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, r3, r9, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          |log'|#_A(x1) = x1 + 7
          |0|_A(x1) = x1 + 6
          |1|_A(x1) = x1 + 15
          |#|_A() = 3
          +_A(x1,x2) = x1 + 4
          -_A(x1,x2) = 4
          not_A(x1) = x1 + 7
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x1 + x2 + 1
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 1
  
    base order 2:
  
      precedence:
      
        not = if > |0| = |1| = + = ge > |log'|# = - = |log'| > |#| > false > true
      
  
    partial status:
  
      pi(|log'|#) = [1]
      pi(|0|) = []
      pi(|1|) = [1]
      pi(|#|) = []
      pi(+) = [1]
      pi(-) = []
      pi(not) = [1]
      pi(true) = []
      pi(false) = []
      pi(if) = [1, 2]
      pi(ge) = [1]
      pi(|log'|) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: |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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: |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, r3, r9, r15, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          |log'|#_A(x1) = x1
          |1|_A(x1) = x1 + 7
          |0|_A(x1) = x1 + 20
          |#|_A() = 4
          +_A(x1,x2) = x1 + 1
          -_A(x1,x2) = 12
          not_A(x1) = 2
          true_A() = 3
          false_A() = 1
          if_A(x1,x2,x3) = x2 + 20
          ge_A(x1,x2) = x1 + 4
          |log'|_A(x1) = x1 + 5
  
    base order 2:
  
      precedence:
      
        if > |log'| > |#| > true > not > |0| > - > |log'|# = |1| = ge > + = false
      
  
    partial status:
  
      pi(|log'|#) = [1]
      pi(|1|) = [1]
      pi(|0|) = [1]
      pi(|#|) = []
      pi(+) = [1]
      pi(-) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = []
      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: +#(+(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, r9, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + x2 + 11
          |0|_A(x1) = x1 + 10
          +_A(x1,x2) = x1 + x2 + 1
          |1|_A(x1) = x1 + 15
          |#|_A() = 3
          -_A(x1,x2) = 4
          not_A(x1) = x1 + 11
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x1 + x2
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 9
  
    base order 2:
  
      precedence:
      
        +# = not > + = if = ge > |0| > |log'| > |#| > false > |1| = true > -
      
  
    partial status:
  
      pi(+#) = []
      pi(|0|) = []
      pi(+) = []
      pi(|1|) = [1]
      pi(|#|) = []
      pi(-) = []
      pi(not) = [1]
      pi(true) = []
      pi(false) = []
      pi(if) = [1]
      pi(ge) = [1]
      pi(|log'|) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: +#(+(x,y),z) -> +#(y,z)
p2: +#(+(x,y),z) -> +#(x,+(y,z))
p3: +#(|1|(x),|1|(y)) -> +#(x,y)
p4: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p5: +#(|0|(x),|1|(y)) -> +#(x,y)
p6: +#(|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 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: +#(+(x,y),z) -> +#(y,z)
p2: +#(|1|(x),|0|(y)) -> +#(x,y)
p3: +#(|0|(x),|1|(y)) -> +#(x,y)
p4: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p5: +#(|1|(x),|1|(y)) -> +#(x,y)
p6: +#(+(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, r9, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + x2 + 2
          +_A(x1,x2) = x1 + x2 + 1
          |1|_A(x1) = x1 + 16
          |0|_A(x1) = x1 + 11
          |#|_A() = 3
          -_A(x1,x2) = 4
          not_A(x1) = x1 + 16
          true_A() = 2
          false_A() = 1
          if_A(x1,x2,x3) = x1 + x2
          ge_A(x1,x2) = x1 + x2 + 1
          |log'|_A(x1) = x1 + 10
  
    base order 2:
  
      precedence:
      
        if > + = |log'| > ge > not > true > +# = |1| = false > |0| > - > |#|
      
  
    partial status:
  
