# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1712395163 -32400
# Node ID 043c698a261da2c3a1409e8d39413ed85400917a
# Parent  315efbf0148f14bac98e25fe0159441571c19f6a
...

diff -r 315efbf0148f -r 043c698a261d hoareBinaryTree1.agda
--- a/hoareBinaryTree1.agda	Fri Apr 05 10:49:49 2024 +0900
+++ b/hoareBinaryTree1.agda	Sat Apr 06 18:19:23 2024 +0900
@@ -833,13 +833,15 @@
                                     (node kp ⟪ Black , vp ⟫ (node kg ⟪ Red , vg ⟫ uncle t) t₂ )
     -- case56 the node is inner, make it outer and rotate grand
     rbr-rotate-lr : {t t₁ uncle : bt (Color ∧ A)} (t₂ t₃ : bt (Color ∧ A)) (kg kp kn : ℕ) {vg vp vn : A}
+         → kp < kn → kn < kg
          → replacedRBTree key value t₁ (node kn ⟪ Red , vn ⟫ t₂ t₃)
          → replacedRBTree key value (node kg ⟪ Black , vg ⟫ (node kp ⟪ Red , vp ⟫ t t₁)     uncle)
                                     (node kn ⟪ Black , vn ⟫ (node kp ⟪ Red , vp ⟫ t t₂)     (node kg ⟪ Red , vg ⟫ t₃ uncle))
     rbr-rotate-rl : {t t₁ uncle : bt (Color ∧ A)} (t₂ t₃ : bt (Color ∧ A)) (kg kp kn : ℕ) {vg vp vn : A}
+         → kg < kn → kn < kp
          → replacedRBTree key value t₁ (node kn ⟪ Red , vn ⟫ t₂ t₃)
-         → replacedRBTree key value (node kg ⟪ Black , vg ⟫ uncle                           (node kp ⟪ Red , vp ⟫ t₁  t))
-                                    (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , vg ⟫ uncle t₂)  (node kp ⟪ Red , vp ⟫ t₃ t))
+         → replacedRBTree key value (node kg ⟪ Black , vg ⟫ uncle                           (node kp ⟪ Red , vp ⟫ t₁ t))
+                                    (node kn ⟪ Black , vn ⟫ (node kg ⟪ Red , vg ⟫ uncle t₂) (node kp ⟪ Red , vp ⟫ t₃ t))
 
 
 --
@@ -920,13 +922,13 @@
 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
+   (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
+   (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
@@ -958,13 +960,13 @@
 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
+   (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
+   (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 
@@ -1196,9 +1198,13 @@
      (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 , _ ⟫ (node kp ⟪ Red , _ ⟫ _ _) _) .(node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₂) (node kg ⟪ Red , _ ⟫ t₃ _)) key value ti (rbr-rotate-lr t₂ t₃ kg kp kn trb) = ?
-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 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 trb) = ?
+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 _)) key value (t-left .kp .kg x x₁ x₂ ti) (rbr-rotate-lr .leaf .leaf kg kp kn lt1 lt2 trb) = 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 , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-left .kp .kg x x₁ x₂ ti) (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)) .leaf) (node kn ⟪ Black , _ ⟫ (node kp ⟪ Red , _ ⟫ _ t₃) (node kg ⟪ Red , _ ⟫ t₄ _)) key value (t-left .kp .kg x x₁ x₂ ti) (rbr-rotate-lr t₃ t₄ kg kp kn lt1 lt2 trb) = ?
+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 , _ ⟫ _ (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) = ?
 
 --
 -- if we consider tree invariant, this may be much simpler and faster