YES

We show the termination of the TRS R:

  app(app(plus(),|0|()),y) -> y
  app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
  app(inc(),xs) -> app(app(map(),app(plus(),app(s(),|0|()))),xs)
  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
p2: app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
p3: app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
p4: app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),|0|()))),xs)
p5: app#(inc(),xs) -> app#(map(),app(plus(),app(s(),|0|())))
p6: app#(inc(),xs) -> app#(plus(),app(s(),|0|()))
p7: app#(inc(),xs) -> app#(s(),|0|())
p8: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
p9: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
p10: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
p11: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)

and R consists of:

r1: app(app(plus(),|0|()),y) -> y
r2: app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
r3: app(inc(),xs) -> app(app(map(),app(plus(),app(s(),|0|()))),xs)
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))

The estimated dependency graph contains the following SCCs:

  {p4, p10, p11}
  {p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
p2: app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
p3: app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),|0|()))),xs)

and R consists of:

r1: app(app(plus(),|0|()),y) -> y
r2: app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
r3: app(inc(),xs) -> app(app(map(),app(plus(),app(s(),|0|()))),xs)
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      app#_A(x1,x2) = x2 + 12
      app_A(x1,x2) = max{x1, x2 + 4}
      map_A = 2
      cons_A = 0
      inc_A = 1
      plus_A = 0
      s_A = 0
      |0|_A = 0
    
    2. max/plus interpretations on natural numbers:
    
      app#_A(x1,x2) = 0
      app_A(x1,x2) = 0
      map_A = 0
      cons_A = 0
      inc_A = 0
      plus_A = 0
      s_A = 0
      |0|_A = 0
    

The next rules are strictly ordered:

  p1, p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),|0|()))),xs)

and R consists of:

r1: app(app(plus(),|0|()),y) -> y
r2: app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
r3: app(inc(),xs) -> app(app(map(),app(plus(),app(s(),|0|()))),xs)
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)

and R consists of:

r1: app(app(plus(),|0|()),y) -> y
r2: app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
r3: app(inc(),xs) -> app(app(map(),app(plus(),app(s(),|0|()))),xs)
r4: app(app(map(),f),nil()) -> nil()
r5: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      app#_A(x1,x2) = max{0, x1 - 2}
      app_A(x1,x2) = max{x1, x2 + 2}
      plus_A = 0
      s_A = 1
    
    2. max/plus interpretations on natural numbers:
    
      app#_A(x1,x2) = 0
      app_A(x1,x2) = 0
      plus_A = 0
      s_A = 0
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.