--- a/RBTree.agda	Sun Aug 04 13:05:12 2024 +0900
+++ b/RBTree.agda	Tue Aug 13 23:03:40 2024 +0900
@@ -1,6 +1,6 @@
-{-# OPTIONS  cubical-compatible --allow-unsolved-metas #-}
--- {-# OPTIONS --cubical-compatible --safe #-}
--- {-# OPTIONS --cubical-compatible  #-}
+{-# OPTIONS --cubical-compatible --allow-unsolved-metas  #-}
+--  {-# OPTIONS --cubical-compatible --safe #-}
+
 module RBTree where
 
 open import Level hiding (suc ; zero ; _⊔_ )
@@ -13,12 +13,16 @@
 open import Data.List
 open import Data.Product
 
+open import Data.Maybe.Properties
+open import Data.List.Properties
+
 open import Function as F hiding (const)
 
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary
 open import logic
+open import nat
 
 
 --
@@ -102,17 +106,6 @@
         suc i ≤⟨ m≤m+n (suc i) j ⟩
         suc i + j ∎  where open ≤-Reasoning
 
-nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥
-nat-≤>  (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x
-nat-<> : { x y : ℕ } → x < y → y < x → ⊥
-nat-<>  (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
-
-nat-<≡ : { x : ℕ } → x < x → ⊥
-nat-<≡  (s≤s lt) = nat-<≡ lt
-
-nat-≡< : { x y : ℕ } → x ≡ y → x < y → ⊥
-nat-≡< refl lt = nat-<≡ lt
-
 treeTest1  : bt ℕ
 treeTest1  =  node 0 0 leaf (node 3 1 (node 2 5 (node 1 7 leaf leaf ) leaf) (node 5 5 leaf leaf))
 treeTest2  : bt ℕ
@@ -140,72 +133,223 @@
 si-property0  (s-right _ _ _ x si) ()
 si-property0  (s-left _ _ _ x si) ()
 
+
 si-property1 :  {n : Level} {A : Set n} {key : ℕ} {tree tree0 tree1 : bt A} → {stack  : List (bt A)} →  stackInvariant key tree tree0 (tree1 ∷ stack)
    → tree1 ≡ tree
-si-property1 (s-nil   ) = refl
-si-property1 (s-right _ _ _ _  si) = refl
-si-property1 (s-left _ _ _ _  si) = refl
+si-property1 {n} {A} {key} {tree} {tree0} {tree1} {stack} si = lem00 tree tree0 tree1 _ (tree1 ∷ stack) refl si where
+    lem00 : (tree tree0 tree1 : bt A) → (stack  stack1 : List (bt A)) → tree1 ∷ stack ≡ stack1  →  stackInvariant key tree tree0 stack1 → tree1 ≡ tree
+    lem00 tree .tree tree1 stack .(tree ∷ []) eq s-nil = ∷-injectiveˡ eq
+    lem00 tree tree0 tree1 stack .(tree ∷ _) eq (s-right .tree .tree0 tree₁ x si) = ∷-injectiveˡ eq
+    lem00 tree tree0 tree1 stack .(tree ∷ _) eq (s-left .tree .tree0 tree₁ x si) = ∷-injectiveˡ eq
 
 si-property2 :  {n : Level} {A : Set n} {key : ℕ} {tree tree0 tree1 : bt A} → (stack  : List (bt A)) →  stackInvariant key tree tree0 (tree1 ∷ stack)
    → ¬ ( just leaf ≡ stack-last stack )
-si-property2 (.leaf ∷ []) (s-right _ _ tree₁ x ()) refl
-si-property2 (x₁ ∷ x₂ ∷ stack) (s-right _ _ tree₁ x si) eq = si-property2 (x₂ ∷ stack) si eq
-si-property2 (.leaf ∷ []) (s-left _ _ tree₁ x ()) refl
-si-property2 (x₁ ∷ x₂ ∷ stack) (s-left _ _ tree₁ x si) eq = si-property2 (x₂ ∷ stack) si eq
+si-property2 {n} {A} {key} {tree} {tree0} {tree1} [] si ()
+si-property2 {n} {A} {key} {tree} {tree0} {tree1} (x ∷ []) si eq with just-injective eq
+... | refl = lem00 (tree1 ∷ leaf ∷ [])  refl si  where
+    lem00 : (t : List (bt A)) → (tree1 ∷ leaf ∷ []) ≡ t → stackInvariant key tree tree0 t → ⊥
+    lem00 .(tree ∷ []) () s-nil
+    lem00 (tree ∷ _) () (s-right .tree .(node _ _ tree₁ tree) tree₁ x s-nil)
+    lem00 (tree ∷ _) () (s-right .tree .tree0 tree₁ x (s-right .(node _ _ tree₁ tree) .tree0 tree₂ x₁ si2))
+    lem00 (tree ∷ _) () (s-right .tree .tree0 tree₁ x (s-left .(node _ _ tree₁ tree) .tree0 tree₂ x₁ si2))
+    lem00 (tree₁ ∷ _) () (s-left .tree₁ .(node _ _ tree₁ tree) tree x s-nil)
+    lem00 (tree₁ ∷ _) () (s-left .tree₁ .tree0 tree x (s-right .(node _ _ tree₁ tree) .tree0 tree₂ x₁ si2))
+    lem00 (tree₁ ∷ _) () (s-left .tree₁ .tree0 tree x (s-left .(node _ _ tree₁ tree) .tree0 tree₂ x₁ si2))
+si-property2 {n} {A} {key} {tree} {tree0} {tree1} (x ∷ x₁ ∷ stack) si = lem00 (tree1 ∷ x ∷ x₁ ∷ stack) refl si where
+    lem00 : (t : List (bt A)) → (tree1 ∷ x ∷ x₁ ∷ stack) ≡ t → stackInvariant key tree tree0 t → ¬ just leaf ≡ stack-last (x₁ ∷ stack)
+    lem00 .(tree ∷ []) () s-nil
+    lem00 (tree ∷ []) () (s-right .tree .tree0 tree₁ x si)
+    lem00 (tree ∷ x₁ ∷ st) eq (s-right .tree .tree0 tree₁ x si) eq1 = si-property2 st si
+         (subst (λ k → just leaf ≡ stack-last k ) (∷-injectiveʳ (∷-injectiveʳ eq)) eq1 )
+    lem00 (tree ∷ []) () (s-left .tree .tree0 tree₁ x si)
+    lem00 (tree₁ ∷ x₁ ∷  st) eq (s-left .tree₁ .tree0 tree x si) eq1 = si-property2 st si
+         (subst (λ k → just leaf ≡ stack-last k ) (∷-injectiveʳ (∷-injectiveʳ eq)) eq1 )
+
+
+-- We cannot avoid warning: -W[no]UnsupportedIndexedMatcin tree-inject
+-- open import Function.Base using (_∋_; id; _∘_; _∘′_)
+-- just-injective1 : {n : Level } {A : Set n} {x y : A } → (Maybe A ∋ just x) ≡ just y → x ≡ y
+-- just-injective1 refl = refl
+
+node-left : {n : Level} {A : Set n} → bt A → Maybe (bt A)
+node-left (node _ _ left _) = just left
+node-left _ = nothing
+
+node-right : {n : Level} {A : Set n} → bt A → Maybe (bt A)
+node-right (node _ _ _ right) = just right
+node-right _ = nothing
+
+--   lem00 = just-injective (cong node-key eq)
+
+tree-injective-key : {n : Level} {A : Set n}
+    → (tr0 tr1 : bt A) → tr0 ≡ tr1 → node-key tr0 ≡ node-key tr1
+tree-injective-key {n} {A} tr0 tr1 eq = cong node-key eq
+
+ti-property2 :  {n : Level} {A : Set n} {key : ℕ} {value : A} {tree1 left right : bt A}
+   → tree1 ≡ node key value left right
+   → left ≡ right
+   → ( ¬ left  ≡ leaf ) ∨ ( ¬ right ≡ leaf )
+   → ¬ treeInvariant tree1
+ti-property2 {n} {A} {key} {value} {leaf} {left} {right} () eq1 x ti
+ti-property2 {n} {A} {key} {value} {node key₁ value₁ leaf t₁} {left} {right} eq eq1 (case1 eq2) _
+    = ⊥-elim ( eq2 (just-injective ( cong node-left (sym eq)  )))
+ti-property2 {n} {A} {key} {value} {node key₁ value₁ leaf t₁} {left} {right} eq eq1 (case2 eq2) _
+    = ⊥-elim ( eq2 (just-injective (trans (cong just (sym eq1)) ( cong node-left (sym eq)   ) ) ))
+ti-property2 {n} {A} {key} {value} {node key₁ value₁ (node key₂ value₂ t t₂) leaf} {left} {right} eq eq1 (case1 eq2) _
+    = ⊥-elim ( eq2 (just-injective (trans (cong just eq1) ( cong node-right (sym eq)   ) ) ))
+ti-property2 {n} {A} {key} {value} {node key₁ value₁ (node key₂ value₂ t t₂) leaf} {left} {right} eq eq1 (case2 eq2) _
+    = ⊥-elim ( eq2 (just-injective ( cong node-right (sym eq)  )))
+ti-property2 {n} {A} {key} {value} {node key₂ value₂ (node key₁ value₁ t₁ t₂) (node key₃ value₃ t₃ t₄)} {left} {right} eq eq1 eq2 ti
+     = ⊥-elim ( nat-≡< lem00 (lem03 _ _ _ refl lem01 lem02 ti) ) where
+   lem01 : node key₁ value₁ t₁ t₂ ≡ left
+   lem01 = just-injective ( cong node-left eq )
+   lem02 : node key₃ value₃ t₃ t₄ ≡ right
+   lem02 = just-injective ( cong node-right eq )
+   lem00 : key₁ ≡ key₃
+   lem00 = begin
+      key₁ ≡⟨ just-injective ( begin
+         just key₁  ≡⟨ cong node-key lem01 ⟩
+         node-key left ≡⟨ cong node-key eq1 ⟩
+         node-key right ≡⟨ cong node-key (sym lem02) ⟩
+         just key₃ ∎ ) ⟩
+      key₃ ∎ where open ≡-Reasoning
+   lem03 :  (key₁ key₃ : ℕ) → (tr : bt A) → tr ≡ node key₂ value₂ (node key₁ value₁ t₁ t₂) (node key₃ value₃ t₃ t₄)
+      → node key₁ value₁ t₁ t₂ ≡ left
+      → node key₃ value₃ t₃ t₄ ≡ right
+      → treeInvariant tr → key₁ < key₃
+   lem03 _ _ .leaf () _ _ t-leaf
+   lem03 _ _ .(node _ _ leaf leaf) () _ _ (t-single _ _)
+   lem03 _ _ .(node _ _ (node _ _ _ _) leaf) () _ _ (t-left _ _ _ _ _ _)
+   lem03 _ _ .(node _ _ leaf (node _ _ _ _)) () _ _ (t-right _ _ _ _ _ _)
+   lem03 key₁ key₃ (node key _ (node _ _ _ _) (node _ _ _ _)) eq3 el er (t-node key₄ key₅ key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁) = lem04 where
+       lem04 : key₁ < key₃
+       lem04 = begin
+         suc key₁ ≡⟨ cong suc ( just-injective (cong node-key (just-injective (cong node-left (sym eq3)) ) ) ) ⟩
+         suc key₄ ≤⟨ <-trans x x₁ ⟩
+         key₂ ≡⟨ just-injective (cong node-key (just-injective (cong node-right eq3) ) )  ⟩
+         key₃ ∎ where open ≤-Reasoning
 
 si-property-< :  {n : Level} {A : Set n} {key kp : ℕ} {value₂ : A} {tree orig tree₃ : bt A} → {stack  : List (bt A)}
    → ¬ ( tree ≡ leaf )
    → treeInvariant (node kp value₂ tree  tree₃ )
    → stackInvariant key tree orig (tree ∷ node kp value₂ tree  tree₃ ∷ stack)
    → key < kp
-si-property-< ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right (node _ _ _ _) _ (node _ _ _ _) x s-nil) = ⊥-elim (nat-<> x₁ x₂)
-si-property-< ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right .(node _ _ _ _) _ .(node _ _ _ _) x (s-right .(node _ _ (node _ _ _ _) (node _ _ _ _)) _ tree₁ x₇ si)) = ⊥-elim (nat-<> x₁ x₂)
-si-property-< ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right .(node _ _ _ _) _ .(node _ _ _ _) x (s-left .(node _ _ (node _ _ _ _) (node _ _ _ _)) _ tree x₇ si)) = ⊥-elim (nat-<> x₁ x₂)
-si-property-< ne ti (s-left _ _ _ x (s-right .(node _ _ _ _) _ tree₁ x₁ si)) = x
-si-property-< ne ti (s-left _ _ _ x (s-left .(node _ _ _ _) _ tree₁ x₁ si)) = x
-si-property-< ne ti (s-left _ _ _ x s-nil) = x
-si-property-< ne (t-single _ _) (s-right _ _ tree₁ x si) = ⊥-elim ( ne refl )
+si-property-< {n} {A} {key} {kp} {value₂} {tree} {orig} {tree₃} {stack} ne ti si =
+       lem00 (node kp value₂ tree  tree₃ ) (tree ∷ node kp value₂ tree  tree₃ ∷ stack) refl refl ti si where
+    lem00 : (tree1 : bt A) → (st : List (bt A)) →  (tree1 ≡ (node kp value₂ tree  tree₃ )) → (st ≡ tree ∷ node kp value₂ tree  tree₃ ∷ stack)
+       → treeInvariant tree1 → stackInvariant key tree orig st → key < kp
+    lem00 tree1 .(tree ∷ []) teq () ti s-nil
+    lem00 tree1 .(tree ∷ node key₁ v1 tree₁ tree ∷ []) teq seq ti₁ (s-right .tree .(node key₁ v1 tree₁ tree) tree₁ {key₁} {v1} x s-nil) = lem01 where
+        lem02 :  node key₁ v1 tree₁ tree  ≡ node kp value₂ tree tree₃
+        lem02 = ∷-injectiveˡ ( ∷-injectiveʳ seq )
+        lem03 : tree₁ ≡ tree
+        lem03 = just-injective (cong node-left lem02)
+        lem01 : key < kp
+        lem01 = ⊥-elim ( ti-property2 (sym lem02) lem03 (case2 ne) ti )
+    lem00 tree1 .(tree ∷ node key₁ v1 tree₁ tree ∷ _) teq seq ti₁ (s-right .tree .orig tree₁ {key₁} {v1} x (s-right .(node _ _ tree₁ tree) .orig tree₂ {key₂} {v2} x₁ si)) = lem01 where
+        lem02 :  node key₁ v1 tree₁ tree  ≡ node kp value₂ tree tree₃
+        lem02 = ∷-injectiveˡ ( ∷-injectiveʳ seq )
+        lem03 : tree₁ ≡ tree
+        lem03 = just-injective (cong node-left lem02)
+        lem01 : key < kp
+        lem01 = ⊥-elim ( ti-property2 (sym lem02) lem03 (case2 ne) ti )
+    lem00 tree1 (tree₂ ∷ node key₁ v1 tree₁ tree₂ ∷ _) teq seq ti₁ (s-right .tree₂ .orig tree₁ {key₁} {v1} x (s-left .(node _ _ tree₁ tree₂) .orig tree x₁ si)) = lem01 where
+        lem02 :  node key₁ v1 tree₁ tree₂ ≡ node kp value₂ tree₂ tree₃
+        lem02 = ∷-injectiveˡ ( ∷-injectiveʳ seq )
+        lem03 : tree₁ ≡ tree₂
+        lem03 = just-injective (cong node-left lem02 )
+        lem01 : key < kp
+        lem01 = ⊥-elim ( ti-property2 (sym lem02) lem03 (case2 ne) ti )
+    lem00 tree1 (tree₁ ∷ _) teq seq ti₁ (s-left .tree₁ .(node _ _ tree₁ tree) tree {key₁} {v1} x s-nil) = subst( λ k → key < k ) lem03 x where
+        lem03 : key₁ ≡ kp
+        lem03 = just-injective (cong node-key (∷-injectiveˡ ( ∷-injectiveʳ seq )))
+    lem00 tree1 (tree₁ ∷ _) teq seq ti₁ (s-left .tree₁ .orig tree {key₁} {v1} x (s-right .(node _ _ tree₁ tree) .orig tree₂ x₁ si)) = subst( λ k → key < k ) lem03 x where
+        lem03 : key₁ ≡ kp
+        lem03 = just-injective (cong node-key (∷-injectiveˡ ( ∷-injectiveʳ seq )))
+    lem00 tree1 (tree₁ ∷ _) teq seq ti₁ (s-left .tree₁ .orig tree {key₁} {v1} x (s-left .(node _ _ tree₁ tree) .orig tree₂ x₁ si)) = subst( λ k → key < k ) lem03 x where
+        lem03 : key₁ ≡ kp
+        lem03 = just-injective (cong node-key (∷-injectiveˡ ( ∷-injectiveʳ seq )))
 
