ (black-depth t₂ ⊔ black-depth t₁)) ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ ( k  ⊔ black-depth t₁)) ) (RB-repl→eq _ _ rbt₁ t) ⟩
+--      suc (black-depth t₄ ⊔ (black-depth t₃ ⊔ black-depth t₁)) ≡⟨ cong (λ k → k ⊔ suc (black-depth t₃ ⊔ black-depth t₁)) ( to-black-eq t₄ x₁ ) ⟩
+--      black-depth (to-black t₄) ⊔ suc (black-depth t₃ ⊔ black-depth t₁) ∎
+--        where open ≡-Reasoning
+--   RB-repl→eq {n} {A} .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) (rb-black _ _ x₃ rbt (rb-red _ _ x₄ x₅ x₆ rbt₁ rbt₂)) (rbr-flip-rr {t₁} {t₂} {t₃} {t₄} x x₁ x₂ t) = begin
+--      suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₂)) ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ k ))) ( RB-repl→eq _ _ rbt₂ t) ⟩
+--      suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₃)) ≡⟨ cong (λ k → k ⊔ suc (black-depth t₁ ⊔ black-depth t₃)) ( to-black-eq t₄ x₁ ) ⟩
+--      black-depth (to-black t₄) ⊔ suc (black-depth t₁ ⊔ black-depth t₃) ∎
+--        where open ≡-Reasoning
+--   RB-repl→eq {n} {A} .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) (rb-black _ _ x₂ (rb-red _ _ x₃ x₄ x₅ rbt rbt₂) rbt₁) (rbr-rotate-ll {t₁} {t₂} {t₃} {t₄} x x₁ t) = begin
+--      suc (black-depth t₂ ⊔ black-depth t₁ ⊔ black-depth t₄) ≡⟨ cong suc ( ⊔-assoc (black-depth t₂) (black-depth t₁) (black-depth t₄)) ⟩
+--      suc (black-depth t₂ ⊔ (black-depth t₁ ⊔ black-depth t₄)) ≡⟨ cong (λ k → suc (k ⊔ (black-depth t₁ ⊔ black-depth t₄)) ) (RB-repl→eq _ _ rbt t) ⟩
+--      suc (black-depth t₃ ⊔ (black-depth t₁ ⊔ black-depth t₄)) ∎
+--        where open ≡-Reasoning
+--   RB-repl→eq {n} {A} .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) (rb-black _ _ x₂ rbt (rb-red _ _ x₃ x₄ x₅ rbt₁ rbt₂)) (rbr-rotate-rr {t₁} {t₂} {t₃} {t₄} x x₁ t) = begin
+--      suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₂)) ≡⟨ cong (λ k → suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ k ))) ( RB-repl→eq _ _ rbt₂ t) ⟩
+--      suc (black-depth t₄ ⊔ (black-depth t₁ ⊔ black-depth t₃)) ≡⟨ cong suc  (sym ( ⊔-assoc (black-depth t₄) (black-depth t₁) (black-depth t₃))) ⟩
+--      suc (black-depth t₄ ⊔ black-depth t₁ ⊔ black-depth t₃) ∎
+--        where open ≡-Reasoning
+--   RB-repl→eq {n} {A} .(node kg ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ _) _) .(node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₂) (node kg ⟪ Red , _ ⟫ t₃ _)) (rb-black .kg _ x₃ (rb-red .kp _ x₄ x₅ x₆ rbt rbt₂) rbt₁) (rbr-rotate-lr {t₀} {t₁} {uncle} t₂ t₃ kg kp kn x x₁ x₂ t) = begin
+--      suc (black-depth t₀ ⊔ black-depth t₁ ⊔ black-depth uncle) ≡⟨ cong suc ( ⊔-assoc (black-depth t₀) (black-depth t₁) (black-depth uncle)) ⟩
+--      suc (black-depth t₀ ⊔ (black-depth t₁ ⊔ black-depth uncle)) ≡⟨ cong (λ k → suc (black-depth t₀ ⊔ (k ⊔ black-depth uncle))) (RB-repl→eq _ _ rbt₂ t) ⟩
+--      suc (black-depth t₀ ⊔ ((black-depth t₂ ⊔ black-depth t₃) ⊔ black-depth uncle)) ≡⟨ cong (λ k → suc (black-depth t₀ ⊔ k )) ( ⊔-assoc (black-depth t₂) (black-depth t₃) (black-depth uncle)) ⟩
+--      suc (black-depth t₀ ⊔ (black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth uncle)))  ≡⟨ cong suc (sym ( ⊔-assoc (black-depth t₀) (black-depth t₂) (black-depth t₃ ⊔ black-depth uncle))) ⟩
+--      suc (black-depth t₀ ⊔ black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth uncle)) ∎
+--        where open ≡-Reasoning
+--   RB-repl→eq {n} {A} .(node kg ⟪ Black , _ ⟫ _ (node kp ⟪ Red , _ ⟫ _ _)) .(node kn ⟪ Black , _ ⟫ (node kg ⟪ Red , _ ⟫ _ t₂) (node kp ⟪ Red , _ ⟫ t₃ _)) (rb-black .kg _ x₃ rbt (rb-red .kp _ x₄ x₅ x₆ rbt₁ rbt₂)) (rbr-rotate-rl {t₀} {t₁} {uncle} t₂ t₃ kg kp kn x x₁ x₂ t) = begin
+--      suc (black-depth uncle ⊔ (black-depth t₁ ⊔ black-depth t₀)) ≡⟨ cong (λ k → suc (black-depth uncle ⊔ (k ⊔ black-depth t₀))) (RB-repl→eq _ _ rbt₁ t) ⟩
+--      suc (black-depth uncle ⊔ ((black-depth t₂ ⊔ black-depth t₃)  ⊔ black-depth t₀)) ≡⟨ cong (λ k → suc (black-depth uncle ⊔ k)) ( ⊔-assoc (black-depth t₂) (black-depth t₃) (black-depth t₀)) ⟩
+--      suc (black-depth uncle ⊔ (black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth t₀))) ≡⟨ cong suc (sym ( ⊔-assoc (black-depth uncle) (black-depth t₂) (black-depth t₃ ⊔ black-depth t₀))) ⟩
+--      suc (black-depth uncle ⊔ black-depth t₂ ⊔ (black-depth t₃ ⊔ black-depth t₀)) ∎
+--        where open ≡-Reasoning
 
 
 RB-repl→ti>  : {n : Level} {A : Set n}  → (tree repl : bt (Color ∧ A)) → (key key₁ : ℕ) → (value : A)
      → replacedRBTree key value tree repl → key₁ < key → tr> key₁ tree → tr> key₁ repl
-RB-repl→ti> .leaf .(node key ⟪ Red , value ⟫ leaf leaf) key key₁ value rbr-leaf lt tr = ⟪ lt , ⟪ tt , tt ⟫ ⟫
-RB-repl→ti> .(node key ⟪ _ , _ ⟫ _ _) .(node key ⟪ _ , value ⟫ _ _) key key₁ value (rbr-node ) lt tr = tr
-RB-repl→ti> .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-right _ x rbt) lt tr
-   = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti> _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-left _ x rbt) lt tr
-   = ⟪ proj1 tr , ⟪ RB-repl→ti> _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-right x _ rbt) lt tr
-   = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti> _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-left x _ rbt) lt tr
-   = ⟪ proj1 tr , ⟪ RB-repl→ti> _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value (rbr-flip-ll x _ _ rbt) lt tr
-   = ⟪ proj1 tr , ⟪ ⟪ proj1 (proj1 (proj2 tr))  , ⟪ RB-repl→ti> _ _ _ _ _ rbt lt (proj1 (proj2 (proj1 (proj2 tr))))
-       , proj2 (proj2 (proj1 (proj2 tr))) ⟫ ⟫  , tr>-to-black (proj2 (proj2 tr)) ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value
-   (rbr-flip-lr x _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5 ⟫ ⟫ = ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti> _ _ _ _ _ rbt lt tr7 ⟫ ⟫  , tr>-to-black tr5 ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
-   (rbr-flip-rl x _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫ = ⟪ tr3 , ⟪ tr>-to-black tr5 , ⟪ tr4 , ⟪  RB-repl→ti> _ _ _ _ _ rbt lt tr6 , tr7 ⟫ ⟫   ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
-   (rbr-flip-rr x _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ ⟫ ⟫ = ⟪ tr3 , ⟪ tr>-to-black tr5 , ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti> _ _ _ _ _ rbt lt tr7 ⟫ ⟫   ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
-   (rbr-rotate-ll x lt2 rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ tr4 , ⟪  RB-repl→ti> _ _ _ _ _ rbt lt tr6 , ⟪ tr3 , ⟪ tr7 , tr5 ⟫ ⟫ ⟫ ⟫
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) key key₁ value
-   (rbr-rotate-rr x lt2 rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ tr4 , ⟪ ⟪ tr3 , ⟪ tr5 , tr6 ⟫ ⟫ , RB-repl→ti> _ _ _ _ _ rbt lt tr7 ⟫ ⟫
-RB-repl→ti> (node kg ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ _) _) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
-   (rbr-rotate-lr left right _ _ kn _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ rr00 , ⟪ ⟪ tr4 , ⟪ tr6 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr3 , ⟪ proj2 (proj2 rr01) , tr5 ⟫ ⟫ ⟫ ⟫ where
-       rr01 : (key₁ < kn) ∧ tr> key₁ left  ∧ tr> key₁ right
-       rr01 = RB-repl→ti> _ _ _ _ _ rbt lt tr7
-       rr00 : key₁ < kn
-       rr00 = proj1 rr01
-RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
-   (rbr-rotate-rl left right kg kp kn _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ rr00 , ⟪  ⟪ tr3 , ⟪ tr5 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr4 , ⟪ proj2 (proj2 rr01) , tr7 ⟫ ⟫ ⟫ ⟫ where
-       rr01 : (key₁ < kn) ∧ tr> key₁ left  ∧ tr> key₁ right
-       rr01 = RB-repl→ti> _ _ _ _ _ rbt lt tr6
-       rr00 : key₁ < kn
-       rr00 = proj1 rr01
+RB-repl→ti> = ?
+--   RB-repl→ti> .leaf .(node key ⟪ Red , value ⟫ leaf leaf) key key₁ value rbr-leaf lt tr = ⟪ lt , ⟪ tt , tt ⟫ ⟫
+--   RB-repl→ti> .(node key ⟪ _ , _ ⟫ _ _) .(node key ⟪ _ , value ⟫ _ _) key key₁ value (rbr-node ) lt tr = tr
+--   RB-repl→ti> .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-right _ x rbt) lt tr
+--      = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti> _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-left _ x rbt) lt tr
+--      = ⟪ proj1 tr , ⟪ RB-repl→ti> _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-right x _ rbt) lt tr
+--      = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti> _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-left x _ rbt) lt tr
+--      = ⟪ proj1 tr , ⟪ RB-repl→ti> _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value (rbr-flip-ll x _ _ rbt) lt tr
+--      = ⟪ proj1 tr , ⟪ ⟪ proj1 (proj1 (proj2 tr))  , ⟪ RB-repl→ti> _ _ _ _ _ rbt lt (proj1 (proj2 (proj1 (proj2 tr))))
+--          , proj2 (proj2 (proj1 (proj2 tr))) ⟫ ⟫  , tr>-to-black (proj2 (proj2 tr)) ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value
+--      (rbr-flip-lr x _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5 ⟫ ⟫ = ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti> _ _ _ _ _ rbt lt tr7 ⟫ ⟫  , tr>-to-black tr5 ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
+--      (rbr-flip-rl x _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫ = ⟪ tr3 , ⟪ tr>-to-black tr5 , ⟪ tr4 , ⟪  RB-repl→ti> _ _ _ _ _ rbt lt tr6 , tr7 ⟫ ⟫   ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
+--      (rbr-flip-rr x _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ ⟫ ⟫ = ⟪ tr3 , ⟪ tr>-to-black tr5 , ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti> _ _ _ _ _ rbt lt tr7 ⟫ ⟫   ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
+--      (rbr-rotate-ll x lt2 rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ tr4 , ⟪  RB-repl→ti> _ _ _ _ _ rbt lt tr6 , ⟪ tr3 , ⟪ tr7 , tr5 ⟫ ⟫ ⟫ ⟫
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) key key₁ value
+--      (rbr-rotate-rr x lt2 rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ tr4 , ⟪ ⟪ tr3 , ⟪ tr5 , tr6 ⟫ ⟫ , RB-repl→ti> _ _ _ _ _ rbt lt tr7 ⟫ ⟫
+--   RB-repl→ti> (node kg ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ _) _) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
+--      (rbr-rotate-lr left right _ _ kn _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ rr00 , ⟪ ⟪ tr4 , ⟪ tr6 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr3 , ⟪ proj2 (proj2 rr01) , tr5 ⟫ ⟫ ⟫ ⟫ where
+--          rr01 : (key₁ < kn) ∧ tr> key₁ left  ∧ tr> key₁ right
+--          rr01 = RB-repl→ti> _ _ _ _ _ rbt lt tr7
+--          rr00 : key₁ < kn
+--          rr00 = proj1 rr01
+--   RB-repl→ti> .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
+--      (rbr-rotate-rl left right kg kp kn _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ rr00 , ⟪  ⟪ tr3 , ⟪ tr5 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr4 , ⟪ proj2 (proj2 rr01) , tr7 ⟫ ⟫ ⟫ ⟫ where
+--          rr01 : (key₁ < kn) ∧ tr> key₁ left  ∧ tr> key₁ right
+--          rr01 = RB-repl→ti> _ _ _ _ _ rbt lt tr6
+--          rr00 : key₁ < kn
+--          rr00 = proj1 rr01
 
 RB-repl→ti<  : {n : Level} {A : Set n}  → (tree repl : bt (Color ∧ A)) → (key key₁ : ℕ) → (value : A)
      → replacedRBTree key value tree repl → key < key₁ → tr< key₁ tree → tr< key₁ repl
-RB-repl→ti< .leaf .(node key ⟪ Red , value ⟫ leaf leaf) key key₁ value rbr-leaf lt tr = ⟪ lt , ⟪ tt , tt ⟫ ⟫
-RB-repl→ti< .(node key ⟪ _ , _ ⟫ _ _) .(node key ⟪ _ , value ⟫ _ _) key key₁ value (rbr-node ) lt tr = tr
-RB-repl→ti< .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-right _ x rbt) lt tr
-   = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti< _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-left _ x rbt) lt tr
-   = ⟪ proj1 tr , ⟪ RB-repl→ti< _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-right x _ rbt) lt tr
-   = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti< _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-left x _ rbt) lt tr
-   = ⟪ proj1 tr , ⟪ RB-repl→ti< _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value (rbr-flip-ll x _ _ rbt) lt tr
-   = ⟪ proj1 tr , ⟪ ⟪ proj1 (proj1 (proj2 tr))  , ⟪ RB-repl→ti< _ _ _ _ _ rbt lt (proj1 (proj2 (proj1 (proj2 tr))))
-       , proj2 (proj2 (proj1 (proj2 tr))) ⟫ ⟫  , tr<-to-black (proj2 (proj2 tr)) ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value
-   (rbr-flip-lr x _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5 ⟫ ⟫ = ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti< _ _ _ _ _ rbt lt tr7 ⟫ ⟫  , tr<-to-black tr5 ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
-   (rbr-flip-rl x _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫ = ⟪ tr3 , ⟪ tr<-to-black tr5 , ⟪ tr4 , ⟪  RB-repl→ti< _ _ _ _ _ rbt lt tr6 , tr7 ⟫ ⟫   ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
-   (rbr-flip-rr x _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ ⟫ ⟫ = ⟪ tr3 , ⟪ tr<-to-black tr5 , ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti< _ _ _ _ _ rbt lt tr7 ⟫ ⟫   ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
-   (rbr-rotate-ll x lt2 rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ tr4 , ⟪  RB-repl→ti< _ _ _ _ _ rbt lt tr6 , ⟪ tr3 , ⟪ tr7 , tr5 ⟫ ⟫ ⟫ ⟫
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) key key₁ value
-   (rbr-rotate-rr x lt2 rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ tr4 , ⟪ ⟪ tr3 , ⟪ tr5 , tr6 ⟫ ⟫ , RB-repl→ti< _ _ _ _ _ rbt lt tr7 ⟫ ⟫
-RB-repl→ti< (node kg ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ _) _) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
-   (rbr-rotate-lr left right _ _ kn _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ rr00 , ⟪ ⟪ tr4 , ⟪ tr6 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr3 , ⟪ proj2 (proj2 rr01) , tr5 ⟫ ⟫ ⟫ ⟫ where
-       rr01 : (kn < key₁ ) ∧ tr< key₁ left  ∧ tr< key₁ right
-       rr01 = RB-repl→ti< _ _ _ _ _ rbt lt tr7
-       rr00 : kn < key₁
-       rr00 = proj1 rr01
-RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
-   (rbr-rotate-rl left right kg kp kn _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ rr00 , ⟪  ⟪ tr3 , ⟪ tr5 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr4 , ⟪ proj2 (proj2 rr01) , tr7 ⟫ ⟫ ⟫ ⟫ where
-       rr01 : (kn < key₁ ) ∧ tr< key₁ left  ∧ tr< key₁ right
-       rr01 = RB-repl→ti< _ _ _ _ _ rbt lt tr6
-       rr00 : kn < key₁
-       rr00 = proj1 rr01
+RB-repl→ti< = ?
