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(app(times(),|0|()),y) -> |0|()
  app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
  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))
  inc() -> app(map(),app(plus(),app(s(),|0|())))
  double() -> app(map(),app(times(),app(s(),app(s(),|0|()))))

-- 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#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
p5: app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
p6: app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
p7: app#(app(times(),app(s(),x)),y) -> app#(times(),x)
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)
p12: inc#() -> app#(map(),app(plus(),app(s(),|0|())))
p13: inc#() -> app#(plus(),app(s(),|0|()))
p14: inc#() -> app#(s(),|0|())
p15: double#() -> app#(map(),app(times(),app(s(),app(s(),|0|()))))
p16: double#() -> app#(times(),app(s(),app(s(),|0|())))
p17: double#() -> app#(s(),app(s(),|0|()))
p18: double#() -> app#(s(),|0|())

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(app(times(),|0|()),y) -> |0|()
r4: app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
r5: app(app(map(),f),nil()) -> nil()
r6: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r7: inc() -> app(map(),app(plus(),app(s(),|0|())))
r8: double() -> app(map(),app(times(),app(s(),app(s(),|0|()))))

The estimated dependency graph contains the following SCCs:

  {p10, p11}
  {p6}
  {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)

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(app(times(),|0|()),y) -> |0|()
r4: app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
r5: app(app(map(),f),nil()) -> nil()
r6: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r7: inc() -> app(map(),app(plus(),app(s(),|0|())))
r8: double() -> app(map(),app(times(),app(s(),app(s(),|0|()))))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^1
    order: standard order
    interpretations:
      app#_A(x1,x2) = x1
      app_A(x1,x2) = x1 + x2
      map_A() = 1
      cons_A() = 1

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: 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(app(times(),|0|()),y) -> |0|()
r4: app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
r5: app(app(map(),f),nil()) -> nil()
r6: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r7: inc() -> app(map(),app(plus(),app(s(),|0|())))
r8: double() -> app(map(),app(times(),app(s(),app(s(),|0|()))))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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(app(times(),|0|()),y) -> |0|()
r4: app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
r5: app(app(map(),f),nil()) -> nil()
r6: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r7: inc() -> app(map(),app(plus(),app(s(),|0|())))
r8: double() -> app(map(),app(times(),app(s(),app(s(),|0|()))))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^1
    order: standard order
    interpretations:
      app#_A(x1,x2) = x2
      app_A(x1,x2) = x1 + x2 + 1
      map_A() = 0
      cons_A() = 0

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: app#(app(times(),app(s(),x)),y) -> app#(app(times(),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(app(times(),|0|()),y) -> |0|()
r4: app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
r5: app(app(map(),f),nil()) -> nil()
r6: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r7: inc() -> app(map(),app(plus(),app(s(),|0|())))
r8: double() -> app(map(),app(times(),app(s(),app(s(),|0|()))))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^1
    order: standard order
    interpretations:
      app#_A(x1,x2) = x1
      app_A(x1,x2) = x1 + x2 + 1
      times_A() = 0
      s_A() = 0

The next rules are strictly ordered:

  p1

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

-- 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(app(times(),|0|()),y) -> |0|()
r4: app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
r5: app(app(map(),f),nil()) -> nil()
r6: app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
r7: inc() -> app(map(),app(plus(),app(s(),|0|())))
r8: double() -> app(map(),app(times(),app(s(),app(s(),|0|()))))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^1
    order: standard order
    interpretations:
      app#_A(x1,x2) = x1
      app_A(x1,x2) = x1 + x2 + 1
      plus_A() = 0
      s_A() = 0

The next rules are strictly ordered:

  p1

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