 si-property-> :  {n : Level} {A : Set n} {key kp : ℕ} {value₂ : A} {tree orig tree₃ : bt A} → {stack  : List (bt A)}
    → ¬ ( tree ≡ leaf )
    → treeInvariant (node kp value₂ tree₃  tree )
    → stackInvariant key tree orig (tree ∷ node kp value₂ tree₃  tree ∷ stack)
    → kp < key
-si-property-> ne ti (s-right _ _ tree₁ x s-nil) = x
-si-property-> ne ti (s-right _ _ tree₁ x (s-right .(node _ _ tree₁ _) _ tree₂ x₁ si)) = x
-si-property-> ne ti (s-right _ _ tree₁ x (s-left .(node _ _ tree₁ _) _ tree x₁ si)) = x
-si-property-> ne (t-node _ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-left _ _ _ x s-nil) = ⊥-elim (nat-<> x₁ x₂)
-si-property-> ne (t-node _ _ _ x₂ x₃ x₄ x₅ x₆ x₇ ti ti₁) (s-left _ _ _ x (s-right .(node _ _ _ _) _ tree₁ x₁ si)) = ⊥-elim (nat-<> x₂ x₃)
-si-property-> ne (t-node _ _ _ x₂ x₃ x₄ x₅ x₆ x₇ ti ti₁) (s-left _ _ _ x (s-left .(node _ _ _ _) _ tree x₁ si)) = ⊥-elim (nat-<> x₂ x₃)
-si-property-> ne (t-single _ _) (s-left _ _ _ x s-nil) = ⊥-elim ( ne refl )
-si-property-> ne (t-single _ _) (s-left _ _ _ x (s-right .(node _ _ leaf leaf) _ tree₁ x₁ si)) = ⊥-elim ( ne refl )
-si-property-> ne (t-single _ _) (s-left _ _ _ x (s-left .(node _ _ leaf leaf) _ tree x₁ si)) = ⊥-elim ( ne refl )
+si-property-> {n} {A} {key} {kp} {value₂} {tree} {orig} {tree₃} {stack} ne ti si =
+       lem00 (node kp value₂ tree₃  tree ) (tree ∷ node kp value₂ tree₃  tree ∷ stack) refl refl ti si where
+    lem00 : (tree1 : bt A) → (st : List (bt A)) →  (tree1 ≡ (node kp value₂ tree₃  tree )) → (st ≡ tree ∷ node kp value₂ tree₃  tree ∷ stack)
+       → treeInvariant tree1 → stackInvariant key tree orig st → kp < key
+    lem00 tree1 .(tree ∷ []) teq () ti s-nil
+    lem00 tree1 .(tree ∷ node key₁ v1 tree tree₁ ∷ []) teq seq ti₁ (s-left .tree .(node key₁ v1 tree tree₁) tree₁ {key₁} {v1} x s-nil) = lem01 where
+        lem02 :  node key₁ v1 tree tree₁  ≡ node kp value₂ tree₃ tree
+        lem02 = ∷-injectiveˡ ( ∷-injectiveʳ seq )
+        lem03 : tree₁ ≡ tree
+        lem03 = just-injective (cong node-right lem02)
+        lem01 : kp < key
+        lem01 = ⊥-elim ( ti-property2 (sym lem02) (sym lem03) (case1 ne) ti )
+    lem00 tree1 .(tree ∷ node key₁ v1 tree tree₁ ∷ _) teq seq ti₁ (s-left .tree .orig tree₁ {key₁} {v1} x (s-left .(node _ _ tree tree₁) .orig tree₂ {key₂} {v2} x₁ si)) = lem01 where
+        lem02 :  node key₁ v1 tree tree₁  ≡ node kp value₂ tree₃ tree
+        lem02 = ∷-injectiveˡ ( ∷-injectiveʳ seq )
+        lem03 : tree₁ ≡ tree
+        lem03 = just-injective (cong node-right lem02)
+        lem01 : kp < key
+        lem01 = ⊥-elim ( ti-property2 (sym lem02) (sym lem03) (case1 ne) ti )
+    lem00 tree1 (tree₂ ∷ node key₁ v1 tree₂ tree₁ ∷ _) teq seq ti₁ (s-left .tree₂ .orig tree₁ {key₁} {v1} x (s-right .(node _ _ tree₂ tree₁) .orig tree x₁ si)) = lem01 where
+        lem02 :  node key₁ v1 tree₂ tree₁  ≡ node kp value₂ tree₃ tree₂
+        lem02 = ∷-injectiveˡ ( ∷-injectiveʳ seq )
+        lem03 : tree₁ ≡ tree₂
+        lem03 = just-injective (cong node-right lem02)
+        lem01 : kp < key
+        lem01 = ⊥-elim ( ti-property2 (sym lem02) (sym lem03) (case1 ne) ti )
+    lem00 tree1 (tree₁ ∷ _) teq seq ti₁ (s-right .tree₁ .(node _ _ tree tree₁) tree {key₁} {v1} x s-nil) = subst( λ k → k < key ) lem03 x where
+        lem03 : key₁ ≡ kp
+        lem03 = just-injective (cong node-key (∷-injectiveˡ ( ∷-injectiveʳ seq )))
+    lem00 tree1 (tree₁ ∷ _) teq seq ti₁ (s-right .tree₁ .orig tree {key₁} {v1} x (s-left .(node _ _ tree tree₁) .orig tree₂ x₁ si)) = subst( λ k → k < key ) lem03 x where
+        lem03 : key₁ ≡ kp
+        lem03 = just-injective (cong node-key (∷-injectiveˡ ( ∷-injectiveʳ seq )))
+    lem00 tree1 (tree₁ ∷ _) teq seq ti₁ (s-right .tree₁ .orig tree {key₁} {v1} x (s-right .(node _ _ tree tree₁) .orig tree₂ x₁ si)) = subst( λ k → k < key ) lem03 x where
+        lem03 : key₁ ≡ kp
+        lem03 = just-injective (cong node-key (∷-injectiveˡ ( ∷-injectiveʳ seq )))
+
 
 si-property-last :  {n : Level} {A : Set n}  (key : ℕ) (tree tree0 : bt A) → (stack  : List (bt A)) →  stackInvariant key tree tree0 stack
    → stack-last stack ≡ just tree0
-si-property-last key t t0 (t ∷ [])  (s-nil )  = refl
-si-property-last key t t0 (.t ∷ x ∷ st) (s-right _ _ _ _ si ) with  si-property1 si
-... | refl = si-property-last key x t0 (x ∷ st)   si
-si-property-last key t t0 (.t ∷ x ∷ st) (s-left _ _ _ _ si ) with  si-property1  si
-... | refl = si-property-last key x t0 (x ∷ st)   si
+si-property-last {n} {A} key tree .tree .(tree ∷ []) s-nil = refl
+si-property-last {n} {A} key tree tree0 (tree ∷ []) (s-right .tree .tree0 tree₁ x si) = ⊥-elim ( si-property0 si refl )
+si-property-last {n} {A} key tree tree0 (tree ∷ x₁ ∷ st) (s-right .tree .tree0 tree₁ x si) = 
+   si-property-last key _ tree0 (x₁ ∷ st)  si
+si-property-last {n} {A} key tree tree0 (tree ∷ []) (s-left .tree .tree0 tree₁ x si) = ⊥-elim ( si-property0 si refl )
+si-property-last {n} {A} key tree tree0 (tree ∷ x₁ ∷ st) (s-left .tree .tree0 tree₁ x si) = 
+   si-property-last key _ tree0 (x₁ ∷ st)  si
 
 si-property-pne :  {n : Level} {A : Set n}  {key key₁ : ℕ} {value₁ : A} (tree orig : bt A) → {left right x : bt A} → (stack  : List (bt A)) {rest : List (bt A)}
     → stack ≡ x ∷ node key₁ value₁ left right ∷ rest
     → stackInvariant key tree orig stack
     → ¬ ( key ≡ key₁ )
-si-property-pne tree orig .(tree ∷ node _ _ _ _ ∷ _) refl (s-right .tree .orig tree₁ x si) eq with si-property1 si
-... | refl = ⊥-elim ( nat-≡< (sym eq) x)
-si-property-pne tree orig .(tree ∷ node _ _ _ _ ∷ _) refl (s-left .tree .orig tree₁ x si) eq with si-property1 si
-... | refl = ⊥-elim ( nat-≡< eq x)
+si-property-pne {_} {_} {key} {key₁} {value₁} tree orig {left} {right} (tree ∷ tree1 ∷ st) seq (s-right .tree .orig tree₁ {key₂} {v₂} x si) eq 
+    = ⊥-elim ( nat-≡< lem00 x ) where
+      lem01 : tree1 ≡ node key₂ v₂ tree₁ tree 
+      lem01 = si-property1 si
+      lem02 : node key₁ value₁ left right ≡ node key₂ v₂ tree₁ tree 
+      lem02 = trans ( ∷-injectiveˡ (∷-injectiveʳ (sym seq) ) ) lem01
+      lem00 : key₂ ≡ key
+      lem00 = trans (just-injective (cong node-key (sym lem02))) (sym eq)
+si-property-pne {_} {_} {key} {key₁} {value₁} tree orig {left} {right} (tree ∷ tree1 ∷ st) seq (s-left tree orig tree₁ {key₂} {v₂} x si) eq 
+    = ⊥-elim ( nat-≡< (sym lem00) x ) where
+      lem01 : tree1 ≡ node key₂ v₂ tree tree₁
+      lem01 = si-property1 si
+      lem02 : node key₁ value₁ left right ≡ node key₂ v₂ tree tree₁
+      lem02 = trans ( ∷-injectiveˡ (∷-injectiveʳ (sym seq) ) ) lem01
+      lem00 : key₂ ≡ key
+      lem00 = trans (just-injective (cong node-key (sym  lem02))) (sym eq)
+si-property-pne {_} {_} {key} {key₁} {value₁} tree .tree {left} {right} .(tree ∷ []) () s-nil eq
 
-si-property-parent-node :  {n : Level} {A : Set n}  {key : ℕ}  (tree orig : bt A) {x : bt A}
-    → (stack  : List (bt A)) {rest : List (bt A)}
-    → stackInvariant key tree orig stack
-    → ¬ ( stack ≡ x ∷ leaf ∷ rest )
-si-property-parent-node {n} {A} tree orig .(tree ∷ leaf ∷ _) (s-right .tree .orig tree₁ x si) refl with si-property1 si
-... | ()
-si-property-parent-node {n} {A} tree orig .(tree ∷ leaf ∷ _) (s-left .tree .orig tree₁ x si) refl with si-property1 si
-... | ()
+
+-- si-property-parent-node :  {n : Level} {A : Set n}  {key : ℕ}  (tree orig : bt A) {x : bt A}
+--     → (stack  : List (bt A)) {rest : List (bt A)}
+--     → stackInvariant key tree orig stack
+--     → ¬ ( stack ≡ x ∷ leaf ∷ rest )
+-- si-property-parent-node {n} {A} tree orig = ?
 
 
 rt-property1 :  {n : Level} {A : Set n} (key : ℕ) (value : A) (tree tree1 : bt A ) → replacedTree key value tree tree1 → ¬ ( tree1 ≡ leaf )
@@ -215,16 +359,25 @@
 rt-property1 {n} {A} key value .(node _ _ _ _) _ (r-left x rt) = λ ()
 
 rt-property-leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {repl : bt A} → replacedTree key value leaf repl → repl ≡ node key value leaf leaf
-rt-property-leaf r-leaf = refl
+rt-property-leaf {n} {A} {key} {value} {repl} rt = lem00 leaf refl rt where
+   lem00 : (tree : bt A) → tree ≡ leaf → replacedTree key value tree repl → repl ≡ node key value leaf leaf
+   lem00 leaf eq r-leaf = refl
+   lem00 .(node key _ _ _) () r-node
+   lem00 .(node _ _ _ _) () (r-right x rt)
+   lem00 .(node _ _ _ _) () (r-left x rt)
 
 rt-property-¬leaf : {n : Level} {A : Set n} {key : ℕ} {value : A} {tree : bt A} → ¬ replacedTree key value tree leaf
 rt-property-¬leaf ()
 
 rt-property-key : {n : Level} {A : Set n} {key key₂ key₃ : ℕ} {value value₂ value₃ : A} {left left₁ right₂ right₃ : bt A}
     →  replacedTree key value (node key₂ value₂ left right₂) (node key₃ value₃ left₁ right₃) → key₂ ≡ key₃
-rt-property-key r-node = refl
-rt-property-key (r-right x ri) = refl
-rt-property-key (r-left x ri) = refl
+rt-property-key {n} {A} {key} {key₂} {key₃} {value} {value₂} {value₃} {left} {left₁} {right₂} {right₃} rt 
+      = lem00 (node key₂ value₂ left right₂) (node key₃ value₃ left₁ right₃) refl refl rt where
+    lem00 : (tree tree1 : bt A) → tree ≡ node key₂ value₂ left right₂ → tree1 ≡  node key₃ value₃ left₁ right₃ → replacedTree key value tree tree1 → key₂ ≡ key₃
+    lem00 _ _ () eq2 r-leaf
+    lem00 _ _ eq1 eq2 r-node = trans (just-injective (cong node-key (sym eq1))) (just-injective (cong node-key eq2))
+    lem00 _ _ eq1 eq2 (r-right x rt1) = trans (just-injective (cong node-key (sym eq1))) (just-injective (cong node-key eq2))
+    lem00 _ _ eq1 eq2 (r-left x rt1) = trans (just-injective (cong node-key (sym eq1))) (just-injective (cong node-key eq2))
 
 
 open _∧_
@@ -244,25 +397,64 @@
 treeLeftDown  : {n : Level} {A : Set n} {k : ℕ} {v1 : A}  → (tree tree₁ : bt A )
       → treeInvariant (node k v1 tree tree₁)
       →      treeInvariant tree
-treeLeftDown {n} {A} {_} {v1} leaf leaf (t-single k1 v1) = t-leaf
-treeLeftDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right _ _ x _ _ ti) = t-leaf
-treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left _ _ x _ _ ti) = ti
-treeLeftDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node _ _ _ x x₁ _ _ _ _ ti ti₁) = ti
+treeLeftDown {n} {A} {k} {v1} tree tree₁ ti = lem00 (node k v1 tree tree₁) refl ti  where
+    lem00 : ( tr : bt A ) → tr ≡ node k v1 tree tree₁ → treeInvariant tr → treeInvariant tree
+    lem00 .leaf () t-leaf
+    lem00 .(node key value leaf leaf) eq (t-single key value) = subst (λ k → treeInvariant k ) (just-injective (cong node-left eq)) t-leaf
+    lem00 .(node key _ leaf (node key₁ _ _ _)) eq (t-right key key₁ x x₁ x₂ ti) = subst (λ k → treeInvariant k ) (just-injective (cong node-left eq)) t-leaf
+    lem00 .(node key₁ _ (node key _ _ _) leaf) eq (t-left key key₁ x x₁ x₂ ti) = subst (λ k → treeInvariant k ) (just-injective (cong node-left eq)) ti
+    lem00 .(node key₁ _ (node key _ _ _) (node key₂ _ _ _)) eq (t-node key key₁ key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁) = subst (λ k → treeInvariant k ) (just-injective (cong node-left eq)) ti
 
 treeRightDown  : {n : Level} {A : Set n} {k : ℕ} {v1 : A}  → (tree tree₁ : bt A )
       → treeInvariant (node k v1 tree tree₁)
       →      treeInvariant tree₁
-treeRightDown {n} {A} {_} {v1} .leaf .leaf (t-single _ .v1) = t-leaf
-treeRightDown {n} {A} {_} {v1} .leaf .(node _ _ _ _) (t-right _ _ x _ _ ti) = ti
-treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .leaf (t-left _ _ x _ _ ti) = t-leaf
-treeRightDown {n} {A} {_} {v1} .(node _ _ _ _) .(node _ _ _ _) (t-node _ _ _ x x₁ _ _ _ _ ti ti₁) = ti₁
+treeRightDown {n} {A} {k} {v1} tree tree₁ ti = lem00 (node k v1 tree tree₁) refl ti  where
+    lem00 : ( tr : bt A ) → tr ≡ node k v1 tree tree₁ → treeInvariant tr → treeInvariant tree₁
+    lem00 .leaf () t-leaf
+    lem00 .(node key value leaf leaf) eq (t-single key value) = subst (λ k → treeInvariant k ) (just-injective (cong node-right eq)) t-leaf
+    lem00 .(node key _ leaf (node key₁ _ _ _)) eq (t-right key key₁ x x₁ x₂ ti) = subst (λ k → treeInvariant k ) (just-injective (cong node-right eq)) ti
+    lem00 .(node key₁ _ (node key _ _ _) leaf) eq (t-left key key₁ x x₁ x₂ ti) = subst (λ k → treeInvariant k ) (just-injective (cong node-right eq)) t-leaf
+    lem00 .(node key₁ _ (node key _ _ _) (node key₂ _ _ _)) eq (t-node key key₁ key₂ x x₁ x₂ x₃ x₄ x₅ ti ti₁) = subst (λ k → treeInvariant k ) (just-injective (cong node-right eq)) ti₁
+
 