+--    RB-repl→ti< .leaf .(node key ⟪ Red , value ⟫ leaf leaf) key key₁ value rbr-leaf lt tr = ⟪ lt , ⟪ tt , tt ⟫ ⟫
+--    RB-repl→ti< .(node key ⟪ _ , _ ⟫ _ _) .(node key ⟪ _ , value ⟫ _ _) key key₁ value (rbr-node ) lt tr = tr
+--    RB-repl→ti< .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-right _ x rbt) lt tr
+--       = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti< _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ _ , _ ⟫ _ _) .(node _ ⟪ _ , _ ⟫ _ _) key key₁ value (rbr-left _ x rbt) lt tr
+--       = ⟪ proj1 tr , ⟪ RB-repl→ti< _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-right x _ rbt) lt tr
+--       = ⟪ proj1 tr , ⟪ proj1 (proj2 tr) , RB-repl→ti< _ _ _ _ _ rbt lt (proj2 (proj2 tr)) ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ _) .(node _ ⟪ Black , _ ⟫ _ _) key key₁ value (rbr-black-left x _ rbt) lt tr
+--       = ⟪ proj1 tr , ⟪ RB-repl→ti< _ _ _ _ _ rbt lt (proj1 (proj2 tr)) , proj2 (proj2 tr) ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value (rbr-flip-ll x _ _ rbt) lt tr
+--       = ⟪ proj1 tr , ⟪ ⟪ proj1 (proj1 (proj2 tr))  , ⟪ RB-repl→ti< _ _ _ _ _ rbt lt (proj1 (proj2 (proj1 (proj2 tr))))
+--           , proj2 (proj2 (proj1 (proj2 tr))) ⟫ ⟫  , tr<-to-black (proj2 (proj2 tr)) ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ _ _) (to-black _)) key key₁ value
+--       (rbr-flip-lr x _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5 ⟫ ⟫ = ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti< _ _ _ _ _ rbt lt tr7 ⟫ ⟫  , tr<-to-black tr5 ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
+--       (rbr-flip-rl x _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫ = ⟪ tr3 , ⟪ tr<-to-black tr5 , ⟪ tr4 , ⟪  RB-repl→ti< _ _ _ _ _ rbt lt tr6 , tr7 ⟫ ⟫   ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Red , _ ⟫ (to-black _) (node _ ⟪ Black , _ ⟫ _ _)) key key₁ value
+--       (rbr-flip-rr x _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ ⟫ ⟫ = ⟪ tr3 , ⟪ tr<-to-black tr5 , ⟪ tr4 , ⟪ tr6 ,  RB-repl→ti< _ _ _ _ _ rbt lt tr7 ⟫ ⟫   ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
+--       (rbr-rotate-ll x lt2 rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ tr4 , ⟪  RB-repl→ti< _ _ _ _ _ rbt lt tr6 , ⟪ tr3 , ⟪ tr7 , tr5 ⟫ ⟫ ⟫ ⟫
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) _) key key₁ value
+--       (rbr-rotate-rr x lt2 rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ tr4 , ⟪ ⟪ tr3 , ⟪ tr5 , tr6 ⟫ ⟫ , RB-repl→ti< _ _ _ _ _ rbt lt tr7 ⟫ ⟫
+--    RB-repl→ti< (node kg ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ _) _) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
+--       (rbr-rotate-lr left right _ _ kn _ _ _ rbt) lt ⟪ tr3 , ⟪ ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫ , tr5  ⟫ ⟫  = ⟪ rr00 , ⟪ ⟪ tr4 , ⟪ tr6 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr3 , ⟪ proj2 (proj2 rr01) , tr5 ⟫ ⟫ ⟫ ⟫ where
+--           rr01 : (kn < key₁ ) ∧ tr< key₁ left  ∧ tr< key₁ right
+--           rr01 = RB-repl→ti< _ _ _ _ _ rbt lt tr7
+--           rr00 : kn < key₁
+--           rr00 = proj1 rr01
+--    RB-repl→ti< .(node _ ⟪ Black , _ ⟫ _ (node _ ⟪ Red , _ ⟫ _ _)) .(node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node _ ⟪ Red , _ ⟫ _ _)) key key₁ value
+--       (rbr-rotate-rl left right kg kp kn _ _ _ rbt) lt ⟪ tr3 , ⟪ tr5 , ⟪ tr4 , ⟪ tr6 , tr7 ⟫ ⟫  ⟫ ⟫  = ⟪ rr00 , ⟪  ⟪ tr3 , ⟪ tr5 , proj1 (proj2 rr01) ⟫ ⟫ , ⟪ tr4 , ⟪ proj2 (proj2 rr01) , tr7 ⟫ ⟫ ⟫ ⟫ where
+--           rr01 : (kn < key₁ ) ∧ tr< key₁ left  ∧ tr< key₁ right
+--           rr01 = RB-repl→ti< _ _ _ _ _ rbt lt tr6
+--           rr00 : kn < key₁
+--           rr00 = proj1 rr01
 
 RB-repl→ti : {n : Level} {A : Set n} → (tree repl : bt (Color ∧ A) ) → (key : ℕ ) → (value : A) → treeInvariant tree
        → replacedRBTree key value tree repl → treeInvariant repl
-RB-repl→ti .leaf .(node key ⟪ Red , value ⟫ leaf leaf) key value ti rbr-leaf = t-single key ⟪ Red , value ⟫
-RB-repl→ti .(node key ⟪ _ , _ ⟫ leaf leaf) .(node key ⟪ _ , value ⟫ leaf leaf) key value (t-single .key .(⟪ _ , _ ⟫)) (rbr-node ) = t-single key ⟪ _ , value ⟫
-RB-repl→ti .(node key ⟪ _ , _ ⟫ leaf (node key₁ _ _ _)) .(node key ⟪ _ , value ⟫ leaf (node key₁ _ _ _)) key value
-   (t-right .key key₁ x x₁ x₂ ti) (rbr-node ) = t-right key key₁ x x₁ x₂ ti
-RB-repl→ti .(node key ⟪ _ , _ ⟫ (node key₁ _ _ _) leaf) .(node key ⟪ _ , value ⟫ (node key₁ _ _ _) leaf) key value
-   (t-left key₁ .key x x₁ x₂ ti) (rbr-node ) = t-left key₁ key x x₁ x₂ ti
-RB-repl→ti .(node key ⟪ _ , _ ⟫ (node key₁ _ _ _) (node key₂ _ _ _)) .(node key ⟪ _ , value ⟫ (node key₁ _ _ _) (node key₂ _ _ _)) key value
-   (t-node key₁ .key key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁) (rbr-node ) = t-node key₁ key key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁
-RB-repl→ti (node key₁ ⟪ ca , v1 ⟫ leaf leaf) (node key₁ ⟪ ca , v1 ⟫ leaf tree@(node key₂ value₁ t t₁)) key value
-   (t-single key₁ ⟪ ca , v1 ⟫) (rbr-right _ x trb) = t-right _ _  (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₁ < key₂ ) ∧ tr> key₁ t ∧ tr> key₁ t₁
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x tt
-RB-repl→ti (node _ ⟪ _ , _ ⟫ leaf (node key₁ _ _ _)) (node key₂ ⟪ ca , v1 ⟫ leaf (node key₃ value₁ t t₁)) key value
-   (t-right _ key₁ x₁ x₂ x₃ ti) (rbr-right _ x trb) = t-right _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
-        rr00 : (key₂ < key₃) ∧ tr> key₂ t ∧ tr> key₂ t₁
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
-RB-repl→ti .(node key₂ ⟪ ca , v1 ⟫ (node key₁ value₁ t t₁) leaf) (node key₂ ⟪ ca , v1 ⟫ (node key₁ value₁ t t₁) (node key₃ value₂ t₂ t₃)) key value
-   (t-left key₁ _ x₁ x₂ x₃ ti) (rbr-right _ x trb) = t-node _ _ _ x₁ (proj1 rr00) x₂ x₃ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₂ < key₃) ∧ tr> key₂ t₂ ∧ tr> key₂ t₃
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x tt
-RB-repl→ti .(node key₃ ⟪ ca , v1 ⟫ (node key₁ v2 t₁ t₂) (node key₂ _ _ _)) (node key₃ ⟪ ca , v1 ⟫ (node key₁ v2 t₁ t₂) (node key₄ value₁ t₃ t₄)) key value
-   (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-right _ x trb) = t-node _ _ _ x₁
-     (proj1 rr00) x₃ x₄ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ ti₁ trb) where
-        rr00 : (key₃ < key₄) ∧ tr> key₃ t₃ ∧ tr> key₃ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x ⟪ x₂ , ⟪ x₅ , x₆ ⟫ ⟫
-RB-repl→ti .(node key₁ ⟪ _ , _ ⟫ leaf leaf) (node key₁ ⟪ _ , _ ⟫ (node key₂ value₁ left left₁) leaf) key value
-   (t-single _ .(⟪ _ , _ ⟫)) (rbr-left _ x trb) = t-left _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₂ < key₁) ∧ tr< key₁ left ∧ tr< key₁ left₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x tt
-RB-repl→ti .(node key₂ ⟪ _ , _ ⟫ leaf (node key₁ _ t₁ t₂)) (node key₂ ⟪ _ , _ ⟫ (node key₃ value₁ t t₃) (node key₁ _ t₁ t₂)) key value
-   (t-right _ key₁ x₁ x₂ x₃ ti) (rbr-left _ x trb) = t-node _ _ _ (proj1 rr00) x₁  (proj1 (proj2 rr00))(proj2 (proj2 rr00)) x₂ x₃ rr01 ti where
-        rr00 : (key₃ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₃
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x tt
-        rr01 : treeInvariant (node key₃ value₁ t t₃)
-        rr01 = RB-repl→ti _ _ _ _ t-leaf trb
-RB-repl→ti .(node _ ⟪ _ , _ ⟫ (node key₁ _ _ _) leaf) (node key₃ ⟪ _ , _ ⟫ (node key₂ value₁ t t₁) leaf) key value
-    (t-left key₁ _ x₁ x₂ x₃ ti) (rbr-left _ x trb) = t-left key₂ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
-        rr00 : (key₂ < key₃) ∧ tr< key₃ t ∧ tr< key₃ t₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
-RB-repl→ti .(node key₃ ⟪ _ , _ ⟫ (node key₁ _ _ _) (node key₂ _ t₁ t₂)) (node key₃ ⟪ _ , _ ⟫ (node key₄ value₁ t t₃) (node key₂ _ t₁ t₂)) key value
-    (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-left _ x trb) = t-node _ _ _ (proj1 rr00) x₂ (proj1 (proj2 rr00))  (proj2 (proj2 rr00))  x₅ x₆ (RB-repl→ti _ _ _ _ ti trb) ti₁ where
-        rr00 : (key₄ < key₃) ∧ tr< key₃ t ∧ tr< key₃ t₃
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x ⟪ x₁ , ⟪ x₃ , x₄ ⟫ ⟫
-RB-repl→ti .(node x₁ ⟪ Black , c ⟫ leaf leaf) (node x₁ ⟪ Black , c ⟫ leaf (node key₁ value₁ t t₁)) key value
-    (t-single x₂ .(⟪ Black , c ⟫)) (rbr-black-right x x₄ trb) = t-right _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (x₁ < key₁) ∧ tr> x₁ t ∧ tr> x₁ t₁
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x₄ tt
-RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ leaf (node key₁ _ _ _)) (node key₂ ⟪ Black , _ ⟫ leaf (node key₃ value₁ t₂ t₃)) key value
-    (t-right _ key₁ x₁ x₂ x₃ ti) (rbr-black-right  x x₄ trb) = t-right _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
-        rr00 : (key₂ < key₃) ∧ tr> key₂ t₂ ∧ tr> key₂ t₃
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x₄ ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
-RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) leaf) (node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₃ value₁ t₂ t₃)) key value (t-left key₁ _ x₁ x₂ x₃ ti) (rbr-black-right x x₄ trb) = t-node _ _ _ x₁ (proj1 rr00) x₂ x₃ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₂ < key₃) ∧ tr> key₂ t₂ ∧ tr> key₂ t₃
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x₄ tt
-RB-repl→ti .(node key₃ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₂ _ _ _)) (node key₃ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₄ value₁ t₂ t₃)) key value
-      (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-black-right x x₇ trb) = t-node _ _ _ x₁ (proj1 rr00) x₃ x₄ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ ti₁ trb) where
-        rr00 : (key₃ < key₄) ∧ tr> key₃ t₂ ∧ tr> key₃ t₃
-        rr00 = RB-repl→ti> _ _ _ _ _ trb x₇ ⟪ x₂ , ⟪ x₅ , x₆ ⟫ ⟫
-RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ leaf leaf) (node key₂ ⟪ Black , _ ⟫ (node key₁ value₁ t t₁) .leaf) key value
-       (t-single .key₂ .(⟪ Black , _ ⟫)) (rbr-black-left x x₇ trb) = t-left _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₁ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ tt
-RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ leaf (node key₁ _ _ _)) (node key₂ ⟪ Black , _ ⟫ (node key₃ value₁ t t₁) .(node key₁ _ _ _)) key value
-       (t-right .key₂ key₁ x₁ x₂ x₃ ti) (rbr-black-left x x₇ trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00))  x₂ x₃ (RB-repl→ti _ _ _ _ t-leaf trb) ti where
-        rr00 : (key₃ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ tt
-RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) leaf) (node key₂ ⟪ Black , _ ⟫ (node key₃ value₁ t t₁) .leaf) key value
-       (t-left key₁ .key₂ x₁ x₂ x₃ ti) (rbr-black-left x x₇ trb) = t-left _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
-        rr00 : (key₃ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
-RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₃ _ _ _)) (node key₂ ⟪ Black , _ ⟫ (node key₄ value₁ t t₁) .(node key₃ _ _ _)) key value
-     (t-node key₁ .key₂ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-black-left x x₇ trb) = t-node _ _ _ (proj1 rr00) x₂ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) x₅ x₆ (RB-repl→ti _ _ _ _ ti trb) ti₁ where
-        rr00 : (key₄ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ ⟪ x₁ , ⟪ x₃ , x₄ ⟫ ⟫
-RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ t₁) leaf) (node key₂ ⟪ Red , value₁ ⟫ (node key₁ ⟪ Black , value₂ ⟫ t t₁) .(to-black leaf)) key value
-      (t-left _ .key₂ x₁ x₂ x₃ ti) (rbr-flip-ll x _ lt trb) = t-left _ _ x₁ rr00 x₃ (RTtoTI0 _ _ _ _ rr02 r-node ) where
-        rr00 : tr< key₂ t
-        rr00 = RB-repl→ti< _ _ _ _ _ trb (<-trans lt x₁) x₂
-        rr02 : treeInvariant (node key₁ ⟪ Red , value₂  ⟫ t t₁)
-        rr02 = RB-repl→ti _ _ _ _ ti (rbr-left _ lt trb)
-RB-repl→ti (node key₂ ⟪ Black , _ ⟫ (node key₁ ⟪ Red , _ ⟫ t₀ t₁) (node key₃ ⟪ c1 , v1 ⟫  left right)) (node key₂ ⟪ Red , value₁ ⟫ (node _ ⟪ Black , value₂ ⟫ t t₁) (node key₃ ⟪ Black , v1 ⟫  left  right)) key value
-       (t-node _ .key₂ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-ll x _ lt trb) = t-node _ _ _ x₁ x₂ rr00 x₄ x₅ x₆ (RTtoTI0 _ _ _ _ rr02 r-node ) (RTtoTI0 _ _ _ _ ti₁ r-node ) where
-        rr00 : tr< key₂ t
-        rr00 = RB-repl→ti< _ _ _ _ _ trb (<-trans lt x₁)  x₃
-        rr02 : treeInvariant (node key₁ ⟪ Red , value₂ ⟫ t t₁)
-        rr02  = RB-repl→ti _ _ _ _ ti (rbr-left _ lt trb)
-RB-repl→ti (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ left right) leaf) (node key₂ ⟪ Red , v1 ⟫ (node key₃ ⟪ Black , v2 ⟫ left right₁) leaf) key value
-      (t-left _ _ x₁ x₂ x₃ ti) (rbr-flip-lr x _ lt lt2 trb) = t-left _ _ x₁ x₂ rr00 (RTtoTI0 _ _ _ _ rr02 r-node ) where
-        rr00 : tr< key₂ right₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-        rr02 : treeInvariant (node key₃ ⟪ Red , v2 ⟫ left right₁ )
-        rr02 = RB-repl→ti _ _ _ _ ti (rbr-right _ lt trb)
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ Red , v2 ⟫ t t₁) (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ t t₄) .(to-black (node key₃ ⟪ c3 , _ ⟫ _ _))) key value
-      (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-lr x _ lt lt2 trb) = t-node _ _ _ x₁ x₂ x₃ rr00 x₅ x₆ (RTtoTI0 _ _ _ _ rr02 r-node ) (RTtoTI0 _ _ _ _ ti₁ r-node ) where
-        rr00 : tr< key₁ t₄
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 x₄
-        rr02 : treeInvariant (node key₂ ⟪ Red , v2 ⟫ t t₄)
-        rr02 = RB-repl→ti _ _ _ _ ti (rbr-right _ lt trb)
-RB-repl→ti (node _ ⟪ Black , _ ⟫ leaf (node _ ⟪ Red , _ ⟫ t t₁)) (node key₁ ⟪ Red , v1 ⟫ .(to-black leaf) (node key₂ ⟪ Black , v2 ⟫ t₂ t₁)) key value
-      (t-right _ _ x₁ x₂ x₃ ti) (rbr-flip-rl x _ lt lt2 trb) = t-right _ _ x₁ rr00 x₃ (RTtoTI0 _ _ _ _ rr02 r-node ) where
-        rr00 : tr> key₁ t₂
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt x₂
-        rr02 : treeInvariant (node key₂ ⟪ Red , v2 ⟫ t₂ t₁)
-        rr02 = RB-repl→ti _ _ _ _ ti (rbr-left _ lt2 trb)
-RB-repl→ti (node _ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node _ ⟪ Red , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , _ ⟫  _ _)) (node key₃ ⟪ Black , _ ⟫ t₄ t₃)) key value
-      (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-rl x _ lt lt2 trb) = t-node key₂ _ _ x₁ x₂ x₃ x₄ rr00 x₆ (RTtoTI0 _ _ _ _ ti r-node ) (RTtoTI0 _ _ _ _ rr02 r-node ) where
-        rr00 : tr> key₁ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt x₅
-        rr02 : treeInvariant (node key₃ ⟪ Red , v3 ⟫ t₄ t₃)
-        rr02 = RB-repl→ti _ _ _ _ ti₁ (rbr-left _ lt2 trb)
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ t t₁)) (node _ ⟪ Red , _ ⟫ .(to-black leaf) (node _ ⟪ Black , v2 ⟫ t t₂)) key value
-    (t-right _ _ x₁ x₂ x₃ ti) (rbr-flip-rr x _ lt trb) = t-right _ _ x₁ x₂ rr00 (RTtoTI0 _ _ _ _ rr02 r-node ) where
-        rr00 : tr> key₁ t₂
-        rr00 = RB-repl→ti> _ _ _ _ _ trb (<-trans x₁ lt )  x₃
-        rr02 : treeInvariant (node key₂ ⟪ Red , v2 ⟫ t t₂)
-        rr02 = RB-repl→ti _ _ _ _ ti (rbr-right _ lt trb)
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node key₃ ⟪ Red , c3 ⟫ t₂ t₃)) (node _ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , _ ⟫ _ _)) (node _ ⟪ Black , c3 ⟫ t₂ t₄)) key value
-    (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-rr x _ lt trb) = t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ rr00 (RTtoTI0 _ _ _ _ ti r-node ) (RTtoTI0 _ _ _ _ rr02 r-node ) where
-        rr00 : tr> key₁ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb (<-trans x₂ lt) x₆
-        rr02 : treeInvariant (node key₃ ⟪ Red , c3 ⟫ t₂ t₄)
-        rr02 = RB-repl→ti _ _ _ _ ti₁ (rbr-right _ lt trb)
-RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ .leaf .leaf) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ .leaf leaf)) key value (t-left _ _ x₁ x₂ x₃
-    (t-single .k2 .(⟪ Red , c2 ⟫))) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10)) (proj2 (proj2 rr10)) tt tt (RB-repl→ti _ _ _ _ t-leaf trb) (t-single _ _ ) where
-        rr10 : (key₁ < k2 ) ∧ tr< k2 t₂ ∧ tr< k2 t₃
-        rr10 = RB-repl→ti< _ _ _ _ _ trb lt tt
-RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ .leaf .(node key₂ _ _ _)) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ (node key₂ value₂ t₁ t₄) leaf)) key value
-    (t-left _ _ x₁ x₂ x₃ (t-right .k2 key₂ x₄ x₅ x₆ ti)) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10)) (proj2 (proj2 rr10)) ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫  tt rr05 rr04 where
-        rr10 : (key₁ < k2 ) ∧ tr< k2 t₂ ∧ tr< k2 t₃
-        rr10 = RB-repl→ti< _ _ _ _ _ trb lt tt
