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

  r3

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1 + 2
          fmap_1_A(x1) = x1 + 2
          fmap_2#_A(x1,x2) = x1
          cons_2_A(x1,x2) = x1 + x2 + 7
          t_f_0_A() = 1
          twice_1_A(x1) = x1 + 7
          twice_2#_A(x1,x2) = x1 + 3
          map_1_A(x1) = x1 + 4
          map_2#_A(x1,x2) = x1 + 3
          app_A(x1,x2) = x2 + 5
          fmap_0_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          app#_A(x1,x2) = max{2, x1 - 9}
          fmap_1_A(x1) = x1 + 3
          fmap_2#_A(x1,x2) = max{2, x1 - 6}
          cons_2_A(x1,x2) = max{4, x1 - 3, x2 - 3}
          t_f_0_A = 3
          twice_1_A(x1) = max{8, x1 + 4}
          twice_2#_A(x1,x2) = max{2, x1 - 9}
          map_1_A(x1) = x1 + 10
          map_2#_A(x1,x2) = max{2, x1 + 1}
          app_A(x1,x2) = max{x1 - 13, x2 + 3}
          fmap_0_A = 10
          app#!_A(x1,x2) = max{5, x1 - 12}
          fmap_1!_A(x1) = 2
          fmap_2#!_A(x1,x2) = max{6, x1 - 8, x2 - 4}
          cons_2!_A(x1,x2) = max{5, x1 - 12, x2 - 4}
          t_f_0!_A = 4
          twice_1!_A(x1) = max{2, x1 - 28}
          twice_2#!_A(x1,x2) = max{3, x1 - 13}
          map_1!_A(x1) = max{6, x1 - 27}
          map_2#!_A(x1,x2) = max{1, x1 - 13}
          app!_A(x1,x2) = max{6, x1 - 11, x2 - 13}
          fmap_0!_A = 1
  
    partial status:
  
      pi(app#) = [1]
      pi(fmap_1) = [1]
      pi(fmap_2#) = []
      pi(cons_2) = [1, 2]
      pi(t_f_0) = []
      pi(twice_1) = [1]
      pi(twice_2#) = []
      pi(map_1) = []
      pi(map_2#) = [1]
      pi(app) = []
      pi(fmap_0) = []

The following rules are removed:

  p1, p2, p3, p4, p5, p7, p8



-- SCC decomposition.

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 estimated dependency graph contains the following SCCs:

  {p1}


-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          map_2#_A(x1,x2) = x2
          cons_2_A(x1,x2) = x1 + x2 + 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          map_2#_A(x1,x2) = 0
          cons_2_A(x1,x2) = max{5, x1 - 3, x2 - 4}
          map_2#!_A(x1,x2) = max{0, x2 - 14}
          cons_2!_A(x1,x2) = max{7, x2 - 14}
  
    partial status:
  
      pi(map_2#) = []
      pi(cons_2) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.