YES
We show the termination of the TRS R:

  |0|(|#|()) -> |#|()
  +(|#|(),x) -> x
  +(x,|#|()) -> x
  +(|0|(x),|0|(y)) -> |0|(+(x,y))
  +(|0|(x),|1|(y)) -> |1|(+(x,y))
  +(|1|(x),|0|(y)) -> |1|(+(x,y))
  +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
  +(+(x,y),z) -> +(x,+(y,z))
  -(|#|(),x) -> |#|()
  -(x,|#|()) -> x
  -(|0|(x),|0|(y)) -> |0|(-(x,y))
  -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
  -(|1|(x),|0|(y)) -> |1|(-(x,y))
  -(|1|(x),|1|(y)) -> |0|(-(x,y))
  not(true()) -> false()
  not(false()) -> true()
  if(true(),x,y) -> x
  if(false(),x,y) -> y
  ge(|0|(x),|0|(y)) -> ge(x,y)
  ge(|0|(x),|1|(y)) -> not(ge(y,x))
  ge(|1|(x),|0|(y)) -> ge(x,y)
  ge(|1|(x),|1|(y)) -> ge(x,y)
  ge(x,|#|()) -> true()
  ge(|#|(),|0|(x)) -> ge(|#|(),x)
  ge(|#|(),|1|(x)) -> false()
  log(x) -> -(|log'|(x),|1|(|#|()))
  |log'|(|#|()) -> |#|()
  |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
  |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())


-- SCC decomposition.

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

p1: +#(|0|(x),|0|(y)) -> |0|#(+(x,y))
p2: +#(|0|(x),|0|(y)) -> +#(x,y)
p3: +#(|0|(x),|1|(y)) -> +#(x,y)
p4: +#(|1|(x),|0|(y)) -> +#(x,y)
p5: +#(|1|(x),|1|(y)) -> |0|#(+(+(x,y),|1|(|#|())))
p6: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p7: +#(|1|(x),|1|(y)) -> +#(x,y)
p8: +#(+(x,y),z) -> +#(x,+(y,z))
p9: +#(+(x,y),z) -> +#(y,z)
p10: -#(|0|(x),|0|(y)) -> |0|#(-(x,y))
p11: -#(|0|(x),|0|(y)) -> -#(x,y)
p12: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|()))
p13: -#(|0|(x),|1|(y)) -> -#(x,y)
p14: -#(|1|(x),|0|(y)) -> -#(x,y)
p15: -#(|1|(x),|1|(y)) -> |0|#(-(x,y))
p16: -#(|1|(x),|1|(y)) -> -#(x,y)
p17: ge#(|0|(x),|0|(y)) -> ge#(x,y)
p18: ge#(|0|(x),|1|(y)) -> not#(ge(y,x))
p19: ge#(|0|(x),|1|(y)) -> ge#(y,x)
p20: ge#(|1|(x),|0|(y)) -> ge#(x,y)
p21: ge#(|1|(x),|1|(y)) -> ge#(x,y)
p22: ge#(|#|(),|0|(x)) -> ge#(|#|(),x)
p23: log#(x) -> -#(|log'|(x),|1|(|#|()))
p24: log#(x) -> |log'|#(x)
p25: |log'|#(|1|(x)) -> +#(|log'|(x),|1|(|#|()))
p26: |log'|#(|1|(x)) -> |log'|#(x)
p27: |log'|#(|0|(x)) -> if#(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())
p28: |log'|#(|0|(x)) -> ge#(x,|1|(|#|()))
p29: |log'|#(|0|(x)) -> +#(|log'|(x),|1|(|#|()))
p30: |log'|#(|0|(x)) -> |log'|#(x)

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The estimated dependency graph contains the following SCCs:

  {p26, p30}
  {p2, p3, p4, p6, p7, p8, p9}
  {p11, p12, p13, p14, p16}
  {p17, p19, p20, p21}
  {p22}


-- Reduction pair.