-        rr04 : treeInvariant (node k1 ⟪ Red , c1 ⟫ (node key₂ value₂ t₁ t₄) leaf)
-        rr04 = RTtoTI0 _ _ _ _ (t-left key₂ _ {_} {⟪ Red , c1 ⟫} {t₁} {t₄} (proj1 x₃) (proj1 (proj2 x₃)) (proj2 (proj2 x₃)) ti) r-node
-        rr05 : treeInvariant (node key₁ value₁ t₂ t₃)
-        rr05 = RB-repl→ti _ _ _ _ t-leaf trb
-RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ (node key₂ value₂ t₁ t₄) .leaf) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ .leaf leaf)) key value
-   (t-left _ _ x₁ x₂ x₃ (t-left key₂ .k2 x₄ x₅ x₆ ti)) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10)) (proj2 (proj2 rr10)) tt tt (RB-repl→ti _ _ _ _ ti trb) (t-single _ _) where
-        rr10 : (key₁ < k2 ) ∧ tr< k2 t₂ ∧ tr< k2 t₃
-        rr10 = RB-repl→ti< _ _ _ _ _ trb lt ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫
-RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ (node key₂ value₃ left right) (node key₃ value₂ t₄ t₅)) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ .(node key₃ _ _ _) leaf)) key value
-     (t-left _ _ x₁ x₂ x₃ (t-node key₂ .k2 key₃ x₄ x₅ x₆ x₇ x₈ x₉ ti ti₁)) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10))  (proj2 (proj2 rr10))  ⟪ x₅ , ⟪ x₈ ,  x₉ ⟫ ⟫ tt rr05 rr04 where
-        rr06 : key < k2
-        rr06 = lt
-        rr10 : (key₁ < k2) ∧ tr< k2 t₂ ∧ tr< k2 t₃
-        rr10 = RB-repl→ti< _ _ _ _ _ trb rr06 ⟪ x₄ , ⟪ x₆ , x₇ ⟫ ⟫
-        rr04 : treeInvariant (node k1 ⟪ Red , c1 ⟫ (node key₃ value₂ t₄ t₅) leaf)
-        rr04 = RTtoTI0 _ _ _ _ (t-left _ _ (proj1 x₃) (proj1 (proj2 x₃)) (proj2 (proj2 x₃)) ti₁ ) (r-left (proj1 x₃) r-node)
-        rr05 : treeInvariant (node key₁ value₁ t₂ t₃)
-        rr05 = RB-repl→ti _ _ _ _ ti trb
-RB-repl→ti (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ .leaf .leaf) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ .leaf (node key₃ _ _ _))) key value
-   (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-single .key₂ .(⟪ Red , c2 ⟫)) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00))  (proj2 (proj2 rr00))  tt ⟪ <-trans x₁ x₂ , ⟪ <-tr> x₅ x₁ , <-tr> x₆ x₁  ⟫ ⟫  rr02 rr03 where
-       rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
-       rr00 = RB-repl→ti< _ _ _ _ _ trb lt tt
-       rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
-       rr02 = RB-repl→ti _ _ _ _ t-leaf trb
-       rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ leaf (node key₃ v3 t₂ t₃))
-       rr03 = RTtoTI0 _ _ _ _ (t-right _ _ {v3} {_} x₂ x₅ x₆ ti₁) r-node
-RB-repl→ti (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ leaf (node key₅ _ _ _)) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ (node key₅ value₂ t₁ t₆) (node key₃ _ _ _))) key value
-    (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-right .key₂ key₅ x₇ x₈ x₉ ti) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫  ⟪ <-trans x₁ x₂  , ⟪ <-tr> x₅ x₁ , <-tr> x₆ x₁ ⟫ ⟫ rr02 rr03 where
-       rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
-       rr00 = RB-repl→ti< _ _ _ _ _ trb lt tt
-       rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
-       rr02 = RB-repl→ti _ _ _ _ t-leaf trb
-       rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ (node key₅ value₂ t₁ t₆) (node key₃ v3 t₂ t₃))
-       rr03 = RTtoTI0 _ _ _ _ (t-node _ _ _ {_} {v3} {_} {_} {_} {_} {_} (proj1 x₄) x₂ (proj1 (proj2 x₄)) (proj2 (proj2 x₄)) x₅ x₆ ti ti₁ ) r-node
-RB-repl→ti (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ .(node key₅ _ _ _) .leaf) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ .leaf (node key₃ _ _ _))) key value (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆
-    (t-left key₅ .key₂ x₇ x₈ x₉ ti) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00))  tt ⟪ <-trans x₁ x₂  , ⟪ <-tr> x₅ x₁  , <-tr> x₆ x₁ ⟫ ⟫  rr02 rr04 where
-        rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫
-        rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
-        rr02 = RB-repl→ti _ _ _ _ ti trb
-        rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ (node key₅ _ _ _) (node key₃ v3 t₂ t₃))
-        rr03 = RTtoTI0 _ _ _ _ (t-node _ _ _ {_} {v3} {_} {_} {_} {_} {_} (proj1 x₃) x₂ (proj1 (proj2 x₃)) (proj2 (proj2 x₃)) x₅ x₆ ti ti₁) r-node
-        rr04 :  treeInvariant (node key₁ ⟪ Red , c1 ⟫ leaf (node key₃ v3 t₂ t₃))
-        rr04 = RTtoTI0 _ _ _ _ (t-right _ _ {v3} {_} x₂ x₅ x₆ ti₁) r-node
-RB-repl→ti {_} {A} (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ .(node key₅ _ _ _) (node key₆ value₆ t₆ t₇)) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ .(node key₆ _ _ _) (node key₃ _ _ _))) key value
-  (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-node key₅ .key₂ key₆ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti ti₂) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ⟪ x₈ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ <-trans x₁ x₂ , ⟪ rr05 , <-tr> x₆ x₁ ⟫ ⟫ rr02 rr03 where
-        rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₉ , x₁₀ ⟫ ⟫
-        rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
-        rr02 = RB-repl→ti _ _ _ _ ti trb
-        rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ (node key₆ value₆ t₆ t₇) (node key₃ v3 t₂ t₃))
-        rr03 = RTtoTI0 _ _ _ _(t-node _ _ _ {_} {value₁} {_} {_} {_} {_} {_} (proj1 x₄) x₂ (proj1 (proj2 x₄)) (proj2 (proj2 x₄)) x₅ x₆ ti₂ ti₁) r-node
-        rr05 : tr> key₂ t₂
-        rr05 = <-tr> x₅ x₁
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ leaf leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₃ value₁ t₃ t₄)) key value
-    (t-right .key₁ .key₂ x₁ x₂ x₃ (t-single .key₂ .(⟪ Red , _ ⟫))) (rbr-rotate-rr x lt trb)
-      = t-node _ _ _ x₁ (proj1 rr00) tt tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-single _ _) (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₂ < key₃) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ leaf (node key₃ value₃ t t₁))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₄ value₁ t₃ t₄)) key value
-   (t-right .key₁ .key₂ x₁ x₂ x₃ (t-right .key₂ key₃ x₄ x₅ x₆ ti)) (rbr-rotate-rr x lt trb)
-      = t-node _ _ _ x₁ (proj1 rr00) tt tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-single _ _ ) (RB-repl→ti _ _ _ _ ti trb) where
-        rr00 : (key₂ < key₄) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ (node key₃ _ _ _) leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₄ value₁ t₃ t₄)) key value
-    (t-right .key₁ .key₂ x₁ x₂ x₃ (t-left key₃ .key₂ x₄ x₅ x₆ ti)) (rbr-rotate-rr x lt trb)
-       = t-node _ _ _ x₁ (proj1 rr00) tt ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-right _ _ (proj1 x₂) (proj1 (proj2 x₂)) (proj2 (proj2 x₂)) ti)  (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₂ < key₄) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ (node key₃ _ _ _) (node key₄ _ _ _))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₅ value₁ t₃ t₄)) key value
-    (t-right .key₁ .key₂ x₁ x₂ x₃ (t-node key₃ .key₂ key₄ x₄ x₅ x₆ x₇ x₈ x₉ ti ti₁)) (rbr-rotate-rr x lt trb) = t-node _ _ _ x₁ (proj1 rr00) tt ⟪ x₄ , ⟪ x₆ , x₇ ⟫ ⟫ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-right _ _ (proj1 x₂) (proj1 (proj2 x₂)) (proj2 (proj2 x₂)) ti) (RB-repl→ti _ _ _ _ ti₁ trb) where
-        rr00 : (key₂ < key₅) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₅ , ⟪ x₈ , x₉ ⟫ ⟫
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ .(node key₃ _ _ _) (node key₂ ⟪ Red , v2 ⟫ .leaf .leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₄ value₁ t₃ t₄)) key value
-   (t-node key₃ .key₁ .key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-single .key₂ .(⟪ Red , v2 ⟫))) (rbr-rotate-rr x lt trb)
-     = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂ , >-tr< x₄ x₂ ⟫ ⟫  tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-left _ _ x₁ x₃ x₄ ti) (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₂ < key₄) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₃ v3 t t₁) (node key₂ ⟪ Red , v2 ⟫ leaf (node key₄ _ _ _))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₅ value₁ t₃ t₄)) key value
-   (t-node key₃ .key₁ .key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-right .key₂ key₄ x₇ x₈ x₉ ti₁)) (rbr-rotate-rr x lt trb)
-      = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂ , >-tr< x₄ x₂ ⟫ ⟫ tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-left _ _ x₁ x₃ x₄ ti) (RB-repl→ti _ _ _ _ ti₁ trb) where
-        rr00 : (key₂ < key₅) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₃ _ _ _) (node key₂ ⟪ Red , v2 ⟫ (node key₄ _ _ _) leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₅ value₁ t₃ t₄)) key value
-   (t-node key₃ key₁ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-left key₄ key₂ x₇ x₈ x₉ ti₁)) (rbr-rotate-rr x lt trb)
-      = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂  , >-tr< x₄ x₂ ⟫ ⟫ ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-node _ _ _  x₁ (proj1 x₅) x₃ x₄ (proj1 (proj2 x₅)) (proj2 (proj2 x₅)) ti ti₁) (RB-repl→ti _ _ _ _ t-leaf trb) where
-        rr00 : (key₂ < key₅) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₃ _ _ _) (node key₂ ⟪ Red , v2 ⟫ (node key₄ _ left right) (node key₅ _ _ _))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₆ value₁ t₃ t₄)) key value
-   (t-node key₃ key₁ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-node key₄ key₂ key₅ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti₁ ti₂)) (rbr-rotate-rr x lt trb)
-     = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂ , >-tr< x₄ x₂ ⟫ ⟫  ⟪ x₇ , ⟪ x₉ , x₁₀ ⟫ ⟫  (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RTtoTI0 _ _ _ _ (t-node _ _ _ {_} {value₁} x₁ (proj1 x₅) x₃ x₄  (proj1 (proj2 x₅)) (proj2 (proj2 x₅)) ti ti₁ ) r-node )
-     (RB-repl→ti _ _ _ _ ti₂ trb) where
-        rr00 : (key₂ < key₆) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₈ , ⟪ x₁₁ , x₁₂ ⟫ ⟫
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ leaf leaf) .leaf) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ leaf _)) .kn _ (t-left .kp .kg x x₁ x₂ ti) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ (node key₁ value₁ t t₁) leaf) .leaf) (node kn ⟪ Black , v3 ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ leaf _)) .kn .v3 (t-left .kp .kg x x₁ x₂ (t-left .key₁ .kp x₃ x₄ x₅ ti)) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) =
-    t-node _ _ _ lt1 lt2 ⟪ <-trans x₃ lt1  , ⟪ >-tr< x₄ lt1  , >-tr< x₅ lt1 ⟫ ⟫  tt tt tt (t-left _ _ x₃ x₄ x₅ ti) (t-single _ _)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ .(⟪ Red , _ ⟫) .leaf .leaf)) .leaf) (node .key₁ ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ leaf _)) .key₁ _ (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .leaf .leaf kg kp .key₁ _ lt1 lt2 (rbr-node )) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .leaf) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ (node key₂ value₂ t₄ t₅) t₆)) key value (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .leaf .(node key₂ value₂ t₄ t₅) kg kp kn _ lt1 lt2 trb) =
-   t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt rr03 tt (t-single _ _) (t-left _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) (treeRightDown _ _ ( RB-repl→ti _ _ _ _ ti trb))) where
-        rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₂) ∧ tr> kp t₄ ∧ tr> kp t₅ )
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
-        rr01 : (kn < kg) ∧ ⊤ ∧ ((key₂ < kg ) ∧ tr< kg t₄ ∧ tr< kg t₅ ) -- tr< kg (node key₂ value₂ t₄ t₅)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
-        rr02 = proj2 (proj2 rr01)
-        rr03 : (kn < key₂) ∧ tr> kn t₄ ∧ tr> kn t₅
-        rr03 with RB-repl→ti _ _ _ _ ti trb
-        ... | t-right .kn .key₂ x x₁ x₂ t = ⟪ x , ⟪ x₁ , x₂ ⟫ ⟫
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .leaf) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ (node key₂ value₂ t₃ t₅)) (node kg ⟪ Red , _ ⟫ leaf _)) key value (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .(node key₂ value₂ t₃ t₅) .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
-... | t-left .key₂ .kn x₆ x₇ x₈ t =
-   t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫ tt tt (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-single _ _) where
-        rr00 : (kp < kn) ∧ ((kp < key₂) ∧ tr> kp t₃ ∧ tr> kp t₅) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₂ < kg) ∧ tr< kg t₃ ∧ tr< kg t₅) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .leaf) (node kn ⟪ Black , value₄ ⟫ (node kp ⟪ Red , _ ⟫ _ (node key₂ value₂ t₃ t₅)) (node kg ⟪ Red , _ ⟫ (node key₃ value₃ t₄ t₆) _)) key value (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .(node key₂ value₂ t₃ t₅) .(node key₃ value₃ t₄ t₆) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
-... | t-node .key₂ .kn .key₃ x₆ x₇ x₈ x₉ x₁₀ x₁₁ t t₇ = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫ ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t₇) where
-        rr00 : (kp < kn) ∧ ((kp < key₂) ∧ tr> kp t₃ ∧ tr> kp t₅) ∧ ((kp < key₃) ∧ tr> kp t₄ ∧ tr> kp t₆ )
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₂ < kg) ∧ tr< kg t₃ ∧ tr< kg t₅) ∧ ((key₃ < kg) ∧ tr< kg t₄ ∧ tr< kg t₆ )
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
-        rr03 = proj2 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ (node key₂ value₂ t₅ t₆) (node key₁ value₁ t₁ t₂)) leaf) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-left .kp .kg x x₁ x₂ (t-node key₂ .kp .key₁ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-lr t₃ t₄ kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
-... | t-single .kn .(⟪ Red , _ ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00)  , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  tt tt tt (t-left _ _ x₃ x₅ x₆ ti) (t-single _ _) where
-        rr00 : (kp < kn) ∧ ⊤ ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr01 : (kn < kg) ∧ ⊤ ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
-... | t-right .kn key₃ {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00) , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  tt ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt (t-left _ _ x₃ x₅ x₆ ti) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti₁ trb))) where
-        rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) -- tr> kp (node key₃ v3 t₇ t₈)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ⊤ ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) -- tr< kg (node key₃ v3 t₇ t₈)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
-        rr03 = proj2 (proj2 rr01)
-... | t-left key₃ .kn {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00) , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt tt (t-node key₂ kp key₃ x₃ (proj1 rr02) x₅ x₆ (proj1 (proj2 rr02))  (proj2 (proj2 rr02)) ti (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti₁ trb))) (t-single _ _) where
-        rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
-... | t-node key₃ .kn key₄ {v0} {v1} {v2} {t₇} {t₈} {t₉}  {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₃ = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00)  , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  ⟪ x₉ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt (t-node _ _ _ x₃ (proj1 rr02) x₅ x₆ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti₁ trb))) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t₃) where
-        rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ((kp < key₄) ∧ tr> kp t₉ ∧ tr> kp t₁₀)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ((key₄ < kg) ∧ tr< kg t₉ ∧ tr< kg t₁₀)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
-        rr03 = proj2 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf leaf) (node key₂ value₂ t₅ t₆)) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ .leaf) (node kg ⟪ Red , _ ⟫ .leaf _)) .kn _ (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-single .kp .(⟪ Red , v2 ⟫)) ti₁) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt ⟪ <-trans lt2 x₁  , ⟪ <-tr> x₄ lt2  , <-tr> x₅ lt2 ⟫ ⟫  (t-single _ _) (t-right _ _ x₁ x₄ x₅ ti₁)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .(node key _ _ _) leaf) (node key₂ value₂ t₅ t₆)) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ .leaf) (node kg ⟪ Red , _ ⟫ .leaf _)) .kn _ (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-left key .kp x₆ x₇ x₈ ti) ti₁) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 ⟪ <-trans x₆ lt1 , ⟪ >-tr< x₇ lt1 , >-tr< x₈ lt1 ⟫ ⟫  tt tt ⟪ <-trans lt2 x₁ , ⟪ <-tr> x₄ lt2 , <-tr> x₅ lt2 ⟫ ⟫ (t-left _ _ x₆ x₇ x₈ ti) (t-right _ _ x₁ x₄ x₅ ti₁)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .(node key₂ _ _ _)) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-right .kp .key₁ x₆ x₇ x₈ ti) ti₁) (rbr-rotate-lr t₃ t₄ kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