      pi(+#) = []
      pi(+) = [1, 2]
      pi(|1|) = []
      pi(|0|) = [1]
      pi(|#|) = []
      pi(-) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = []
      pi(|log'|) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: +#(|1|(x),|0|(y)) -> +#(x,y)
p2: +#(+(x,y),z) -> +#(x,+(y,z))
p3: +#(|1|(x),|1|(y)) -> +#(x,y)
p4: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p5: +#(|0|(x),|1|(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, r9, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + x2 + 16
          |1|_A(x1) = x1 + 15
          |0|_A(x1) = x1 + 10
          +_A(x1,x2) = x1 + x2 + 1
          |#|_A() = 3
          -_A(x1,x2) = 4
          not_A(x1) = x1 + 11
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x1 + x2
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 9
  
    base order 2:
  
      precedence:
      
        +# = + = not = if > |0| > |log'| > |1| = ge > - > |#| > true > false
      
  
    partial status:
  
      pi(+#) = [1]
      pi(|1|) = [1]
      pi(|0|) = [1]
      pi(+) = []
      pi(|#|) = []
      pi(-) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = [1]
      pi(|log'|) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: +#(+(x,y),z) -> +#(x,+(y,z))
p2: +#(|1|(x),|1|(y)) -> +#(x,y)
p3: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p4: +#(|0|(x),|1|(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 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: +#(+(x,y),z) -> +#(x,+(y,z))
p2: +#(|0|(x),|1|(y)) -> +#(x,y)
p3: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p4: +#(|1|(x),|1|(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, r9, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + x2 + 16
          +_A(x1,x2) = x1 + x2 + 1
          |0|_A(x1) = x1 + 10
          |1|_A(x1) = x1 + 15
          |#|_A() = 3
          -_A(x1,x2) = 4
          not_A(x1) = x1 + 11
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x1 + x2
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 9
  
    base order 2:
  
      precedence:
      
        +# = + = if > |0| > |log'| > |1| = not = ge > - > |#| > true = false
      
  
    partial status:
  
      pi(+#) = [1]
      pi(+) = []
      pi(|0|) = [1]
      pi(|1|) = [1]
      pi(|#|) = []
      pi(-) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = [1]
      pi(|log'|) = [1]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: +#(+(x,y),z) -> +#(x,+(y,z))
p2: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p3: +#(|1|(x),|1|(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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: +#(+(x,y),z) -> +#(x,+(y,z))
p2: +#(|1|(x),|1|(y)) -> +#(x,y)
p3: +#(|1|(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, r2, r3, r4, r5, r6, r7, r8, r9, r10, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + x2 + 18
          +_A(x1,x2) = x1 + x2 + 7
          |1|_A(x1) = x1 + 17
          |#|_A() = 3
          |0|_A(x1) = x1 + 6
          -_A(x1,x2) = x1 + 5
          not_A(x1) = x1 + 7
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x1 + x2
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 18
  
    base order 2:
  
      precedence:
      
        + = if > |log'| > ge > +# = |1| > not > |0| = - > |#| > false > true
      
  
    partial status:
  
      pi(+#) = []
      pi(+) = []
      pi(|1|) = []
      pi(|#|) = []
      pi(|0|) = [1]
      pi(-) = [1]
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = []
      pi(|log'|) = [1]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: +#(+(x,y),z) -> +#(x,+(y,z))
p2: +#(|1|(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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: +#(+(x,y),z) -> +#(x,+(y,z))
p2: +#(|1|(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, r2, r3, r4, r5, r6, r7, r8, r15, r16, r17, r18, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + x2 + 1
          +_A(x1,x2) = x1 + x2 + 7
          |1|_A(x1) = x1 + 18
          |#|_A() = 4
          |0|_A(x1) = x1 + 6
          not_A(x1) = x1 + 2
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x2 + x3 + 1
          ge_A(x1,x2) = 3
          |log'|_A(x1) = x1 + 19
  
    base order 2:
  
      precedence:
      
