YES

We show the termination of the TRS R:

  a__f(X) -> a__if(mark(X),c(),f(true()))
  a__if(true(),X,Y) -> mark(X)
  a__if(false(),X,Y) -> mark(Y)
  mark(f(X)) -> a__f(mark(X))
  mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3)
  mark(c()) -> c()
  mark(true()) -> true()
  mark(false()) -> false()
  a__f(X) -> f(X)
  a__if(X1,X2,X3) -> if(X1,X2,X3)

-- SCC decomposition.

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

p1: a__f#(X) -> a__if#(mark(X),c(),f(true()))
p2: a__f#(X) -> mark#(X)
p3: a__if#(true(),X,Y) -> mark#(X)
p4: a__if#(false(),X,Y) -> mark#(Y)
p5: mark#(f(X)) -> a__f#(mark(X))
p6: mark#(f(X)) -> mark#(X)
p7: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),mark(X2),X3)
p8: mark#(if(X1,X2,X3)) -> mark#(X1)
p9: mark#(if(X1,X2,X3)) -> mark#(X2)

and R consists of:

r1: a__f(X) -> a__if(mark(X),c(),f(true()))
r2: a__if(true(),X,Y) -> mark(X)
r3: a__if(false(),X,Y) -> mark(Y)
r4: mark(f(X)) -> a__f(mark(X))
r5: mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3)
r6: mark(c()) -> c()
r7: mark(true()) -> true()
r8: mark(false()) -> false()
r9: a__f(X) -> f(X)
r10: a__if(X1,X2,X3) -> if(X1,X2,X3)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

p1: a__f#(X) -> a__if#(mark(X),c(),f(true()))
p2: a__if#(false(),X,Y) -> mark#(Y)
p3: mark#(if(X1,X2,X3)) -> mark#(X2)
p4: mark#(if(X1,X2,X3)) -> mark#(X1)
p5: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),mark(X2),X3)
p6: a__if#(true(),X,Y) -> mark#(X)
p7: mark#(f(X)) -> mark#(X)
p8: mark#(f(X)) -> a__f#(mark(X))
p9: a__f#(X) -> mark#(X)

and R consists of:

r1: a__f(X) -> a__if(mark(X),c(),f(true()))
r2: a__if(true(),X,Y) -> mark(X)
r3: a__if(false(),X,Y) -> mark(Y)
r4: mark(f(X)) -> a__f(mark(X))
r5: mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3)
r6: mark(c()) -> c()
r7: mark(true()) -> true()
r8: mark(false()) -> false()
r9: a__f(X) -> f(X)
r10: a__if(X1,X2,X3) -> if(X1,X2,X3)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__f#_A(x1) = ((1,0),(1,1)) x1 + (7,11)
            a__if#_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + x3 + (1,2)
            mark_A(x1) = ((1,0),(1,1)) x1
            c_A() = (0,1)
            f_A(x1) = ((1,0),(1,1)) x1 + (6,1)
            true_A() = (0,3)
            false_A() = (2,1)
            mark#_A(x1) = ((1,0),(1,1)) x1 + (1,4)
            if_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + x3
            a__f_A(x1) = ((1,0),(1,1)) x1 + (6,6)
            a__if_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + x3
    
      precedence:
      
        mark > c = a__f > false = mark# > a__if# = a__if > if > a__f# = true > f
      
      partial status:
    
        pi(a__f#) = [1]
        pi(a__if#) = []
        pi(mark) = []
        pi(c) = []
        pi(f) = [1]
        pi(true) = []
        pi(false) = []
        pi(mark#) = []
        pi(if) = []
        pi(a__f) = [1]
        pi(a__if) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__f#_A(x1) = (5,6)
            a__if#_A(x1,x2,x3) = (4,5)
            mark_A(x1) = (2,2)
            c_A() = (1,1)
            f_A(x1) = (3,7)
            true_A() = (2,2)
            false_A() = (1,1)
            mark#_A(x1) = (3,4)
            if_A(x1,x2,x3) = (1,3)
            a__f_A(x1) = (4,3)
            a__if_A(x1,x2,x3) = (2,2)
    
      precedence:
      
        a__f# = mark = true = false = mark# = if = a__if > a__f > a__if# = c = f
      
      partial status:
    
        pi(a__f#) = []
        pi(a__if#) = []
        pi(mark) = []
        pi(c) = []
        pi(f) = []
        pi(true) = []
        pi(false) = []
        pi(mark#) = []
        pi(if) = []
        pi(a__f) = []
        pi(a__if) = []
    

The next rules are strictly ordered:

  p1, p5, p8

We remove them from the problem.

-- SCC decomposition.

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

p1: a__if#(false(),X,Y) -> mark#(Y)
p2: mark#(if(X1,X2,X3)) -> mark#(X2)
p3: mark#(if(X1,X2,X3)) -> mark#(X1)
p4: a__if#(true(),X,Y) -> mark#(X)
p5: mark#(f(X)) -> mark#(X)
p6: a__f#(X) -> mark#(X)

and R consists of:

r1: a__f(X) -> a__if(mark(X),c(),f(true()))
r2: a__if(true(),X,Y) -> mark(X)
r3: a__if(false(),X,Y) -> mark(Y)
r4: mark(f(X)) -> a__f(mark(X))
r5: mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3)
r6: mark(c()) -> c()
r7: mark(true()) -> true()
r8: mark(false()) -> false()
r9: a__f(X) -> f(X)
r10: a__if(X1,X2,X3) -> if(X1,X2,X3)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p5}


-- Reduction pair.

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

p1: mark#(if(X1,X2,X3)) -> mark#(X2)
p2: mark#(f(X)) -> mark#(X)
p3: mark#(if(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__f(X) -> a__if(mark(X),c(),f(true()))
r2: a__if(true(),X,Y) -> mark(X)
r3: a__if(false(),X,Y) -> mark(Y)
r4: mark(f(X)) -> a__f(mark(X))
r5: mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3)
r6: mark(c()) -> c()
r7: mark(true()) -> true()
r8: mark(false()) -> false()
r9: a__f(X) -> f(X)
r10: a__if(X1,X2,X3) -> if(X1,X2,X3)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
            if_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + x3 + (2,3)
            f_A(x1) = x1 + (2,1)
    
      precedence:
      
        mark# = if = f
      
      partial status:
    
        pi(mark#) = []
        pi(if) = []
        pi(f) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            mark#_A(x1) = x1
            if_A(x1,x2,x3) = (1,1)
            f_A(x1) = (1,1)
    
      precedence:
      
        mark# > if > f
      
      partial status:
    
        pi(mark#) = [1]
        pi(if) = []
        pi(f) = []
    

The next rules are strictly ordered:

  p1, p2, p3

We remove them from the problem.  Then no dependency pair remains.