-... | t-single .kn .(⟪ Red , value₃ ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt tt ⟪ <-trans (proj1 rr01) x₁  , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-single _ _) (t-right _ _ x₁ x₄ x₅ ti₁) where
-        rr00 : (kp < kn) ∧ ⊤ ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr01 : (kn < kg) ∧ ⊤ ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-... | t-right .kn key₃ {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-single _ _) (t-node _ _ _ (proj1 rr03) x₁ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₄ x₅  (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti trb)) ti₁) where
-        rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ⊤ ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-        rr03 = proj2 (proj2 rr01)
-... | t-left key₃ .kn {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₉ , ⟪ x₁₀  , x₁₁ ⟫ ⟫ tt ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-right _ _  (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) (treeLeftDown _ _ (RB-repl→ti _ _ _ _  ti trb))) (t-right _ _ x₁ x₄ x₅ ti₁) where
-        rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-        rr03 = proj1 (proj2 rr01)
-... | t-node key₃ .kn key₄ {v0} {v1} {v2} {t₇} {t₈} {t₉}  {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₃ = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₉ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁  , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01)  ⟫ ⟫ (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02))  (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti trb))) (t-node _ _ _ (proj1 rr03) x₁ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₄ x₅ t₃ ti₁)  where
-        rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ((kp < key₄) ∧ tr> kp t₉ ∧ tr> kp t₁₀)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ((key₄ < kg) ∧ tr< kg t₉ ∧ tr< kg t₁₀)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-        rr03 = proj2 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .(node key₃ _ _ _) (node key₁ value₁ t₁ t₂)) .(node key₂ _ _ _)) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-node key₃ .kp .key₁ x₆ x₇ x₈ x₉ x₁₀ x₁₁ ti ti₂) ti₁) (rbr-rotate-lr t₃ t₄ kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₂ trb
-... | t-single .kn .(⟪ Red , value₃ ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00)  , ⟪ >-tr< x₈ (proj1 rr00) , >-tr< x₉ (proj1 rr00) ⟫ ⟫  tt tt ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-left _ _ x₆ x₈ x₉ ti) (t-right _ _ x₁ x₄ x₅ ti₁) where
-        rr00 : (kp < kn) ∧ ⊤ ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
-        rr01 : (kn < kg) ∧ ⊤ ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-... | t-right .kn key₄ {v1} {v3} {t₇} {t₈} x₁₂ x₁₃ x₁₄ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00) , ⟪ >-tr< x₈ (proj1 rr00) , >-tr< x₉ (proj1 rr00) ⟫ ⟫  tt ⟪ x₁₂ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-left _ _ x₆ x₈ x₉ ti) (t-node _ _ _ (proj1 rr03) x₁ (proj1 (proj2 rr03))  (proj2 (proj2 rr03)) x₄ x₅ (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti₂ trb)) ti₁ ) where
-        rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₄) ∧ tr> kp t₇ ∧ tr> kp t₈)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
-        rr02 = proj2 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ⊤ ∧ ((key₄ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-        rr03 = proj2 (proj2 rr01)
-... | t-left key₄ .kn {v1} {v3} {t₇} {t₈} x₁₂ x₁₃ x₁₄ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00) , ⟪ >-tr< x₈ (proj1 rr00) , >-tr< x₉ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-node _ _ _ x₆ (proj1 rr02) x₈ x₉ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti₂ trb)) )  (t-right _ _ x₁ x₄ x₅ ti₁) where
-        rr00 : (kp < kn) ∧ ((kp < key₄) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₄ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-        rr03 = proj1 (proj2 rr01)
-... | t-node key₄ .kn key₅ {v0} {v1} {v2} {t₇} {t₈} {t₉}  {t₁₀}x₁₂ x₁₃ x₁₄ x₁₅ x₁₆ x₁₇ t t₃ = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00)  , ⟪ >-tr< x₈ (proj1 rr00)   , >-tr< x₉ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₄ , x₁₅ ⟫ ⟫  ⟪ x₁₃ , ⟪ x₁₆ , x₁₇ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01)  ⟫ ⟫  (t-node _ _ _ x₆ (proj1 rr02) x₈ x₉ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti t ) (t-node _ _ _ (proj1 rr04) x₁ (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) x₄ x₅ (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti₂ trb)) ti₁ ) where
-        rr00 : (kp < kn) ∧ ((kp < key₄) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ((kp < key₅) ∧ tr> kp t₉ ∧ tr> kp t₁₀)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
-        rr02 = proj1 (proj2 rr00)
-        rr05 = proj2 (proj2 rr00)
-        rr01 : (kn < kg) ∧ ((key₄ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ((key₅ < kg) ∧ tr< kg t₉ ∧ tr< kg t₁₀)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
-        rr03 = proj1 (proj2 rr01)
-        rr04 = proj2 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-right .kg .kp x x₁ x₂ ti) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node kn ⟪ Red , _ ⟫ leaf leaf) leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-right .kg .kp x x₁ x₂ ti) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 (rbr-node )) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf (node key₁ value₁ t₅ t₆))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-right .kg .kp x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt ⟪ <-trans lt2 x₃ , ⟪ <-tr> x₄ lt2 , <-tr> x₅ lt2 ⟫ ⟫  (t-single _ _) (t-right _ _ x₃ x₄ x₅ ti)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) (node key₁ value₁ t₅ t₆))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-right .kg .kp x x₁ x₂ (t-node key₂ .kp .key₁ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 trb) = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt tt ⟪ <-trans (proj1 rr01) x₄ , ⟪ <-tr> x₇ (proj1 rr01) , <-tr> x₈ (proj1 rr01) ⟫ ⟫  (t-single _ _) (t-right _ _ x₄ x₇ x₈ ti₁) where
-        rr00 : (kg < kn) ∧ ⊤ ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
-        rr01 : (kn < kp) ∧ ⊤ ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf (node key₁ value₁ t₂ t₃)) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-right .kg .kp x x₁ x₂ (t-left key₂ .kp x₃ x₄ x₅ ti)) (rbr-rotate-rl .(node key₁ value₁ t₂ t₃) .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
-... | t-left .key₁ .kn x₆ x₇ x₈ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫ tt tt  (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-single _ _) where
-        rr00 : (kg < kn) ∧ ((kg < key₁) ∧ tr> kg t₂ ∧ tr> kg t₃) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₁ < kp) ∧ tr< kp t₂ ∧ tr< kp t₃) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .(node key₃ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf (node key₁ value₁ t₂ t₃)) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-right .kg .kp x x₁ x₂ (t-node key₂ .kp key₃ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-rl .(node key₁ value₁ t₂ t₃) .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
-... | t-left .key₁ .kn x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt ⟪ <-trans (proj1 rr01) x₄  , ⟪ <-tr> x₇ (proj1 rr01)  , <-tr> x₈ (proj1 rr01) ⟫ ⟫   (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-right _ _ x₄ x₇ x₈ ti₁) where
-        rr00 : (kg < kn) ∧ ((kg < key₁) ∧ tr> kg t₂ ∧ tr> kg t₃) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₁ < kp) ∧ tr< kp t₂ ∧ tr< kp t₃) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) .leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-single .kp .(⟪ Red , vp ⟫))) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 ⟪ <-trans x lt1 , ⟪ >-tr< x₂ lt1 , >-tr< x₃ lt1 ⟫ ⟫  tt tt tt (t-left _ _ x x₂ x₃ ti) (t-single _ _)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf .(node key₂ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) .leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-right .kp key₂ x₆ x₇ x₈ ti₁)) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 ⟪ <-trans x lt1 , ⟪ >-tr< x₂ lt1 , >-tr< x₃ lt1 ⟫ ⟫  tt tt ⟪ <-trans lt2 x₆ , ⟪ <-tr> x₇ lt2 , <-tr> x₈ lt2 ⟫ ⟫   (t-left _ _ x x₂ x₃ ti) (t-right _ _ x₆ x₇ x₈ ti₁)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-left key₂ .kp x₆ x₇ x₈ ti₁)) (rbr-rotate-rl t₂ .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
-... | t-single .kn .(⟪ Red , vn ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  tt tt tt (t-left _ _ x x₂ x₃ ti) (t-single _ _) where
-        rr00 : (kg < kn) ∧ ⊤ ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr01 : (kn < kp) ∧ ⊤ ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-... | t-left key₃ .kn {v1} {v3} {t₇} {t₃} x₉ x₁₀ x₁₁ ti₀ = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫ ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt tt (t-node _ _ _  x (proj1 rr02) x₂ x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti ti₀) (t-single _ _) where
-        rr00 : (kg < kn) ∧ ((kg < key₃) ∧ tr> kg t₇ ∧ tr> kg t₃) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₃ < kp) ∧ tr< kp t₇ ∧ tr< kp t₃) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr03 = proj1 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .(node key₃ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-node key₂ .kp key₃ x₆ x₇ x₈ x₉ x₁₀ x₁₁ ti₁ ti₂)) (rbr-rotate-rl t₂ .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
-... | t-single .kn .(⟪ Red , vn ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  tt tt  ⟪ <-trans (proj1 rr01)  x₇ , ⟪ <-tr> x₁₀ (proj1 rr01)   , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫  (t-left _ _ x x₂ x₃ ti) (t-right _ _ x₇ x₁₀ x₁₁ ti₂) where
-        rr00 : (kg < kn) ∧ ⊤ ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ⊤ ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
-... | t-left key₄ .kn {v1} {v3} {t₇} {t₃} x₁₂ x₁₃ x₁₄ ti₀ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt ⟪ <-trans (proj1 rr01) x₇ , ⟪ <-tr> x₁₀ (proj1 rr01) , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫  (t-node _ _ _ x (proj1 rr02) x₂ x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti ti₀) (t-right _ _ x₇  x₁₀ x₁₁ ti₂) where
-        rr00 : (kg < kn) ∧ ((kg < key₄) ∧ tr> kg t₇ ∧ tr> kg t₃) ∧ ⊤
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr02 = proj1 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₄ < kp) ∧ tr< kp t₇ ∧ tr< kp t₃) ∧ ⊤
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-right .kg .kp x x₁ x₂ (t-left key₂ .kp x₃ x₄ x₅ ti)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
-... | t-right .kn .key₁ x₆ x₇ x₈ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt tt  ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫  tt  (t-single _ _) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t) where
-        rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
-        rr02 = proj2 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
-        rr03 = proj2 (proj2 rr01)
-... | t-node key₃ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₆ x₇ x₈ x₉ x₁₀ x₁₁ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫  ⟪ x₇ , ⟪ x₁₀   , x₁₁ ⟫ ⟫ tt   (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t₁) where
-        rr00 : (kg < kn) ∧ ((kg < key₃) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
-        rr02 = proj1 (proj2 rr00)
-        rr04 = proj2 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₃ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
-        rr03 = proj2 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .(node key₃ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-right .kg .kp x x₁ x₂ (t-node key₂ .kp key₃ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
-... | t-right .kn .key₁ x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt tt  ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₄ , ⟪ <-tr> x₇ (proj1 rr01) , <-tr> x₈ (proj1 rr01)  ⟫ ⟫  (t-single _ _) (t-node _ _ _ (proj1 rr03) x₄ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₇ x₈ t ti₁ ) where
-        rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
-        rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
-        rr03 = proj2 (proj2 rr01)
-... | t-node key₄ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt ⟪ x₉ , ⟪  x₁₁ , x₁₂ ⟫ ⟫ ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₄ , ⟪ <-tr> x₇ (proj1 rr01) , <-tr> x₈ (proj1 rr01) ⟫ ⟫  (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-node _ _ _ (proj1 rr04) x₄ (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) x₇ x₈ t₁ ti₁ ) where
-        rr00 : (kg < kn) ∧ ((kg < key₄) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
-        rr02 = proj1 (proj2 rr00)
-        rr05 = proj2 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₄ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
-        rr03 = proj1 (proj2 rr01)
-        rr04 = proj2 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₃ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₂ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-node key₂ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-left key₃ .kp x₆ x₇ x₈ ti₁)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
-... | t-right .kn .key₁ x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00)  , ⟪ >-tr< x₂ (proj1 rr00)  , >-tr< x₃ (proj1 rr00) ⟫ ⟫ tt  ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫  tt  (t-left _ _ x  x₂ x₃ ti ) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t) where
-        rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr02 = proj2 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr03 = proj2 (proj2 rr01)
-... | t-node key₄ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  ⟪ x₉ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt (t-node _ _ _ x (proj1 rr02) x₂ x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti t) (t-left _ _ (proj1 rr04) (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) t₁) where
-        rr00 : (kg < kn) ∧ ((kg < key₄) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr02 = proj1 (proj2 rr00)
-        rr05 = proj2 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₄ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
-        rr03 = proj1 (proj2 rr01)
-        rr04 = proj2 (proj2 rr01)
-RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₃ _ _ _) .(node key₄ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₂ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-node key₂ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-node key₃ .kp key₄ x₆ x₇ x₈ x₉ x₁₀ x₁₁ ti₁ ti₂)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
-... | t-right .kn .key₁ x₁₂ x₁₃ x₁₄ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)   ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫ tt  ⟪  x₁₂  , ⟪  x₁₃  , x₁₄  ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₇ , ⟪ <-tr> x₁₀ (proj1 rr01) , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫  (t-left _ _ x x₂ x₃ ti) (t-node _ _ _ (proj1 rr03) x₇ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₁₀ x₁₁ t ti₂ ) where
-        rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr02 = proj2 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
-        rr03 = proj2 (proj2 rr01)
-... | t-node key₅ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₁₂ x₁₃ x₁₄ x₁₅ x₁₆ x₁₇ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00)  , >-tr< x₃ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₄ , x₁₅ ⟫ ⟫  ⟪ x₁₃ , ⟪ x₁₆ , x₁₇ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₇  , ⟪ <-tr> x₁₀ (proj1 rr01) , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫   (t-node _ _ _ x (proj1 rr02) x₂  x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti t ) (t-node _ _ _ (proj1 rr04) x₇ (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) x₁₀ x₁₁ t₁ ti₂ ) where
-        rr00 : (kg < kn) ∧ ((kg < key₅) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
-        rr02 = proj1 (proj2 rr00)
-        rr05 = proj2 (proj2 rr00)
-        rr01 : (kn < kp) ∧ ((key₅ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
-        rr03 = proj1 (proj2 rr01)
-        rr04 = proj2 (proj2 rr01)
+RB-repl→ti = ?