        + = if > |log'| > +# = |1| > |#| > ge > not = false > |0| = true
      
  
    partial status:
  
      pi(+#) = []
      pi(+) = []
      pi(|1|) = [1]
      pi(|#|) = []
      pi(|0|) = [1]
      pi(not) = [1]
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = []
      pi(|log'|) = [1]

The following rules are removed:

  p2



-- 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, r9, r15, r17, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          +#_A(x1,x2) = x1 + 7
          +_A(x1,x2) = x1 + x2 + 6
          |0|_A(x1) = x1 + 5
          |#|_A() = 3
          |1|_A(x1) = x1 + 15
          -_A(x1,x2) = 4
          not_A(x1) = x1
          true_A() = 2
          false_A() = 1
          if_A(x1,x2,x3) = x1 + x2
          ge_A(x1,x2) = x1
          |log'|_A(x1) = x1 + 20
  
    base order 2:
  
      precedence:
      
        +# = + = if > false = |log'| > not > |0| = |1| > - > |#| > true > ge
      
  
    partial status:
  
      pi(+#) = []
      pi(+) = []
      pi(|0|) = [1]
      pi(|#|) = []
      pi(|1|) = [1]
      pi(-) = []
      pi(not) = [1]
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = []
      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, r3, r9, r10, r11, r12, r13, r14, r15, r16, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          -#_A(x1,x2) = x1 + x2 + 14
          |0|_A(x1) = x1 + 13
          |1|_A(x1) = x1 + 7
          -_A(x1,x2) = x1 + x2 + 2
          |#|_A() = 3
          +_A(x1,x2) = x1 + 1
          not_A(x1) = x1 + 2
          true_A() = 1
          false_A() = 2
          ge_A(x1,x2) = x1 + 4
          |log'|_A(x1) = x1 + 1
  
    base order 2:
  
      precedence:
      
        - > |1| > |0| = not = ge > |log'| > + > |#| > -# = false > true
      
  
    partial status:
  
      pi(-#) = []
      pi(|0|) = [1]
      pi(|1|) = []
      pi(-) = []
      pi(|#|) = []
      pi(+) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(ge) = [1]
      pi(|log'|) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: -#(|1|(x),|1|(y)) -> -#(x,y)
p2: -#(|1|(x),|0|(y)) -> -#(x,y)
p3: -#(|0|(x),|1|(y)) -> -#(x,y)
p4: -#(|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 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: -#(|1|(x),|1|(y)) -> -#(x,y)
p2: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|()))
p3: -#(|0|(x),|1|(y)) -> -#(x,y)
p4: -#(|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, r3, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          -#_A(x1,x2) = x1 + x2
          |1|_A(x1) = x1 + 6
          |0|_A(x1) = x1 + 6
          -_A(x1,x2) = x1
          |#|_A() = 3
          +_A(x1,x2) = x1 + 1
          not_A(x1) = x1 + 3
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x2 + x3 + 3
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 1
  
    base order 2:
  
      precedence:
      
        -# > - > ge > |#| > false > |1| = true > not = if > |0| > |log'| > +
      
  
    partial status:
  
      pi(-#) = [1]
      pi(|1|) = [1]
      pi(|0|) = [1]
      pi(-) = [1]
      pi(|#|) = []
      pi(+) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = []
      pi(|log'|) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|()))
p2: -#(|0|(x),|1|(y)) -> -#(x,y)
p3: -#(|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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|()))
p2: -#(|0|(x),|1|(y)) -> -#(x,y)
p3: -#(|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, r3, r9, r10, r11, r12, r13, r14, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          -#_A(x1,x2) = x1 + 4
          |0|_A(x1) = x1 + 6
          |1|_A(x1) = x1 + 6
          -_A(x1,x2) = x1
          |#|_A() = 3
          +_A(x1,x2) = x1 + 1
          not_A(x1) = x1 + 2
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x2 + 6
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        - = if > |0| = |1| > -# = |log'| > + = ge > |#| > false > true > not
      
  
    partial status:
  
      pi(-#) = [1]
      pi(|0|) = []
      pi(|1|) = [1]
      pi(-) = [1]
      pi(|#|) = []
      pi(+) = [1]
      pi(not) = [1]
      pi(true) = []
      pi(false) = []
      pi(if) = [2]
      pi(ge) = [1]
      pi(|log'|) = [1]

