--- a/hoareBinaryTree1.agda	Fri May 03 14:33:41 2024 +0900
+++ b/hoareBinaryTree1.agda	Fri May 03 18:19:33 2024 +0900
@@ -1228,7 +1228,7 @@
         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
+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 ⟫ ⟫
@@ -1254,9 +1254,10 @@
         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 ⟫ t leaf) .(node key₂ _ _ _)) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁) (rbr-rotate-lr t₃ t₄ kg kp kn lt1 lt2 trb) = ?
-RB-repl→ti (node kg ⟪ Black , v1 ⟫ (node kp ⟪ Red , v2 ⟫ t (node key₁ value₁ t₁ t₂)) .(node key₂ _ _ _)) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-node .kp .kg key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁) (rbr-rotate-lr t₃ t₄ kg kp kn lt1 lt2 trb) = ?
+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 ⟫ t (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 ti ti₁) (rbr-rotate-lr t₃ t₄ kg kp kn lt1 lt2 trb) with RB-repl→ti _ _ _ _ ti₁ trb
+... | t = ?
 RB-repl→ti .(node kg ⟪ Black , _ ⟫ _ (node kp ⟪ Red , _ ⟫ _ _)) .(node kn ⟪ Black , _ ⟫ (node kg ⟪ Red , _ ⟫ _ t₂) (node kp ⟪ Red , _ ⟫ leaf _)) key value ti (rbr-rotate-rl t₂ leaf kg kp kn lt1 lt2 trb) = ?
 RB-repl→ti .(node kg ⟪ Black , _ ⟫ _ (node kp ⟪ Red , _ ⟫ _ _)) .(node kn ⟪ Black , _ ⟫ (node kg ⟪ Red , _ ⟫ _ t₂) (node kp ⟪ Red , _ ⟫ (node key₁ value₁ t₃ t₄) _)) key value ti (rbr-rotate-rl t₂ (node key₁ value₁ t₃ t₄) kg kp kn lt1 lt2 trb) = ?