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

p1: |log'|#(|0|(x)) -> |log'|#(x)
p2: |log'|#(|1|(x)) -> |log'|#(x)

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The set of usable rules consists of

  r1

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          |log'|#_A(x1) = x1
          |0|_A(x1) = x1 + 16
          |1|_A(x1) = x1 + 5
          |#|_A() = 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            |log'|#_A(x1) = x1
            |0|_A(x1) = x1 + 6
            |1|_A(x1) = x1 + 2
            |#|_A() = 11
            |log'|#!_A(x1) = 0
            |0|!_A(x1) = 7
            |1|!_A(x1) = 6
            |#|!_A() = 8
  
    partial status:
  
      pi(|log'|#) = []
      pi(|0|) = []
      pi(|1|) = []
      pi(|#|) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

p1: +#(|0|(x),|0|(y)) -> +#(x,y)
p2: +#(+(x,y),z) -> +#(y,z)
p3: +#(+(x,y),z) -> +#(x,+(y,z))
p4: +#(|1|(x),|1|(y)) -> +#(x,y)
p5: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p6: +#(|0|(x),|1|(y)) -> +#(x,y)
p7: +#(|1|(x),|0|(y)) -> +#(x,y)

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + x2 + 16
          |0|_A(x1) = x1 + 5
          +_A(x1,x2) = x1 + x2 + 6
          |1|_A(x1) = x1 + 15
          |#|_A() = 3
          |log'|_A(x1) = x1 + 20
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            +#_A(x1,x2) = 0
            |0|_A(x1) = 7
            +_A(x1,x2) = x1 + x2 + 12
            |1|_A(x1) = 1
            |#|_A() = 4
            |log'|_A(x1) = x1 + 1
            +#!_A(x1,x2) = 14
            |0|!_A(x1) = x1 + 14
            +!_A(x1,x2) = x1 + x2 + 11
            |1|!_A(x1) = 18
            |#|!_A() = 12
            |log'|!_A(x1) = 19
  
    partial status:
  
      pi(+#) = []
      pi(|0|) = []
      pi(+) = []
      pi(|1|) = []
      pi(|#|) = []
      pi(|log'|) = [1]

The following rules are removed:

  p1, p2, p4, p5, p6, p7



-- SCC decomposition.

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

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

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + 4
          +_A(x1,x2) = x1 + x2 + 3
          |0|_A(x1) = 4
          |#|_A() = 3
          |1|_A(x1) = 5
          |log'|_A(x1) = x1 + 8
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            +#_A(x1,x2) = x1 + x2
            +_A(x1,x2) = x1 + x2 + 20
            |0|_A(x1) = 5
            |#|_A() = 4
            |1|_A(x1) = 2
            |log'|_A(x1) = x1 + 1
            +#!_A(x1,x2) = x1
            +!_A(x1,x2) = 43
            |0|!_A(x1) = x1 + 24
            |#|!_A() = 40
            |1|!_A(x1) = x1 + 34
            |log'|!_A(x1) = x1 + 37
  
    partial status:
  
      pi(+#) = []
      pi(+) = [1]
      pi(|0|) = []
      pi(|#|) = []
      pi(|1|) = []
      pi(|log'|) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: -#(|0|(x),|0|(y)) -> -#(x,y)
p2: -#(|1|(x),|1|(y)) -> -#(x,y)
p3: -#(|1|(x),|0|(y)) -> -#(x,y)
p4: -#(|0|(x),|1|(y)) -> -#(x,y)
p5: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|()))

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The set of usable rules consists of

  r1, r9, r10, r11, r12, r13, r14, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          -#_A(x1,x2) = x1
          |0|_A(x1) = x1
          |1|_A(x1) = x1
          -_A(x1,x2) = x1
          |#|_A() = 2
          if_A(x1,x2,x3) = x2 + 1
          true_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            -#_A(x1,x2) = 0
            |0|_A(x1) = x1 + 7
            |1|_A(x1) = x1 + 7
            -_A(x1,x2) = x1
            |#|_A() = 3
            if_A(x1,x2,x3) = x2
            true_A() = 10
            -#!_A(x1,x2) = x1
            |0|!_A(x1) = x1 + 13
            |1|!_A(x1) = x1 + 8
            -!_A(x1,x2) = x1 + 7
            |#|!_A() = 15
            if!_A(x1,x2,x3) = x1 + 11
            true!_A() = 10
  
    partial status:
  
      pi(-#) = [1]
      pi(|0|) = [1]
      pi(|1|) = [1]
      pi(-) = [1]
      pi(|#|) = []
      pi(if) = [2]
      pi(true) = []

The following rules are removed:

  p1, p2, p3, p4, p5

Then no dependency pair remains.