 ti-property1 :  {n : Level} {A : Set n} {key₁ : ℕ} {value₂ : A} {left right : bt A} → treeInvariant (node key₁ value₂  left right ) →   tr< key₁ left ∧  tr> key₁ right
-ti-property1 {n} {A} {key₁} {value₂} {.leaf} {.leaf} (t-single .key₁ .value₂) = ⟪ tt , tt ⟫
-ti-property1 {n} {A} {key₁} {value₂} {.leaf} {.(node key₂ _ _ _)} (t-right .key₁ key₂ x x₁ x₂ t) = ⟪ tt , ⟪ x , ⟪ x₁ , x₂ ⟫ ⟫ ⟫
-ti-property1 {n} {A} {key₁} {value₂} {.(node key _ _ _)} {.leaf} (t-left key .key₁ x x₁ x₂ t) = ⟪ ⟪ x , ⟪ x₁ , x₂ ⟫ ⟫ , tt ⟫
-ti-property1 {n} {A} {key₁} {value₂} {.(node key _ _ _)} {.(node key₂ _ _ _)} (t-node key .key₁ key₂ x x₁ x₂ x₃ x₄ x₅ t t₁)
-     = ⟪ ⟪ x , ⟪ x₂ , x₃ ⟫ ⟫ , ⟪ x₁ , ⟪ x₄ , x₅ ⟫ ⟫ ⟫
+ti-property1 {n} {A} {key₁} {value₂} {left} {right} ti = lem00 key₁ (node key₁ value₂ left right) refl ti where
+    lem00 : (key₁ : ℕ) → ( tree : bt A ) → tree ≡ node key₁ value₂ left right → treeInvariant tree → tr< key₁ left ∧  tr> key₁ right
+    lem00 - .leaf () t-leaf
+    lem00 key₁ .(node key value leaf leaf) eq (t-single key value) = subst₂ (λ j k → tr< key₁ j ∧ tr> key₁ k ) lem01 lem02 ⟪ tt , tt ⟫ where
+         lem01 : leaf ≡ left
+         lem01 = just-injective (cong node-left eq)
+         lem02 : leaf ≡ right
+         lem02 = just-injective (cong node-right eq)
+    lem00 key₂ .(node key _ leaf (node key₁ _ _ _)) eq (t-right key key₁ {value} {value₁} {t₁} {t₂} x x₁ x₂ ti) 
+         = subst₂ (λ j k → tr< key₂ j ∧ tr> key₂ k ) lem01 lem02 ⟪ tt , ⟪ subst (λ k → k < key₁) lem04 x , 
+              ⟪ subst (λ k → tr> k t₁) lem04 x₁ , subst (λ k → tr> k t₂) lem04 x₂ ⟫  ⟫ ⟫ where
+         lem01 : leaf ≡ left
+         lem01 = just-injective (cong node-left eq)
+         lem02 : node key₁ value₁ t₁ t₂ ≡ right
+         lem02 = just-injective (cong node-right eq)
+         lem04 : key ≡ key₂
+         lem04 = just-injective (cong node-key eq)
+    lem00 key₂ .(node key₁ _ (node key _ _ _) leaf) eq (t-left key key₁ {value} {value₁} {t₁} {t₂} x x₁ x₂ ti) 
+         = subst₂ (λ j k → tr< key₂ j ∧ tr> key₂ k ) lem02 lem01 ⟪ ⟪ subst (λ k → key < k) lem04 x , 
+              ⟪ subst (λ k → tr< k t₁) lem04 x₁  , subst (λ k → tr< k t₂) lem04 x₂  ⟫  ⟫ , tt ⟫ where
+         lem01 : leaf ≡ right
+         lem01 = just-injective (cong node-right eq)
+         lem02 : node key value t₁ t₂ ≡ left
+         lem02 = just-injective (cong node-left eq)
+         lem04 : key₁ ≡ key₂
+         lem04 = just-injective (cong node-key eq)
+    lem00 key₂ .(node key₁ _ (node key _ _ _) (node key₃ _ _ _)) eq (t-node key key₁ key₃ {value} {value₁} {value₂} 
+      {t₁} {t₂} {t₃} {t₄} x x₁ x₂ x₃ x₄ x₅ ti ti₁) 
+        = subst₂ (λ j k → tr< key₂ j ∧ tr> key₂ k ) lem01 lem02 ⟪ ⟪ subst (λ k → key < k) lem04 x , 
+             ⟪ subst (λ k → tr< k t₁) lem04 x₂ , subst (λ k → tr< k t₂) lem04 x₃ ⟫ ⟫   
+           , ⟪ subst (λ k → k < key₃) lem04 x₁  , ⟪  subst (λ k → tr> k t₃) lem04 x₄   , subst (λ k → tr> k t₄) lem04 x₅  ⟫  ⟫ ⟫ where
+         lem01 : node key value t₁ t₂  ≡ left
+         lem01 = just-injective (cong node-left eq)
+         lem02 : node key₃ value₂ t₃ t₄  ≡ right
+         lem02 = just-injective (cong node-right eq)
+         lem04 : key₁ ≡ key₂
+         lem04 = just-injective (cong node-key eq)
 
 
 findP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree tree0 : bt A ) → (stack : List (bt A))
@@ -276,24 +468,14 @@
 findP {n} {_} {A} key (node key₁ v1 tree tree₁) tree0 st  Pre next _ | tri< a ¬b ¬c = next tree (tree ∷ st)
        ⟪ treeLeftDown tree tree₁ (proj1 Pre)  , findP1 a st (proj2 Pre) ⟫ depth-1< where
    findP1 : key < key₁ → (st : List (bt A)) →  stackInvariant key (node key₁ v1 tree tree₁) tree0 st → stackInvariant key tree tree0 (tree ∷ st)
-   findP1 a (x ∷ st) si = s-left _ _ _ a si
+   findP1 = ? -- a (x ∷ st) si = s-left _ _ _ a si
 findP key n@(node key₁ v1 tree tree₁) tree0 st Pre next _ | tri> ¬a ¬b c = next tree₁ (tree₁ ∷ st) ⟪ treeRightDown tree tree₁ (proj1 Pre) , s-right _ _ _ c (proj2 Pre) ⟫ depth-2<
 
 replaceTree1 : {n  : Level} {A : Set n} {t t₁ : bt A } → ( k : ℕ ) → (v1 value : A ) →  treeInvariant (node k v1 t t₁) → treeInvariant (node k value t t₁)
-replaceTree1 k v1 value (t-single .k .v1) = t-single k value
-replaceTree1 k v1 value (t-right _ _ x a b t) = t-right _ _ x a b t
-replaceTree1 k v1 value (t-left _ _ x a b t) = t-left _ _ x a b t
-replaceTree1 k v1 value (t-node _ _ _ x x₁ a b c d t t₁) = t-node _ _ _ x x₁ a b c d t t₁
+replaceTree1 k v1 value = ?
 
 open import Relation.Binary.Definitions
 
-lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥
-lemma3 refl ()
-lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥
-lemma5 (s≤s z≤n) ()
-¬x<x : {x : ℕ} → ¬ (x < x)
-¬x<x (s≤s lt) = ¬x<x lt
-
 child-replaced :  {n : Level} {A : Set n} (key : ℕ)   (tree : bt A) → bt A
 child-replaced key leaf = leaf
 child-replaced key (node key₁ value left right) with <-cmp key key₁
@@ -306,30 +488,27 @@
    → child-replaced key (node key₁ value left right) ≡ left
 child-replaced-left {n} {A} {key} {key₁} {value} {left} {right} lt = ch00 (node key₁ value left right) refl lt where
      ch00 : (tree : bt A) → tree ≡ node key₁ value left right → key < key₁ → child-replaced key tree ≡ left
-     ch00 (node key₁ value tree tree₁) refl lt1 with <-cmp key key₁
-     ... | tri< a ¬b ¬c = refl
-     ... | tri≈ ¬a b ¬c = ⊥-elim (¬a lt1)
-     ... | tri> ¬a ¬b c = ⊥-elim (¬a lt1)
+     ch00 = ?
 
 child-replaced-right :  {n : Level} {A : Set n} {key key₁ : ℕ}  {value : A}  {left right : bt A}
    → key₁ < key
    → child-replaced key (node key₁ value left right) ≡ right
 child-replaced-right {n} {A} {key} {key₁} {value} {left} {right} lt = ch00 (node key₁ value left right) refl lt where
      ch00 : (tree : bt A) → tree ≡ node key₁ value left right → key₁ < key → child-replaced key tree ≡ right
-     ch00 (node key₁ value tree tree₁) refl lt1 with <-cmp key key₁
-     ... | tri< a ¬b ¬c = ⊥-elim (¬c lt1)
-     ... | tri≈ ¬a b ¬c = ⊥-elim (¬c lt1)
-     ... | tri> ¬a ¬b c = refl
+     ch00 tree eq lt1 with <-cmp key key₁
+     ... | tri< a ¬b ¬c = ⊥-elim (¬c ?)
+     ... | tri≈ ¬a b ¬c = ⊥-elim (¬c ?)
+     ... | tri> ¬a ¬b c = ?
 
 child-replaced-eq :  {n : Level} {A : Set n} {key key₁ : ℕ}  {value : A}  {left right : bt A}
    → key₁ ≡ key
    → child-replaced key (node key₁ value left right) ≡ node key₁ value left right
 child-replaced-eq {n} {A} {key} {key₁} {value} {left} {right} lt = ch00 (node key₁ value left right) refl lt where
      ch00 : (tree : bt A) → tree ≡ node key₁ value left right → key₁ ≡ key → child-replaced key tree ≡ node key₁ value left right
-     ch00 (node key₁ value tree tree₁) refl lt1 with <-cmp key key₁
-     ... | tri< a ¬b ¬c = ⊥-elim (¬b (sym lt1))
-     ... | tri≈ ¬a b ¬c = refl
-     ... | tri> ¬a ¬b c = ⊥-elim (¬b (sym lt1))
+     ch00 tree eq lt1 with <-cmp key key₁
+     ... | tri< a ¬b ¬c = ⊥-elim (¬b (sym ?))
+     ... | tri≈ ¬a b ¬c = ?
+     ... | tri> ¬a ¬b c = ⊥-elim (¬b (sym ?))
 
 open _∧_
 
@@ -357,62 +536,47 @@
 
 RTtoTI0  : {n : Level} {A : Set n}  → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant tree
      → replacedTree key value tree repl → treeInvariant repl
-RTtoTI0 .leaf .(node key value leaf leaf) key value ti r-leaf = t-single key value
-RTtoTI0 .(node key _ leaf leaf) .(node key value leaf leaf) key value (t-single .key _) r-node = t-single key value
-RTtoTI0 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right _ _ x a b ti) r-node = t-right _ _ x a b ti
-RTtoTI0 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left _ _ x a b ti) r-node = t-left _ _ x a b ti
-RTtoTI0 .(node key _ (node _ _ _ _) (node _ _ _ _)) .(node key value (node _ _ _ _) (node _ _ _ _)) key value (t-node _ _ _ x x₁ a b c d ti ti₁) r-node = t-node _ _ _ x x₁ a b c d ti ti₁
--- r-right case
-RTtoTI0 (node _ _ leaf leaf) (node _ _ leaf .(node key value leaf leaf)) key value (t-single _ _) (r-right x r-leaf) = t-right _ _ x _ _ (t-single key value)
-RTtoTI0 (node _ _ leaf right@(node _ _ _ _)) (node key₁ value₁ leaf leaf) key value (t-right _ _ x₁ a b ti) (r-right x ri) = t-single key₁ value₁
-RTtoTI0 (node key₁ _ leaf right@(node key₂ _ left₁ right₁)) (node key₁ value₁ leaf right₃@(node key₃ _ left₂ right₂)) key value (t-right key₄ key₅ x₁ a b ti) (r-right x ri) =
-      t-right _ _ (subst (λ k → key₁ < k ) (rt-property-key ri) x₁) (rr00 ri a ) (rr02 ri b) (RTtoTI0 right right₃ key value ti ri) where
-         rr00 : replacedTree key value (node key₂ _ left₁ right₁) (node key₃ _ left₂ right₂) → tr> key₁ left₁ → tr> key₁ left₂
-         rr00 r-node tb = tb
-         rr00 (r-right x ri) tb = tb
-         rr00 (r-left x₂ ri) tb = ri-tr> _ _ _ _ _ ri x tb
-         rr02 : replacedTree key value (node key₂ _ left₁ right₁) (node key₃ _ left₂ right₂) → tr> key₁ right₁ → tr> key₁ right₂
-         rr02 r-node tb = tb
-         rr02 (r-right x₂ ri) tb = ri-tr> _ _ _ _ _ ri x tb
-         rr02 (r-left x ri) tb = tb
-RTtoTI0 (node key₁ _ (node _ _ _ _) leaf) (node key₁ _ (node key₃ value left right) leaf) key value₁ (t-left _ _ x₁ a b ti) (r-right x ())
-RTtoTI0 (node key₁ _ (node key₃ _ _ _) leaf) (node key₁ _ (node key₃ value₃ _ _) (node key value leaf leaf)) key value (t-left _ _ x₁ a b ti) (r-right x r-leaf) =
-      t-node _ _ _ x₁ x a b tt tt ti (t-single key value)
-RTtoTI0 (node key₁ _ (node _ _ left₀ right₀) (node key₂ _ left₁ right₁)) (node key₁ _ (node _ _ left₂ right₂) (node key₃ _ left₃ right₃)) key value (t-node _ _ _ x₁ x₂ a b c d ti ti₁) (r-right x ri) =
-      t-node _ _ _ x₁ (subst (λ k → key₁ < k ) (rt-property-key ri) x₂) a b (rr00 ri c) (rr02 ri d) ti (RTtoTI0 _ _ key value ti₁ ri) where
-         rr00 : replacedTree key value (node key₂ _ _ _) (node key₃ _ _ _) → tr> key₁ left₁ → tr> key₁ left₃
-         rr00 r-node tb = tb
-         rr00 (r-right x₃ ri) tb = tb
-         rr00 (r-left x₃ ri) tb = ri-tr> _ _ _ _ _ ri x tb
-         rr02 : replacedTree key value (node key₂ _ _ _) (node key₃ _ _ _) → tr> key₁ right₁ → tr> key₁ right₃
-         rr02 r-node tb = tb
-         rr02 (r-right x₃ ri) tb = ri-tr> _ _ _ _ _ ri x tb
-         rr02 (r-left x₃ ri) tb = tb
--- r-left case
-RTtoTI0 .(node _ _ leaf leaf) .(node _ _ (node key value leaf leaf) leaf) key value (t-single _ _) (r-left x r-leaf) = t-left _ _ x tt tt (t-single _ _ )
-RTtoTI0 .(node _ _ leaf (node _ _ _ _)) (node key₁ value₁ (node key value leaf leaf) (node _ _ _ _)) key value (t-right _ _ x₁ a b ti) (r-left x r-leaf) =
-      t-node _ _ _ x x₁ tt tt a b (t-single key value) ti
-RTtoTI0 (node key₃ _ (node key₂ _ left₁ right₁) leaf) (node key₃ _ (node key₁ value₁ left₂ right₂) leaf) key value (t-left _ _ x₁ a b ti) (r-left x ri) =
-      t-left _ _ (subst (λ k → k < key₃ ) (rt-property-key ri) x₁) (rr00 ri a) (rr02 ri b) (RTtoTI0 _ _ key value ti ri) where -- key₁ < key₃
-         rr00 : replacedTree key value (node key₂ _ left₁ right₁) (node key₁ _ left₂ right₂) → tr< key₃ left₁ → tr< key₃ left₂
-         rr00 r-node tb = tb
-         rr00 (r-right x₂ ri) tb = tb
-         rr00 (r-left x₂ ri) tb = ri-tr< _ _ _ _ _ ri x tb
-         rr02 : replacedTree key value (node key₂ _ left₁ right₁) (node key₁ _ left₂ right₂) → tr< key₃ right₁ → tr< key₃ right₂
-         rr02 r-node tb = tb
-         rr02 (r-right x₃ ri) tb = ri-tr< _ _ _ _ _ ri x tb
-         rr02 (r-left x₃ ri) tb = tb
-RTtoTI0 (node key₁ _ (node key₂ _ left₂ right₂) (node key₃ _ left₃ right₃)) (node key₁ _ (node key₄ _ left₄ right₄) (node key₅ _ left₅ right₅)) key value (t-node _ _ _ x₁ x₂ a b c d ti ti₁) (r-left x ri) =
-      t-node _ _ _ (subst (λ k → k < key₁ ) (rt-property-key ri) x₁) x₂  (rr00 ri a) (rr02 ri b) c d (RTtoTI0 _ _ key value ti ri) ti₁ where
-         rr00 : replacedTree key value (node key₂ _ left₂ right₂) (node key₄ _ left₄ right₄) → tr< key₁ left₂ → tr< key₁ left₄
-         rr00 r-node tb = tb
-         rr00 (r-right x₃ ri) tb = tb
-         rr00 (r-left x₃ ri) tb = ri-tr< _ _ _ _ _ ri x tb
-         rr02 : replacedTree key value (node key₂ _ left₂ right₂) (node key₄ _ left₄ right₄) → tr< key₁ right₂ → tr< key₁ right₄
-         rr02 r-node tb = tb
-         rr02 (r-right x₃ ri) tb = ri-tr< _ _ _ _ _ ri x tb
-         rr02 (r-left x₃ ri) tb = tb
+RTtoTI0 = ?
 
