YES
We show the termination of the TRS R:

  D(t()) -> |1|()
  D(constant()) -> |0|()
  D(+(x,y)) -> +(D(x),D(y))
  D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
  D(-(x,y)) -> -(D(x),D(y))
  D(minus(x)) -> minus(D(x))
  D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
  D(ln(x)) -> div(D(x),x)
  D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))


-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(+(x,y)) -> D#(y)
p3: D#(*(x,y)) -> D#(x)
p4: D#(*(x,y)) -> D#(y)
p5: D#(-(x,y)) -> D#(x)
p6: D#(-(x,y)) -> D#(y)
p7: D#(minus(x)) -> D#(x)
p8: D#(div(x,y)) -> D#(x)
p9: D#(div(x,y)) -> D#(y)
p10: D#(ln(x)) -> D#(x)
p11: D#(pow(x,y)) -> D#(x)
p12: D#(pow(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

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: D#(+(x,y)) -> D#(x)
p2: D#(pow(x,y)) -> D#(y)
p3: D#(pow(x,y)) -> D#(x)
p4: D#(ln(x)) -> D#(x)
p5: D#(div(x,y)) -> D#(y)
p6: D#(div(x,y)) -> D#(x)
p7: D#(minus(x)) -> D#(x)
p8: D#(-(x,y)) -> D#(y)
p9: D#(-(x,y)) -> D#(x)
p10: D#(*(x,y)) -> D#(y)
p11: D#(*(x,y)) -> D#(x)
p12: D#(+(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = max{6, x1}
        +_A(x1,x2) = max{165, x1 + 130, x2 + 130}
        pow_A(x1,x2) = max{x1 + 137, x2 + 158}
        ln_A(x1) = max{6, x1 + 4}
        div_A(x1,x2) = max{x1 + 17, x2 + 173}
        minus_A(x1) = x1 + 6
        -_A(x1,x2) = max{174, x1 + 165, x2 + 11}
        *_A(x1,x2) = max{x1 + 59, x2 + 138}
        D_A(x1) = max{135, x1 + 6}
        t_A = 143
        |1|_A = 142
        constant_A = 137
        |0|_A = 136
  
    base order 2:
  
      precedence:
      
        + > - = * > pow = t > div = |1| > ln = D = constant > minus = |0| > D#
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = []
      pi(pow) = [1, 2]
      pi(ln) = [1]
      pi(div) = []
      pi(minus) = [1]
      pi(-) = []
      pi(*) = []
      pi(D) = []
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p12



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(pow(x,y)) -> D#(y)
p3: D#(pow(x,y)) -> D#(x)
p4: D#(ln(x)) -> D#(x)
p5: D#(div(x,y)) -> D#(y)
p6: D#(div(x,y)) -> D#(x)
p7: D#(minus(x)) -> D#(x)
p8: D#(-(x,y)) -> D#(y)
p9: D#(-(x,y)) -> D#(x)
p10: D#(*(x,y)) -> D#(y)
p11: D#(*(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(*(x,y)) -> D#(x)
p3: D#(*(x,y)) -> D#(y)
p4: D#(-(x,y)) -> D#(x)
p5: D#(-(x,y)) -> D#(y)
p6: D#(minus(x)) -> D#(x)
p7: D#(div(x,y)) -> D#(x)
p8: D#(div(x,y)) -> D#(y)
p9: D#(ln(x)) -> D#(x)
p10: D#(pow(x,y)) -> D#(x)
p11: D#(pow(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = max{7, x1}
        +_A(x1,x2) = max{133, x1 + 65, x2 + 67}
        *_A(x1,x2) = max{62, x1 + 8, x2 + 10}
        -_A(x1,x2) = max{x1 + 127, x2 + 60}
        minus_A(x1) = max{66, x1 + 8}
        div_A(x1,x2) = max{7, x1 + 5, x2 + 1}
        ln_A(x1) = x1 + 66
        pow_A(x1,x2) = max{x1 + 63, x2 + 27}
        D_A(x1) = max{66, x1}
        t_A = 7
        |1|_A = 6
        constant_A = 1
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        * > div = ln = pow > D# > minus > + = - = D = t = constant > |1| = |0|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = []
      pi(*) = [1]
      pi(-) = [1]
      pi(minus) = []
      pi(div) = [1]
      pi(ln) = [1]
      pi(pow) = [1, 2]
      pi(D) = [1]
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p11



