redBlackInSomeState : { m : Level } (a : Set Level.zero) (n : Maybe (Node a @$\mathbb{N}$@)) {t : Set m} @$\rightarrow$@ RedBlackTree {Level.zero} {m} {t} a @$\mathbb{N}$@
redBlackInSomeState {m} a n {t} = record { root = n ; nodeStack = emptySingleLinkedStack ; compare = compare2 }

putTest1 :{ m : Level } (n : Maybe (Node @$\mathbb{N}$@ @$\mathbb{N}$@))
         @$\rightarrow$@ (k : @$\mathbb{N}$@) (x : @$\mathbb{N}$@)
         @$\rightarrow$@ putTree1 {_} {_} {@$\mathbb{N}$@} {@$\mathbb{N}$@} (redBlackInSomeState {_} @$\mathbb{N}$@ n {Set Level.zero}) k x
         (\ t @$\rightarrow$@ getRedBlackTree t k (\ t x1 @$\rightarrow$@ check2 x1 x  @$\equiv$@ True))
putTest1 n k x with n
...  | Just n1 = lemma2 ( record { top = Nothing } )
   where
     lemma2 : (s : SingleLinkedStack (Node @$\mathbb{N}$@ @$\mathbb{N}$@) ) @$\rightarrow$@ putTree1 (record { root = Just n1 ; nodeStack = s ; compare = compare2 }) k x (@$\lambda$@ t @$\rightarrow$@
         GetRedBlackTree.checkNode t k (@$\lambda$@ t@$\_{1}$@ x1 @$\rightarrow$@ check2 x1 x @$\equiv$@ True) (root t))
     lemma2 s with compare2 k (key n1)
     ... |  EQ = lemma3 {!!}
        where
           lemma3 : compare2 k (key n1) @$\equiv$@  EQ @$\rightarrow$@ getRedBlackTree {_} {_} {@$\mathbb{N}$@} {@$\mathbb{N}$@} {Set Level.zero} ( record {  root = Just ( record {
               key   = key n1 ; value = x ; right = right n1 ; left  = left n1 ; color = Black
               } ) ; nodeStack = s ; compare = @$\lambda$@ x@$\_{1}$@ y @$\rightarrow$@ compare2 x@$\_{1}$@ y  } ) k ( \ t x1 @$\rightarrow$@ check2 x1 x  @$\equiv$@ True)
           lemma3 eq with compare2 x x | putTest1Lemma2 x
           ... | EQ | refl with compare2 k (key n1)  | eq
           ...              | EQ | refl with compare2 x x | putTest1Lemma2 x
           ...                    | EQ | refl = refl
     ... |  GT = {!!}
     ... |  LT = {!!}

...  | Nothing =  lemma1
   where
     lemma1 : getRedBlackTree {_} {_} {@$\mathbb{N}$@} {@$\mathbb{N}$@} {Set Level.zero} ( record {  root = Just ( record {
               key   = k ; value = x ; right = Nothing ; left  = Nothing ; color = Red
        } ) ; nodeStack = record { top = Nothing } ; compare = @$\lambda$@ x@$\_{1}$@ y @$\rightarrow$@ compare2 x@$\_{1}$@ y  } ) k
        ( \ t x1 @$\rightarrow$@ check2 x1 x  @$\equiv$@ True)
     lemma1 with compare2 k k | putTest1Lemma2 k
     ... | EQ | refl with compare2 x x | putTest1Lemma2 x
     ...              | EQ | refl = refl