+--   RTtoTI0 .(node key _ leaf leaf) .(node key value leaf leaf) key value = ?
+--   RTtoTI0 .(node key _ leaf (node _ _ _ _)) .(node key value leaf (node _ _ _ _)) key value (t-right _ _ x a b ti) r-node = t-right _ _ x a b ti
+--   RTtoTI0 .(node key _ (node _ _ _ _) leaf) .(node key value (node _ _ _ _) leaf) key value (t-left _ _ x a b ti) r-node = t-left _ _ x a b ti
+--   RTtoTI0 .(node key _ (node _ _ _ _) (node _ _ _ _)) .(node key value (node _ _ _ _) (node _ _ _ _)) key value (t-node _ _ _ x x₁ a b c d ti ti₁) r-node = t-node _ _ _ x x₁ a b c d ti ti₁
+--   -- r-right case
+--   RTtoTI0 (node _ _ leaf leaf) (node _ _ leaf .(node key value leaf leaf)) key value (t-single _ _) (r-right x r-leaf) = t-right _ _ x _ _ (t-single key value)
+--   RTtoTI0 (node _ _ leaf right@(node _ _ _ _)) (node key₁ value₁ leaf leaf) key value (t-right _ _ x₁ a b ti) (r-right x ri) = t-single key₁ value₁
+--   RTtoTI0 (node key₁ _ leaf right@(node key₂ _ left₁ right₁)) (node key₁ value₁ leaf right₃@(node key₃ _ left₂ right₂)) key value (t-right key₄ key₅ x₁ a b ti) (r-right x ri) =
+--         t-right _ _ (subst (λ k → key₁ < k ) (rt-property-key ri) x₁) (rr00 ri a ) (rr02 ri b) (RTtoTI0 right right₃ key value ti ri) where
+--            rr00 : replacedTree key value (node key₂ _ left₁ right₁) (node key₃ _ left₂ right₂) → tr> key₁ left₁ → tr> key₁ left₂
+--            rr00 = ?
+--            rr02 : replacedTree key value (node key₂ _ left₁ right₁) (node key₃ _ left₂ right₂) → tr> key₁ right₁ → tr> key₁ right₂
+--            rr02 = ?
+--   RTtoTI0 (node key₁ _ (node _ _ _ _) leaf) (node key₁ _ (node key₃ value left right) leaf) key value₁ (t-left _ _ x₁ a b ti) (r-right x ())
+--   RTtoTI0 (node key₁ _ (node key₃ _ _ _) leaf) (node key₁ _ (node key₃ value₃ _ _) (node key value leaf leaf)) key value (t-left _ _ x₁ a b ti) (r-right x r-leaf) =
+--         t-node _ _ _ x₁ x a b tt tt ti (t-single key value)
+--   RTtoTI0 (node key₁ _ (node _ _ left₀ right₀) (node key₂ _ left₁ right₁)) (node key₁ _ (node _ _ left₂ right₂) (node key₃ _ left₃ right₃)) key value (t-node _ _ _ x₁ x₂ a b c d ti ti₁) (r-right x ri) =
+--         t-node _ _ _ x₁ (subst (λ k → key₁ < k ) (rt-property-key ri) x₂) a b (rr00 ri c) (rr02 ri d) ti (RTtoTI0 _ _ key value ti₁ ri) where
+--            rr00 : replacedTree key value (node key₂ _ _ _) (node key₃ _ _ _) → tr> key₁ left₁ → tr> key₁ left₃
+--            rr00 = ?
+--            rr02 : replacedTree key value (node key₂ _ _ _) (node key₃ _ _ _) → tr> key₁ right₁ → tr> key₁ right₃
+--            rr02 = ?
+--   -- r-left case
+--   RTtoTI0 .(node _ _ leaf leaf) .(node _ _ (node key value leaf leaf) leaf) key value (t-single _ _) (r-left x r-leaf) = t-left _ _ x tt tt (t-single _ _ )
+--   RTtoTI0 .(node _ _ leaf (node _ _ _ _)) (node key₁ value₁ (node key value leaf leaf) (node _ _ _ _)) key value (t-right _ _ x₁ a b ti) (r-left x r-leaf) =
+--         t-node _ _ _ x x₁ tt tt a b (t-single key value) ti
+--   RTtoTI0 (node key₃ _ (node key₂ _ left₁ right₁) leaf) (node key₃ _ (node key₁ value₁ left₂ right₂) leaf) key value (t-left _ _ x₁ a b ti) (r-left x ri) =
+--         t-left _ _ (subst (λ k → k < key₃ ) (rt-property-key ri) x₁) (rr00 ri a) (rr02 ri b) (RTtoTI0 _ _ key value ti ri) where -- key₁ < key₃
+--            rr00 : replacedTree key value (node key₂ _ left₁ right₁) (node key₁ _ left₂ right₂) → tr< key₃ left₁ → tr< key₃ left₂
+--            rr00 = ?
+--            rr02 : replacedTree key value (node key₂ _ left₁ right₁) (node key₁ _ left₂ right₂) → tr< key₃ right₁ → tr< key₃ right₂
+--            rr02 = ?
+--   RTtoTI0 (node key₁ _ (node key₂ _ left₂ right₂) (node key₃ _ left₃ right₃)) (node key₁ _ (node key₄ _ left₄ right₄) (node key₅ _ left₅ right₅)) key value (t-node _ _ _ x₁ x₂ a b c d ti ti₁) (r-left x ri) =
+--         t-node _ _ _ (subst (λ k → k < key₁ ) (rt-property-key ri) x₁) x₂  (rr00 ri a) (rr02 ri b) c d (RTtoTI0 _ _ key value ti ri) ti₁ where
+--            rr00 : replacedTree key value (node key₂ _ left₂ right₂) (node key₄ _ left₄ right₄) → tr< key₁ left₂ → tr< key₁ left₄
+--            rr00 = ?
+--            rr02 : replacedTree key value (node key₂ _ left₂ right₂) (node key₄ _ left₄ right₄) → tr< key₁ right₂ → tr< key₁ right₄
+--            rr02 = ?
+--
 -- RTtoTI1  : {n : Level} {A : Set n}  → (tree repl : bt A) → (key : ℕ) → (value : A) → treeInvariant repl
 --      → replacedTree key value tree repl → treeInvariant tree
 -- RTtoTI1 .leaf .(node key value leaf leaf) key value ti r-leaf = t-leaf
@@ -439,50 +603,14 @@
            → ( tree parent orig : bt A) →  (key : ℕ)
            → stackInvariant key tree  orig ( tree ∷ parent ∷ rest )
            → stackInvariant key parent orig (parent ∷ rest )
-popStackInvariant rest tree parent orig key (s-right .tree .orig tree₁ x si) = sc00 where
-     sc00 : stackInvariant key parent orig (parent ∷ rest )
-     sc00 with si-property1 si
-     ... | refl = si
-popStackInvariant rest tree parent orig key (s-left .tree .orig tree₁ x si) = sc00 where
-     sc00 : stackInvariant key parent orig (parent ∷ rest )
-     sc00 with si-property1 si
-     ... | refl = si
-
+popStackInvariant rest tree parent orig key = ?
 
 siToTreeinvariant : {n : Level} {A : Set n} → (rest : List ( bt A))
            → ( tree orig : bt A) →  (key : ℕ)
            → treeInvariant orig
            → stackInvariant key tree  orig ( tree ∷ rest )
            → treeInvariant tree
-siToTreeinvariant .[] tree .tree key ti s-nil = ti
-siToTreeinvariant .(node _ _ leaf leaf ∷ []) .leaf .(node _ _ leaf leaf) key (t-single _ _) (s-right .leaf .(node _ _ leaf leaf) .leaf x s-nil) = t-leaf
-siToTreeinvariant .(node _ _ leaf (node key₁ _ _ _) ∷ []) .(node key₁ _ _ _) .(node _ _ leaf (node key₁ _ _ _)) key (t-right _ key₁ x₁ x₂ x₃ ti) (s-right .(node key₁ _ _ _) .(node _ _ leaf (node key₁ _ _ _)) .leaf x s-nil) = ti
-siToTreeinvariant .(node _ _ (node key₁ _ _ _) leaf ∷ []) .leaf .(node _ _ (node key₁ _ _ _) leaf) key (t-left key₁ _ x₁ x₂ x₃ ti) (s-right .leaf .(node _ _ (node key₁ _ _ _) leaf) .(node key₁ _ _ _) x s-nil) = t-leaf
-siToTreeinvariant .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _) ∷ []) .(node key₂ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) key (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-right .(node key₂ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) .(node key₁ _ _ _) x s-nil) = ti₁
-siToTreeinvariant .(node _ _ tree₁ tree ∷ _) tree orig key ti (s-right .tree .orig tree₁ x si2@(s-right .(node _ _ tree₁ tree) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2
-... | t-single _ _ = t-leaf
-... | t-right _ key₁ x₂ x₃ x₄ ti₁ = ti₁
-... | t-left key₁ _ x₂ x₃ x₄ ti₁ = t-leaf
-... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ ti₁ ti₂ = ti₂
-siToTreeinvariant .(node _ _ tree₁ tree ∷ _) tree orig key ti (s-right .tree .orig tree₁ x si2@(s-left .(node _ _ tree₁ tree) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2
-... | t-single _ _ = t-leaf
-... | t-right _ key₁ x₂ x₃ x₄ ti₁ = ti₁
-... | t-left key₁ _ x₂ x₃ x₄ ti₁ = t-leaf
-... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ ti₁ ti₂ = ti₂
-siToTreeinvariant .(node _ _ leaf leaf ∷ []) .leaf .(node _ _ leaf leaf) key (t-single _ _) (s-left .leaf .(node _ _ leaf leaf) .leaf x s-nil) = t-leaf
-siToTreeinvariant .(node _ _ leaf (node key₁ _ _ _) ∷ []) .leaf .(node _ _ leaf (node key₁ _ _ _)) key (t-right _ key₁ x₁ x₂ x₃ ti) (s-left .leaf .(node _ _ leaf (node key₁ _ _ _)) .(node key₁ _ _ _) x s-nil) = t-leaf
-siToTreeinvariant .(node _ _ (node key₁ _ _ _) leaf ∷ []) .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) leaf) key (t-left key₁ _ x₁ x₂ x₃ ti) (s-left .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) leaf) .leaf x s-nil) = ti
-siToTreeinvariant .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _) ∷ []) .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) key (t-node key₁ _ key₂ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (s-left .(node key₁ _ _ _) .(node _ _ (node key₁ _ _ _) (node key₂ _ _ _)) .(node key₂ _ _ _) x s-nil) = ti
-siToTreeinvariant .(node _ _ tree tree₁ ∷ _) tree orig key ti (s-left .tree .orig tree₁ x si2@(s-right .(node _ _ tree tree₁) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2
-... | t-single _ _ = t-leaf
-... | t-right _ key₁ x₂ x₃ x₄ t = t-leaf
-... | t-left key₁ _ x₂ x₃ x₄ t = t
-... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ t t₁ = t
-siToTreeinvariant .(node _ _ tree tree₁ ∷ _) tree orig key ti (s-left .tree .orig tree₁ x si2@(s-left .(node _ _ tree tree₁) .orig tree₂ x₁ si)) with siToTreeinvariant _ _ _ _ ti si2
-... | t-single _ _ = t-leaf
-... | t-right _ key₁ x₂ x₃ x₄ t = t-leaf
-... | t-left key₁ _ x₂ x₃ x₄ t = t
-... | t-node key₁ _ key₂ x₂ x₃ x₄ x₅ x₆ x₇ t t₁ = t
+siToTreeinvariant = ?
 
 record replacePR {n : Level} {A : Set n} (key : ℕ) (value : A) (tree repl : bt A ) (stack : List (bt A)) (C : bt A → bt A → List (bt A) → Set n) : Set n where
    field
@@ -496,15 +624,16 @@
 replaceNodeP : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → (tree : bt A)
     → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key )
     → (treeInvariant tree ) → ((tree1 : bt A) → treeInvariant tree1 →  replacedTree key value (child-replaced key tree) tree1 → t) → t
-replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf
-replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P)
-     (subst (λ j → replacedTree k v1 j  (node k v1 t t₁) ) repl00 r-node) where
-         repl00 : node k value t t₁ ≡ child-replaced k (node k value t t₁)
-         repl00 with <-cmp k k
-         ... | tri< a ¬b ¬c = ⊥-elim (¬b refl)
-         ... | tri≈ ¬a b ¬c = refl
-         ... | tri> ¬a ¬b c = ⊥-elim (¬b refl)
-
+replaceNodeP = ?
+--   replaceNodeP k v1 leaf C P next = next (node k v1 leaf leaf) (t-single k v1 ) r-leaf
+--   replaceNodeP k v1 (node .k value t t₁) (case2 refl) P next = next (node k v1 t t₁) (replaceTree1 k value v1 P)
+--        (subst (λ j → replacedTree k v1 j  (node k v1 t t₁) ) repl00 r-node) where
+--            repl00 : node k value t t₁ ≡ child-replaced k (node k value t t₁)
+--            repl00 with <-cmp k k
+--            ... | tri< a ¬b ¬c = ⊥-elim (¬b refl)
+--            ... | tri≈ ¬a b ¬c = refl
+--            ... | tri> ¬a ¬b c = ⊥-elim (¬b refl)
+--
 
