YES
We show the termination of the TRS R:

  app(app(twice(),f),x) -> app(f,app(f,x))
  app(app(map(),f),nil()) -> nil()
  app(app(map(),f),app(app(cons(),h),t)) -> app(app(cons(),app(f,h)),app(app(map(),f),t))
  app(app(fmap(),nil()),x) -> nil()
  app(app(fmap(),app(app(cons(),f),t_f())),x) -> app(app(cons(),app(f,x)),app(app(fmap(),t_f()),x))

Instead we prove termination of the uncurried system:

  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(fmap_0(),y) -> fmap_1(y)
  app(fmap_1(x1),y) -> fmap_2(x1,y)
  app(map_0(),y) -> map_1(y)
  app(map_1(x1),y) -> map_2(x1,y)
  app(twice_0(),y) -> twice_1(y)
  app(twice_1(x1),y) -> twice_2(x1,y)
  twice_2(f,x) -> app(f,app(f,x))
  map_2(f,nil_0()) -> nil_0()
  map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
  fmap_2(nil_0(),x) -> nil_0()
  fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))


-- SCC decomposition.

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

p1: app#(fmap_1(x1),y) -> fmap_2#(x1,y)
p2: app#(map_1(x1),y) -> map_2#(x1,y)
p3: app#(twice_1(x1),y) -> twice_2#(x1,y)
p4: twice_2#(f,x) -> app#(f,app(f,x))
p5: twice_2#(f,x) -> app#(f,x)
p6: map_2#(f,cons_2(h,t)) -> app#(f,h)
p7: map_2#(f,cons_2(h,t)) -> map_2#(f,t)
p8: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x)
p9: fmap_2#(cons_2(f,t_f_0()),x) -> fmap_2#(t_f_0(),x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

p1: app#(fmap_1(x1),y) -> fmap_2#(x1,y)
p2: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x)
p3: app#(twice_1(x1),y) -> twice_2#(x1,y)
p4: twice_2#(f,x) -> app#(f,x)
p5: app#(map_1(x1),y) -> map_2#(x1,y)
p6: map_2#(f,cons_2(h,t)) -> map_2#(f,t)
p7: map_2#(f,cons_2(h,t)) -> app#(f,h)
p8: twice_2#(f,x) -> app#(f,app(f,x))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The set of usable rules consists of

  r1, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 2
        fmap_1_A(x1) = x1 + 21
        fmap_2#_A(x1,x2) = x1 + 20
        cons_2_A(x1,x2) = max{6, x1 - 16, x2 - 18}
        t_f_0_A = 1
        twice_1_A(x1) = max{11, x1 + 4}
        twice_2#_A(x1,x2) = max{5, x1 + 3}
        map_1_A(x1) = x1 + 5
        map_2#_A(x1,x2) = x1 + 4
        app_A(x1,x2) = max{10, x1 - 38, x2 - 11}
        cons_0_A = 1
        cons_1_A(x1) = 2
        fmap_2_A(x1,x2) = max{9, x1 - 16, x2 - 12}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        cons_1 > app = cons_0 > twice_1 = map_1 = fmap_2 > fmap_1 = cons_2 = nil_0 > fmap_2# > app# > t_f_0 = twice_2# = map_2#
      
  
    partial status:
  
      pi(app#) = [1]
      pi(fmap_1) = []
      pi(fmap_2#) = []
      pi(cons_2) = []
      pi(t_f_0) = []
      pi(twice_1) = []
      pi(twice_2#) = []
      pi(map_1) = [1]
      pi(map_2#) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(fmap_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: fmap_2#(cons_2(f,t_f_0()),x) -> app#(f,x)
p2: app#(twice_1(x1),y) -> twice_2#(x1,y)
p3: twice_2#(f,x) -> app#(f,x)
p4: app#(map_1(x1),y) -> map_2#(x1,y)
p5: map_2#(f,cons_2(h,t)) -> map_2#(f,t)
p6: map_2#(f,cons_2(h,t)) -> app#(f,h)
p7: twice_2#(f,x) -> app#(f,app(f,x))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p5, p6, p7}


-- Reduction pair.