+--   RB-repl→ti .leaf .(node key ⟪ Red , value ⟫ leaf leaf) key value ti rbr-leaf = t-single key ⟪ Red , value ⟫
+--   RB-repl→ti .(node key ⟪ _ , _ ⟫ leaf leaf) .(node key ⟪ _ , value ⟫ leaf leaf) key value (t-single .key .(⟪ _ , _ ⟫)) (rbr-node ) = t-single key ⟪ _ , value ⟫
+--   RB-repl→ti .(node key ⟪ _ , _ ⟫ leaf (node key₁ _ _ _)) .(node key ⟪ _ , value ⟫ leaf (node key₁ _ _ _)) key value
+--      (t-right .key key₁ x x₁ x₂ ti) (rbr-node ) = t-right key key₁ x x₁ x₂ ti
+--   RB-repl→ti .(node key ⟪ _ , _ ⟫ (node key₁ _ _ _) leaf) .(node key ⟪ _ , value ⟫ (node key₁ _ _ _) leaf) key value
+--      (t-left key₁ .key x x₁ x₂ ti) (rbr-node ) = t-left key₁ key x x₁ x₂ ti
+--   RB-repl→ti .(node key ⟪ _ , _ ⟫ (node key₁ _ _ _) (node key₂ _ _ _)) .(node key ⟪ _ , value ⟫ (node key₁ _ _ _) (node key₂ _ _ _)) key value
+--      (t-node key₁ .key key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁) (rbr-node ) = t-node key₁ key key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁
+--   RB-repl→ti (node key₁ ⟪ ca , v1 ⟫ leaf leaf) (node key₁ ⟪ ca , v1 ⟫ leaf tree@(node key₂ value₁ t t₁)) key value
+--      (t-single key₁ ⟪ ca , v1 ⟫) (rbr-right _ x trb) = t-right _ _  (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₁ < key₂ ) ∧ tr> key₁ t ∧ tr> key₁ t₁
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x tt
+--   RB-repl→ti (node _ ⟪ _ , _ ⟫ leaf (node key₁ _ _ _)) (node key₂ ⟪ ca , v1 ⟫ leaf (node key₃ value₁ t t₁)) key value
+--      (t-right _ key₁ x₁ x₂ x₃ ti) (rbr-right _ x trb) = t-right _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
+--           rr00 : (key₂ < key₃) ∧ tr> key₂ t ∧ tr> key₂ t₁
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
+--   RB-repl→ti .(node key₂ ⟪ ca , v1 ⟫ (node key₁ value₁ t t₁) leaf) (node key₂ ⟪ ca , v1 ⟫ (node key₁ value₁ t t₁) (node key₃ value₂ t₂ t₃)) key value
+--      (t-left key₁ _ x₁ x₂ x₃ ti) (rbr-right _ x trb) = t-node _ _ _ x₁ (proj1 rr00) x₂ x₃ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₂ < key₃) ∧ tr> key₂ t₂ ∧ tr> key₂ t₃
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x tt
+--   RB-repl→ti .(node key₃ ⟪ ca , v1 ⟫ (node key₁ v2 t₁ t₂) (node key₂ _ _ _)) (node key₃ ⟪ ca , v1 ⟫ (node key₁ v2 t₁ t₂) (node key₄ value₁ t₃ t₄)) key value
+--      (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-right _ x trb) = t-node _ _ _ x₁
+--        (proj1 rr00) x₃ x₄ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ ti₁ trb) where
+--           rr00 : (key₃ < key₄) ∧ tr> key₃ t₃ ∧ tr> key₃ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x ⟪ x₂ , ⟪ x₅ , x₆ ⟫ ⟫
+--   RB-repl→ti .(node key₁ ⟪ _ , _ ⟫ leaf leaf) (node key₁ ⟪ _ , _ ⟫ (node key₂ value₁ left left₁) leaf) key value
+--      (t-single _ .(⟪ _ , _ ⟫)) (rbr-left _ x trb) = t-left _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₂ < key₁) ∧ tr< key₁ left ∧ tr< key₁ left₁
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x tt
+--   RB-repl→ti .(node key₂ ⟪ _ , _ ⟫ leaf (node key₁ _ t₁ t₂)) (node key₂ ⟪ _ , _ ⟫ (node key₃ value₁ t t₃) (node key₁ _ t₁ t₂)) key value
+--      (t-right _ key₁ x₁ x₂ x₃ ti) (rbr-left _ x trb) = t-node _ _ _ (proj1 rr00) x₁  (proj1 (proj2 rr00))(proj2 (proj2 rr00)) x₂ x₃ rr01 ti where
+--           rr00 : (key₃ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₃
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x tt
+--           rr01 : treeInvariant (node key₃ value₁ t t₃)
+--           rr01 = RB-repl→ti _ _ _ _ t-leaf trb
+--   RB-repl→ti .(node _ ⟪ _ , _ ⟫ (node key₁ _ _ _) leaf) (node key₃ ⟪ _ , _ ⟫ (node key₂ value₁ t t₁) leaf) key value
+--       (t-left key₁ _ x₁ x₂ x₃ ti) (rbr-left _ x trb) = t-left key₂ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
+--           rr00 : (key₂ < key₃) ∧ tr< key₃ t ∧ tr< key₃ t₁
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
+--   RB-repl→ti .(node key₃ ⟪ _ , _ ⟫ (node key₁ _ _ _) (node key₂ _ t₁ t₂)) (node key₃ ⟪ _ , _ ⟫ (node key₄ value₁ t t₃) (node key₂ _ t₁ t₂)) key value
+--       (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-left _ x trb) = t-node _ _ _ (proj1 rr00) x₂ (proj1 (proj2 rr00))  (proj2 (proj2 rr00))  x₅ x₆ (RB-repl→ti _ _ _ _ ti trb) ti₁ where
+--           rr00 : (key₄ < key₃) ∧ tr< key₃ t ∧ tr< key₃ t₃
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x ⟪ x₁ , ⟪ x₃ , x₄ ⟫ ⟫
+--   RB-repl→ti .(node x₁ ⟪ Black , c ⟫ leaf leaf) (node x₁ ⟪ Black , c ⟫ leaf (node key₁ value₁ t t₁)) key value
+--       (t-single x₂ .(⟪ Black , c ⟫)) (rbr-black-right x x₄ trb) = t-right _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (x₁ < key₁) ∧ tr> x₁ t ∧ tr> x₁ t₁
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x₄ tt
+--   RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ leaf (node key₁ _ _ _)) (node key₂ ⟪ Black , _ ⟫ leaf (node key₃ value₁ t₂ t₃)) key value
+--       (t-right _ key₁ x₁ x₂ x₃ ti) (rbr-black-right  x x₄ trb) = t-right _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
+--           rr00 : (key₂ < key₃) ∧ tr> key₂ t₂ ∧ tr> key₂ t₃
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x₄ ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
+--   RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) leaf) (node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₃ value₁ t₂ t₃)) key value (t-left key₁ _ x₁ x₂ x₃ ti) (rbr-black-right x x₄ trb) = t-node _ _ _ x₁ (proj1 rr00) x₂ x₃ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₂ < key₃) ∧ tr> key₂ t₂ ∧ tr> key₂ t₃
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x₄ tt
+--   RB-repl→ti .(node key₃ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₂ _ _ _)) (node key₃ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₄ value₁ t₂ t₃)) key value
+--         (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-black-right x x₇ trb) = t-node _ _ _ x₁ (proj1 rr00) x₃ x₄ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti (RB-repl→ti _ _ _ _ ti₁ trb) where
+--           rr00 : (key₃ < key₄) ∧ tr> key₃ t₂ ∧ tr> key₃ t₃
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb x₇ ⟪ x₂ , ⟪ x₅ , x₆ ⟫ ⟫
+--   RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ leaf leaf) (node key₂ ⟪ Black , _ ⟫ (node key₁ value₁ t t₁) .leaf) key value
+--          (t-single .key₂ .(⟪ Black , _ ⟫)) (rbr-black-left x x₇ trb) = t-left _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₁ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ tt
+--   RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ leaf (node key₁ _ _ _)) (node key₂ ⟪ Black , _ ⟫ (node key₃ value₁ t t₁) .(node key₁ _ _ _)) key value
+--          (t-right .key₂ key₁ x₁ x₂ x₃ ti) (rbr-black-left x x₇ trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00))  x₂ x₃ (RB-repl→ti _ _ _ _ t-leaf trb) ti where
+--           rr00 : (key₃ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ tt
+--   RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) leaf) (node key₂ ⟪ Black , _ ⟫ (node key₃ value₁ t t₁) .leaf) key value
+--          (t-left key₁ .key₂ x₁ x₂ x₃ ti) (rbr-black-left x x₇ trb) = t-left _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ ti trb) where
+--           rr00 : (key₃ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ ⟪ x₁ , ⟪ x₂ , x₃ ⟫ ⟫
+--   RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node key₁ _ _ _) (node key₃ _ _ _)) (node key₂ ⟪ Black , _ ⟫ (node key₄ value₁ t t₁) .(node key₃ _ _ _)) key value
+--        (t-node key₁ .key₂ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-black-left x x₇ trb) = t-node _ _ _ (proj1 rr00) x₂ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) x₅ x₆ (RB-repl→ti _ _ _ _ ti trb) ti₁ where
+--           rr00 : (key₄ < key₂) ∧ tr< key₂ t ∧ tr< key₂ t₁
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb x₇ ⟪ x₁ , ⟪ x₃ , x₄ ⟫ ⟫
+--   RB-repl→ti .(node key₂ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ t₁) leaf) (node key₂ ⟪ Red , value₁ ⟫ (node key₁ ⟪ Black , value₂ ⟫ t t₁) .(to-black leaf)) key value
+--         (t-left _ .key₂ x₁ x₂ x₃ ti) (rbr-flip-ll x _ lt trb) = t-left _ _ x₁ rr00 x₃ (RTtoTI0 _ _ _ _ rr02 r-node ) where
+--           rr00 : tr< key₂ t
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb (<-trans lt x₁) x₂
+--           rr02 : treeInvariant (node key₁ ⟪ Red , value₂  ⟫ t t₁)
+--           rr02 = RB-repl→ti _ _ _ _ ti (rbr-left _ lt trb)
+--   RB-repl→ti (node key₂ ⟪ Black , _ ⟫ (node key₁ ⟪ Red , _ ⟫ t₀ t₁) (node key₃ ⟪ c1 , v1 ⟫  left right)) (node key₂ ⟪ Red , value₁ ⟫ (node _ ⟪ Black , value₂ ⟫ t t₁) (node key₃ ⟪ Black , v1 ⟫  left  right)) key value
+--          (t-node _ .key₂ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-ll x _ lt trb) = t-node _ _ _ x₁ x₂ rr00 x₄ x₅ x₆ (RTtoTI0 _ _ _ _ rr02 r-node ) (RTtoTI0 _ _ _ _ ti₁ r-node ) where
+--           rr00 : tr< key₂ t
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb (<-trans lt x₁)  x₃
+--           rr02 : treeInvariant (node key₁ ⟪ Red , value₂ ⟫ t t₁)
+--           rr02  = RB-repl→ti _ _ _ _ ti (rbr-left _ lt trb)
+--   RB-repl→ti (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ left right) leaf) (node key₂ ⟪ Red , v1 ⟫ (node key₃ ⟪ Black , v2 ⟫ left right₁) leaf) key value
+--         (t-left _ _ x₁ x₂ x₃ ti) (rbr-flip-lr x _ lt lt2 trb) = t-left _ _ x₁ x₂ rr00 (RTtoTI0 _ _ _ _ rr02 r-node ) where
+--           rr00 : tr< key₂ right₁
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--           rr02 : treeInvariant (node key₃ ⟪ Red , v2 ⟫ left right₁ )
+--           rr02 = RB-repl→ti _ _ _ _ ti (rbr-right _ lt trb)
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ Red , v2 ⟫ t t₁) (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ t t₄) .(to-black (node key₃ ⟪ c3 , _ ⟫ _ _))) key value
+--         (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-lr x _ lt lt2 trb) = t-node _ _ _ x₁ x₂ x₃ rr00 x₅ x₆ (RTtoTI0 _ _ _ _ rr02 r-node ) (RTtoTI0 _ _ _ _ ti₁ r-node ) where
+--           rr00 : tr< key₁ t₄
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 x₄
+--           rr02 : treeInvariant (node key₂ ⟪ Red , v2 ⟫ t t₄)
+--           rr02 = RB-repl→ti _ _ _ _ ti (rbr-right _ lt trb)
+--   RB-repl→ti (node _ ⟪ Black , _ ⟫ leaf (node _ ⟪ Red , _ ⟫ t t₁)) (node key₁ ⟪ Red , v1 ⟫ .(to-black leaf) (node key₂ ⟪ Black , v2 ⟫ t₂ t₁)) key value
+--         (t-right _ _ x₁ x₂ x₃ ti) (rbr-flip-rl x _ lt lt2 trb) = t-right _ _ x₁ rr00 x₃ (RTtoTI0 _ _ _ _ rr02 r-node ) where
+--           rr00 : tr> key₁ t₂
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt x₂
+--           rr02 : treeInvariant (node key₂ ⟪ Red , v2 ⟫ t₂ t₁)
+--           rr02 = RB-repl→ti _ _ _ _ ti (rbr-left _ lt2 trb)
+--   RB-repl→ti (node _ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node _ ⟪ Red , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , _ ⟫  _ _)) (node key₃ ⟪ Black , _ ⟫ t₄ t₃)) key value
+--         (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-rl x _ lt lt2 trb) = t-node key₂ _ _ x₁ x₂ x₃ x₄ rr00 x₆ (RTtoTI0 _ _ _ _ ti r-node ) (RTtoTI0 _ _ _ _ rr02 r-node ) where
+--           rr00 : tr> key₁ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt x₅
+--           rr02 : treeInvariant (node key₃ ⟪ Red , v3 ⟫ t₄ t₃)
+--           rr02 = RB-repl→ti _ _ _ _ ti₁ (rbr-left _ lt2 trb)
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ t t₁)) (node _ ⟪ Red , _ ⟫ .(to-black leaf) (node _ ⟪ Black , v2 ⟫ t t₂)) key value
+--       (t-right _ _ x₁ x₂ x₃ ti) (rbr-flip-rr x _ lt trb) = t-right _ _ x₁ x₂ rr00 (RTtoTI0 _ _ _ _ rr02 r-node ) where
+--           rr00 : tr> key₁ t₂
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb (<-trans x₁ lt )  x₃
+--           rr02 : treeInvariant (node key₂ ⟪ Red , v2 ⟫ t t₂)
+--           rr02 = RB-repl→ti _ _ _ _ ti (rbr-right _ lt trb)
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node key₃ ⟪ Red , c3 ⟫ t₂ t₃)) (node _ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , _ ⟫ _ _)) (node _ ⟪ Black , c3 ⟫ t₂ t₄)) key value
+--       (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-rr x _ lt trb) = t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ rr00 (RTtoTI0 _ _ _ _ ti r-node ) (RTtoTI0 _ _ _ _ rr02 r-node ) where
+--           rr00 : tr> key₁ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb (<-trans x₂ lt) x₆
+--           rr02 : treeInvariant (node key₃ ⟪ Red , c3 ⟫ t₂ t₄)
+--           rr02 = RB-repl→ti _ _ _ _ ti₁ (rbr-right _ lt trb)
+--   RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ .leaf .leaf) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ .leaf leaf)) key value (t-left _ _ x₁ x₂ x₃
+--       (t-single .k2 .(⟪ Red , c2 ⟫))) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10)) (proj2 (proj2 rr10)) tt tt (RB-repl→ti _ _ _ _ t-leaf trb) (t-single _ _ ) where
+--           rr10 : (key₁ < k2 ) ∧ tr< k2 t₂ ∧ tr< k2 t₃
+--           rr10 = RB-repl→ti< _ _ _ _ _ trb lt tt
+--   RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ .leaf .(node key₂ _ _ _)) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ (node key₂ value₂ t₁ t₄) leaf)) key value
+--       (t-left _ _ x₁ x₂ x₃ (t-right .k2 key₂ x₄ x₅ x₆ ti)) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10)) (proj2 (proj2 rr10)) ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫  tt rr05 rr04 where
+--           rr10 : (key₁ < k2 ) ∧ tr< k2 t₂ ∧ tr< k2 t₃
+--           rr10 = RB-repl→ti< _ _ _ _ _ trb lt tt
+--           rr04 : treeInvariant (node k1 ⟪ Red , c1 ⟫ (node key₂ value₂ t₁ t₄) leaf)
+--           rr04 = RTtoTI0 _ _ _ _ (t-left key₂ _ {_} {⟪ Red , c1 ⟫} {t₁} {t₄} (proj1 x₃) (proj1 (proj2 x₃)) (proj2 (proj2 x₃)) ti) r-node
+--           rr05 : treeInvariant (node key₁ value₁ t₂ t₃)
+--           rr05 = RB-repl→ti _ _ _ _ t-leaf trb
+--   RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ (node key₂ value₂ t₁ t₄) .leaf) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ .leaf leaf)) key value