 replaceP : {n m : Level} {A : Set n} {t : Set m}
      → (key : ℕ) → (value : A) → {tree : bt A} ( repl : bt A)
@@ -514,15 +643,16 @@
      → (exit : (tree1 repl : bt A) → treeInvariant repl ∧ replacedTree key value tree1 repl → t) → t
 replaceP key value {tree}  repl [] Pre next exit = ⊥-elim ( si-property0 (replacePR.si Pre) refl ) -- can't happen
 replaceP key value {tree}  repl (leaf ∷ []) Pre next exit with  si-property-last  _ _ _ _  (replacePR.si Pre)-- tree0 ≡ leaf
-... | refl  =  exit (replacePR.tree0 Pre) (node key value leaf leaf) ⟪ t-single _ _  ,  r-leaf ⟫
+... | eq  =  exit (replacePR.tree0 Pre) (node key value leaf leaf) ⟪ t-single _ _  ,  ? ⟫
 replaceP key value {tree}  repl (node key₁ value₁ left right ∷ []) Pre next exit with <-cmp key key₁
 ... | tri< a ¬b ¬c = exit (replacePR.tree0 Pre) (node key₁ value₁ repl right )
         ⟪ RTtoTI0 _ _ _ _ (replacePR.ti Pre) repl01 , repl01 ⟫ where
     repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ repl right )
-    repl01 with si-property1 (replacePR.si Pre) | si-property-last  _ _ _ _  (replacePR.si Pre)
-    repl01 | refl | refl = subst (λ k → replacedTree key value  (node key₁ value₁ k right ) (node key₁ value₁ repl right )) repl02 (r-left a repl03) where
+    -- repl01 = ? --with si-property1 (replacePR.si Pre) | si-property-last  _ _ _ _  (replacePR.si Pre)
+    repl01 = ? where
+    -- | refl | refl = subst (λ k → replacedTree key value  (node key₁ value₁ k right ) (node key₁ value₁ repl right )) repl02 (r-left a repl03) where
         repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl
-        repl03 = replacePR.ri Pre
+        repl03 = replacePR.ri ?
         repl02 : child-replaced key (node key₁ value₁ left right) ≡ left
         repl02 with <-cmp key key₁
         ... | tri< a ¬b ¬c = refl
@@ -530,8 +660,9 @@
         ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a)
 ... | tri≈ ¬a b ¬c = exit (replacePR.tree0 Pre) repl ⟪  RTtoTI0 _ _ _ _ (replacePR.ti Pre) repl01 , repl01 ⟫ where
     repl01 : replacedTree key value (replacePR.tree0 Pre) repl
-    repl01 with si-property1 (replacePR.si Pre) | si-property-last  _ _ _ _  (replacePR.si Pre)
-    repl01 | refl | refl = subst (λ k → replacedTree key value k repl) repl02 (replacePR.ri Pre) where
+    --repl01 with si-property1 (replacePR.si Pre) | si-property-last  _ _ _ _  (replacePR.si Pre)
+    repl01 = ? where
+    --repl01 | refl | refl = subst (λ k → replacedTree key value k repl) repl02 (replacePR.ri Pre) where
         repl02 : child-replaced key (node key₁ value₁ left right) ≡ node key₁ value₁ left right
         repl02 with <-cmp key key₁
         ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b)
@@ -539,10 +670,11 @@
         ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b)
 ... | tri> ¬a ¬b c = exit (replacePR.tree0 Pre) (node key₁ value₁ left repl  ) ⟪  RTtoTI0 _ _ _ _ (replacePR.ti Pre) repl01 , repl01 ⟫ where
     repl01 : replacedTree key value (replacePR.tree0 Pre) (node key₁ value₁ left repl )
-    repl01 with si-property1 (replacePR.si Pre) | si-property-last  _ _ _ _  (replacePR.si Pre)
-    repl01 | refl | refl = subst (λ k → replacedTree key value  (node key₁ value₁ left k ) (node key₁ value₁ left repl )) repl02 (r-right c repl03) where
+    -- repl01 with si-property1 (replacePR.si Pre) | si-property-last  _ _ _ _  (replacePR.si Pre)
+    repl01 = ? where
+    -- repl01 | refl | refl = subst (λ k → replacedTree key value  (node key₁ value₁ left k ) (node key₁ value₁ left repl )) repl02 (r-right c repl03) where
         repl03 : replacedTree key value ( child-replaced key (node key₁ value₁ left right)) repl
-        repl03 = replacePR.ri Pre
+        repl03 = replacePR.ri ?
         repl02 : child-replaced key (node key₁ value₁ left right) ≡ right
         repl02 with <-cmp key key₁
         ... | tri< a ¬b ¬c = ⊥-elim ( ¬c c)
@@ -550,216 +682,218 @@
         ... | tri> ¬a ¬b c = refl
 replaceP {n} {_} {A} key value  {tree}  repl (leaf ∷ st@(tree1 ∷ st1)) Pre next exit = next key value repl st Post ≤-refl where
     Post :  replacePR key value tree1 repl (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤)
-    Post with replacePR.si Pre
-    ... | s-right  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ;
-           rti = replacePR.rti Pre ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 tree₁ leaf
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl07 : child-replaced key (node key₂ v1 tree₁ leaf) ≡ leaf
-        repl07 with <-cmp key key₂
-        ... |  tri< a ¬b ¬c = ⊥-elim (¬c x)
-        ... |  tri≈ ¬a b ¬c = ⊥-elim (¬c x)
-        ... |  tri> ¬a ¬b c = refl
-        repl12 : replacedTree key value (child-replaced key  tree1  ) repl
-        repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf
-    ... | s-left  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = replacePR.rti Pre ; ri = repl12 ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 leaf tree₁
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl07 : child-replaced key (node key₂ v1 leaf tree₁ ) ≡ leaf
-        repl07 with <-cmp key key₂
-        ... |  tri< a ¬b ¬c = refl
-        ... |  tri≈ ¬a b ¬c = ⊥-elim (¬a x)
-        ... |  tri> ¬a ¬b c = ⊥-elim (¬a x)
-        repl12 : replacedTree key value (child-replaced key  tree1  ) repl
-        repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07) ) (sym (rt-property-leaf (replacePR.ri Pre ))) r-leaf
+    Post = ? -- where with replacePR.si Pre
+--   -- ... | s-right  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; ri = repl12 ;
+--          rti = replacePR.rti Pre ; ci = lift tt } where
+--       repl09 : tree1 ≡ node key₂ v1 tree₁ leaf
+--       repl09 = si-property1 si
+--       repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--       repl10 with si-property1 si
+--       ... | refl = si
+--       repl07 : child-replaced key (node key₂ v1 tree₁ leaf) ≡ leaf
+--       repl07 with <-cmp key key₂
+--       ... |  tri< a ¬b ¬c = ⊥-elim (¬c x)
+--       ... |  tri≈ ¬a b ¬c = ⊥-elim (¬c x)
+--       ... |  tri> ¬a ¬b c = refl
+--       repl12 : replacedTree key value (child-replaced key  tree1  ) repl
+--       repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07 ) ) (sym (rt-property-leaf (replacePR.ri Pre))) r-leaf
+--   ... | s-left  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = replacePR.rti Pre ; ri = repl12 ; ci = lift tt } where
+--       repl09 : tree1 ≡ node key₂ v1 leaf tree₁
+--       repl09 = si-property1 si
+--       repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--       repl10 with si-property1 si
+--       ... | refl = si
+--       repl07 : child-replaced key (node key₂ v1 leaf tree₁ ) ≡ leaf
+--       repl07 with <-cmp key key₂
+--       ... |  tri< a ¬b ¬c = refl
+--       ... |  tri≈ ¬a b ¬c = ⊥-elim (¬a x)
+--       ... |  tri> ¬a ¬b c = ⊥-elim (¬a x)
+--       repl12 : replacedTree key value (child-replaced key  tree1  ) repl
+--       repl12 = subst₂ (λ j k → replacedTree key value j k ) (sym (subst (λ k → child-replaced key k ≡ leaf) (sym repl09) repl07) ) (sym (rt-property-leaf (replacePR.ri Pre ))) r-leaf
 replaceP {n} {_} {A} key value {tree}  repl (node key₁ value₁ left right ∷ st@(tree1 ∷ st1)) Pre next exit  with <-cmp key key₁
 ... | tri< a ¬b ¬c = next key value (node key₁ value₁ repl right ) st Post ≤-refl where
     Post : replacePR key value tree1 (node key₁ value₁ repl right ) st (λ _ _ _ → Lift n ⊤)
-    Post with replacePR.si Pre
-    ... | s-right _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10
-      ; rti = repl13
-      ; ri = repl12 ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right)
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl11 : stackInvariant key (node key₂ v1 tree₁ (node key₁ value₁ left right)) (replacePR.tree0 Pre) (node key₂ v1 tree₁ (node key₁ value₁ left right)∷ st1)
-        repl11 = subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10
-        repl03 : child-replaced key (node key₁ value₁ left right) ≡ left
-        repl03 with <-cmp key key₁
-        ... | tri< a1 ¬b ¬c = refl
-        ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a)
-        ... | tri> ¬a ¬b c = ⊥-elim (¬a a)
-        repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right
-        repl02 with repl09 | <-cmp key key₂
-        ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt)
-        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt)
-        ... | refl | tri> ¬a ¬b c = refl
-        repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1
-        repl04  = begin
-            node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03  ⟩
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
-        repl05  = begin
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ repl right)
-        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04  (r-left a (replacePR.ri Pre))
-        repl13 : treeInvariant (node key₁ value₁ repl right)
-        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeRightDown _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) repl11 ) )) repl12
-    ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl03 : child-replaced key (node key₁ value₁ left right) ≡ left
-        repl03 with <-cmp key key₁
-        ... | tri< a1 ¬b ¬c = refl
-        ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a)
-        ... | tri> ¬a ¬b c = ⊥-elim (¬a a)
-        repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right
-        repl02 with repl09 | <-cmp key key₂
-        ... | refl | tri< a ¬b ¬c = refl
-        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt)
-        ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt)
-        repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1
-        repl04  = begin
-            node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03  ⟩
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
-        repl05  = begin
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ repl right)
-        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04  (r-left a (replacePR.ri Pre))
-        repl11 = subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10
-        repl13 :  treeInvariant (node key₁ value₁ repl right)
-        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeLeftDown _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) repl11 ) )) repl12
+    Post = ? -- with replacePR.si Pre
+--    ... | s-right _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10
+--      ; rti = repl13
+--      ; ri = repl12 ; ci = lift tt } where
+--        repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right)
+--        repl09 = si-property1 si
+--        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--        repl10 with si-property1 si
+--        ... | refl = si
+--        repl11 : stackInvariant key (node key₂ v1 tree₁ (node key₁ value₁ left right)) (replacePR.tree0 Pre) (node key₂ v1 tree₁ (node key₁ value₁ left right)∷ st1)
+--        repl11 = subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10
+--        repl03 : child-replaced key (node key₁ value₁ left right) ≡ left
+--        repl03 with <-cmp key key₁
+--        ... | tri< a1 ¬b ¬c = refl
+--        ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a)
+--        ... | tri> ¬a ¬b c = ⊥-elim (¬a a)
+--        repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right
+--        repl02 with repl09 | <-cmp key key₂
+--        ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt)
+--        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt)
+--        ... | refl | tri> ¬a ¬b c = refl
+--        repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1
+--        repl04  = begin
+--            node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03  ⟩
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
+--        repl05  = begin
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ repl right)
+--        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04  (r-left a (replacePR.ri Pre))
+--        repl13 : treeInvariant (node key₁ value₁ repl right)
+--        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeRightDown _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) repl11 ) )) repl12
+--    ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 ; ci = lift tt } where
+--        repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁
+--        repl09 = si-property1 si
+--        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--        repl10 with si-property1 si
+--        ... | refl = si
+--        repl03 : child-replaced key (node key₁ value₁ left right) ≡ left
+--        repl03 with <-cmp key key₁
+--        ... | tri< a1 ¬b ¬c = refl
+--        ... | tri≈ ¬a b ¬c = ⊥-elim (¬a a)
+--        ... | tri> ¬a ¬b c = ⊥-elim (¬a a)
+--        repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right
+--        repl02 with repl09 | <-cmp key key₂
+--        ... | refl | tri< a ¬b ¬c = refl
+--        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt)
+--        ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt)
+--        repl04 : node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡ child-replaced key tree1
+--        repl04  = begin
+--            node key₁ value₁ (child-replaced key (node key₁ value₁ left right)) right ≡⟨ cong (λ k → node key₁ value₁ k right) repl03  ⟩
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
+--        repl05  = begin
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ repl right)
+--        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ repl right) ) repl04  (r-left a (replacePR.ri Pre))
+--        repl11 = subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10
+--        repl13 :  treeInvariant (node key₁ value₁ repl right)
+--        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeLeftDown _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) repl11 ) )) repl12
 ... | tri≈ ¬a b ¬c = next key value (node key₁ value  left right ) st Post ≤-refl where
     Post :  replacePR key value tree1 (node key₁ value left right ) (tree1 ∷ st1) (λ _ _ _ → Lift n ⊤)
-    Post with replacePR.si Pre
-    ... | s-right  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 b ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 tree₁ tree -- (node key₁ value₁  left right)
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl07 : child-replaced key (node key₂ v1 tree₁ tree) ≡ tree
-        repl07 with <-cmp key key₂
-        ... |  tri< a ¬b ¬c = ⊥-elim (¬c x)
-        ... |  tri≈ ¬a b ¬c = ⊥-elim (¬c x)
-        ... |  tri> ¬a ¬b c = refl
-        repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key  tree1  ) (node key₁ value left right )
-        repl12 refl with repl09
-        ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node
-        repl13 : treeInvariant (node key₁ value left right)
-        repl13 = RTtoTI0 _ _ _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) (replacePR.si Pre))  r-node
-    ... | s-left  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 b ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 tree tree₁
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl07 : child-replaced key (node key₂ v1 tree tree₁ ) ≡ tree
-        repl07 with <-cmp key key₂
-        ... |  tri< a ¬b ¬c = refl
-        ... |  tri≈ ¬a b ¬c = ⊥-elim (¬a x)
-        ... |  tri> ¬a ¬b c = ⊥-elim (¬a x)
-        repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key  tree1  ) (node key₁ value left right )
-        repl12 refl with repl09
-        ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node
-        repl13 : treeInvariant (node key₁ value left right)
-        repl13 = RTtoTI0 _ _ _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) (replacePR.si Pre))  r-node
+    Post = ?
+--    with replacePR.si Pre
+--    ... | s-right  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 b ; ci = lift tt } where
+--        repl09 : tree1 ≡ node key₂ v1 tree₁ tree -- (node key₁ value₁  left right)
+--        repl09 = si-property1 si
+--        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--        repl10 with si-property1 si
+--        ... | refl = si
+--        repl07 : child-replaced key (node key₂ v1 tree₁ tree) ≡ tree
+--        repl07 with <-cmp key key₂
+--        ... |  tri< a ¬b ¬c = ⊥-elim (¬c x)
+--        ... |  tri≈ ¬a b ¬c = ⊥-elim (¬c x)
+--        ... |  tri> ¬a ¬b c = refl
+--        repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key  tree1  ) (node key₁ value left right )
+--        repl12 refl with repl09
+--        ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node
+--        repl13 : treeInvariant (node key₁ value left right)
+--        repl13 = RTtoTI0 _ _ _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) (replacePR.si Pre))  r-node
+--    ... | s-left  _ _ tree₁ {key₂} {v1} x si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 b ; ci = lift tt } where
+--        repl09 : tree1 ≡ node key₂ v1 tree tree₁
+--        repl09 = si-property1 si
+--        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--        repl10 with si-property1 si
+--        ... | refl = si
+--        repl07 : child-replaced key (node key₂ v1 tree tree₁ ) ≡ tree
+--        repl07 with <-cmp key key₂
+--        ... |  tri< a ¬b ¬c = refl
+--        ... |  tri≈ ¬a b ¬c = ⊥-elim (¬a x)
+--        ... |  tri> ¬a ¬b c = ⊥-elim (¬a x)
+--        repl12 : (key ≡ key₁) → replacedTree key value (child-replaced key  tree1  ) (node key₁ value left right )
+--        repl12 refl with repl09
+--        ... | refl = subst (λ k → replacedTree key value k (node key₁ value left right )) (sym repl07) r-node
+--        repl13 : treeInvariant (node key₁ value left right)
+--        repl13 = RTtoTI0 _ _ _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) (replacePR.si Pre))  r-node
 ... | tri> ¬a ¬b c = next key value (node key₁ value₁ left repl ) st Post ≤-refl where
     Post : replacePR key value tree1 (node key₁ value₁ left repl ) st (λ _ _ _ → Lift n ⊤)
-    Post with replacePR.si Pre
-    ... | s-right _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right)
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl03 : child-replaced key (node key₁ value₁ left right) ≡ right
-        repl03 with <-cmp key key₁
-        ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c)
-        ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c)
-        ... | tri> ¬a ¬b c = refl
-        repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right
-        repl02 with repl09 | <-cmp key key₂
-        ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt)
-        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt)
-        ... | refl | tri> ¬a ¬b c = refl
-        repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1
-        repl04  = begin
-            node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
-        repl05  = begin
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ left repl)
-        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre))
-        repl11 = subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10
-        repl13 : treeInvariant (node key₁ value₁ left repl)
-        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeRightDown _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) repl11 ) )) repl12
-    ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 ; ci = lift tt } where
-        repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁
-        repl09 = si-property1 si
-        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
-        repl10 with si-property1 si
-        ... | refl = si
-        repl03 : child-replaced key (node key₁ value₁ left right) ≡ right
-        repl03 with <-cmp key key₁
-        ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c)
-        ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c)
-        ... | tri> ¬a ¬b c = refl
-        repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right
-        repl02 with repl09 | <-cmp key key₂
-        ... | refl | tri< a ¬b ¬c = refl
-        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt)
-        ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt)
-        repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1
-        repl04  = begin
-            node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
-        repl05  = begin
-            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
-            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
-            child-replaced key tree1 ∎  where open ≡-Reasoning
-        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ left repl)
-        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04  (r-right c (replacePR.ri Pre))
-        repl13 : treeInvariant (node key₁ value₁ left repl)
-        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeLeftDown _ _
-           (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) (subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10)) )) repl12
+    Post = ?
+    -- with replacePR.si Pre
+--    ... | s-right _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 ; ci = lift tt } where
+--        repl09 : tree1 ≡ node key₂ v1 tree₁ (node key₁ value₁ left right)
+--        repl09 = si-property1 si
+--        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--        repl10 with si-property1 si
+--        ... | refl = si
+--        repl03 : child-replaced key (node key₁ value₁ left right) ≡ right
+--        repl03 with <-cmp key key₁
+--        ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c)
+--        ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c)
+--        ... | tri> ¬a ¬b c = refl
+--        repl02 : child-replaced key (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡ node key₁ value₁ left right
+--        repl02 with repl09 | <-cmp key key₂
+--        ... | refl | tri< a ¬b ¬c = ⊥-elim (¬c lt)
+--        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬c lt)
+--        ... | refl | tri> ¬a ¬b c = refl
+--        repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1
+--        repl04  = begin
+--            node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
+--        repl05  = begin
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 tree₁ (node key₁ value₁ left right) ) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ left repl)
+--        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04 (r-right c (replacePR.ri Pre))
+--        repl11 = subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10
+--        repl13 : treeInvariant (node key₁ value₁ left repl)
+--        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeRightDown _ _ (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) repl11 ) )) repl12
+--    ... | s-left _ _ tree₁ {key₂} {v1} lt si = record { tree0 = replacePR.tree0 Pre ; ti = replacePR.ti Pre ; si = repl10 ; rti = repl13 ; ri = repl12 ; ci = lift tt } where
+--        repl09 : tree1 ≡ node key₂ v1 (node key₁ value₁ left right) tree₁
+--        repl09 = si-property1 si
+--        repl10 : stackInvariant key tree1 (replacePR.tree0 Pre) (tree1 ∷ st1)
+--        repl10 with si-property1 si
+--        ... | refl = si
+--        repl03 : child-replaced key (node key₁ value₁ left right) ≡ right
+--        repl03 with <-cmp key key₁
+--        ... | tri< a1 ¬b ¬c = ⊥-elim (¬c c)
+--        ... | tri≈ ¬a b ¬c = ⊥-elim (¬c c)
+--        ... | tri> ¬a ¬b c = refl
+--        repl02 : child-replaced key (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡ node key₁ value₁ left right
+--        repl02 with repl09 | <-cmp key key₂
+--        ... | refl | tri< a ¬b ¬c = refl
+--        ... | refl | tri≈ ¬a b ¬c = ⊥-elim (¬a lt)
+--        ... | refl | tri> ¬a ¬b c = ⊥-elim (¬a lt)
+--        repl04 : node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡ child-replaced key tree1
+--        repl04  = begin
+--            node key₁ value₁ left (child-replaced key (node key₁ value₁ left right)) ≡⟨ cong (λ k → node key₁ value₁ left k ) repl03 ⟩
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl05 : node key₁ value₁ left right ≡ child-replaced key tree1
+--        repl05  = begin
+--            node key₁ value₁ left right ≡⟨ sym repl02 ⟩
+--            child-replaced key  (node key₂ v1 (node key₁ value₁ left right) tree₁) ≡⟨ cong (λ k → child-replaced key k ) (sym repl09) ⟩
+--            child-replaced key tree1 ∎  where open ≡-Reasoning
+--        repl12 : replacedTree key value (child-replaced key  tree1  ) (node key₁ value₁ left repl)
+--        repl12 = subst (λ k → replacedTree key value k (node key₁ value₁ left repl) ) repl04  (r-right c (replacePR.ri Pre))
+--        repl13 : treeInvariant (node key₁ value₁ left repl)
+--        repl13 = RTtoTI0 _ _ _ _ (subst (λ k → treeInvariant k) repl05 (treeLeftDown _ _
+--           (siToTreeinvariant _ _ _ _ (replacePR.ti Pre) (subst (λ k → stackInvariant key k (replacePR.tree0 Pre) (k ∷ st1)) repl09 repl10)) )) repl12
 
 TerminatingLoopS : {l m : Level} {t : Set l} (Index : Set m ) → {Invraiant : Index → Set m } → ( reduce : Index → ℕ)
    → (r : Index) → (p : Invraiant r)
    → (loop : (r : Index)  → Invraiant r → (next : (r1 : Index)  → Invraiant r1 → reduce r1 < reduce r  → t ) → t) → t
 TerminatingLoopS {_} {_} {t} Index {Invraiant} reduce r p loop with <-cmp 0 (reduce r)
-... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim (lemma3 b lt) )
+... | tri≈ ¬a b ¬c = loop r p (λ r1 p1 lt → ⊥-elim ? ) --
 ... | tri< a ¬b ¬c = loop r p (λ r1 p1 lt1 → TerminatingLoop1 (reduce r) r r1 (m≤n⇒m≤1+n lt1) p1 lt1 ) where
     TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j  → Invraiant r1 →  reduce r1 < reduce r → t
-    TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim (lemma5 n≤j lt1))
+    TerminatingLoop1 zero r r1 n≤j p1 lt = loop r1 p1 (λ r2 p1 lt1 → ⊥-elim ?) -- (lemma5 n≤j lt1))
     TerminatingLoop1 (suc j) r r1  n≤j p1 lt with <-cmp (reduce r1) (suc j)
     ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a p1 lt
     ... | tri≈ ¬a b ¬c = loop r1 p1 (λ r2 p2 lt1 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b lt1 ) p2 lt1 )
@@ -849,57 +983,44 @@
  →  (left right : bt (Color ∧ A))
  → RBtreeInvariant (node key ⟪ c , value ⟫ left right)
  → RBtreeInvariant left
-RBtreeLeftDown left right (rb-red _ _ x x₁ x₂ rb rb₁) = rb
-RBtreeLeftDown left right (rb-black _ _ x rb rb₁) = rb
+RBtreeLeftDown left right = ?
 
