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(sumwith(),f),nil()) -> nil()
  app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),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#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
p5: app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(plus(),app(f,x))
p6: app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x)
p7: app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),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(sumwith(),f),nil()) -> nil()
r4: app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))

The estimated dependency graph contains the following SCCs:

  {p6, p7}
  {p2}


-- Reduction pair.

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

p1: app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x)
p2: app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),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(sumwith(),f),nil()) -> nil()
r4: app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    app#_A(x1,x2) = max{1, x1, x2}
    app_A(x1,x2) = max{x1, x2 + 2}
    sumwith_A = 0
    cons_A = 2

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(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),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(sumwith(),f),nil()) -> nil()
r4: app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))

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(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),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(sumwith(),f),nil()) -> nil()
r4: app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    app#_A(x1,x2) = x2 + 4
    app_A(x1,x2) = max{x1 - 2, x2 + 1}
    sumwith_A = 0
    cons_A = 2

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(sumwith(),f),nil()) -> nil()
r4: app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    app#_A(x1,x2) = max{1, x1}
    app_A(x1,x2) = max{x1, x2 + 3}
    plus_A = 2
    s_A = 0

The next rules are strictly ordered:

  p1

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