YES

We show the termination of the TRS R:

  f(X) -> if(X,c(),n__f(n__true()))
  if(true(),X,Y) -> X
  if(false(),X,Y) -> activate(Y)
  f(X) -> n__f(X)
  true() -> n__true()
  activate(n__f(X)) -> f(activate(X))
  activate(n__true()) -> true()
  activate(X) -> X

-- SCC decomposition.

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

p1: f#(X) -> if#(X,c(),n__f(n__true()))
p2: if#(false(),X,Y) -> activate#(Y)
p3: activate#(n__f(X)) -> f#(activate(X))
p4: activate#(n__f(X)) -> activate#(X)
p5: activate#(n__true()) -> true#()

and R consists of:

r1: f(X) -> if(X,c(),n__f(n__true()))
r2: if(true(),X,Y) -> X
r3: if(false(),X,Y) -> activate(Y)
r4: f(X) -> n__f(X)
r5: true() -> n__true()
r6: activate(n__f(X)) -> f(activate(X))
r7: activate(n__true()) -> true()
r8: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: f#(X) -> if#(X,c(),n__f(n__true()))
p2: if#(false(),X,Y) -> activate#(Y)
p3: activate#(n__f(X)) -> activate#(X)
p4: activate#(n__f(X)) -> f#(activate(X))

and R consists of:

r1: f(X) -> if(X,c(),n__f(n__true()))
r2: if(true(),X,Y) -> X
r3: if(false(),X,Y) -> activate(Y)
r4: f(X) -> n__f(X)
r5: true() -> n__true()
r6: activate(n__f(X)) -> f(activate(X))
r7: activate(n__true()) -> true()
r8: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1) = ((1,0),(0,0)) x1 + (9,4)
            if#_A(x1,x2,x3) = x1 + x3 + (1,2)
            c_A() = (0,1)
            n__f_A(x1) = x1 + (7,0)
            n__true_A() = (0,1)
            false_A() = (4,4)
            activate#_A(x1) = x1 + (5,5)
            activate_A(x1) = ((1,0),(1,0)) x1 + (2,3)
            if_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + ((1,0),(0,0)) x3 + (0,4)
            true_A() = (1,2)
            f_A(x1) = x1 + (7,6)
    
      precedence:
      
        activate# > f# > if# > false = if > activate > n__true = f > n__f > c = true
      
      partial status:
    
        pi(f#) = []
        pi(if#) = []
        pi(c) = []
        pi(n__f) = [1]
        pi(n__true) = []
        pi(false) = []
        pi(activate#) = [1]
        pi(activate) = []
        pi(if) = [2]
        pi(true) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            f#_A(x1) = (10,6)
            if#_A(x1,x2,x3) = (9,1)
            c_A() = (1,1)
            n__f_A(x1) = (5,5)
            n__true_A() = (1,2)
            false_A() = (9,3)
            activate#_A(x1) = ((0,0),(1,0)) x1 + (8,0)
            activate_A(x1) = (7,4)
            if_A(x1,x2,x3) = (0,0)
            true_A() = (2,1)
            f_A(x1) = (6,3)
    
      precedence:
      
        if > f# = c > if# = activate > activate# = true > f > false > n__f = n__true
      
      partial status:
    
        pi(f#) = []
        pi(if#) = []
        pi(c) = []
        pi(n__f) = []
        pi(n__true) = []
        pi(false) = []
        pi(activate#) = []
        pi(activate) = []
        pi(if) = []
        pi(true) = []
        pi(f) = []
    

The next rules are strictly ordered:

  p3, p4

We remove them from the problem.

-- SCC decomposition.

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

p1: f#(X) -> if#(X,c(),n__f(n__true()))
p2: if#(false(),X,Y) -> activate#(Y)

and R consists of:

r1: f(X) -> if(X,c(),n__f(n__true()))
r2: if(true(),X,Y) -> X
r3: if(false(),X,Y) -> activate(Y)
r4: f(X) -> n__f(X)
r5: true() -> n__true()
r6: activate(n__f(X)) -> f(activate(X))
r7: activate(n__true()) -> true()
r8: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)