YES

We show the termination of the TRS R:

  dx(X) -> one()
  dx(a()) -> zero()
  dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
  dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
  dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
  dx(neg(ALPHA)) -> neg(dx(ALPHA))
  dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
  dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
  dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

-- SCC decomposition.

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

p1: dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
p2: dx#(plus(ALPHA,BETA)) -> dx#(BETA)
p3: dx#(times(ALPHA,BETA)) -> dx#(ALPHA)
p4: dx#(times(ALPHA,BETA)) -> dx#(BETA)
p5: dx#(minus(ALPHA,BETA)) -> dx#(ALPHA)
p6: dx#(minus(ALPHA,BETA)) -> dx#(BETA)
p7: dx#(neg(ALPHA)) -> dx#(ALPHA)
p8: dx#(div(ALPHA,BETA)) -> dx#(ALPHA)
p9: dx#(div(ALPHA,BETA)) -> dx#(BETA)
p10: dx#(ln(ALPHA)) -> dx#(ALPHA)
p11: dx#(exp(ALPHA,BETA)) -> dx#(ALPHA)
p12: dx#(exp(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
p2: dx#(exp(ALPHA,BETA)) -> dx#(BETA)
p3: dx#(exp(ALPHA,BETA)) -> dx#(ALPHA)
p4: dx#(ln(ALPHA)) -> dx#(ALPHA)
p5: dx#(div(ALPHA,BETA)) -> dx#(BETA)
p6: dx#(div(ALPHA,BETA)) -> dx#(ALPHA)
p7: dx#(neg(ALPHA)) -> dx#(ALPHA)
p8: dx#(minus(ALPHA,BETA)) -> dx#(BETA)
p9: dx#(minus(ALPHA,BETA)) -> dx#(ALPHA)
p10: dx#(times(ALPHA,BETA)) -> dx#(BETA)
p11: dx#(times(ALPHA,BETA)) -> dx#(ALPHA)
p12: dx#(plus(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

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:
            dx#_A(x1) = x1 + (2,1)
            plus_A(x1,x2) = x1 + x2 + (3,2)
            exp_A(x1,x2) = x1 + x2 + (3,2)
            ln_A(x1) = ((1,0),(0,0)) x1 + (1,2)
            div_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (3,2)
            neg_A(x1) = ((1,0),(0,0)) x1 + (1,2)
            minus_A(x1,x2) = x1 + x2 + (3,0)
            times_A(x1,x2) = x1 + x2 + (3,2)
    
      precedence:
      
        times > div > plus = neg > minus > dx# > exp > ln
      
      partial status:
    
        pi(dx#) = [1]
        pi(plus) = []
        pi(exp) = []
        pi(ln) = []
        pi(div) = []
        pi(neg) = []
        pi(minus) = [1, 2]
        pi(times) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            dx#_A(x1) = (0,0)
            plus_A(x1,x2) = (1,1)
            exp_A(x1,x2) = (1,1)
            ln_A(x1) = (1,1)
            div_A(x1,x2) = (1,1)
            neg_A(x1) = (1,1)
            minus_A(x1,x2) = x1 + x2 + (1,1)
            times_A(x1,x2) = (1,1)
    
      precedence:
      
        plus = div > times > dx# = exp = ln = neg = minus
      
      partial status:
    
        pi(dx#) = []
        pi(plus) = []
        pi(exp) = []
        pi(ln) = []
        pi(div) = []
        pi(neg) = []
        pi(minus) = [1, 2]
        pi(times) = []
    

The next rules are strictly ordered:

  p4, p5, p6, p7, p8, p9

We remove them from the problem.

-- SCC decomposition.

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

p1: dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
p2: dx#(exp(ALPHA,BETA)) -> dx#(BETA)
p3: dx#(exp(ALPHA,BETA)) -> dx#(ALPHA)
p4: dx#(times(ALPHA,BETA)) -> dx#(BETA)
p5: dx#(times(ALPHA,BETA)) -> dx#(ALPHA)
p6: dx#(plus(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
p2: dx#(plus(ALPHA,BETA)) -> dx#(BETA)
p3: dx#(times(ALPHA,BETA)) -> dx#(ALPHA)
p4: dx#(times(ALPHA,BETA)) -> dx#(BETA)
p5: dx#(exp(ALPHA,BETA)) -> dx#(ALPHA)
p6: dx#(exp(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

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:
            dx#_A(x1) = ((1,0),(1,1)) x1 + (1,1)
            plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,2)
            times_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,2)
            exp_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,2)
    
      precedence:
      
        plus > times > dx# = exp
      
      partial status:
    
        pi(dx#) = [1]
        pi(plus) = []
        pi(times) = [2]
        pi(exp) = [2]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            dx#_A(x1) = (0,0)
            plus_A(x1,x2) = (1,0)
            times_A(x1,x2) = ((1,0),(0,0)) x2 + (1,0)
            exp_A(x1,x2) = (1,0)
    
      precedence:
      
        times > exp > dx# = plus
      
      partial status:
    
        pi(dx#) = []
        pi(plus) = []
        pi(times) = []
        pi(exp) = []
    

The next rules are strictly ordered:

  p2, p3, p5, p6

We remove them from the problem.

-- SCC decomposition.

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

p1: dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
p2: dx#(times(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
p2: dx#(times(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

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:
            dx#_A(x1) = ((1,0),(0,0)) x1
            plus_A(x1,x2) = x1 + x2 + (1,1)
            times_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (1,1)
    
      precedence:
      
        dx# = plus = times
      
      partial status:
    
        pi(dx#) = []
        pi(plus) = [1, 2]
        pi(times) = [1]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            dx#_A(x1) = (0,0)
            plus_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (1,1)
            times_A(x1,x2) = (1,1)
    
      precedence:
      
        dx# = plus = times
      
      partial status:
    
        pi(dx#) = []
        pi(plus) = [2]
        pi(times) = []
    

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: dx#(times(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: dx#(times(ALPHA,BETA)) -> dx#(BETA)

and R consists of:

r1: dx(X) -> one()
r2: dx(a()) -> zero()
r3: dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
r4: dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
r5: dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
r6: dx(neg(ALPHA)) -> neg(dx(ALPHA))
r7: dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
r8: dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
r9: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA))))

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:
            dx#_A(x1) = x1 + (1,2)
            times_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,1)
    
      precedence:
      
        times > dx#
      
      partial status:
    
        pi(dx#) = [1]
        pi(times) = [1]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            dx#_A(x1) = (0,0)
            times_A(x1,x2) = (1,1)
    
      precedence:
      
        dx# > times
      
      partial status:
    
        pi(dx#) = []
        pi(times) = []
    

The next rules are strictly ordered:

  p1

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