 
 RBtreeRightDown : {n : Level} {A : Set n} { key : ℕ} {value : A} {c : Color}
  → (left right : bt (Color ∧ A))
  → RBtreeInvariant (node key ⟪ c , value ⟫ left right)
  → RBtreeInvariant right
-RBtreeRightDown left right (rb-red _ _ x x₁ x₂ rb rb₁) = rb₁
-RBtreeRightDown left right (rb-black _ _ x rb rb₁) = rb₁
+RBtreeRightDown left right = ?
 
 RBtreeEQ : {n : Level} {A : Set n} {key : ℕ} {value : A} {c : Color}
  → {left right : bt (Color ∧ A)}
  → RBtreeInvariant (node key ⟪ c , value ⟫ left right)
  → black-depth left ≡ black-depth right
-RBtreeEQ  (rb-red _ _ x x₁ x₂ rb rb₁) = x₂
-RBtreeEQ  (rb-black _ _ x rb rb₁) = x
+RBtreeEQ  = ?
 
 RBtreeToBlack : {n : Level} {A : Set n}
  → (tree : bt (Color ∧ A))
  → RBtreeInvariant tree
  → RBtreeInvariant (to-black tree)
-RBtreeToBlack leaf rb-leaf = rb-leaf
-RBtreeToBlack (node key ⟪ Red , value ⟫ left right) (rb-red _ _ x x₁ x₂ rb rb₁) = rb-black key value x₂ rb rb₁
-RBtreeToBlack (node key ⟪ Black , value ⟫ left right) (rb-black _ _ x rb rb₁) = rb-black key value x rb rb₁
+RBtreeToBlack = ?
 
 RBtreeToBlackColor : {n : Level} {A : Set n}
  → (tree : bt (Color ∧ A))
  → RBtreeInvariant tree
  → color (to-black tree) ≡ Black
-RBtreeToBlackColor leaf rb-leaf = refl
-RBtreeToBlackColor (node key ⟪ Red , value ⟫ left right) (rb-red _ _ x x₁ x₂ rb rb₁) = refl
-RBtreeToBlackColor (node key ⟪ Black , value ⟫ left right) (rb-black _ _ x rb rb₁) = refl
+RBtreeToBlackColor = ?
 
 RBtreeParentColorBlack : {n : Level} {A : Set n} → (left right  : bt (Color ∧ A)) { value : A} {key : ℕ} { c : Color}
  → RBtreeInvariant (node key ⟪ c , value ⟫ left right)
  → (color left ≡ Red) ∨ (color right ≡ Red)
  → c ≡ Black
-RBtreeParentColorBlack leaf leaf (rb-red _ _ x₁ x₂ x₃ x₄ x₅) (case1 ())
-RBtreeParentColorBlack leaf leaf (rb-red _ _ x₁ x₂ x₃ x₄ x₅) (case2 ())
-RBtreeParentColorBlack (node key ⟪ Red , proj4 ⟫ left left₁) right (rb-red _ _ () x₁ x₂ rb rb₁) (case1 x₃)
-RBtreeParentColorBlack (node key ⟪ Black , proj4 ⟫ left left₁) right (rb-red _ _ x x₁ x₂ rb rb₁) (case1 ())
-RBtreeParentColorBlack left (node key ⟪ Red , proj4 ⟫ right right₁) (rb-red _ _ x () x₂ rb rb₁) (case2 x₃)
-RBtreeParentColorBlack left (node key ⟪ Black , proj4 ⟫ right right₁) (rb-red _ _ x x₁ x₂ rb rb₁) (case2 ())
-RBtreeParentColorBlack left right (rb-black _ _ x rb rb₁) x₃ = refl
+RBtreeParentColorBlack = ?
 
 RBtreeChildrenColorBlack : {n : Level} {A : Set n} → (left right  : bt (Color ∧ A)) { value : Color ∧ A} {key : ℕ}
  → RBtreeInvariant (node key value left right)
  → proj1 value  ≡ Red
  → (color left ≡ Black) ∧  (color right ≡ Black)
-RBtreeChildrenColorBlack left right (rb-red _ _ x x₁ x₂ rbi rbi₁) refl = ⟪ x , x₁ ⟫
+RBtreeChildrenColorBlack left right = ?
 
 --
 --  findRBT exit with replaced node
@@ -1082,80 +1203,80 @@
    → c ≡ Red
    → RBtreeInvariant tree
    → (black-depth tree ≡ black-depth left ) ∧ (black-depth tree ≡ black-depth right)
-red-children-eq {n} {A} {.(node key₁ ⟪ Red , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Red} refl eq₁ rb2@(rb-red key₁ value₁ x x₁ x₂ rb rb₁) =
-  ⟪ ( begin
-        black-depth left ⊔ black-depth right ≡⟨ cong (λ k → black-depth left ⊔ k) (sym (RBtreeEQ rb2)) ⟩
-        black-depth left ⊔ black-depth left ≡⟨ ⊔-idem _ ⟩
-        black-depth left ∎  ) ,  (
-     begin
-        black-depth left ⊔ black-depth right ≡⟨ cong (λ k → k ⊔ black-depth right ) (RBtreeEQ rb2) ⟩
-        black-depth right ⊔ black-depth right ≡⟨ ⊔-idem _ ⟩
-        black-depth right ∎ ) ⟫ where open ≡-Reasoning
-red-children-eq {n} {A} {tree} {left} {right} {key} {value} {Black} eq () rb
+red-children-eq {n} {A} = ?
+
+
+
+
+
+
+
+
+
 
 black-children-eq : {n : Level} {A : Set n}  {tree left right : bt (Color ∧ A)} → {key : ℕ} → {value : A} → {c : Color}
    → tree ≡ node key ⟪ c , value ⟫ left right
    → c ≡ Black
    → RBtreeInvariant tree
    → (black-depth tree ≡ suc (black-depth left) ) ∧ (black-depth tree ≡ suc (black-depth right))
-black-children-eq {n} {A} {.(node key₁ ⟪ Black , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Black} refl eq₁ rb2@(rb-black key₁ value₁ x rb rb₁) =
-  ⟪ ( begin
-        suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (black-depth left ⊔ k)) (sym (RBtreeEQ rb2)) ⟩
-        suc (black-depth left ⊔ black-depth left) ≡⟨ ⊔-idem _ ⟩
-        suc (black-depth left) ∎  ) ,  (
-     begin
-        suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (k ⊔ black-depth right)) (RBtreeEQ rb2) ⟩
-        suc (black-depth right ⊔ black-depth right) ≡⟨ ⊔-idem _ ⟩
-        suc (black-depth right) ∎ ) ⟫ where open ≡-Reasoning
-black-children-eq {n} {A} {tree} {left} {right} {key} {value} {Red} eq () rb
+black-children-eq {n} {A} = ?
+
+
+
+
+
+
+
+
+
 
 black-depth-cong  : {n : Level} (A : Set n)  {tree tree₁ : bt (Color ∧ A)}
    → tree ≡ tree₁ → black-depth tree ≡ black-depth tree₁
-black-depth-cong  {n} A  {leaf} {leaf} refl = refl
-black-depth-cong {n} A {node key ⟪ Red , value ⟫ left right} {node .key ⟪ Red , .value ⟫ .left .right} refl
-   = cong₂ (λ j k → j ⊔ k ) (black-depth-cong A {left} {left} refl) (black-depth-cong A {right} {right} refl)
-black-depth-cong {n} A {node key ⟪ Black , value ⟫ left right} {node .key ⟪ Black , .value ⟫ .left .right} refl
-   = cong₂ (λ j k → suc (j ⊔ k) ) (black-depth-cong A {left} {left} refl) (black-depth-cong A {right} {right} refl)
+black-depth-cong  {n} A  = ?
+
+
+
+
 
 black-depth-resp  : {n : Level} (A : Set n)   (tree tree₁ : bt (Color ∧ A)) → {c1 c2 : Color}  { l1 l2 r1 r2 : bt (Color ∧ A)} {key1 key2 : ℕ} {value1 value2 : A}
    → tree ≡ node key1 ⟪ c1 , value1 ⟫ l1 r1 → tree₁ ≡ node key2 ⟪ c2 , value2 ⟫ l2 r2
    → color tree  ≡ color tree₁
    → black-depth l1 ≡ black-depth l2 → black-depth r1 ≡ black-depth r2
    → black-depth tree ≡ black-depth tree₁
-black-depth-resp  A _ _ {Black} {Black} refl refl refl eql eqr = cong₂ (λ j k → suc (j ⊔ k) ) eql eqr
-black-depth-resp  A _ _ {Red} {Red} refl refl refl eql eqr = cong₂ (λ j k → j ⊔ k ) eql eqr
+black-depth-resp  A _ _ = ?
+
 
 red-children-eq1 : {n : Level} {A : Set n}  {tree left right : bt (Color ∧ A)} → {key : ℕ} → {value : A} → {c : Color}
    → tree ≡ node key ⟪ c , value ⟫ left right
    → color tree ≡ Red
    → RBtreeInvariant tree
    → (black-depth tree ≡ black-depth left ) ∧ (black-depth tree ≡ black-depth right)
-red-children-eq1 {n} {A} {.(node key₁ ⟪ Red , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Red} refl eq₁ rb2@(rb-red key₁ value₁ x x₁ x₂ rb rb₁) =
-  ⟪ ( begin
-        black-depth left ⊔ black-depth right ≡⟨ cong (λ k → black-depth left ⊔ k) (sym (RBtreeEQ rb2)) ⟩
-        black-depth left ⊔ black-depth left ≡⟨ ⊔-idem _ ⟩
-        black-depth left ∎  ) ,  (
-     begin
-        black-depth left ⊔ black-depth right ≡⟨ cong (λ k → k ⊔ black-depth right ) (RBtreeEQ rb2) ⟩
-        black-depth right ⊔ black-depth right ≡⟨ ⊔-idem _ ⟩
-        black-depth right ∎ ) ⟫ where open ≡-Reasoning
-red-children-eq1 {n} {A} {.(node key ⟪ Black , value ⟫ left right)} {left} {right} {key} {value} {Black} refl () rb
+red-children-eq1 {n} {A} = ?
+
+
+
+
+
+
+
+
+
 
 black-children-eq1 : {n : Level} {A : Set n}  {tree left right : bt (Color ∧ A)} → {key : ℕ} → {value : A} → {c : Color}
    → tree ≡ node key ⟪ c , value ⟫ left right
    → color tree ≡ Black
    → RBtreeInvariant tree
    → (black-depth tree ≡ suc (black-depth left) ) ∧ (black-depth tree ≡ suc (black-depth right))
-black-children-eq1 {n} {A} {.(node key₁ ⟪ Black , value₁ ⟫ left right)} {left} {right} {.key₁} {.value₁} {Black} refl eq₁ rb2@(rb-black key₁ value₁ x rb rb₁) =
-  ⟪ ( begin
-        suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (black-depth left ⊔ k)) (sym (RBtreeEQ rb2)) ⟩
-        suc (black-depth left ⊔ black-depth left) ≡⟨ ⊔-idem _ ⟩
-        suc (black-depth left) ∎  ) ,  (
-     begin
-        suc (black-depth left ⊔ black-depth right) ≡⟨ cong (λ k → suc (k ⊔ black-depth right)) (RBtreeEQ rb2) ⟩
-        suc (black-depth right ⊔ black-depth right) ≡⟨ ⊔-idem _ ⟩
-        suc (black-depth right) ∎ ) ⟫ where open ≡-Reasoning
-black-children-eq1 {n} {A} {.(node key ⟪ Red , value ⟫ left right)} {left} {right} {key} {value} {Red} refl () rb
+black-children-eq1 {n} {A} = ?
+
+
+
+
+
+
+
+
+
 
 
 rbi-from-red-black : {n : Level } {A : Set n} → (n1 rp-left : bt (Color ∧ A)) → (kp : ℕ) → (vp : Color ∧ A)
@@ -1195,60 +1316,60 @@
 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 {n} {A} = ?
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 RB-repl→ti>  : {n : Level} {A : Set n}  → (tree repl : bt (Color ∧ A)) → (key key₁ : ℕ) → (value : A)
@@ -1329,491 +1450,7 @@
 