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(*(x,y)) -> D#(x)
p3: D#(*(x,y)) -> D#(y)
p4: D#(-(x,y)) -> D#(x)
p5: D#(-(x,y)) -> D#(y)
p6: D#(minus(x)) -> D#(x)
p7: D#(div(x,y)) -> D#(x)
p8: D#(div(x,y)) -> D#(y)
p9: D#(ln(x)) -> D#(x)
p10: D#(pow(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(pow(x,y)) -> D#(x)
p3: D#(ln(x)) -> D#(x)
p4: D#(div(x,y)) -> D#(y)
p5: D#(div(x,y)) -> D#(x)
p6: D#(minus(x)) -> D#(x)
p7: D#(-(x,y)) -> D#(y)
p8: D#(-(x,y)) -> D#(x)
p9: D#(*(x,y)) -> D#(y)
p10: D#(*(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = max{2, x1}
        +_A(x1,x2) = max{x1 + 177, x2 + 18}
        pow_A(x1,x2) = max{x1 + 162, x2 + 88}
        ln_A(x1) = x1 + 90
        div_A(x1,x2) = max{65, x1 + 4, x2 + 46}
        minus_A(x1) = max{2, x1 + 1}
        -_A(x1,x2) = max{180, x1 + 105, x2 + 38}
        *_A(x1,x2) = max{x1 + 9, x2 + 89}
        D_A(x1) = max{102, x1 + 17}
        t_A = 143
        |1|_A = 142
        constant_A = 104
        |0|_A = 103
  
    base order 2:
  
      precedence:
      
        + > - = * > div > pow > ln = D = t = constant > minus = |1| = |0| > D#
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = []
      pi(pow) = []
      pi(ln) = [1]
      pi(div) = []
      pi(minus) = [1]
      pi(-) = [2]
      pi(*) = [2]
      pi(D) = []
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p10



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(pow(x,y)) -> D#(x)
p3: D#(ln(x)) -> D#(x)
p4: D#(div(x,y)) -> D#(y)
p5: D#(div(x,y)) -> D#(x)
p6: D#(minus(x)) -> D#(x)
p7: D#(-(x,y)) -> D#(y)
p8: D#(-(x,y)) -> D#(x)
p9: D#(*(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

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: D#(+(x,y)) -> D#(x)
p2: D#(*(x,y)) -> D#(y)
p3: D#(-(x,y)) -> D#(x)
p4: D#(-(x,y)) -> D#(y)
p5: D#(minus(x)) -> D#(x)
p6: D#(div(x,y)) -> D#(x)
p7: D#(div(x,y)) -> D#(y)
p8: D#(ln(x)) -> D#(x)
p9: D#(pow(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = max{8, x1}
        +_A(x1,x2) = max{x1 + 13, x2 + 17}
        *_A(x1,x2) = max{x1 + 34, x2 + 33}
        -_A(x1,x2) = max{8, x1 + 6, x2 + 3}
        minus_A(x1) = x1 + 8
        div_A(x1,x2) = max{8, x1 + 7, x2 + 7}
        ln_A(x1) = x1 + 54
        pow_A(x1,x2) = max{14, x1 + 9, x2 + 11}
        D_A(x1) = max{2, x1 - 51}
        constant_A = 1
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        + = minus > * = - = pow = D = constant > div = |0| > ln > D#
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = []
      pi(*) = [1]
      pi(-) = []
      pi(minus) = [1]
      pi(div) = [1]
      pi(ln) = [1]
      pi(pow) = []
      pi(D) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p9



