YES

Problem:
 merge(nil(),y) -> y
 merge(x,nil()) -> x
 merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
 ++(nil(),y) -> y
 ++(.(x,y),z) -> .(x,++(y,z))
 if(true(),x,y) -> x
 if(false(),x,y) -> x

Proof:
 DP Processor:
  DPs:
   merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v)
   merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v))
   merge#(.(x,y),.(u,v)) -> if#(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
   ++#(.(x,y),z) -> ++#(y,z)
  TRS:
   merge(nil(),y) -> y
   merge(x,nil()) -> x
   merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
   ++(nil(),y) -> y
   ++(.(x,y),z) -> .(x,++(y,z))
   if(true(),x,y) -> x
   if(false(),x,y) -> x
  TDG Processor:
   DPs:
    merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v)
    merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v))
    merge#(.(x,y),.(u,v)) -> if#(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
    ++#(.(x,y),z) -> ++#(y,z)
   TRS:
    merge(nil(),y) -> y
    merge(x,nil()) -> x
    merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
    ++(nil(),y) -> y
    ++(.(x,y),z) -> .(x,++(y,z))
    if(true(),x,y) -> x
    if(false(),x,y) -> x
   graph:
    ++#(.(x,y),z) -> ++#(y,z) -> ++#(.(x,y),z) -> ++#(y,z)
    merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v) ->
    merge#(.(x,y),.(u,v)) -> if#(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
    merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v) ->
    merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v))
    merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v) ->
    merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v)
    merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v)) ->
    merge#(.(x,y),.(u,v)) -> if#(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
    merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v)) ->
    merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v))
    merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v)) -> merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v)
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 7/16
    DPs:
     merge#(.(x,y),.(u,v)) -> merge#(.(x,y),v)
     merge#(.(x,y),.(u,v)) -> merge#(y,.(u,v))
    TRS:
     merge(nil(),y) -> y
     merge(x,nil()) -> x
     merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
     ++(nil(),y) -> y
     ++(.(x,y),z) -> .(x,++(y,z))
     if(true(),x,y) -> x
     if(false(),x,y) -> x
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [merge#](x0, x1) = x0 + x1 + 3,
      
      [false] = 4,
      
      [true] = 3,
      
      [++](x0, x1) = x0 + 6x1 + 1,
      
      [if](x0, x1, x2) = 5x0 + x1 + 3,
      
      [<](x0, x1) = 0,
      
      [.](x0, x1) = x1 + 1,
      
      [merge](x0, x1) = 4x0 + x1 + 7,
      
      [nil] = 0
     orientation:
      merge#(.(x,y),.(u,v)) = v + y + 5 >= v + y + 4 = merge#(.(x,y),v)
      
      merge#(.(x,y),.(u,v)) = v + y + 5 >= v + y + 4 = merge#(y,.(u,v))
      
      merge(nil(),y) = y + 7 >= y = y
      
      merge(x,nil()) = 4x + 7 >= x = x
      
      merge(.(x,y),.(u,v)) = v + 4y + 12 >= v + 4y + 12 = if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
      
      ++(nil(),y) = 6y + 1 >= y = y
      
      ++(.(x,y),z) = y + 6z + 2 >= y + 6z + 2 = .(x,++(y,z))
      
      if(true(),x,y) = x + 18 >= x = x
      
      if(false(),x,y) = x + 23 >= x = x
     problem:
      DPs:
       
      TRS:
       merge(nil(),y) -> y
       merge(x,nil()) -> x
       merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
       ++(nil(),y) -> y
       ++(.(x,y),z) -> .(x,++(y,z))
       if(true(),x,y) -> x
       if(false(),x,y) -> x
     Qed
    
    DPs:
     ++#(.(x,y),z) -> ++#(y,z)
    TRS:
     merge(nil(),y) -> y
     merge(x,nil()) -> x
     merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
     ++(nil(),y) -> y
     ++(.(x,y),z) -> .(x,++(y,z))
     if(true(),x,y) -> x
     if(false(),x,y) -> x
    KBO Processor:
     argument filtering:
      pi(nil) = []
      pi(merge) = [0,1]
      pi(.) = [0,1]
      pi(<) = 1
      pi(if) = 1
      pi(++) = [0,1]
      pi(true) = []
      pi(false) = []
      pi(++#) = 0
     weight function:
      w0 = 1
      w(false) = w(true) = w(nil) = 1
      w(++#) = w(++) = w(if) = w(<) = w(.) = w(merge) = 0
     precedence:
      true ~ ++ ~ merge ~ nil > ++# ~ false ~ if ~ < ~ .
     problem:
      DPs:
       
      TRS:
       merge(nil(),y) -> y
       merge(x,nil()) -> x
       merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
       ++(nil(),y) -> y
       ++(.(x,y),z) -> .(x,++(y,z))
       if(true(),x,y) -> x
       if(false(),x,y) -> x
     Qed