 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 , value₁ ⟫ (node key₁ ⟪ Red , value₂ ⟫ _ 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)
-RB-repl→ti (node key₂ ⟪ Black , _ ⟫ (node key₁ ⟪ Red , _ ⟫ t₀ leaf) (node key₃ ⟪ c1 , v1 ⟫ left right)) (node key₂ ⟪ Red , value₁ ⟫ (node _ ⟪ Black , value₂ ⟫ (node key₄ value₃ t t₁) .leaf) (node key₃ ⟪ Black , v1 ⟫ left right)) key value (t-node _ .key₂ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-ll x y lt trb) = t-node _ _ _ x₁ x₂ ⟪ <-trans (proj1 rr00) x₁  , ⟪ >-tr< (proj1 (proj2 rr00)) x₁ , >-tr< (proj2 (proj2 rr00)) x₁ ⟫ ⟫ tt x₅ x₆ (t-left _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ (treeLeftDown _ _ ti) trb)) (replaceTree1 _ _ _ ti₁) where
-        rr00 : (key₄ < key₁) ∧ tr< key₁ t ∧ tr< key₁ t₁
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt (proj1 (ti-property1 ti))
-RB-repl→ti (node key₂ ⟪ Black , _ ⟫ (node key₁ ⟪ Red , _ ⟫ .leaf (node key₄ value₃ t₁ t₂)) (node key₃ ⟪ c0 , v1 ⟫ left right)) (node key₂ ⟪ Red , value₁ ⟫ (node _ ⟪ Black , value₂ ⟫ (node key₅ value₄ t t₃) .(node key₄ value₃ t₁ t₂)) (node key₃ ⟪ Black , v1 ⟫ left right)) key value (t-node _ .key₂ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-right .key₁ .key₄ x₇ x₈ x₉ ti) ti₁) (rbr-flip-ll x y lt trb)
-    = t-node _ _ _ x₁ x₂ ⟪ <-trans (proj1 rr00) x₁ , ⟪ >-tr< (proj1 (proj2 rr00)) x₁  , >-tr< (proj2 (proj2 rr00)) x₁  ⟫ ⟫  x₄ x₅ x₆ (t-node _ _ _ (proj1 rr00) x₇ (proj1 (proj2 rr00)) (proj2 (proj2 (rr00))) x₈ x₉ (RB-repl→ti _ _ _ _ t-leaf trb) ti) (replaceTree1 _ _ _ ti₁) where
-        rr00 : (key₅ < key₁) ∧ tr< key₁ t ∧ tr< key₁ t₃
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt tt
-RB-repl→ti (node key₂ ⟪ Black , _ ⟫ (node key₁ ⟪ Red , _ ⟫ .(node key₆ _ _ _) (node key₄ value₃ t₁ t₂)) (node key₃ ⟪ c0 , v1 ⟫ left right)) (node key₂ ⟪ Red , value₁ ⟫ (node _ ⟪ Black , value₂ ⟫ (node key₅ value₄ t t₃) .(node key₄ value₃ t₁ t₂)) (node key₃ ⟪ Black , v1 ⟫ left right)) key value (t-node _ .key₂ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-node key₆ .key₁ .key₄ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti ti₂) ti₁) (rbr-flip-ll x y lt trb)
-   = t-node _ _ _ x₁ x₂  ⟪ <-trans (proj1 rr00) x₁ , ⟪ >-tr< (proj1 (proj2 rr00)) x₁ , >-tr< (proj2 (proj2 rr00)) x₁ ⟫ ⟫
-     x₄ x₅ x₆ (t-node _ _ _ (proj1 rr00) x₈ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) x₁₁ x₁₂ (RB-repl→ti _ _ _ _ ti trb) ti₂) (replaceTree1 _ _ _ ti₁) where
-        rr00 : (key₅ < key₁) ∧ tr< key₁ t ∧ tr< key₁ t₃
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₉ , x₁₀ ⟫ ⟫
-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)
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ Red , v2 ⟫ leaf leaf) (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ .leaf (node key₄ value₁ t₄ t₅)) .(to-black (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃))) key value (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-lr x y lt lt2 trb)
-  = t-node _ _ _ x₁ x₂ tt rr00 x₅ x₆ (t-right _ _ (proj1 rr01) (proj1 (proj2 rr01)) (proj2 (proj2 rr01)) (RB-repl→ti _ _ _ _ t-leaf trb)) (replaceTree1 _ _ _ ti₁) where
-       rr00 : (key₄ < key₁) ∧ tr< key₁ t₄ ∧ tr< key₁ t₅
-       rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 tt
-       rr01 : (key₂ < key₄) ∧ tr> key₂ t₄ ∧ tr> key₂ t₅
-       rr01 = RB-repl→ti> _ _ _ _ _ trb lt tt
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ Red , v2 ⟫ leaf (node key₅ value₂ t₁ t₆)) (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ .leaf (node key₄ value₁ t₄ t₅)) .(to-black (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃))) key value (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-right .key₂ .key₅ x₇ x₈ x₉ ti) ti₁) (rbr-flip-lr x y lt lt2 trb) = t-node _ _ _ x₁ x₂ tt rr00 x₅ x₆ (t-right _ _ (proj1 rr01) (proj1 (proj2 rr01)) (proj2 (proj2 rr01)) (RB-repl→ti _ _ _ _ ti trb)) (replaceTree1 _ _ _ ti₁) where
-       rr00 : (key₄ < key₁) ∧ tr< key₁ t₄ ∧ tr< key₁ t₅
-       rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 x₄
-       rr01 : (key₂ < key₄) ∧ tr> key₂ t₄ ∧ tr> key₂ t₅
-       rr01 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₇ , ⟪ x₈ , x₉ ⟫ ⟫
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ Red , v2 ⟫ (node key₄ value₁ t t₅) .leaf) (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ .(node key₄ value₁ t t₅) (node key₅ value₂ t₄ t₆)) .(to-black (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃))) key value (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-left .key₄ .key₂ x₇ x₈ x₉ ti) ti₁) (rbr-flip-lr x y lt lt2 trb)
-    = t-node _ _ _ x₁ x₂ x₃ rr00 x₅ x₆ (t-node _ _ _ x₇ (proj1 rr01) x₈ x₉ (proj1 (proj2 rr01)) (proj2 (proj2 rr01)) ti (RB-repl→ti _ _ _ _ t-leaf trb)) (replaceTree1 _ _ _ ti₁) where
-       rr00 : (key₅ < key₁) ∧ tr< key₁ t₄ ∧ tr< key₁ t₆
-       rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 tt
-       rr01 : (key₂ < key₅) ∧ tr> key₂ t₄ ∧ tr> key₂ t₆
-       rr01 = RB-repl→ti> _ _ _ _ _ trb lt tt
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ Red , v2 ⟫ (node key₄ value₁ t t₅) .(node key₆ _ _ _)) (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃)) (node key₁ ⟪ Red , _ ⟫ (node _ ⟪ Black , _ ⟫ .(node key₄ value₁ t t₅) (node key₅ value₂ t₄ t₆)) .(to-black (node key₃ ⟪ c3 , v3 ⟫ t₂ t₃))) key value (t-node _ _ key₃ x₁ x₂ x₃ x₄ x₅ x₆ (t-node .key₄ .key₂ key₆ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti ti₂) ti₁) (rbr-flip-lr x y lt lt2 trb)
-    = t-node _ _ _ x₁ x₂ x₃ rr00 x₅ x₆ (t-node _ _ _ x₇ (proj1 rr01) x₉ x₁₀ (proj1 (proj2 rr01)) (proj2 (proj2 rr01)) ti (RB-repl→ti _ _ _ _ ti₂ trb)) (replaceTree1 _ _ _ ti₁) where
-        rr00 : (key₅ < key₁) ∧ tr< key₁ t₄ ∧ tr< key₁ t₆
-        rr00 = RB-repl→ti< _ _ _ _ _ trb lt2 x₄
-        rr01 : (key₂ < key₅) ∧ tr> key₂ t₄ ∧ tr> key₂ t₆
-        rr01 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₈ , ⟪ x₁₁ , x₁₂ ⟫ ⟫
-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)
-RB-repl→ti (node _ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node _ ⟪ Red , v3 ⟫ t₂ leaf)) (node key₁ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , v2 ⟫ t t₁)) (node key₃ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) .leaf)) key value (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-rl x y lt lt2 trb)
-   = t-node _ _ _ x₁ x₂ x₃ x₄ rr00 tt (replaceTree1 _ _ _ ti) (t-left _ _ (proj1 rr01) (proj1 (proj2 rr01)) (proj2 (proj2 rr01)) (RB-repl→ti _ _ _ _ (treeLeftDown _ _ ti₁) trb)) where
-        rr00 : (key₁ < key₄) ∧ tr> key₁ t₄ ∧ tr> key₁ t₅
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt x₅
-        rr01 : (key₄ < key₃) ∧ tr< key₃ t₄ ∧ tr< key₃ t₅
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 (proj1 (ti-property1 ti₁))
-RB-repl→ti (node _ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node _ ⟪ Red , v3 ⟫ .leaf (node key₅ value₂ t₃ t₆))) (node key₁ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , v2 ⟫ t t₁)) (node key₃ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) .(node key₅ value₂ t₃ t₆))) key value (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-right .key₃ .key₅ x₇ x₈ x₉ ti₁)) (rbr-flip-rl x y lt lt2 trb) = t-node _ _ _ x₁ x₂ x₃ x₄ rr00 rr02 (replaceTree1 _ _ _ ti) (t-node _ _ _ (proj1 rr01) x₇ (proj1 (proj2 rr01)) (proj2 (proj2 rr01)) x₈ x₉ (RB-repl→ti _ _ _ _ t-leaf trb) ti₁) where
-        rr00 : (key₁ < key₄) ∧ tr> key₁ t₄ ∧ tr> key₁ t₅
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt x₅
-        rr01 : (key₄ < key₃) ∧ tr< key₃ t₄ ∧ tr< key₃ t₅
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 tt
-        rr02 : (key₁ < key₅) ∧ tr> key₁ t₃ ∧ tr> key₁ t₆
-        rr02 = ⟪ <-trans x₂ x₇ , ⟪ <-tr> x₈ x₂  , <-tr> x₉ x₂ ⟫ ⟫
-RB-repl→ti (node _ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node _ ⟪ Red , v3 ⟫ .(node key₆ _ _ _) (node key₅ value₂ t₃ t₆))) (node key₁ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , v2 ⟫ t t₁)) (node key₃ ⟪ Black , _ ⟫ (node key₄ value₁ t₄ t₅) .(node key₅ value₂ t₃ t₆))) key value (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-node key₆ .key₃ .key₅ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti₁ ti₂)) (rbr-flip-rl x y lt lt2 trb)
-   = t-node _ _ _ x₁ x₂ x₃ x₄ rr00 rr02 (replaceTree1 _ _ _ ti) (t-node _ _ _ (proj1 rr01) x₈ (proj1 (proj2 rr01))  (proj2 (proj2 rr01))  x₁₁ x₁₂ (RB-repl→ti _ _ _ _ ti₁ trb) ti₂) where
-        rr00 : (key₁ < key₄) ∧ tr> key₁ t₄ ∧ tr> key₁ t₅
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt x₅
-        rr01 : (key₄ < key₃ ) ∧ tr< key₃  t₄ ∧ tr< key₃ t₅
-        rr01 = RB-repl→ti< _ _ _ _ _ trb lt2 ⟪ x₇ , ⟪ x₉ , x₁₀ ⟫ ⟫
-        rr02 : (key₁ < key₅) ∧ tr> key₁ t₃ ∧ tr> key₁ t₆
-        rr02 = ⟪ proj1 x₆ , ⟪  <-tr> x₁₁ x₂ , <-tr> x₁₂ x₂  ⟫ ⟫
-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)
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node key₃ ⟪ Red , c3 ⟫ leaf t₃)) (node _ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , v2 ⟫ t t₁)) (node _ ⟪ Black , c3 ⟫ .leaf (node key₄ value₁ t₄ t₅))) key value (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti ti₁) (rbr-flip-rr x y lt trb) = t-node _ _ _ x₁ x₂ x₃ x₄ x₅ rr01 (replaceTree1 _ _ _ ti) (t-right _ _ (proj1 rr00) (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) (RB-repl→ti _ _ _ _ (treeRightDown _ _ ti₁) trb)) where
-        rr00 : (key₃ < key₄) ∧ tr> key₃ t₄ ∧ tr> key₃ t₅
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt (proj2 (ti-property1 ti₁))
-        rr01 : (key₁ < key₄) ∧ tr> key₁ t₄ ∧ tr> key₁ t₅
-        rr01 = RB-repl→ti> _ _ _ _ _ trb (<-trans x₂ lt) x₆
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node key₃ ⟪ Red , c3 ⟫ (node key₄ value₁ t₂ t₅) .leaf)) (node _ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , v2 ⟫ t t₁)) (node _ ⟪ Black , c3 ⟫ .(node key₄ value₁ t₂ t₅) (node key₅ value₂ t₄ t₆))) key value (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-left .key₄ .key₃ x₇ x₈ x₉ ti₁)) (rbr-flip-rr x y lt trb) = t-node _ _ _ x₁ x₂ x₃ x₄ x₅ rr02 (replaceTree1 _ _ _ ti) (t-node _ _ _ x₇ (proj1 rr00) x₈ x₉ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti₁ (RB-repl→ti _ _ _ _ t-leaf trb)) where
-        rr00 : (key₃ < key₅) ∧ tr> key₃ t₄ ∧ tr> key₃ t₆
-        rr00 = RB-repl→ti> _ _ _ _ _ trb lt tt
-        rr02 : (key₁ < key₅) ∧ tr> key₁ t₄ ∧ tr> key₁ t₆
-        rr02 = ⟪ <-trans x₂ (proj1 rr00) , ⟪ <-tr> (proj1 (proj2 rr00)) x₂  , <-tr> (proj2 (proj2 rr00)) x₂ ⟫ ⟫
-RB-repl→ti (node key₁ ⟪ Black , v1 ⟫ (node key₂ ⟪ c2 , v2 ⟫ t t₁) (node key₃ ⟪ Red , c3 ⟫ (node key₄ value₁ t₂ t₅) .(node key₆ _ _ _))) (node _ ⟪ Red , _ ⟫ .(to-black (node key₂ ⟪ c2 , v2 ⟫ t t₁)) (node _ ⟪ Black , c3 ⟫ .(node key₄ value₁ t₂ t₅) (node key₅ value₂ t₄ t₆))) key value (t-node key₂ _ _ x₁ x₂ x₃ x₄ x₅ x₆ ti (t-node .key₄ .key₃ key₆ x₇ x₈ x₉ x₁₀ x₁₁ x₁₂ ti₁ ti₂)) (rbr-flip-rr x y lt trb) = t-node _ _ _ x₁ x₂ x₃ x₄ x₅ rr02 (replaceTree1 _ _ _ ti) (t-node _ _ _ x₇ (proj1 rr00) x₉ x₁₀ (proj1 (proj2 rr00)) (proj2 (proj2 rr00)) ti₁ (RB-repl→ti _ _ _ _ ti₂ trb))  where
-         rr00 : (key₃ < key₅) ∧ tr> key₃ t₄ ∧ tr> key₃ t₆
-         rr00 = RB-repl→ti> _ _ _ _ _ trb lt ⟪ x₈ , ⟪ x₁₁ , x₁₂ ⟫ ⟫
-         rr02 : (key₁ < key₅) ∧ tr> key₁ t₄ ∧ tr> key₁ t₆
-         rr02 = ⟪ <-trans x₂ (proj1 rr00) , ⟪ <-tr> (proj1 (proj2 rr00)) x₂  , <-tr> (proj2 (proj2 rr00)) x₂ ⟫ ⟫
-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 = ?
 
 --
 -- if we consider tree invariant, this may be much simpler and faster
@@ -1821,38 +1458,12 @@
 stackToPG : {n : Level} {A : Set n} → {key : ℕ } → (tree orig : bt A )
            →  (stack : List (bt A)) → stackInvariant key tree orig stack
            → ( stack ≡ orig ∷ [] ) ∨ ( stack ≡ tree ∷ orig ∷ [] ) ∨ PG A key tree stack
-stackToPG {n} {A} {key} tree .tree .(tree ∷ []) s-nil = case1 refl
-stackToPG {n} {A} {key} tree .(node _ _ _ tree) .(tree ∷ node _ _ _ tree ∷ []) (s-right _ _ _ x s-nil) = case2 (case1 refl)
-stackToPG {n} {A} {key} tree .(node k2 v2 t5 (node k1 v1 t2 tree)) (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ [])) (s-right tree (node k2 v2 t5 (node k1 v1 t2 tree)) t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) (node k2 v2 t5 (node k1 v1 t2 tree)) t5 {k2} {v2} x₁ s-nil)) = case2 (case2
-    record {  parent = node k1 v1 t2 tree ;  grand = _ ; pg = s2-1s2p  x refl refl  ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ _)) (s-right tree orig t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) orig t5 {k2} {v2} x₁ (s-right _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 t2 tree ;  grand = _ ; pg = s2-1s2p  x  refl refl  ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig (tree ∷ node _ _ _ tree ∷ .(node k2 v2 t5 (node k1 v1 t2 tree) ∷ _)) (s-right tree orig t2 {k1} {v1} x (s-right (node k1 v1 t2 tree) orig t5 {k2} {v2} x₁ (s-left _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 t2 tree ;  grand = _ ; pg = s2-1s2p  x refl refl  ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree .(node k2 v2 (node k1 v1 t1 tree) t2) .(tree ∷ node k1 v1 t1 tree ∷ node k2 v2 (node k1 v1 t1 tree) t2 ∷ []) (s-right _ _ t1 {k1} {v1} x (s-left _ _ t2 {k2} {v2} x₁ s-nil)) = case2 (case2
-    record {  parent = node k1 v1 t1 tree ;  grand = _ ; pg = s2-1sp2 x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 t1 tree ∷ node k2 v2 (node k1 v1 t1 tree) t2 ∷ _) (s-right _ _ t1 {k1} {v1} x (s-left _ _ t2 {k2} {v2} x₁ (s-right _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 t1 tree ;  grand = _ ; pg = s2-1sp2 x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 t1 tree ∷ node k2 v2 (node k1 v1 t1 tree) t2 ∷ _) (s-right _ _ t1 {k1} {v1} x (s-left _ _ t2 {k2} {v2} x₁ (s-left _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 t1 tree ;  grand = _ ; pg = s2-1sp2 x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree .(node _ _ tree _) .(tree ∷ node _ _ tree _ ∷ []) (s-left _ _ t1 {k1} {v1} x s-nil) = case2 (case1 refl)
-stackToPG {n} {A} {key} tree .(node _ _ _ (node k1 v1 tree t1)) .(tree ∷ node k1 v1 tree t1 ∷ node _ _ _ (node k1 v1 tree t1) ∷ []) (s-left _ _ t1 {k1} {v1} x (s-right _ _ _ x₁ s-nil)) = case2 (case2
-    record {  parent = node k1 v1 tree t1 ;  grand = _ ; pg =  s2-s12p x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ _ (node k1 v1 tree t1) ∷ _) (s-left _ _ t1 {k1} {v1} x (s-right _ _ _ x₁ (s-right _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 tree t1 ;  grand = _ ; pg =  s2-s12p x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ _ (node k1 v1 tree t1) ∷ _) (s-left _ _ t1 {k1} {v1} x (s-right _ _ _ x₁ (s-left _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 tree t1 ;  grand = _ ; pg =  s2-s12p x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree .(node _ _ (node k1 v1 tree t1) _) .(tree ∷ node k1 v1 tree t1 ∷ node _ _ (node k1 v1 tree t1) _ ∷ []) (s-left _ _ t1 {k1} {v1} x (s-left _ _ _ x₁ s-nil)) = case2 (case2
-    record {  parent = node k1 v1 tree t1 ;  grand = _ ; pg =  s2-s1p2 x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ (node k1 v1 tree t1) _ ∷ _) (s-left _ _ t1 {k1} {v1} x (s-left _ _ _ x₁ (s-right _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 tree t1 ;  grand = _ ; pg =  s2-s1p2 x refl refl ; rest = _ ; stack=gp = refl } )
-stackToPG {n} {A} {key} tree orig .(tree ∷ node k1 v1 tree t1 ∷ node _ _ (node k1 v1 tree t1) _ ∷ _) (s-left _ _ t1 {k1} {v1} x (s-left _ _ _ x₁ (s-left _ _ _ x₂ si))) = case2 (case2
-    record {  parent = node k1 v1 tree t1 ;  grand = _ ; pg =  s2-s1p2 x refl refl ; rest = _ ; stack=gp = refl } )
+stackToPG {n} {A} {key} = ?
 
 stackCase1 : {n : Level} {A : Set n} → {key : ℕ } → {tree orig : bt A }
            →  {stack : List (bt A)} → stackInvariant key tree orig stack
            →  stack ≡ orig ∷ [] → tree ≡ orig
-stackCase1 s-nil refl = refl
+stackCase1 = ?
 
 pg-prop-1 : {n : Level} (A : Set n) → (key : ℕ) → (tree orig : bt A )
            →  (stack : List (bt A)) → (pg : PG A key tree stack)
@@ -1871,12 +1482,7 @@
            →  (stack : List (bt (Color ∧ A)))  → (si : stackInvariant key tree orig stack )
            →  RBtreeInvariant tree
 PGtoRBinvariant1 tree .tree rb .(tree ∷ []) s-nil = rb
-PGtoRBinvariant1 tree orig rb (tree ∷ rest) (s-right .tree .orig tree₁ x si) with PGtoRBinvariant1 _ orig rb _ si
-... | rb-red _ value x₁ x₂ x₃ t t₁ = t₁
-... | rb-black _ value x₁ t t₁ = t₁
-PGtoRBinvariant1 tree orig rb (tree ∷ rest) (s-left .tree .orig tree₁ x si) with PGtoRBinvariant1 _ orig rb _ si
-... | rb-red _ value x₁ x₂ x₃ t t₁ = t
-... | rb-black _ value x₁ t t₁ = t
+PGtoRBinvariant1 tree orig rb = ?
 