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(*(x,y)) -> D#(y)
p3: D#(-(x,y)) -> D#(x)
p4: D#(-(x,y)) -> D#(y)
p5: D#(minus(x)) -> D#(x)
p6: D#(div(x,y)) -> D#(x)
p7: D#(div(x,y)) -> D#(y)
p8: D#(ln(x)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

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: D#(+(x,y)) -> D#(x)
p2: D#(ln(x)) -> D#(x)
p3: D#(div(x,y)) -> D#(y)
p4: D#(div(x,y)) -> D#(x)
p5: D#(minus(x)) -> D#(x)
p6: D#(-(x,y)) -> D#(y)
p7: D#(-(x,y)) -> D#(x)
p8: D#(*(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = x1 + 16
        +_A(x1,x2) = max{58, x1 + 17, x2 - 12}
        ln_A(x1) = x1 + 17
        div_A(x1,x2) = max{x1 + 121, x2 + 19}
        minus_A(x1) = max{164, x1 + 58}
        -_A(x1,x2) = max{109, x1 + 60, x2 + 58}
        *_A(x1,x2) = max{x1 - 9, x2 + 18}
        D_A(x1) = max{105, x1 + 48}
        t_A = 52
        |1|_A = 51
        constant_A = 107
        |0|_A = 106
  
    base order 2:
  
      precedence:
      
        - > div = minus > + = D = t > ln = * = |1| = constant > D# = |0|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = [1]
      pi(ln) = [1]
      pi(div) = []
      pi(minus) = [1]
      pi(-) = [2]
      pi(*) = []
      pi(D) = []
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p8



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(ln(x)) -> D#(x)
p3: D#(div(x,y)) -> D#(y)
p4: D#(div(x,y)) -> D#(x)
p5: D#(minus(x)) -> D#(x)
p6: D#(-(x,y)) -> D#(y)
p7: D#(-(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(-(x,y)) -> D#(x)
p3: D#(-(x,y)) -> D#(y)
p4: D#(minus(x)) -> D#(x)
p5: D#(div(x,y)) -> D#(x)
p6: D#(div(x,y)) -> D#(y)
p7: D#(ln(x)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = max{10, x1}
        +_A(x1,x2) = max{x1 + 22, x2 + 12}
        -_A(x1,x2) = max{x1 + 46, x2 + 16}
        minus_A(x1) = x1 + 23
        div_A(x1,x2) = max{10, x1 + 5, x2 + 3}
        ln_A(x1) = x1 + 10
        D_A(x1) = max{22, x1 - 55}
        constant_A = 1
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        - > + = minus = div = ln > D# = D = constant > |0|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = [2]
      pi(-) = [1, 2]
      pi(minus) = [1]
      pi(div) = []
      pi(ln) = [1]
      pi(D) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p7



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(-(x,y)) -> D#(x)
p3: D#(-(x,y)) -> D#(y)
p4: D#(minus(x)) -> D#(x)
p5: D#(div(x,y)) -> D#(x)
p6: D#(div(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

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: D#(+(x,y)) -> D#(x)
p2: D#(div(x,y)) -> D#(y)
p3: D#(div(x,y)) -> D#(x)
p4: D#(minus(x)) -> D#(x)
p5: D#(-(x,y)) -> D#(y)
p6: D#(-(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = max{31, x1}
        +_A(x1,x2) = max{x1 + 77, x2 + 71}
        div_A(x1,x2) = max{x1 + 57, x2 + 33}
        minus_A(x1) = max{37, x1 + 32}
        -_A(x1,x2) = max{31, x1 + 26, x2 + 14}
        D_A(x1) = max{4, x1 - 138}
        t_A = 1
        |1|_A = 0
        constant_A = 137
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        - > + = div = minus = t > D# > D = constant > |1| = |0|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = [1, 2]
      pi(div) = [2]
      pi(minus) = [1]
      pi(-) = []
      pi(D) = []
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p6



