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
  Usable Rule 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:
    f11(x,y) -> x
    f11(x,y) -> y
    merge(x,nil()) -> x
    merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
    merge(nil(),y) -> y
   EDG 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:
     f11(x,y) -> x
     f11(x,y) -> y
     merge(x,nil()) -> x
     merge(.(x,y),.(u,v)) -> if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
     merge(nil(),y) -> y
    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)) -> 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)) -> 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#(.(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)) -> if#(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
    Restore Modifier:
     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
     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:
       dimension: 1
       interpretation:
        [merge#](x0, x1) = x0,
        
        [false] = 0,
        
        [true] = 0,
        
        [++](x0, x1) = x0 + x1 + 1,
        
        [if](x0, x1, x2) = x1,
        
        [<](x0, x1) = 0,
        
        [.](x0, x1) = x1 + 1,
        
        [merge](x0, x1) = x0 + x1,
        
        [nil] = 0
       orientation:
        merge#(.(x,y),.(u,v)) = y + 1 >= y + 1 = merge#(.(x,y),v)
        
        merge#(.(x,y),.(u,v)) = y + 1 >= y = merge#(y,.(u,v))
        
        merge(nil(),y) = y >= y = y
        
        merge(x,nil()) = x >= x = x
        
        merge(.(x,y),.(u,v)) = v + y + 2 >= v + y + 2 = if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
        
        ++(nil(),y) = y + 1 >= y = y
        
        ++(.(x,y),z) = y + z + 2 >= y + z + 2 = .(x,++(y,z))
        
        if(true(),x,y) = x >= x = x
        
        if(false(),x,y) = x >= x = x
       problem:
        DPs:
         merge#(.(x,y),.(u,v)) -> merge#(.(x,y),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:
        dimension: 1
        interpretation:
         [merge#](x0, x1) = x1 + 1,
         
         [false] = 0,
         
         [true] = 0,
         
         [++](x0, x1) = x0 + x1 + 1,
         
         [if](x0, x1, x2) = x1,
         
         [<](x0, x1) = 0,
         
         [.](x0, x1) = x1 + 1,
         
         [merge](x0, x1) = x0 + x1 + 1,
         
         [nil] = 0
        orientation:
         merge#(.(x,y),.(u,v)) = v + 2 >= v + 1 = merge#(.(x,y),v)
         
         merge(nil(),y) = y + 1 >= y = y
         
         merge(x,nil()) = x + 1 >= x = x
         
         merge(.(x,y),.(u,v)) = v + y + 3 >= v + y + 3 = if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
         
         ++(nil(),y) = y + 1 >= y = y
         
         ++(.(x,y),z) = y + z + 2 >= y + z + 2 = .(x,++(y,z))
         
         if(true(),x,y) = x >= x = x
         
         if(false(),x,y) = x >= 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
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [++#](x0, x1) = x0,
        
        [false] = 0,
        
        [true] = 0,
        
        [++](x0, x1) = x0 + x1,
        
        [if](x0, x1, x2) = x1,
        
        [<](x0, x1) = 0,
        
        [.](x0, x1) = x1 + 1,
        
        [merge](x0, x1) = x0 + x1 + 1,
        
        [nil] = 0
       orientation:
        ++#(.(x,y),z) = y + 1 >= y = ++#(y,z)
        
        merge(nil(),y) = y + 1 >= y = y
        
        merge(x,nil()) = x + 1 >= x = x
        
        merge(.(x,y),.(u,v)) = v + y + 3 >= v + y + 3 = if(<(x,u),.(x,merge(y,.(u,v))),.(u,merge(.(x,y),v)))
        
        ++(nil(),y) = y >= y = y
        
        ++(.(x,y),z) = y + z + 1 >= y + z + 1 = .(x,++(y,z))
        
        if(true(),x,y) = x >= x = x
        
        if(false(),x,y) = x >= 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