YES
We show the termination of the TRS R:

  app(app(F(),app(app(F(),f),x)),x) -> app(app(F(),app(G(),app(app(F(),f),x))),app(f,x))

Instead we prove termination of the uncurried system:

  app(F_0(),y) -> F_1(y)
  app(F_1(x1),y) -> F_2(x1,y)
  app(G_0(),y) -> G_1(y)
  F_2(F_2(f,x),x) -> F_2(G_1(F_2(f,x)),app(f,x))


-- SCC decomposition.

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

p1: app#(F_1(x1),y) -> F_2#(x1,y)
p2: F_2#(F_2(f,x),x) -> F_2#(G_1(F_2(f,x)),app(f,x))
p3: F_2#(F_2(f,x),x) -> app#(f,x)

and R consists of:

r1: app(F_0(),y) -> F_1(y)
r2: app(F_1(x1),y) -> F_2(x1,y)
r3: app(G_0(),y) -> G_1(y)
r4: F_2(F_2(f,x),x) -> F_2(G_1(F_2(f,x)),app(f,x))

The estimated dependency graph contains the following SCCs:

  {p1, p3}


-- Reduction pair.

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

p1: app#(F_1(x1),y) -> F_2#(x1,y)
p2: F_2#(F_2(f,x),x) -> app#(f,x)

and R consists of:

r1: app(F_0(),y) -> F_1(y)
r2: app(F_1(x1),y) -> F_2(x1,y)
r3: app(G_0(),y) -> G_1(y)
r4: F_2(F_2(f,x),x) -> F_2(G_1(F_2(f,x)),app(f,x))

The set of usable rules consists of

  r1, r3

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          app#_A(x1,x2) = x1
          F_1_A(x1) = x1 + 2
          F_2#_A(x1,x2) = x1
          F_2_A(x1,x2) = x1 + x2 + 4
          app_A(x1,x2) = x2 + 3
          F_0_A() = 1
          G_0_A() = 1
          G_1_A(x1) = x1
  
    base order 2:
  
      precedence:
      
        F_2 > F_2# = app > app# = F_1 = G_1 > F_0 = G_0
      
  
    partial status:
  
      pi(app#) = []
      pi(F_1) = [1]
      pi(F_2#) = [1]
      pi(F_2) = [2]
      pi(app) = []
      pi(F_0) = []
      pi(G_0) = []
      pi(G_1) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: F_2#(F_2(f,x),x) -> app#(f,x)

and R consists of:

r1: app(F_0(),y) -> F_1(y)
r2: app(F_1(x1),y) -> F_2(x1,y)
r3: app(G_0(),y) -> G_1(y)
r4: F_2(F_2(f,x),x) -> F_2(G_1(F_2(f,x)),app(f,x))

The estimated dependency graph contains the following SCCs:

  (no SCCs)