YES
We show the termination of the TRS R:

  app(app(app(comp(),f),g),x) -> app(f,app(g,x))
  app(twice(),f) -> app(app(comp(),f),f)

Instead we prove termination of the uncurried system:

  app(comp_0(),y) -> comp_1(y)
  app(comp_1(x1),y) -> comp_2(x1,y)
  app(comp_2(x1,x2),y) -> comp_3(x1,x2,y)
  app(twice_0(),y) -> twice_1(y)
  comp_3(f,g,x) -> app(f,app(g,x))
  twice_1(f) -> comp_2(f,f)


-- SCC decomposition.

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

p1: app#(comp_2(x1,x2),y) -> comp_3#(x1,x2,y)
p2: app#(twice_0(),y) -> twice_1#(y)
p3: comp_3#(f,g,x) -> app#(f,app(g,x))
p4: comp_3#(f,g,x) -> app#(g,x)

and R consists of:

r1: app(comp_0(),y) -> comp_1(y)
r2: app(comp_1(x1),y) -> comp_2(x1,y)
r3: app(comp_2(x1,x2),y) -> comp_3(x1,x2,y)
r4: app(twice_0(),y) -> twice_1(y)
r5: comp_3(f,g,x) -> app(f,app(g,x))
r6: twice_1(f) -> comp_2(f,f)

The estimated dependency graph contains the following SCCs:

  {p1, p3, p4}


-- Reduction pair.

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

p1: app#(comp_2(x1,x2),y) -> comp_3#(x1,x2,y)
p2: comp_3#(f,g,x) -> app#(g,x)
p3: comp_3#(f,g,x) -> app#(f,app(g,x))

and R consists of:

r1: app(comp_0(),y) -> comp_1(y)
r2: app(comp_1(x1),y) -> comp_2(x1,y)
r3: app(comp_2(x1,x2),y) -> comp_3(x1,x2,y)
r4: app(twice_0(),y) -> twice_1(y)
r5: comp_3(f,g,x) -> app(f,app(g,x))
r6: twice_1(f) -> comp_2(f,f)

The set of usable rules consists of

  r1

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{5, x1 + 1}
        comp_2_A(x1,x2) = max{6, x1 + 4, x2 + 2}
        comp_3#_A(x1,x2,x3) = max{6, x1 + 4, x2 + 2}
        app_A(x1,x2) = max{3, x1 - 7, x2 - 4}
        comp_0_A = 6
        comp_1_A(x1) = max{3, x1 - 4}
  
    base order 2:
  
      precedence:
      
        app# = comp_2 > comp_3# > app > comp_1 > comp_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(comp_2) = []
      pi(comp_3#) = []
      pi(app) = []
      pi(comp_0) = []
      pi(comp_1) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: app#(comp_2(x1,x2),y) -> comp_3#(x1,x2,y)
p2: comp_3#(f,g,x) -> app#(g,x)

and R consists of:

r1: app(comp_0(),y) -> comp_1(y)
r2: app(comp_1(x1),y) -> comp_2(x1,y)
r3: app(comp_2(x1,x2),y) -> comp_3(x1,x2,y)
r4: app(twice_0(),y) -> twice_1(y)
r5: comp_3(f,g,x) -> app(f,app(g,x))
r6: twice_1(f) -> comp_2(f,f)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: app#(comp_2(x1,x2),y) -> comp_3#(x1,x2,y)
p2: comp_3#(f,g,x) -> app#(g,x)

and R consists of:

r1: app(comp_0(),y) -> comp_1(y)
r2: app(comp_1(x1),y) -> comp_2(x1,y)
r3: app(comp_2(x1,x2),y) -> comp_3(x1,x2,y)
r4: app(twice_0(),y) -> twice_1(y)
r5: comp_3(f,g,x) -> app(f,app(g,x))
r6: twice_1(f) -> comp_2(f,f)

The set of usable rules consists of

  r1

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 - 4}
        comp_2_A(x1,x2) = max{x1 + 9, x2 + 2}
        comp_3#_A(x1,x2,x3) = max{x1 + 4, x2 - 3}
        app_A(x1,x2) = max{2, x1 - 12, x2 - 3}
        comp_0_A = 1
        comp_1_A(x1) = max{2, x1 - 3}
  
    base order 2:
  
      precedence:
      
        comp_2 > comp_3# = app > app# = comp_1 > comp_0
      
  
    partial status:
  
      pi(app#) = []
      pi(comp_2) = [1, 2]
      pi(comp_3#) = []
      pi(app) = []
      pi(comp_0) = []
      pi(comp_1) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: app#(comp_2(x1,x2),y) -> comp_3#(x1,x2,y)

and R consists of:

r1: app(comp_0(),y) -> comp_1(y)
r2: app(comp_1(x1),y) -> comp_2(x1,y)
r3: app(comp_2(x1,x2),y) -> comp_3(x1,x2,y)
r4: app(twice_0(),y) -> twice_1(y)
r5: comp_3(f,g,x) -> app(f,app(g,x))
r6: twice_1(f) -> comp_2(f,f)

The estimated dependency graph contains the following SCCs:

  (no SCCs)