YES
We show the termination of the TRS R:

  app(app(fmap(),fnil()),x) -> nil()
  app(app(fmap(),app(app(fcons(),f),t)),x) -> app(app(cons(),app(f,x)),app(app(fmap(),t),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(fcons_0(),y) -> fcons_1(y)
  app(fcons_1(x1),y) -> fcons_2(x1,y)
  app(fmap_0(),y) -> fmap_1(y)
  app(fmap_1(x1),y) -> fmap_2(x1,y)
  fmap_2(fnil_0(),x) -> nil_0()
  fmap_2(fcons_2(f,t),x) -> cons_2(app(f,x),fmap_2(t,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: fmap_2#(fcons_2(f,t),x) -> app#(f,x)
p3: fmap_2#(fcons_2(f,t),x) -> fmap_2#(t,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(fcons_0(),y) -> fcons_1(y)
r4: app(fcons_1(x1),y) -> fcons_2(x1,y)
r5: app(fmap_0(),y) -> fmap_1(y)
r6: app(fmap_1(x1),y) -> fmap_2(x1,y)
r7: fmap_2(fnil_0(),x) -> nil_0()
r8: fmap_2(fcons_2(f,t),x) -> cons_2(app(f,x),fmap_2(t,x))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- 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#(fcons_2(f,t),x) -> fmap_2#(t,x)
p3: fmap_2#(fcons_2(f,t),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(fcons_0(),y) -> fcons_1(y)
r4: app(fcons_1(x1),y) -> fcons_2(x1,y)
r5: app(fmap_0(),y) -> fmap_1(y)
r6: app(fmap_1(x1),y) -> fmap_2(x1,y)
r7: fmap_2(fnil_0(),x) -> nil_0()
r8: fmap_2(fcons_2(f,t),x) -> cons_2(app(f,x),fmap_2(t,x))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. 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 + 3
              fmap_2#_A(x1,x2) = x1
              fcons_2_A(x1,x2) = x1 + x2 + 5
              app#!_A(x1,x2) = 4
              fmap_1!_A(x1) = 3
              fmap_2#!_A(x1,x2) = 4
              fcons_2!_A(x1,x2) = 5
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            app#_A(x1,x2) = max{0, x1 - 15, x2 - 1}
            fmap_1_A(x1) = x1 + 1
            fmap_2#_A(x1,x2) = max{0, x1 - 14, x2 - 1}
            fcons_2_A(x1,x2) = max{x1 + 11, x2 + 13}
            app#!_A(x1,x2) = max{10, x1 - 2}
            fmap_1!_A(x1) = max{5, x1 + 3}
            fmap_2#!_A(x1,x2) = max{6, x1 - 2}
            fcons_2!_A(x1,x2) = max{4, x2 + 2}
        
  
    partial status:
  
      pi(app#) = [2]
      pi(fmap_1) = []
      pi(fmap_2#) = [2]
      pi(fcons_2) = [1, 2]

The following rules are removed:

  p1, p2, p3

Then no dependency pair remains.