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

p1: app#(twice_1(x1),y) -> twice_2#(x1,y)
p2: twice_2#(f,x) -> app#(f,app(f,x))
p3: app#(map_1(x1),y) -> map_2#(x1,y)
p4: map_2#(f,cons_2(h,t)) -> app#(f,h)
p5: map_2#(f,cons_2(h,t)) -> map_2#(f,t)
p6: twice_2#(f,x) -> app#(f,x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The set of usable rules consists of

  r1, r3, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{1, x1}
        twice_1_A(x1) = max{16, x1 + 3}
        twice_2#_A(x1,x2) = x1 + 2
        app_A(x1,x2) = max{15, x1 - 17, x2 - 17}
        map_1_A(x1) = max{3, x1 + 2}
        map_2#_A(x1,x2) = max{3, x1 + 2}
        cons_2_A(x1,x2) = max{5, x1 - 9, x2 - 9}
        cons_0_A = 3
        cons_1_A(x1) = max{4, x1 - 18}
        fmap_0_A = 18
        fmap_1_A(x1) = max{14, x1 - 18}
        fmap_2_A(x1,x2) = max{16, x1 - 4, x2 - 1}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        fmap_2 > app = cons_2 = nil_0 > app# = twice_1 = map_1 = cons_1 = fmap_1 > twice_2# = map_2# = cons_0 = fmap_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(twice_1) = []
      pi(twice_2#) = []
      pi(app) = []
      pi(map_1) = [1]
      pi(map_2#) = [1]
      pi(cons_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(fmap_0) = []
      pi(fmap_1) = []
      pi(fmap_2) = []
      pi(nil_0) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: app#(twice_1(x1),y) -> twice_2#(x1,y)
p2: twice_2#(f,x) -> app#(f,app(f,x))
p3: map_2#(f,cons_2(h,t)) -> app#(f,h)
p4: map_2#(f,cons_2(h,t)) -> map_2#(f,t)
p5: twice_2#(f,x) -> app#(f,x)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The estimated dependency graph contains the following SCCs:

  {p4}
  {p1, p2, p5}


-- Reduction pair.

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

p1: map_2#(f,cons_2(h,t)) -> map_2#(f,t)

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The set of usable rules consists of

  r1, r3, r5, r7, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{x1 + 5, x2 + 12}
        cons_2_A(x1,x2) = max{3, x1 - 7, x2 + 1}
        app_A(x1,x2) = max{5, x1 - 14, x2 - 6}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 7}
        fmap_0_A = 3
        fmap_1_A(x1) = max{4, x1 - 7}
        map_0_A = 2
        map_1_A(x1) = max{3, x1 - 7}
        twice_0_A = 3
        twice_1_A(x1) = max{4, x1 - 7}
        fmap_2_A(x1,x2) = max{6, x1 - 6, x2 - 3}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        fmap_2 > map_2# = app > cons_2 = fmap_1 = twice_1 > cons_1 = fmap_0 = map_1 = twice_0 = nil_0 > cons_0 = map_0
      
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(fmap_0) = []
      pi(fmap_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(twice_0) = []
      pi(twice_1) = []
      pi(fmap_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: app#(twice_1(x1),y) -> twice_2#(x1,y)
p2: twice_2#(f,x) -> app#(f,x)
p3: twice_2#(f,x) -> app#(f,app(f,x))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The set of usable rules consists of

  r1, r3, r5, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 2
        twice_1_A(x1) = max{2, x1}
        twice_2#_A(x1,x2) = max{4, x1 + 2}
        app_A(x1,x2) = 3
        cons_0_A = 2
        cons_1_A(x1) = 1
        fmap_0_A = 2
        fmap_1_A(x1) = 1
        map_0_A = 3
        map_1_A(x1) = 2
        fmap_2_A(x1,x2) = max{8, x1 - 10, x2 - 3}
        nil_0_A = 0
  
    base order 2:
  
      precedence:
      
        fmap_2 > app# = twice_2# > twice_1 > app = cons_0 = fmap_0 = map_0 = nil_0 > cons_1 = fmap_1 = map_1
      
  
    partial status:
  
      pi(app#) = [1]
      pi(twice_1) = [1]
      pi(twice_2#) = [1]
      pi(app) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(fmap_0) = []
      pi(fmap_1) = []
      pi(map_0) = []
      pi(map_1) = []
      pi(fmap_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: twice_2#(f,x) -> app#(f,x)
p2: twice_2#(f,x) -> app#(f,app(f,x))

and R consists of:

r1: app(cons_0(),y) -> cons_1(y)
r2: app(cons_1(x1),y) -> cons_2(x1,y)
r3: app(fmap_0(),y) -> fmap_1(y)
r4: app(fmap_1(x1),y) -> fmap_2(x1,y)
r5: app(map_0(),y) -> map_1(y)
r6: app(map_1(x1),y) -> map_2(x1,y)
r7: app(twice_0(),y) -> twice_1(y)
r8: app(twice_1(x1),y) -> twice_2(x1,y)
r9: twice_2(f,x) -> app(f,app(f,x))
r10: map_2(f,nil_0()) -> nil_0()
r11: map_2(f,cons_2(h,t)) -> cons_2(app(f,h),map_2(f,t))
r12: fmap_2(nil_0(),x) -> nil_0()
r13: fmap_2(cons_2(f,t_f_0()),x) -> cons_2(app(f,x),fmap_2(t_f_0(),x))

The estimated dependency graph contains the following SCCs:

  (no SCCs)