+--      (t-left _ _ x₁ x₂ x₃ (t-left key₂ .k2 x₄ x₅ x₆ ti)) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10)) (proj2 (proj2 rr10)) tt tt (RB-repl→ti _ _ _ _ ti trb) (t-single _ _) where
+--           rr10 : (key₁ < k2 ) ∧ tr< k2 t₂ ∧ tr< k2 t₃
+--           rr10 = RB-repl→ti< _ _ _ _ _ trb lt ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫
+--   RB-repl→ti {_} {A} (node k1 ⟪ Black , c1 ⟫ (node k2 ⟪ Red , c2 ⟫ (node key₂ value₃ left right) (node key₃ value₂ t₄ t₅)) leaf) (node _ ⟪ Black , _ ⟫ (node key₁ value₁ t₂ t₃) (node _ ⟪ Red , _ ⟫ .(node key₃ _ _ _) leaf)) key value
+--        (t-left _ _ x₁ x₂ x₃ (t-node key₂ .k2 key₃ x₄ x₅ x₆ x₇ x₈ x₉ ti ti₁)) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr10) x₁ (proj1 (proj2 rr10))  (proj2 (proj2 rr10))  ⟪ x₅ , ⟪ x₈ ,  x₉ ⟫ ⟫ tt rr05 rr04 where
+--           rr06 : key < k2
+--           rr06 = lt
+--           rr10 : (key₁ < k2) ∧ tr< k2 t₂ ∧ tr< k2 t₃
+--           rr10 = RB-repl→ti< _ _ _ _ _ trb rr06 ⟪ x₄ , ⟪ x₆ , x₇ ⟫ ⟫
+--           rr04 : treeInvariant (node k1 ⟪ Red , c1 ⟫ (node key₃ value₂ t₄ t₅) leaf)
+--           rr04 = RTtoTI0 _ _ _ _ (t-left _ _ (proj1 x₃) (proj1 (proj2 x₃)) (proj2 (proj2 x₃)) ti₁ ) (r-left (proj1 x₃) r-node)
+--           rr05 : treeInvariant (node key₁ value₁ t₂ t₃)
+--           rr05 = RB-repl→ti _ _ _ _ ti trb
+--   RB-repl→ti (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ .leaf .leaf) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ .leaf (node key₃ _ _ _))) key value
+--      (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-single .key₂ .(⟪ Red , c2 ⟫)) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00))  (proj2 (proj2 rr00))  tt ⟪ <-trans x₁ x₂ , ⟪ <-tr> x₅ x₁ , <-tr> x₆ x₁  ⟫ ⟫  rr02 rr03 where
+--          rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
+--          rr00 = RB-repl→ti< _ _ _ _ _ trb lt tt
+--          rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
+--          rr02 = RB-repl→ti _ _ _ _ t-leaf trb
+--          rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ leaf (node key₃ v3 t₂ t₃))
+--          rr03 = RTtoTI0 _ _ _ _ (t-right _ _ {v3} {_} x₂ x₅ x₆ ti₁) r-node
+--   RB-repl→ti (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ leaf (node key₅ _ _ _)) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ (node key₅ value₂ t₁ t₆) (node key₃ _ _ _))) key value
+--       (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-right .key₂ key₅ x₇ x₈ x₉ ti) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫  ⟪ <-trans x₁ x₂  , ⟪ <-tr> x₅ x₁ , <-tr> x₆ x₁ ⟫ ⟫ rr02 rr03 where
+--          rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
+--          rr00 = RB-repl→ti< _ _ _ _ _ trb lt tt
+--          rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
+--          rr02 = RB-repl→ti _ _ _ _ t-leaf trb
+--          rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ (node key₅ value₂ t₁ t₆) (node key₃ v3 t₂ t₃))
+--          rr03 = RTtoTI0 _ _ _ _ (t-node _ _ _ {_} {v3} {_} {_} {_} {_} {_} (proj1 x₄) x₂ (proj1 (proj2 x₄)) (proj2 (proj2 x₄)) x₅ x₆ ti ti₁ ) r-node
+--   RB-repl→ti (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ .(node key₅ _ _ _) .leaf) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ .leaf (node key₃ _ _ _))) key value (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆
+--       (t-left key₅ .key₂ x₇ x₈ x₉ ti) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00))  tt ⟪ <-trans x₁ x₂  , ⟪ <-tr> x₅ x₁  , <-tr> x₆ x₁ ⟫ ⟫  rr02 rr04 where
+--           rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫
+--           rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
+--           rr02 = RB-repl→ti _ _ _ _ ti trb
+--           rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ (node key₅ _ _ _) (node key₃ v3 t₂ t₃))
+--           rr03 = RTtoTI0 _ _ _ _ (t-node _ _ _ {_} {v3} {_} {_} {_} {_} {_} (proj1 x₃) x₂ (proj1 (proj2 x₃)) (proj2 (proj2 x₃)) x₅ x₆ ti ti₁) r-node
+--           rr04 :  treeInvariant (node key₁ ⟪ Red , c1 ⟫ leaf (node key₃ v3 t₂ t₃))
+--           rr04 = RTtoTI0 _ _ _ _ (t-right _ _ {v3} {_} x₂ x₅ x₆ ti₁) r-node
+--   RB-repl→ti {_} {A} (node key₁ ⟪ Black , c1 ⟫ (node key₂ ⟪ Red , c2 ⟫ .(node key₅ _ _ _) (node key₆ value₆ t₆ t₇)) (node key₃ v3 t₂ t₃)) (node _ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) (node _ ⟪ Red , _ ⟫ .(node key₆ _ _ _) (node key₃ _ _ _))) key value
+--     (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-node key₅ .key₂ key₆ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti ti₂) ti₁) (rbr-rotate-ll x lt trb) = t-node _ _ _ (proj1 rr00) x₁ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ⟪ x₈ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ <-trans x₁ x₂ , ⟪ rr05 , <-tr> x₆ x₁ ⟫ ⟫ rr02 rr03 where
+--           rr00 : (key₄ < key₂) ∧ tr< key₂ t₄ ∧ tr< key₂ t₅
+--           rr00 = RB-repl→ti< _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₉ , x₁₀ ⟫ ⟫
+--           rr02 : treeInvariant (node key₄ value₁ t₄ t₅)
+--           rr02 = RB-repl→ti _ _ _ _ ti trb
+--           rr03 : treeInvariant (node key₁ ⟪ Red , c1 ⟫ (node key₆ value₆ t₆ t₇) (node key₃ v3 t₂ t₃))
+--           rr03 = RTtoTI0 _ _ _ _(t-node _ _ _ {_} {value₁} {_} {_} {_} {_} {_} (proj1 x₄) x₂ (proj1 (proj2 x₄)) (proj2 (proj2 x₄)) x₅ x₆ ti₂ ti₁) r-node
+--           rr05 : tr> key₂ t₂
+--           rr05 = <-tr> x₅ x₁
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ leaf leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₃ value₁ t₃ t₄)) key value
+--       (t-right .key₁ .key₂ x₁ x₂ x₃ (t-single .key₂ .(⟪ Red , _ ⟫))) (rbr-rotate-rr x lt trb)
+--         = t-node _ _ _ x₁ (proj1 rr00) tt tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-single _ _) (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₂ < key₃) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ leaf (node key₃ value₃ t t₁))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₄ value₁ t₃ t₄)) key value
+--      (t-right .key₁ .key₂ x₁ x₂ x₃ (t-right .key₂ key₃ x₄ x₅ x₆ ti)) (rbr-rotate-rr x lt trb)
+--         = t-node _ _ _ x₁ (proj1 rr00) tt tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-single _ _ ) (RB-repl→ti _ _ _ _ ti trb) where
+--           rr00 : (key₂ < key₄) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ (node key₃ _ _ _) leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₄ value₁ t₃ t₄)) key value
+--       (t-right .key₁ .key₂ x₁ x₂ x₃ (t-left key₃ .key₂ x₄ x₅ x₆ ti)) (rbr-rotate-rr x lt trb)
+--          = t-node _ _ _ x₁ (proj1 rr00) tt ⟪ x₄ , ⟪ x₅ , x₆ ⟫ ⟫ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-right _ _ (proj1 x₂) (proj1 (proj2 x₂)) (proj2 (proj2 x₂)) ti)  (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₂ < key₄) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ leaf (node key₂ ⟪ Red , v2 ⟫ (node key₃ _ _ _) (node key₄ _ _ _))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₅ value₁ t₃ t₄)) key value
+--       (t-right .key₁ .key₂ x₁ x₂ x₃ (t-node key₃ .key₂ key₄ x₄ x₅ x₆ x₇ x₈ x₉ ti ti₁)) (rbr-rotate-rr x lt trb) = t-node _ _ _ x₁ (proj1 rr00) tt ⟪ x₄ , ⟪ x₆ , x₇ ⟫ ⟫ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-right _ _ (proj1 x₂) (proj1 (proj2 x₂)) (proj2 (proj2 x₂)) ti) (RB-repl→ti _ _ _ _ ti₁ trb) where
+--           rr00 : (key₂ < key₅) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₅ , ⟪ x₈ , x₉ ⟫ ⟫
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ .(node key₃ _ _ _) (node key₂ ⟪ Red , v2 ⟫ .leaf .leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₄ value₁ t₃ t₄)) key value
+--      (t-node key₃ .key₁ .key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-single .key₂ .(⟪ Red , v2 ⟫))) (rbr-rotate-rr x lt trb)
+--        = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂ , >-tr< x₄ x₂ ⟫ ⟫  tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-left _ _ x₁ x₃ x₄ ti) (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₂ < key₄) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₃ v3 t t₁) (node key₂ ⟪ Red , v2 ⟫ leaf (node key₄ _ _ _))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₅ value₁ t₃ t₄)) key value
+--      (t-node key₃ .key₁ .key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-right .key₂ key₄ x₇ x₈ x₉ ti₁)) (rbr-rotate-rr x lt trb)
+--         = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂ , >-tr< x₄ x₂ ⟫ ⟫ tt (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-left _ _ x₁ x₃ x₄ ti) (RB-repl→ti _ _ _ _ ti₁ trb) where
+--           rr00 : (key₂ < key₅) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₃ _ _ _) (node key₂ ⟪ Red , v2 ⟫ (node key₄ _ _ _) leaf)) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₅ value₁ t₃ t₄)) key value
+--      (t-node key₃ key₁ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-left key₄ key₂ x₇ x₈ x₉ ti₁)) (rbr-rotate-rr x lt trb)
+--         = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂  , >-tr< x₄ x₂ ⟫ ⟫ ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (t-node _ _ _  x₁ (proj1 x₅) x₃ x₄ (proj1 (proj2 x₅)) (proj2 (proj2 x₅)) ti ti₁) (RB-repl→ti _ _ _ _ t-leaf trb) where
+--           rr00 : (key₂ < key₅) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
+--   RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₃ _ _ _) (node key₂ ⟪ Red , v2 ⟫ (node key₄ _ left right) (node key₅ _ _ _))) (node _ ⟪ Black , _ ⟫ (node _ ⟪ Red , _ ⟫ _ _) (node key₆ value₁ t₃ t₄)) key value
+--      (t-node key₃ key₁ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-node key₄ key₂ key₅ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti₁ ti₂)) (rbr-rotate-rr x lt trb)
+--        = t-node _ _ _ x₂ (proj1 rr00) ⟪ <-trans x₁ x₂ , ⟪ >-tr< x₃ x₂ , >-tr< x₄ x₂ ⟫ ⟫  ⟪ x₇ , ⟪ x₉ , x₁₀ ⟫ ⟫  (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RTtoTI0 _ _ _ _ (t-node _ _ _ {_} {value₁} x₁ (proj1 x₅) x₃ x₄  (proj1 (proj2 x₅)) (proj2 (proj2 x₅)) ti ti₁ ) r-node )
+--        (RB-repl→ti _ _ _ _ ti₂ trb) where
+--           rr00 : (key₂ < key₆) ∧ tr> key₂ t₃ ∧ tr> key₂ t₄
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₈ , ⟪ x₁₁ , x₁₂ ⟫ ⟫
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ leaf leaf) .leaf) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ leaf _)) .kn _ (t-left .kp .kg x x₁ x₂ ti) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ (node key₁ value₁ t t₁) leaf) .leaf) (node kn ⟪ Black , v3 ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ leaf _)) .kn .v3 (t-left .kp .kg x x₁ x₂ (t-left .key₁ .kp x₃ x₄ x₅ ti)) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) =
+--       t-node _ _ _ lt1 lt2 ⟪ <-trans x₃ lt1  , ⟪ >-tr< x₄ lt1  , >-tr< x₅ lt1 ⟫ ⟫  tt tt tt (t-left _ _ x₃ x₄ x₅ ti) (t-single _ _)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ .(⟪ Red , _ ⟫) .leaf .leaf)) .leaf) (node .key₁ ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ leaf _)) .key₁ _ (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .leaf .leaf kg kp .key₁ _ lt1 lt2 (rbr-node )) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .leaf) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ leaf) (node kg ⟪ Red , _ ⟫ (node key₂ value₂ t₄ t₅) t₆)) key value (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .leaf .(node key₂ value₂ t₄ t₅) kg kp kn _ lt1 lt2 trb) =
+--      t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt rr03 tt (t-single _ _) (t-left _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) (treeRightDown _ _ ( RB-repl→ti _ _ _ _ ti trb))) where
+--           rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₂) ∧ tr> kp t₄ ∧ tr> kp t₅ )
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
+--           rr01 : (kn < kg) ∧ ⊤ ∧ ((key₂ < kg ) ∧ tr< kg t₄ ∧ tr< kg t₅ ) -- tr< kg (node key₂ value₂ t₄ t₅)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
+--           rr02 = proj2 (proj2 rr01)
+--           rr03 : (kn < key₂) ∧ tr> kn t₄ ∧ tr> kn t₅
+--           rr03 with RB-repl→ti _ _ _ _ ti trb
+--           ... | t-right .kn .key₂ x x₁ x₂ t = ⟪ x , ⟪ x₁ , x₂ ⟫ ⟫
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .leaf) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ (node key₂ value₂ t₃ t₅)) (node kg ⟪ Red , _ ⟫ leaf _)) key value (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .(node key₂ value₂ t₃ t₅) .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
+--   ... | t-left .key₂ .kn x₆ x₇ x₈ t =
+--      t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫ tt tt (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-single _ _) where
+--           rr00 : (kp < kn) ∧ ((kp < key₂) ∧ tr> kp t₃ ∧ tr> kp t₅) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₂ < kg) ∧ tr< kg t₃ ∧ tr< kg t₅) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .leaf) (node kn ⟪ Black , value₄ ⟫ (node kp ⟪ Red , _ ⟫ _ (node key₂ value₂ t₃ t₅)) (node kg ⟪ Red , _ ⟫ (node key₃ value₃ t₄ t₆) _)) key value (t-left .kp .kg x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-lr .(node key₂ value₂ t₃ t₅) .(node key₃ value₃ t₄ t₆) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
+--   ... | t-node .key₂ .kn .key₃ x₆ x₇ x₈ x₉ x₁₀ x₁₁ t t₇ = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫ ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t₇) where
+--           rr00 : (kp < kn) ∧ ((kp < key₂) ∧ tr> kp t₃ ∧ tr> kp t₅) ∧ ((kp < key₃) ∧ tr> kp t₄ ∧ tr> kp t₆ )
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₂ < kg) ∧ tr< kg t₃ ∧ tr< kg t₅) ∧ ((key₃ < kg) ∧ tr< kg t₄ ∧ tr< kg t₆ )
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
+--           rr03 = proj2 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ (node key₂ value₂ t₅ t₆) (node key₁ value₁ t₁ t₂)) leaf) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-left .kp .kg x x₁ x₂ (t-node key₂ .kp .key₁ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-lr t₃ t₄ kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