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|()))
p2: -#(|0|(x),|1|(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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: -#(|0|(x),|1|(y)) -> -#(-(x,y),|1|(|#|()))
p2: -#(|0|(x),|1|(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, r3, r9, r10, r17, r23, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          -#_A(x1,x2) = x2
          |0|_A(x1) = x1 + 6
          |1|_A(x1) = x1 + 9
          -_A(x1,x2) = x1 + 1
          |#|_A() = 0
          +_A(x1,x2) = x1 + 1
          if_A(x1,x2,x3) = x2 + 6
          true_A() = 1
          ge_A(x1,x2) = 5
          false_A() = 2
          |log'|_A(x1) = x1 + 4
  
    base order 2:
  
      precedence:
      
        if > |log'| > |0| = + = ge > false > |1| = - = true > -# > |#|
      
  
    partial status:
  
      pi(-#) = []
      pi(|0|) = [1]
      pi(|1|) = [1]
      pi(-) = [1]
      pi(|#|) = []
      pi(+) = []
      pi(if) = []
      pi(true) = []
      pi(ge) = []
      pi(false) = []
      pi(|log'|) = [1]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: -#(|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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: -#(|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, r3, r9, r10, r11, r12, r13, r14, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          -#_A(x1,x2) = x1 + 4
          |0|_A(x1) = x1 + 7
          |1|_A(x1) = x1 + 7
          -_A(x1,x2) = x1
          |#|_A() = 3
          +_A(x1,x2) = x1 + 4
          not_A(x1) = x1 + 8
          true_A() = 2
          false_A() = 1
          if_A(x1,x2,x3) = x2 + 4
          ge_A(x1,x2) = x1 + 3
          |log'|_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        - = not = if > |0| > -# = + > |1| > |log'| > ge > |#| > true > false
      
  
    partial status:
  
      pi(-#) = []
      pi(|0|) = []
      pi(|1|) = []
      pi(-) = [1]
      pi(|#|) = []
      pi(+) = [1]
      pi(not) = [1]
      pi(true) = []
      pi(false) = []
      pi(if) = [2]
      pi(ge) = [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: 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, r9, r15, r16, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge#_A(x1,x2) = x1 + x2 + 6
          |0|_A(x1) = x1 + 7
          |1|_A(x1) = x1 + 12
          |#|_A() = 3
          -_A(x1,x2) = 4
          not_A(x1) = x1 + 8
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x1 + x2 + 17
          ge_A(x1,x2) = x1 + 2
          |log'|_A(x1) = x1 + 17
  
    base order 2:
  
      precedence:
      
        if > |log'| > ge# > |0| = |1| = not > - > |#| > true > false = ge
      
  
    partial status:
  
      pi(ge#) = []
      pi(|0|) = []
      pi(|1|) = [1]
      pi(|#|) = []
      pi(-) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = [1, 2]
      pi(ge) = []
      pi(|log'|) = [1]

The following rules are removed:

  p4



-- SCC decomposition.

