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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__f#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          a__if#_A(x1,x2,x3) = x1 + x2 + x3 + (1,2)
          mark_A(x1) = x1
          c_A() = (0,0)
          f_A(x1) = ((1,0),(1,1)) x1
          true_A() = (0,0)
          false_A() = (0,1)
          mark#_A(x1) = x1 + (1,2)
          if_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3
          a__f_A(x1) = ((1,0),(1,1)) x1
          a__if_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3
  
    precedence:
    
      a__f# = a__if# = mark# > mark = c = f = true = false = if = a__f = a__if
    
    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:

  p2

We remove them from the problem.

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

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


-- 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#(true(),X,Y) -> mark#(X)
p3: mark#(f(X)) -> a__f#(mark(X))
p4: a__f#(X) -> mark#(X)
p5: mark#(f(X)) -> mark#(X)
p6: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),mark(X2),X3)
p7: mark#(if(X1,X2,X3)) -> mark#(X1)
p8: 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 set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__f#_A(x1) = x1 + (1,8)
          a__if#_A(x1,x2,x3) = x1 + x2 + ((0,0),(1,0)) x3 + (0,2)
          mark_A(x1) = x1
          c_A() = (0,1)
          f_A(x1) = x1 + (2,4)
          true_A() = (0,2)
          mark#_A(x1) = x1 + (0,3)
          if_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + ((1,0),(1,0)) x3
          a__f_A(x1) = x1 + (2,4)
          a__if_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + ((1,0),(1,0)) x3
          false_A() = (1,1)
  
    precedence:
    
      mark = if = a__f = a__if > c = true = false > f > a__f# = a__if# > mark#
    
    partial status:
  
      pi(a__f#) = [1]
      pi(a__if#) = []
      pi(mark) = []
      pi(c) = []
      pi(f) = [1]
      pi(true) = []
      pi(mark#) = [1]
      pi(if) = []
      pi(a__f) = []
      pi(a__if) = []
      pi(false) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- 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: mark#(f(X)) -> a__f#(mark(X))
p3: a__f#(X) -> mark#(X)
p4: mark#(f(X)) -> mark#(X)
p5: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),mark(X2),X3)
p6: mark#(if(X1,X2,X3)) -> mark#(X1)
p7: 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:

  {p2, p3, p4, p6, p7}


-- Reduction pair.

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

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

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

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: a__f#(X) -> mark#(X)
p2: mark#(if(X1,X2,X3)) -> mark#(X2)
p3: mark#(if(X1,X2,X3)) -> mark#(X1)
p4: mark#(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, p4}


-- 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:

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

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(if(X1,X2,X3)) -> mark#(X2)
p2: 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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: mark#(if(X1,X2,X3)) -> mark#(X2)
p2: 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 monotone reduction pair:

  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) = x1 + x2 + x3 + (2,1)
  
    precedence:
    
      mark# > if
    
    partial status:
  
      pi(mark#) = [1]
      pi(if) = [1, 2, 3]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: 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}


-- Reduction pair.

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

p1: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  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),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
  
    precedence:
    
      mark# > if
    
    partial status:
  
      pi(mark#) = [1]
      pi(if) = [2, 3]

The next rules are strictly ordered:

  p1

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