+--   ... | t-single .kn .(⟪ Red , _ ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00)  , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  tt tt tt (t-left _ _ x₃ x₅ x₆ ti) (t-single _ _) where
+--           rr00 : (kp < kn) ∧ ⊤ ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr01 : (kn < kg) ∧ ⊤ ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
+--   ... | t-right .kn key₃ {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00) , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  tt ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt (t-left _ _ x₃ x₅ x₆ ti) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti₁ trb))) where
+--           rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) -- tr> kp (node key₃ v3 t₇ t₈)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ⊤ ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) -- tr< kg (node key₃ v3 t₇ t₈)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
+--           rr03 = proj2 (proj2 rr01)
+--   ... | t-left key₃ .kn {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00) , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt tt (t-node key₂ kp key₃ x₃ (proj1 rr02) x₅ x₆ (proj1 (proj2 rr02))  (proj2 (proj2 rr02)) ti (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti₁ trb))) (t-single _ _) where
+--           rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
+--   ... | t-node key₃ .kn key₄ {v0} {v1} {v2} {t₇} {t₈} {t₉}  {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₃ = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₃ (proj1 rr00)  , ⟪ >-tr< x₅ (proj1 rr00) , >-tr< x₆ (proj1 rr00) ⟫ ⟫  ⟪ x₉ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt (t-node _ _ _ x₃ (proj1 rr02) x₅ x₆ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti₁ trb))) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t₃) where
+--           rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ((kp < key₄) ∧ tr> kp t₉ ∧ tr> kp t₁₀)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₄ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ((key₄ < kg) ∧ tr< kg t₉ ∧ tr< kg t₁₀)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₂
+--           rr03 = proj2 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf leaf) (node key₂ value₂ t₅ t₆)) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ .leaf) (node kg ⟪ Red , _ ⟫ .leaf _)) .kn _ (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-single .kp .(⟪ Red , v2 ⟫)) ti₁) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt ⟪ <-trans lt2 x₁  , ⟪ <-tr> x₄ lt2  , <-tr> x₅ lt2 ⟫ ⟫  (t-single _ _) (t-right _ _ x₁ x₄ x₅ ti₁)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .(node key _ _ _) leaf) (node key₂ value₂ t₅ t₆)) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ .leaf) (node kg ⟪ Red , _ ⟫ .leaf _)) .kn _ (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-left key .kp x₆ x₇ x₈ ti) ti₁) (rbr-rotate-lr .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 ⟪ <-trans x₆ lt1 , ⟪ >-tr< x₇ lt1 , >-tr< x₈ lt1 ⟫ ⟫  tt tt ⟪ <-trans lt2 x₁ , ⟪ <-tr> x₄ lt2 , <-tr> x₅ lt2 ⟫ ⟫ (t-left _ _ x₆ x₇ x₈ ti) (t-right _ _ x₁ x₄ x₅ ti₁)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .leaf (node key₁ value₁ t₁ t₂)) .(node key₂ _ _ _)) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-right .kp .key₁ x₆ x₇ x₈ ti) ti₁) (rbr-rotate-lr t₃ t₄ kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
+--   ... | t-single .kn .(⟪ Red , value₃ ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt tt ⟪ <-trans (proj1 rr01) x₁  , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-single _ _) (t-right _ _ x₁ x₄ x₅ ti₁) where
+--           rr00 : (kp < kn) ∧ ⊤ ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr01 : (kn < kg) ∧ ⊤ ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--   ... | t-right .kn key₃ {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-single _ _) (t-node _ _ _ (proj1 rr03) x₁ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₄ x₅  (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti trb)) ti₁) where
+--           rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ⊤ ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--           rr03 = proj2 (proj2 rr01)
+--   ... | t-left key₃ .kn {v1} {v3} {t₇} {t₈} x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₉ , ⟪ x₁₀  , x₁₁ ⟫ ⟫ tt ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-right _ _  (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) (treeLeftDown _ _ (RB-repl→ti _ _ _ _  ti trb))) (t-right _ _ x₁ x₄ x₅ ti₁) where
+--           rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--           rr03 = proj1 (proj2 rr01)
+--   ... | t-node key₃ .kn key₄ {v0} {v1} {v2} {t₇} {t₈} {t₉}  {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₃ = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₉ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁  , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01)  ⟫ ⟫ (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02))  (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti trb))) (t-node _ _ _ (proj1 rr03) x₁ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₄ x₅ t₃ ti₁)  where
+--           rr00 : (kp < kn) ∧ ((kp < key₃) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ((kp < key₄) ∧ tr> kp t₉ ∧ tr> kp t₁₀)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₃ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ((key₄ < kg) ∧ tr< kg t₉ ∧ tr< kg t₁₀)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--           rr03 = proj2 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ .(node key₃ _ _ _) (node key₁ value₁ t₁ t₂)) .(node key₂ _ _ _)) (node kn ⟪ Black , value₃ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ (t-node key₃ .kp .key₁ x₆ x₇ x₈ x₉ x₁₀ x₁₁ ti ti₂) ti₁) (rbr-rotate-lr t₃ t₄ kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₂ trb
+--   ... | t-single .kn .(⟪ Red , value₃ ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00)  , ⟪ >-tr< x₈ (proj1 rr00) , >-tr< x₉ (proj1 rr00) ⟫ ⟫  tt tt ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01)  , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-left _ _ x₆ x₈ x₉ ti) (t-right _ _ x₁ x₄ x₅ ti₁) where
+--           rr00 : (kp < kn) ∧ ⊤ ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
+--           rr01 : (kn < kg) ∧ ⊤ ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--   ... | t-right .kn key₄ {v1} {v3} {t₇} {t₈} x₁₂ x₁₃ x₁₄ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00) , ⟪ >-tr< x₈ (proj1 rr00) , >-tr< x₉ (proj1 rr00) ⟫ ⟫  tt ⟪ x₁₂ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-left _ _ x₆ x₈ x₉ ti) (t-node _ _ _ (proj1 rr03) x₁ (proj1 (proj2 rr03))  (proj2 (proj2 rr03)) x₄ x₅ (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti₂ trb)) ti₁ ) where
+--           rr00 : (kp < kn) ∧ ⊤ ∧ ((kp < key₄) ∧ tr> kp t₇ ∧ tr> kp t₈)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
+--           rr02 = proj2 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ⊤ ∧ ((key₄ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--           rr03 = proj2 (proj2 rr01)
+--   ... | t-left key₄ .kn {v1} {v3} {t₇} {t₈} x₁₂ x₁₃ x₁₄ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00) , ⟪ >-tr< x₈ (proj1 rr00) , >-tr< x₉ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01) ⟫ ⟫  (t-node _ _ _ x₆ (proj1 rr02) x₈ x₉ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti (treeLeftDown _ _ (RB-repl→ti _ _ _ _ ti₂ trb)) )  (t-right _ _ x₁ x₄ x₅ ti₁) where
+--           rr00 : (kp < kn) ∧ ((kp < key₄) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₄ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--           rr03 = proj1 (proj2 rr01)
+--   ... | t-node key₄ .kn key₅ {v0} {v1} {v2} {t₇} {t₈} {t₉}  {t₁₀}x₁₂ x₁₃ x₁₄ x₁₅ x₁₆ x₁₇ t t₃ = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x₆ (proj1 rr00)  , ⟪ >-tr< x₈ (proj1 rr00)   , >-tr< x₉ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₄ , x₁₅ ⟫ ⟫  ⟪ x₁₃ , ⟪ x₁₆ , x₁₇ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₁ , ⟪ <-tr> x₄ (proj1 rr01) , <-tr> x₅ (proj1 rr01)  ⟫ ⟫  (t-node _ _ _ x₆ (proj1 rr02) x₈ x₉ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti t ) (t-node _ _ _ (proj1 rr04) x₁ (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) x₄ x₅ (treeRightDown _ _ (RB-repl→ti _ _ _ _ ti₂ trb)) ti₁ ) where
+--           rr00 : (kp < kn) ∧ ((kp < key₄) ∧ tr> kp t₇ ∧ tr> kp t₈) ∧ ((kp < key₅) ∧ tr> kp t₉ ∧ tr> kp t₁₀)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 ⟪ x₇ , ⟪ x₁₀ , x₁₁ ⟫ ⟫
+--           rr02 = proj1 (proj2 rr00)
+--           rr05 = proj2 (proj2 rr00)
+--           rr01 : (kn < kg) ∧ ((key₄ < kg) ∧ tr< kg t₇ ∧ tr< kg t₈) ∧ ((key₅ < kg) ∧ tr< kg t₉ ∧ tr< kg t₁₀)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 x₃
+--           rr03 = proj1 (proj2 rr01)
+--           rr04 = proj2 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-right .kg .kp x x₁ x₂ ti) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node kn ⟪ Red , _ ⟫ leaf leaf) leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-right .kg .kp x x₁ x₂ ti) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 (rbr-node )) = t-node _ _ _ lt1 lt2 tt tt tt tt (t-single _ _) (t-single _ _)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf (node key₁ value₁ t₅ t₆))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-right .kg .kp x x₁ x₂ (t-right .kp .key₁ x₃ x₄ x₅ ti)) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 tt tt tt ⟪ <-trans lt2 x₃ , ⟪ <-tr> x₄ lt2 , <-tr> x₅ lt2 ⟫ ⟫  (t-single _ _) (t-right _ _ x₃ x₄ x₅ ti)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) (node key₁ value₁ t₅ t₆))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf leaf) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-right .kg .kp x x₁ x₂ (t-node key₂ .kp .key₁ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 trb) = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt tt tt ⟪ <-trans (proj1 rr01) x₄ , ⟪ <-tr> x₇ (proj1 rr01) , <-tr> x₈ (proj1 rr01) ⟫ ⟫  (t-single _ _) (t-right _ _ x₄ x₇ x₈ ti₁) where
+--           rr00 : (kg < kn) ∧ ⊤ ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
+--           rr01 : (kn < kp) ∧ ⊤ ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf (node key₁ value₁ t₂ t₃)) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-right .kg .kp x x₁ x₂ (t-left key₂ .kp x₃ x₄ x₅ ti)) (rbr-rotate-rl .(node key₁ value₁ t₂ t₃) .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
+--   ... | t-left .key₁ .kn x₆ x₇ x₈ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫ tt tt  (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-single _ _) where
+--           rr00 : (kg < kn) ∧ ((kg < key₁) ∧ tr> kg t₂ ∧ tr> kg t₃) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₁ < kp) ∧ tr< kp t₂ ∧ tr< kp t₃) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .(node key₃ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf (node key₁ value₁ t₂ t₃)) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-right .kg .kp x x₁ x₂ (t-node key₂ .kp key₃ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-rl .(node key₁ value₁ t₂ t₃) .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
+--   ... | t-left .key₁ .kn x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01) tt ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt ⟪ <-trans (proj1 rr01) x₄  , ⟪ <-tr> x₇ (proj1 rr01)  , <-tr> x₈ (proj1 rr01) ⟫ ⟫   (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-right _ _ x₄ x₇ x₈ ti₁) where
+--           rr00 : (kg < kn) ∧ ((kg < key₁) ∧ tr> kg t₂ ∧ tr> kg t₃) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₁ < kp) ∧ tr< kp t₂ ∧ tr< kp t₃) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) .leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-single .kp .(⟪ Red , vp ⟫))) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 ⟪ <-trans x lt1 , ⟪ >-tr< x₂ lt1 , >-tr< x₃ lt1 ⟫ ⟫  tt tt tt (t-left _ _ x x₂ x₃ ti) (t-single _ _)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .leaf .(node key₂ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) .leaf) (node kp ⟪ Red , _ ⟫ leaf _)) .kn .vn (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-right .kp key₂ x₆ x₇ x₈ ti₁)) (rbr-rotate-rl .leaf .leaf kg kp kn _ lt1 lt2 rbr-leaf) = t-node _ _ _ lt1 lt2 ⟪ <-trans x lt1 , ⟪ >-tr< x₂ lt1 , >-tr< x₃ lt1 ⟫ ⟫  tt tt ⟪ <-trans lt2 x₆ , ⟪ <-tr> x₇ lt2 , <-tr> x₈ lt2 ⟫ ⟫   (t-left _ _ x x₂ x₃ ti) (t-right _ _ x₆ x₇ x₈ ti₁)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-left key₂ .kp x₆ x₇ x₈ ti₁)) (rbr-rotate-rl t₂ .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
+--   ... | t-single .kn .(⟪ Red , vn ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  tt tt tt (t-left _ _ x x₂ x₃ ti) (t-single _ _) where
+--           rr00 : (kg < kn) ∧ ⊤ ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr01 : (kn < kp) ∧ ⊤ ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--   ... | t-left key₃ .kn {v1} {v3} {t₇} {t₃} x₉ x₁₀ x₁₁ ti₀ = t-node _ _ _ (proj1 rr00) (proj1 rr01) ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫ ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫ tt tt (t-node _ _ _  x (proj1 rr02) x₂ x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti ti₀) (t-single _ _) where
+--           rr00 : (kg < kn) ∧ ((kg < key₃) ∧ tr> kg t₇ ∧ tr> kg t₃) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₃ < kp) ∧ tr< kp t₇ ∧ tr< kp t₃) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr03 = proj1 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .(node key₃ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₁ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ leaf _)) key value (t-node key₁ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-node key₂ .kp key₃ x₆ x₇ x₈ x₉ x₁₀ x₁₁ ti₁ ti₂)) (rbr-rotate-rl t₂ .leaf kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