-- Reduction pair.

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

p1: ge#(|0|(x),|0|(y)) -> ge#(x,y)
p2: ge#(|1|(x),|1|(y)) -> ge#(x,y)
p3: ge#(|1|(x),|0|(y)) -> ge#(x,y)
p4: ge#(|0|(x),|1|(y)) -> ge#(y,x)

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The set of usable rules consists of

  r1

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge#_A(x1,x2) = x1 + x2 + 16
          |0|_A(x1) = x1 + 15
          |1|_A(x1) = x1 + 5
          |#|_A() = 3
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            ge#_A(x1,x2) = x1 + x2
            |0|_A(x1) = x1 + 3
            |1|_A(x1) = x1 + 1
            |#|_A() = 4
            ge#!_A(x1,x2) = 3
            |0|!_A(x1) = x1 + 10
            |1|!_A(x1) = 4
            |#|!_A() = 2
  
    partial status:
  
      pi(ge#) = []
      pi(|0|) = []
      pi(|1|) = []
      pi(|#|) = []

The following rules are removed:

  p1, p2, p3, p4

Then no dependency pair remains.

-- Reduction pair.

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

p1: ge#(|#|(),|0|(x)) -> ge#(|#|(),x)

and R consists of:

r1: |0|(|#|()) -> |#|()
r2: +(|#|(),x) -> x
r3: +(x,|#|()) -> x
r4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
r5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
r6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
r7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|#|())))
r8: +(+(x,y),z) -> +(x,+(y,z))
r9: -(|#|(),x) -> |#|()
r10: -(x,|#|()) -> x
r11: -(|0|(x),|0|(y)) -> |0|(-(x,y))
r12: -(|0|(x),|1|(y)) -> |1|(-(-(x,y),|1|(|#|())))
r13: -(|1|(x),|0|(y)) -> |1|(-(x,y))
r14: -(|1|(x),|1|(y)) -> |0|(-(x,y))
r15: not(true()) -> false()
r16: not(false()) -> true()
r17: if(true(),x,y) -> x
r18: if(false(),x,y) -> y
r19: ge(|0|(x),|0|(y)) -> ge(x,y)
r20: ge(|0|(x),|1|(y)) -> not(ge(y,x))
r21: ge(|1|(x),|0|(y)) -> ge(x,y)
r22: ge(|1|(x),|1|(y)) -> ge(x,y)
r23: ge(x,|#|()) -> true()
r24: ge(|#|(),|0|(x)) -> ge(|#|(),x)
r25: ge(|#|(),|1|(x)) -> false()
r26: log(x) -> -(|log'|(x),|1|(|#|()))
r27: |log'|(|#|()) -> |#|()
r28: |log'|(|1|(x)) -> +(|log'|(x),|1|(|#|()))
r29: |log'|(|0|(x)) -> if(ge(x,|1|(|#|())),+(|log'|(x),|1|(|#|())),|#|())

The set of usable rules consists of

  r1

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge#_A(x1,x2) = 3
          |#|_A() = 2
          |0|_A(x1) = 12
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            ge#_A(x1,x2) = 0
            |#|_A() = 2
            |0|_A(x1) = x1 + 5
            ge#!_A(x1,x2) = x2 + 5
            |#|!_A() = 7
            |0|!_A(x1) = 6
  
    partial status:
  
      pi(ge#) = []
      pi(|#|) = []
      pi(|0|) = []

The following rules are removed:

  p1

Then no dependency pair remains.