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(div(x,y)) -> D#(y)
p3: D#(div(x,y)) -> D#(x)
p4: D#(minus(x)) -> D#(x)
p5: D#(-(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(-(x,y)) -> D#(y)
p3: D#(minus(x)) -> D#(x)
p4: D#(div(x,y)) -> D#(x)
p5: D#(div(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = max{2, x1}
        +_A(x1,x2) = max{125, x1 + 27, x2 + 55}
        -_A(x1,x2) = max{16, x1 + 3, x2 + 5}
        minus_A(x1) = max{2, x1 + 1}
        div_A(x1,x2) = max{145, x1 + 135, x2 + 3}
        D_A(x1) = max{10, x1 - 148}
        t_A = 5
        |1|_A = 4
        constant_A = 1
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        + = div > - = D = t = constant > minus = |1| = |0| > D#
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = [2]
      pi(-) = []
      pi(minus) = [1]
      pi(div) = [2]
      pi(D) = []
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(-(x,y)) -> D#(y)
p3: D#(minus(x)) -> D#(x)
p4: D#(div(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

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: D#(+(x,y)) -> D#(x)
p2: D#(div(x,y)) -> D#(x)
p3: D#(minus(x)) -> D#(x)
p4: D#(-(x,y)) -> D#(y)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = x1 + 19
        +_A(x1,x2) = max{57, x1 + 41, x2 - 17}
        div_A(x1,x2) = max{32, x1 + 20, x2 - 17}
        minus_A(x1) = max{38, x1 + 20}
        -_A(x1,x2) = max{x1 + 14, x2 + 20}
        D_A(x1) = max{17, x1 + 14}
        t_A = 36
        |1|_A = 35
        constant_A = 1
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        minus > + = div = - = D = t = constant > D# = |0| > |1|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = []
      pi(div) = [1]
      pi(minus) = [1]
      pi(-) = [2]
      pi(D) = [1]
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(div(x,y)) -> D#(x)
p3: D#(minus(x)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(minus(x)) -> D#(x)
p3: D#(div(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = x1 + 46
        +_A(x1,x2) = max{x1 + 47, x2 - 34}
        minus_A(x1) = max{73, x1 + 67}
        div_A(x1,x2) = max{x1 + 64, x2 - 35}
        D_A(x1) = max{5, x1 - 74}
        t_A = 1
        |1|_A = 0
        constant_A = 1
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        + = div > minus > D# = D = t = constant > |1| = |0|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = [1]
      pi(minus) = [1]
      pi(div) = []
      pi(D) = []
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(minus(x)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: D#(+(x,y)) -> D#(x)
p2: D#(minus(x)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = x1 + 12
        +_A(x1,x2) = max{x1 + 121, x2 - 5}
        minus_A(x1) = x1 + 13
        D_A(x1) = max{4, x1 - 14}
        constant_A = 13
        |0|_A = 3
  
    base order 2:
  
      precedence:
      
        + = minus > D# = D = constant > |0|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = [1]
      pi(minus) = [1]
      pi(D) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: D#(+(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: D#(+(x,y)) -> D#(x)

and R consists of:

r1: D(t()) -> |1|()
r2: D(constant()) -> |0|()
r3: D(+(x,y)) -> +(D(x),D(y))
r4: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
r5: D(-(x,y)) -> -(D(x),D(y))
r6: D(minus(x)) -> minus(D(x))
r7: D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,|2|())))
r8: D(ln(x)) -> div(D(x),x)
r9: D(pow(x,y)) -> +(*(*(y,pow(x,-(y,|1|()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))

The set of usable rules consists of

  r1, r2

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        D#_A(x1) = x1 + 13
        +_A(x1,x2) = max{x1 + 14, x2 - 11}
        D_A(x1) = max{10, x1 - 39}
        t_A = 1
        |1|_A = 0
        constant_A = 1
        |0|_A = 0
  
    base order 2:
  
      precedence:
      
        + > D# = D = t = constant > |1| = |0|
      
  
    partial status:
  
      pi(D#) = [1]
      pi(+) = [1]
      pi(D) = []
      pi(t) = []
      pi(|1|) = []
      pi(constant) = []
      pi(|0|) = []

The following rules are removed:

  p1

Then no dependency pair remains.