+--   ... | t-single .kn .(⟪ Red , vn ⟫) = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  tt tt  ⟪ <-trans (proj1 rr01)  x₇ , ⟪ <-tr> x₁₀ (proj1 rr01)   , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫  (t-left _ _ x x₂ x₃ ti) (t-right _ _ x₇ x₁₀ x₁₁ ti₂) where
+--           rr00 : (kg < kn) ∧ ⊤ ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ⊤ ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
+--   ... | t-left key₄ .kn {v1} {v3} {t₇} {t₃} x₁₂ x₁₃ x₁₄ ti₀ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt ⟪ <-trans (proj1 rr01) x₇ , ⟪ <-tr> x₁₀ (proj1 rr01) , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫  (t-node _ _ _ x (proj1 rr02) x₂ x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti ti₀) (t-right _ _ x₇  x₁₀ x₁₁ ti₂) where
+--           rr00 : (kg < kn) ∧ ((kg < key₄) ∧ tr> kg t₇ ∧ tr> kg t₃) ∧ ⊤
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr02 = proj1 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₄ < kp) ∧ tr< kp t₇ ∧ tr< kp t₃) ∧ ⊤
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-right .kg .kp x x₁ x₂ (t-left key₂ .kp x₃ x₄ x₅ ti)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
+--   ... | t-right .kn .key₁ x₆ x₇ x₈ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt tt  ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫  tt  (t-single _ _) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t) where
+--           rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
+--           rr02 = proj2 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
+--           rr03 = proj2 (proj2 rr01)
+--   ... | t-node key₃ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₆ x₇ x₈ x₉ x₁₀ x₁₁ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫  ⟪ x₇ , ⟪ x₁₀   , x₁₁ ⟫ ⟫ tt   (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t₁) where
+--           rr00 : (kg < kn) ∧ ((kg < key₃) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
+--           rr02 = proj1 (proj2 rr00)
+--           rr04 = proj2 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₃ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₄ , x₅ ⟫ ⟫
+--           rr03 = proj2 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₂ _ _ _) .(node key₃ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .leaf t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-right .kg .kp x x₁ x₂ (t-node key₂ .kp key₃ x₃ x₄ x₅ x₆ x₇ x₈ ti ti₁)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti trb
+--   ... | t-right .kn .key₁ x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt tt  ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₄ , ⟪ <-tr> x₇ (proj1 rr01) , <-tr> x₈ (proj1 rr01)  ⟫ ⟫  (t-single _ _) (t-node _ _ _ (proj1 rr03) x₄ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₇ x₈ t ti₁ ) where
+--           rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
+--           rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
+--           rr03 = proj2 (proj2 rr01)
+--   ... | t-node key₄ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  tt ⟪ x₉ , ⟪  x₁₁ , x₁₂ ⟫ ⟫ ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₄ , ⟪ <-tr> x₇ (proj1 rr01) , <-tr> x₈ (proj1 rr01) ⟫ ⟫  (t-right _ _ (proj1 rr02) (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) t) (t-node _ _ _ (proj1 rr04) x₄ (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) x₇ x₈ t₁ ti₁ ) where
+--           rr00 : (kg < kn) ∧ ((kg < key₄) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₁
+--           rr02 = proj1 (proj2 rr00)
+--           rr05 = proj2 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₄ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₃ , ⟪ x₅ , x₆ ⟫ ⟫
+--           rr03 = proj1 (proj2 rr01)
+--           rr04 = proj2 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₃ _ _ _) .leaf)) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₂ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-node key₂ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-left key₃ .kp x₆ x₇ x₈ ti₁)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
+--   ... | t-right .kn .key₁ x₉ x₁₀ x₁₁ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00)  , ⟪ >-tr< x₂ (proj1 rr00)  , >-tr< x₃ (proj1 rr00) ⟫ ⟫ tt  ⟪ x₉ , ⟪ x₁₀ , x₁₁ ⟫ ⟫  tt  (t-left _ _ x  x₂ x₃ ti ) (t-left _ _ (proj1 rr03) (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) t) where
+--           rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr02 = proj2 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr03 = proj2 (proj2 rr01)
+--   ... | t-node key₄ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₉ x₁₀ x₁₁ x₁₂ x₁₃ x₁₄ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫  ⟪ x₉ , ⟪ x₁₁ , x₁₂ ⟫ ⟫  ⟪ x₁₀ , ⟪ x₁₃ , x₁₄ ⟫ ⟫  tt (t-node _ _ _ x (proj1 rr02) x₂ x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti t) (t-left _ _ (proj1 rr04) (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) t₁) where
+--           rr00 : (kg < kn) ∧ ((kg < key₄) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr02 = proj1 (proj2 rr00)
+--           rr05 = proj2 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₄ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₇ , x₈ ⟫ ⟫
+--           rr03 = proj1 (proj2 rr01)
+--           rr04 = proj2 (proj2 rr01)
+--   RB-repl→ti (node kg ⟪ Black , vg ⟫ _ (node kp ⟪ Red , vp ⟫ .(node key₃ _ _ _) .(node key₄ _ _ _))) (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , _ ⟫ .(node key₂ _ _ _) t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value (t-node key₂ .kg .kp x x₁ x₂ x₃ x₄ x₅ ti (t-node key₃ .kp key₄ x₆ x₇ x₈ x₉ x₁₀ x₁₁ ti₁ ti₂)) (rbr-rotate-rl t₂ .(node key₁ value₁ t₃ t₄) kg kp kn _ lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
+--   ... | t-right .kn .key₁ x₁₂ x₁₃ x₁₄ t = t-node _ _ _ (proj1 rr00) (proj1 rr01)   ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00) , >-tr< x₃ (proj1 rr00) ⟫ ⟫ tt  ⟪  x₁₂  , ⟪  x₁₃  , x₁₄  ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₇ , ⟪ <-tr> x₁₀ (proj1 rr01) , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫  (t-left _ _ x x₂ x₃ ti) (t-node _ _ _ (proj1 rr03) x₇ (proj1 (proj2 rr03)) (proj2 (proj2 rr03)) x₁₀ x₁₁ t ti₂ ) where
+--           rr00 : (kg < kn) ∧ ⊤ ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr02 = proj2 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ⊤ ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
+--           rr03 = proj2 (proj2 rr01)
+--   ... | t-node key₅ .kn .key₁ {v0} {v1} {v2} {t₇} {t₈} {t₁₀} x₁₂ x₁₃ x₁₄ x₁₅ x₁₆ x₁₇ t t₁ = t-node _ _ _ (proj1 rr00) (proj1 rr01)  ⟪ <-trans x (proj1 rr00) , ⟪ >-tr< x₂ (proj1 rr00)  , >-tr< x₃ (proj1 rr00) ⟫ ⟫  ⟪ x₁₂ , ⟪ x₁₄ , x₁₅ ⟫ ⟫  ⟪ x₁₃ , ⟪ x₁₆ , x₁₇ ⟫ ⟫  ⟪ <-trans (proj1 rr01) x₇  , ⟪ <-tr> x₁₀ (proj1 rr01) , <-tr> x₁₁ (proj1 rr01) ⟫ ⟫   (t-node _ _ _ x (proj1 rr02) x₂  x₃ (proj1 (proj2 rr02)) (proj2 (proj2 rr02)) ti t ) (t-node _ _ _ (proj1 rr04) x₇ (proj1 (proj2 rr04)) (proj2 (proj2 rr04)) x₁₀ x₁₁ t₁ ti₂ ) where
+--           rr00 : (kg < kn) ∧ ((kg < key₅) ∧ tr> kg t₇ ∧ tr> kg t₈) ∧ ((kg < key₁) ∧ tr> kg t₃ ∧ tr> kg t₄)
+--           rr00 = RB-repl→ti> _ _ _ _ _ trb lt1 x₄
+--           rr02 = proj1 (proj2 rr00)
+--           rr05 = proj2 (proj2 rr00)
+--           rr01 : (kn < kp) ∧ ((key₅ < kp) ∧ tr< kp t₇ ∧ tr< kp t₈) ∧ ((key₁ < kp) ∧ tr< kp t₃ ∧ tr< kp t₄)
+--           rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₆ , ⟪ x₈ , x₉ ⟫ ⟫
+--           rr03 = proj1 (proj2 rr01)
+--           rr04 = proj2 (proj2 rr01)
 
 
 --
@@ -1522,6 +1526,29 @@
 ... | s2-s12p refl refl = ⟪ (λ () ) , ( λ () ) ⟫
 ... | s2-1s2p refl refl = ⟪ (λ () ) , ( λ () ) ⟫
 
+stackCase2 : {n : Level} {A : Set n} → (rest : List ( bt (Color ∧ A)))
+           → ( tree parent grand orig : bt (Color ∧ A)) →  (key : ℕ)
+           → stackInvariant key tree  orig ( tree ∷ parent ∷ grand ∷ rest )
+           → stackInvariant key grand orig (grand ∷ rest )
+stackCase2 rest tree parent grand orig key (s-right .tree .orig tree₁ x si) = sc01 sc00 where
+     sc00 : stackInvariant key parent orig (parent ∷ grand ∷ rest )
+     sc00 with si-property1 si
+     ... | refl = si
+     sc01 : stackInvariant key parent orig (parent ∷ grand ∷ rest ) → stackInvariant key grand orig (grand ∷ rest )
+     sc01 (s-right .parent .orig tree₁ x si2) with si-property1 si2
+     ... | refl = si2
+     sc01 (s-left .parent .orig tree x si2) with si-property1 si2
+     ... | refl = si2
+stackCase2 rest tree parent grand orig key (s-left .tree .orig tree₁ x si) = sc01 sc00 where
+     sc00 : stackInvariant key parent orig (parent ∷ grand ∷ rest )
+     sc00 with si-property1 si
+     ... | refl = si
+     sc01 : stackInvariant key parent orig (parent ∷ grand ∷ rest ) → stackInvariant key grand orig (grand ∷ rest )
+     sc01 (s-right .parent .orig tree₁ x si2) with si-property1 si2
+     ... | refl = si2
+     sc01 (s-left .parent .orig tree x si2) with si-property1 si2
+     ... | refl = si2
+
 -- PGtoRBinvariant : {n : Level} {A : Set n} → {key d0 ds dp dg : ℕ } → (tree orig : bt (Color ∧ A) )
 --            →  RBtreeInvariant orig
 --            →  (stack : List (bt (Color ∧ A)))  → (pg : PG (Color ∧ A) tree stack )
@@ -1692,49 +1719,47 @@
          → ⊥
        rb06 (.right ∷ node key₁ value₁ left right ∷ []) refl (s-right .right .(node key₁ value₁ left right) .left x s-nil) eq = ⊥-elim ( nat-≡< (sym eq) x)
        rb06 (.left ∷ node key₁ value₁ left right ∷ []) refl (s-left .left .(node key₁ value₁ left right) .right x s-nil) eq = ⊥-elim ( nat-≡< eq x)
-    ... | tri> ¬a ¬b c | cr = insertCase13 value₁ refl pdepth< where
-       rb04 : stackInvariant key (RBI.tree r) orig stack → stack ≡ RBI.tree r ∷ orig ∷ [] → tr0 ≡ orig → right ≡ RBI.tree r
-       rb04 (s-right tree .(node key₁ _ tree₁ tree) tree₁ x s-nil) refl refl = refl
-       rb04 (s-left tree₁ .(node key₁ value₁ left right) tree {key₂} x si) refl refl with si-property1 si
-       ... | refl = ⊥-elim ( nat-<> x c  )
-       --
-       --  RBI key value (node key₁ ⟪ Black , value₄ ⟫ left right) repl stack
-       --
-       insertCase13 : (v : Color ∧ A ) → v ≡ value₁ → black-depth repl ≡ black-depth (child-replaced key (RBI.tree r)) → t
-       insertCase13 ⟪ cl , value₄ ⟫ refl beq with <-cmp key key₁ | child-replaced key (node key₁ ⟪ cl , value₄ ⟫ left right) in creq
-       ... | tri< a ¬b ¬c | cr = ⊥-elim (¬c c)
-       ... | tri≈ ¬a b ¬c | cr = ⊥-elim (¬c c)
-       ... | tri> ¬a ¬b c | cr = exit (node key₁ ⟪ Black , value₄ ⟫ left repl) (orig ∷ [])  refl record {
+    ... | tri> ¬a ¬b c | cr = exit (to-black (node key₁ value₁ left repl)) (orig ∷ [])  refl record {
          tree = orig
          ; origti = RBI.origti r
          ; origrb = RBI.origrb r
          ; treerb = RBI.origrb r
-         ; replrb = {!!}
+         ; replrb = rb-black _ _ (sym rb06) (RBtreeLeftDown _ _ (RBI.origrb r)) (RBI.replrb r)
          ; si = s-nil
-         ; state = top-black {!!} {!!}
+         ; state = top-black refl (rbr-black-right repl-red c (subst (λ k → replacedRBTree key value k repl) (sym (rb04 si eq refl)) rot))
          } where
-           -- rb09 : {n : Level} {A : Set n} →  {key key1 key2 : ℕ}  {value value1  : A} {t1 t2  : bt (Color ∧ A)}
-           --  → RBtreeInvariant (node key ⟪ Red , value ⟫ leaf (node key1 ⟪ Black , value1 ⟫ t1 t2))
-           --  → key < key1
-           -- rb09 (rb-right-red x x0 x2) = x
-           -- rb05 should more general
-           tkey : {n : Level} {A : Set n } → (rbt : bt (Color ∧ A)) → ℕ
-           tkey (node key value t t2) = key
-           tkey leaf = {!!} -- key is none
+       rb04 : stackInvariant key (RBI.tree r) orig stack → stack ≡ RBI.tree r ∷ orig ∷ [] → tr0 ≡ orig → right ≡ RBI.tree r
+       rb04 (s-right tree .(node key₁ _ tree₁ tree) tree₁ x s-nil) refl refl = refl
+       rb04 (s-left tree₁ .(node key₁ value₁ left right) tree {key₂} x si) refl refl with si-property1 si
+       ... | refl = ⊥-elim ( nat-<> x c  )
+       rb05 : child-replaced key (node key₁ value₁ left right) ≡ right
+       rb05 with <-cmp key key₁
+       ... | tri< a ¬b ¬c = ⊥-elim (¬c c)
+       ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c)
+       ... | tri> ¬a ¬b c = refl
+       rb06 : black-depth repl ≡ black-depth left
+       rb06 = begin
+         black-depth repl ≡⟨ sym (RB-repl→eq _ _ (RBI.treerb r) rot) ⟩
+         black-depth (RBI.tree r) ≡⟨ cong black-depth (sym (rb04 si eq refl)) ⟩
+         black-depth right ≡⟨ (sym (RBtreeEQ (RBI.origrb r))) ⟩
+         black-depth left ∎
+            where open ≡-Reasoning  
 ... | case2 (case2 pg) with PG.uncle pg in uneq
 ... | leaf = {!!} -- insertCase5
 ... | node key₁ ⟪ Black , value₁ ⟫ t₁ t₂ = {!!} -- insertCase5
 ... | node key₁ ⟪ Red , value₁ ⟫ t₁ t₂ with PG.pg pg
-... | s2-s1p2 {kp} {kg} {vp} {vg} {n1} {n2} x x₁ = next (to-red (node kg vg (to-black (node kp vp repl n1)) (to-black (PG.uncle pg)))) (PG.rest pg)
+... | s2-s1p2 {kp} {kg} {vp} {vg} {n1} {n2} x x₁ = next (to-red (node kg vg (to-black (node kp vp repl n1)) (to-black (PG.uncle pg)))) (PG.grand pg ∷ (PG.rest pg))
     record {
          tree = PG.grand pg
          ; origti = RBI.origti r
          ; origrb = RBI.origrb r
          ; treerb = {!!}
          ; replrb = {!!}
-         ; si = {!!}
+         ; si = stackCase2 (PG.rest pg) (RBI.tree r) (PG.parent pg) (PG.grand pg) orig _  rb00
          ; state = rotate refl {!!} {!!}
-     }  {!!}
+     }  ? where
+       rb00 : stackInvariant key (RBI.tree r) orig (RBI.tree r ∷ PG.parent pg ∷ PG.grand pg ∷ PG.rest pg)
+       rb00 = subst (λ k → stackInvariant key (RBI.tree r) orig k) (PG.stack=gp pg) (RBI.si r)
 ... | s2-1sp2 {kp} {kg} {vp} {vg} {n1} {n2} x x₁ = {!!}
 ... | s2-s12p {kp} {kg} {vp} {vg} {n1} {n2} x x₁ = {!!}
 ... | s2-1s2p {kp} {kg} {vp} {vg} {n1} {n2} x x₁ = {!!}