 RBI-child-replaced : {n : Level} {A : Set n} (tr : bt (Color ∧ A)) (key : ℕ) →  RBtreeInvariant tr → RBtreeInvariant (child-replaced key tr)
 RBI-child-replaced {n} {A} leaf key rbi = rbi
@@ -1898,85 +1504,7 @@
  → stackInvariant key tree1 tree0 stack
  → (tree1 ≡ leaf ) ∨ ( node-key tree1 ≡ just key )
  → (exit : (stack : List ( bt (Color ∧ A))) (r : RBI key value tree0 stack ) → t ) → t
-createRBT {n} {m} {A} {t} key value orig rbi ti tree (x ∷ []) si P exit = crbt00 orig P refl where
-     crbt00 : (tree1 : bt (Color ∧ A)) → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) → tree1 ≡ orig → t
-     crbt00 leaf P refl = exit (x ∷ []) record {
-         tree = leaf
-         ; repl = node key ⟪ Red , value ⟫ leaf leaf
-         ; origti = ti
-         ; origrb = rbi
-         ; treerb = rb-leaf
-         ; replrb = rb-red key value refl refl refl rb-leaf rb-leaf
-         ; replti = t-single _ _
-         ; si = subst (λ k → stackInvariant key k leaf _ ) crbt01 si
-         ; state = rotate leaf refl rbr-leaf
-         } where
-             crbt01 : tree ≡ leaf
-             crbt01 with si-property-last _ _ _ _ si | si-property1 si
-             ... | refl | refl = refl
-     crbt00 (node key₁ value₁ left right ) (case1 eq) refl with si-property-last _ _ _ _ si | si-property1 si
-     ... | refl | refl = ⊥-elim (bt-ne (sym eq))
-     crbt00 (node key₁ value₁ left right ) (case2 eq) refl with si-property-last _ _ _ _ si | si-property1 si
-     ... | refl | refl = exit (x ∷ []) record {
-         tree = node key₁ value₁ left right
-         ; repl = node key₁ ⟪ proj1 value₁ , value ⟫ left right
-         ; origti = ti
-         ; origrb = rbi
-         ; treerb = rbi
-         ; replrb = crbt03 value₁ rbi
-         ; replti = RTtoTI0 _ _ _ _ (siToTreeinvariant _ _ _ _ ti si) r-node
-         ; si = si
-         ; state = rebuild refl (crbt01 value₁ ) (λ ()) crbt04
-         }  where
-             crbt01 : (value₁ : Color ∧ A) → black-depth (node key₁ ⟪ proj1 value₁ , value ⟫ left right) ≡ black-depth (node key₁ value₁ left right)
-             crbt01 ⟪ Red , proj4 ⟫ = refl
-             crbt01 ⟪ Black , proj4 ⟫ = refl
-             crbt03 : (value₁ : Color ∧ A) → RBtreeInvariant (node key₁ value₁ left right) → RBtreeInvariant (node key₁ ⟪ proj1 value₁ , value ⟫ left right)
-             crbt03 ⟪ Red , proj4 ⟫ (rb-red .key₁ .proj4 x x₁ x₂ rbi rbi₁) = rb-red key₁ _ x x₁ x₂ rbi rbi₁
-             crbt03 ⟪ Black , proj4 ⟫ (rb-black .key₁ .proj4 x rbi rbi₁) = rb-black key₁ _ x rbi rbi₁
-             keq : ( just key₁ ≡ just key ) → key₁ ≡ key
-             keq refl = refl
-             crbt04 : replacedRBTree key value (node key₁ value₁ left right) (node key₁ ⟪ proj1 value₁ , value ⟫ left right)
-             crbt04 = subst (λ k → replacedRBTree k value (node key₁ value₁ left right) (node key₁ ⟪ proj1 value₁ , value ⟫ left right)) (keq eq) rbr-node
-createRBT {n} {m} {A} {t} key value orig rbi ti tree (x ∷ leaf ∷ stack) si P exit = ⊥-elim (si-property-parent-node _ _ _ si refl)
-createRBT {n} {m} {A} {t} key value orig rbi ti tree sp@(x ∷ node kp vp pleft pright ∷ stack) si P exit = crbt00 tree P refl where
-     crbt00 : (tree1 : bt (Color ∧ A)) → (tree ≡ leaf ) ∨ ( node-key tree ≡ just key ) → tree1 ≡ tree → t
-     crbt00 leaf P refl = exit sp record {
-         tree = leaf
-         ; repl = node key ⟪ Red , value ⟫ leaf leaf
-         ; origti = ti
-         ; origrb = rbi
-         ; treerb = rb-leaf
-         ; replrb = rb-red key value refl refl refl rb-leaf rb-leaf
-         ; replti = t-single _ _
-         ; si = si
-         ; state = rotate leaf refl rbr-leaf
-         }
-     crbt00 (node key₁ value₁ left right ) (case1 eq) refl = ⊥-elim (bt-ne (sym eq))
-     crbt00 (node key₁ value₁ left right ) (case2 eq) refl with si-property-last _ _ _ _ si | si-property1 si
-     ... | eq1 | eq2 = exit sp record {
-         tree = node key₁ value₁ left right
-         ; repl = node key₁ ⟪ proj1 value₁ , value ⟫ left right
-         ; origti = ti
-         ; origrb = rbi
-         ; treerb = treerb
-         ; replrb = crbt03 value₁ treerb
-         ; replti = RB-repl→ti _ _ _ _  (siToTreeinvariant _ _ _ _ ti (subst (λ k → stackInvariant _ _ orig (k ∷ _)) eq2 si )) crbt04
-         ; si = si
-         ; state = rebuild refl (crbt01 value₁ ) (λ ()) crbt04
-         }  where
-             crbt01 : (value₁ : Color ∧ A) → black-depth (node key₁ ⟪ proj1 value₁ , value ⟫ left right) ≡ black-depth (node key₁ value₁ left right)
-             crbt01 ⟪ Red , proj4 ⟫ = refl
-             crbt01 ⟪ Black , proj4 ⟫ = refl
-             crbt03 : (value₁ : Color ∧ A) → RBtreeInvariant (node key₁ value₁ left right) → RBtreeInvariant (node key₁ ⟪ proj1 value₁ , value ⟫ left right)
-             crbt03 ⟪ Red , proj4 ⟫ (rb-red .key₁ .proj4 x x₁ x₂ rbi rbi₁) = rb-red key₁ _ x x₁ x₂ rbi rbi₁
-             crbt03 ⟪ Black , proj4 ⟫ (rb-black .key₁ .proj4 x rbi rbi₁) = rb-black key₁ _ x rbi rbi₁
-             keq : ( just key₁ ≡ just key ) → key₁ ≡ key
-             keq refl = refl
-             crbt04 : replacedRBTree key value (node key₁ value₁ left right) (node key₁ ⟪ proj1 value₁ , value ⟫ left right)
-             crbt04 = subst (λ k → replacedRBTree k value (node key₁ value₁ left right) (node key₁ ⟪ proj1 value₁ , value ⟫ left right)) (keq eq) rbr-node
-             treerb : RBtreeInvariant (node key₁ value₁ left right)
-             treerb = PGtoRBinvariant1 _ orig rbi _ si
+createRBT {n} {m} {A} {t} key value orig rbi ti tree = ?
 
 
 ti-to-black : {n : Level} {A : Set n} → {tree : bt (Color ∧ A)} → treeInvariant tree → treeInvariant (to-black tree)
@@ -2015,14 +1543,12 @@
        → t
     insertCase4 leaf eq1 eq rot col si = ⊥-elim (rb04 eq eq1 si) where -- can't happen
        rb04 : {stack : List ( bt ( Color ∧ A))} → stack ≡ RBI.tree r ∷ orig ∷ [] → leaf ≡ orig → stackInvariant key (RBI.tree r) orig stack →   ⊥
-       rb04  refl refl (s-right tree leaf tree₁ x si) = si-property2 _ (s-right tree leaf tree₁ x si) refl
-       rb04  refl refl (s-left tree₁ leaf tree x si) = si-property2 _  (s-left tree₁ leaf tree x si) refl
+       rb04  = ?
     insertCase4  tr0@(node key₁ value₁ left right) refl eq rot col si with <-cmp key key₁
     ... | tri< a ¬b ¬c = rb07 col where
        rb04 : stackInvariant key (RBI.tree r) orig stack → stack ≡ RBI.tree r ∷ orig ∷ [] → tr0 ≡ orig → left ≡ RBI.tree r
-       rb04 (s-left tree₁ .(node key₁ value₁ left right) tree {key₂} x s-nil) refl refl = refl
-       rb04 (s-right tree .(node key₁ _ tree₁ tree) tree₁ x s-nil) refl refl with si-property1 si
-       ... | refl = ⊥-elim ( nat-<> x a  )
+       rb04 = ? -- (s-left tree₁ .(node key₁ value₁ left right) tree {key₂} x s-nil) eq eq1 = ?
+       -- rb04 (s-right tree .(node key₁ _ tree₁ tree) tree₁ x s-nil) eq eq1 = ?
        rb06 : black-depth repl ≡ black-depth right
        rb06 = begin
          black-depth repl ≡⟨ sym (RB-repl→eq _ _ (RBI.treerb r) rot) ⟩
@@ -2067,12 +1593,12 @@
               rb10 : replacedRBTree key value (node key₁ ⟪ Black , proj2 value₁ ⟫ left right) (node key₁ ⟪ Black , proj2 value₁ ⟫ repl right)
               rb10 = rbr-black-left repl-red a (subst (λ k → replacedRBTree key value k repl) (sym (rb04 si eq refl)) rot)
 
-    ... | tri≈ ¬a b ¬c = ⊥-elim ( si-property-pne _ _ stack eq si b ) -- can't happen
+    ... | tri≈ ¬a b ¬c = ? -- ⊥-elim ( si-property-pne _ _ stack eq si b ) -- can't happen
     ... | tri> ¬a ¬b c = rb07 col 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  )
+       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  )
        rb06 : black-depth repl ≡ black-depth left
        rb06 = begin
          black-depth repl ≡⟨ sym (RB-repl→eq _ _ (RBI.treerb r) rot) ⟩
@@ -2150,9 +1676,10 @@
                rb03 : black-depth n1 ≡ black-depth repl
                rb03 = trans (sym (RBtreeEQ (subst (λ k → RBtreeInvariant k) x (treerb pg)))) (RB-repl→eq _ _ (RBI.treerb r) rot)
                rb02 : RBtreeInvariant rb01
-               rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg)
-               ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value (trans (sym ceq) cx) cx₁ (sym rb03) (RBI.replrb r) t₁
-               ... | rb-black .kp value ex t t₁ = rb-black kp value (sym rb03) (RBI.replrb r) t₁
+               rb02 = ?
+               -- with subst (λ k → RBtreeInvariant k) x (treerb pg)
+               -- ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value (trans (sym ceq) cx) cx₁ (sym rb03) (RBI.replrb r) t₁
+               -- ... | rb-black .kp value ex t t₁ = rb-black kp value (sym rb03) (RBI.replrb r) t₁
                rb05 : treeInvariant (node kp vp (RBI.tree r) n1 )
                rb05 = subst (λ k → treeInvariant k) x (rb08 pg)
                rb04 : key < kp
@@ -2180,9 +1707,9 @@
                rb03 : black-depth n1 ≡ black-depth repl
                rb03 = trans (RBtreeEQ (subst (λ k → RBtreeInvariant k) x (treerb pg))) ((RB-repl→eq _ _ (RBI.treerb r) rot))
                rb02 : RBtreeInvariant rb01
-               rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg)
-               ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value cx (trans (sym ceq) cx₁) rb03 t (RBI.replrb r)
-               ... | rb-black .kp value ex t t₁ = rb-black kp value rb03 t (RBI.replrb r)
+               rb02 = ? -- with subst (λ k → RBtreeInvariant k) x (treerb pg)
+               -- ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value cx (trans (sym ceq) cx₁) rb03 t (RBI.replrb r)
+               -- ... | rb-black .kp value ex t t₁ = rb-black kp value rb03 t (RBI.replrb r)
                rb05 : treeInvariant (node kp vp n1 (RBI.tree r) )
                rb05 = subst (λ k → treeInvariant k) x (rb08 pg)
                rb04 : kp < key
@@ -2210,9 +1737,9 @@
                rb03 : black-depth n1 ≡ black-depth repl
                rb03 = trans (sym (RBtreeEQ (subst (λ k → RBtreeInvariant k) x (treerb pg)))) ((RB-repl→eq _ _ (RBI.treerb r) rot))
                rb02 : RBtreeInvariant rb01
-               rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg)
-               ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value (trans (sym ceq) cx) cx₁ (sym rb03) (RBI.replrb r) t₁
-               ... | rb-black .kp value ex t t₁ = rb-black kp value (sym rb03) (RBI.replrb r) t₁
+               rb02 = ? -- with subst (λ k → RBtreeInvariant k) x (treerb pg)
+               -- ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value (trans (sym ceq) cx) cx₁ (sym rb03) (RBI.replrb r) t₁
+               -- ... | rb-black .kp value ex t t₁ = rb-black kp value (sym rb03) (RBI.replrb r) t₁
                rb05 : treeInvariant (node kp vp (RBI.tree r) n1)
                rb05 = subst (λ k → treeInvariant k) x (rb08 pg)
                rb04 : key < kp
@@ -2240,9 +1767,9 @@
                rb03 : black-depth n1 ≡ black-depth repl
                rb03 = trans (RBtreeEQ (subst (λ k → RBtreeInvariant k) x (treerb pg))) ((RB-repl→eq _ _ (RBI.treerb r) rot))
                rb02 : RBtreeInvariant rb01
-               rb02 with subst (λ k → RBtreeInvariant k) x (treerb pg)
-               ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value cx (trans (sym ceq) cx₁) rb03 t (RBI.replrb r)
-               ... | rb-black .kp value ex t t₁ = rb-black kp value rb03 t (RBI.replrb r)
+               rb02 = ? -- with subst (λ k → RBtreeInvariant k) x (treerb pg)
+               -- ... | rb-red .kp value cx cx₁ ex₂ t t₁ = rb-red kp value cx (trans (sym ceq) cx₁) rb03 t (RBI.replrb r)
+               -- ... | rb-black .kp value ex t t₁ = rb-black kp value rb03 t (RBI.replrb r)
                rb05 : treeInvariant (node kp vp n1 (RBI.tree r))
                rb05 = subst (λ k → treeInvariant k) x (rb08 pg)
                rb04 : kp < key
@@ -2261,8 +1788,8 @@
        → color repl ≡ Red → color (PG.uncle pg) ≡ Black → color (PG.parent pg) ≡ Red → t
     insertCase5 leaf eq pg rot repl-red uncle-black pcolor = ⊥-elim ( rb00 repl repl-red (cong color (sym eq))) where
         rb00 : (repl : bt (Color ∧ A)) → color repl ≡ Red → color repl ≡ Black → ⊥
-        rb00 (node key ⟪ Black , proj4 ⟫ repl repl₁) () eq1
-        rb00 (node key ⟪ Red , proj4 ⟫ repl repl₁) eq ()
+        rb00 = ? -- (node key ⟪ Black , proj4 ⟫ repl repl₁) () eq1
+        -- rb00 (node key ⟪ Red , proj4 ⟫ repl repl₁) eq ()
     insertCase5 (node rkey vr rp-left rp-right) eq pg rot repl-red uncle-black pcolor = insertCase51 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)
@@ -2643,8 +2170,8 @@
     ... | rebuild ceq bdepth-eq ¬leaf rot = rebuildCase ceq bdepth-eq ¬leaf rot
     ... | top-black eq rot = exit stack (trans eq (cong (λ k → k ∷ []) rb00)) r where
         rb00 : RBI.tree r ≡ orig
-        rb00 with si-property-last _ _ _ _ (subst (λ k → stackInvariant key (RBI.tree r) orig k) (eq) (RBI.si r))
-        ... | refl = refl
+        rb00 = ? -- with si-property-last _ _ _ _ (subst (λ k → stackInvariant key (RBI.tree r) orig k) (eq) (RBI.si r))
+        -- ... | refl = refl
     ... | rotate _ repl-red rot with stackToPG (RBI.tree r) orig stack (RBI.si r)
     ... | case1 eq  = exit stack eq r     -- no stack, replace top node
     ... | case2 (case1 eq) = insertCase4 orig refl eq rot (case1 repl-red) (RBI.si r)     -- one level stack, orig is parent of repl

--- a/logic.agda	Sun Aug 04 13:05:12 2024 +0900
+++ b/logic.agda	Tue Aug 13 23:03:40 2024 +0900
@@ -91,8 +91,8 @@
   no0  : (A → ⊥) → Dec0 A
 
 open _∧_ 
-∧-injective : {n m : Level} {A : Set n} {B : Set m} → {a b : A ∧ B}  → proj1 a ≡ proj1 b → proj2 a ≡ proj2 b → a ≡ b
-∧-injective refl refl = refl
+∧-uniue : {n m : Level} {A : Set n} {B : Set m} → {a b : A ∧ B}  → proj1 a ≡ proj1 b → proj2 a ≡ proj2 b → a ≡ b
+∧-uniue refl refl = refl
 
 not-not-bool : { b : Bool } → not (not b) ≡ b
 not-not-bool {true} = refl