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)

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, p2, p3}


-- 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),|0|(y)) -> ge#(x,y)
p3: ge#(|1|(x),|1|(y)) -> ge#(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, r3, r9, r10, r15, r17, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge#_A(x1,x2) = x1 + x2 + 49
          |0|_A(x1) = x1 + 31
          |1|_A(x1) = x1 + 48
          |#|_A() = 3
          +_A(x1,x2) = x1 + 9
          -_A(x1,x2) = x1 + 16
          not_A(x1) = x1
          true_A() = 2
          false_A() = 1
          if_A(x1,x2,x3) = x2 + 23
          ge_A(x1,x2) = x1 + 3
          |log'|_A(x1) = x1 + 21
  
    base order 2:
  
      precedence:
      
        if > - = |log'| > ge# > |0| = |1| > + > |#| > true > not = false = ge
      
  
    partial status:
  
      pi(ge#) = [1, 2]
      pi(|0|) = [1]
      pi(|1|) = []
      pi(|#|) = []
      pi(+) = []
      pi(-) = [1]
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = [2]
      pi(ge) = [1]
      pi(|log'|) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: ge#(|1|(x),|0|(y)) -> ge#(x,y)
p2: ge#(|1|(x),|1|(y)) -> ge#(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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: ge#(|1|(x),|0|(y)) -> ge#(x,y)
p2: ge#(|1|(x),|1|(y)) -> ge#(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, r3, r9, r10, r15, r17, r23, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge#_A(x1,x2) = x1 + x2 + 22
          |1|_A(x1) = x1 + 24
          |0|_A(x1) = x1 + 21
          |#|_A() = 3
          +_A(x1,x2) = x1 + 1
          -_A(x1,x2) = x1 + 10
          not_A(x1) = x1
          true_A() = 2
          false_A() = 1
          if_A(x1,x2,x3) = x1 + x2 + 4
          ge_A(x1,x2) = x1 + 17
          |log'|_A(x1) = x1 + 19
  
    base order 2:
  
      precedence:
      
        - = not = if > ge# = |0| > |log'| > |#| > true > ge > |1| = false > +
      
  
    partial status:
  
      pi(ge#) = []
      pi(|1|) = [1]
      pi(|0|) = []
      pi(|#|) = []
      pi(+) = [1]
      pi(-) = [1]
      pi(not) = [1]
      pi(true) = []
      pi(false) = []
      pi(if) = [1, 2]
      pi(ge) = [1]
      pi(|log'|) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: ge#(|1|(x),|1|(y)) -> ge#(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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: ge#(|1|(x),|1|(y)) -> ge#(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, r3, r9, r15, r16, r17, r19, r20, r21, r22, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge#_A(x1,x2) = x1 + x2 + 28
          |1|_A(x1) = x1 + 27
          |0|_A(x1) = x1 + 20
          |#|_A() = 5
          +_A(x1,x2) = x1 + 6
          -_A(x1,x2) = 6
          not_A(x1) = 3
          true_A() = 1
          false_A() = 2
          if_A(x1,x2,x3) = x2 + 15
          ge_A(x1,x2) = 4
          |log'|_A(x1) = x1 + 13
  
    base order 2:
  
      precedence:
      
        if > |log'| > ge# = + > |1| = |0| = - = ge > |#| = not > false > true
      
  
    partial status:
  
      pi(ge#) = [1]
      pi(|1|) = [1]
      pi(|0|) = [1]
      pi(|#|) = []
      pi(+) = []
      pi(-) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(if) = []
      pi(ge) = []
      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: 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, r3, r5, r6, r9, r15, r16, r20, r23, r24, r25, r27

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ge#_A(x1,x2) = x2 + 6
          |#|_A() = 5
          |0|_A(x1) = x1 + 17
          +_A(x1,x2) = x1 + 8
          |1|_A(x1) = 24
          -_A(x1,x2) = 6
          not_A(x1) = 3
          true_A() = 1
          false_A() = 2
          ge_A(x1,x2) = 4
          |log'|_A(x1) = x1 + 8
  
    base order 2:
  
      precedence:
      
        |log'| > |1| > ge# = |0| = + = - > |#| > ge > false > true > not
      
  
    partial status:
  
      pi(ge#) = []
      pi(|#|) = []
      pi(|0|) = [1]
      pi(+) = []
      pi(|1|) = []
      pi(-) = []
      pi(not) = []
      pi(true) = []
      pi(false) = []
      pi(ge) = []
      pi(|log'|) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.