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

  r12

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = 2
        fmap_1_A(x1) = 3
        fmap_2#_A(x1,x2) = 2
        cons_2_A(x1,x2) = 5
        t_f_0_A = 6
        twice_1_A(x1) = max{15, x1 + 1}
        twice_2#_A(x1,x2) = 2
        map_1_A(x1) = max{15, x1 + 14}
        map_2#_A(x1,x2) = 2
        app_A(x1,x2) = max{12, x1 - 12, x2 - 12}
        fmap_2_A(x1,x2) = max{4, x1 - 12}
        nil_0_A = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            app#_A(x1,x2) = x1
            fmap_1_A(x1) = x1 + 2
            fmap_2#_A(x1,x2) = x1
            cons_2_A(x1,x2) = x1 + 2
            t_f_0_A() = 3
            twice_1_A(x1) = x1
            twice_2#_A(x1,x2) = x1
            map_1_A(x1) = x1
            map_2#_A(x1,x2) = x1
            app_A(x1,x2) = x2
            fmap_2_A(x1,x2) = x1 + 1
            nil_0_A() = 2
            app#!_A(x1,x2) = x1
            fmap_1!_A(x1) = x1 + 1
            fmap_2#!_A(x1,x2) = x1 + 1
            cons_2!_A(x1,x2) = x1 + 4
            t_f_0!_A() = 5
            twice_1!_A(x1) = x1 + 1
            twice_2#!_A(x1,x2) = x1
            map_1!_A(x1) = x1 + 4
            map_2#!_A(x1,x2) = x1
            app!_A(x1,x2) = x1 + x2 + 3
            fmap_2!_A(x1,x2) = 1
            nil_0!_A() = 0
  
    partial status:
  
      pi(app#) = []
      pi(fmap_1) = []
      pi(fmap_2#) = []
      pi(cons_2) = []
      pi(t_f_0) = []
      pi(twice_1) = [1]
      pi(twice_2#) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(app) = []
      pi(fmap_2) = []
      pi(nil_0) = []

The following rules are removed:

  p1, p2



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

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


-- 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

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = 6
        twice_1_A(x1) = max{7, x1 - 4}
        twice_2#_A(x1,x2) = 6
        app_A(x1,x2) = max{4, x1 + 3}
        map_1_A(x1) = x1 + 8
        map_2#_A(x1,x2) = 6
        cons_2_A(x1,x2) = max{8, x2 - 5}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            app#_A(x1,x2) = 0
            twice_1_A(x1) = x1 + 3
            twice_2#_A(x1,x2) = 0
            app_A(x1,x2) = x1 + 1
            map_1_A(x1) = x1 + 3
            map_2#_A(x1,x2) = 0
            cons_2_A(x1,x2) = x1 + x2 + 5
            app#!_A(x1,x2) = x1
            twice_1!_A(x1) = 1
            twice_2#!_A(x1,x2) = x1 + 2
            app!_A(x1,x2) = 3
            map_1!_A(x1) = x1 + 4
            map_2#!_A(x1,x2) = x1 + 1
            cons_2!_A(x1,x2) = 3
  
    partial status:
  
      pi(app#) = []
      pi(twice_1) = []
      pi(twice_2#) = []
      pi(app) = []
      pi(map_1) = [1]
      pi(map_2#) = []
      pi(cons_2) = []

The following rules are removed:

  p1, p2, p3, p4, p6



-- 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:
  
      max/plus interpretation on natural numbers:
      
        map_2#_A(x1,x2) = max{5, x2 + 2}
        cons_2_A(x1,x2) = max{6, x1 + 4, x2 + 4}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        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) = x2 + 2
            map_2#!_A(x1,x2) = 4
            cons_2!_A(x1,x2) = 5
  
    partial status:
  
      pi(map_2#) = [2]
      pi(cons_2) = [1, 2]

The following rules are removed:

  p1

Then no dependency pair remains.