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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          |log'|#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          |0|_A(x1) = x1 + (2,2)
          |1|_A(x1) = ((1,0),(0,0)) x1 + (2,2)
  
    precedence:
    
      |log'|# = |0| = |1|
    
    partial status:
  
      pi(|log'|#) = []
      pi(|0|) = [1]
      pi(|1|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: |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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          |log'|#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          |1|_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      |log'|# = |1|
    
    partial status:
  
      pi(|log'|#) = []
      pi(|1|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          +#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (0,2)
          |0|_A(x1) = x1 + (1,1)
          +_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,14)
          |1|_A(x1) = x1 + (5,25)
          |#|_A() = (1,38)
  
    precedence:
    
      + > +# = |0| = |1| = |#|
    
    partial status:
  
      pi(+#) = [1]
      pi(|0|) = [1]
      pi(+) = []
      pi(|1|) = [1]
      pi(|#|) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: +#(|0|(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)
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: +#(|0|(x),|0|(y)) -> +#(x,y)
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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          +#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (7,3)
          |0|_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          |1|_A(x1) = ((1,0),(1,1)) x1 + (6,4)
          +_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,1)
          |#|_A() = (1,5)
  
    precedence:
    
      +# = |1| > + > |0| = |#|
    
    partial status:
  
      pi(+#) = []
      pi(|0|) = [1]
      pi(|1|) = []
      pi(+) = [1, 2]
      pi(|#|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: +#(|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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          +#_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,4)
          |1|_A(x1) = ((1,0),(0,0)) x1 + (6,6)
          |0|_A(x1) = ((1,0),(1,0)) x1 + (1,7)
          +_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,1)
          |#|_A() = (1,9)
  
    precedence:
    
      +# = |1| = |0| = + = |#|
    
    partial status:
  
      pi(+#) = []
      pi(|1|) = []
      pi(|0|) = []
      pi(+) = [1, 2]
      pi(|#|) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          +#_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (8,13)
          |1|_A(x1) = ((1,0),(0,0)) x1 + (7,2)
          |0|_A(x1) = ((1,0),(0,0)) x1 + (1,12)
          +_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (4,3)
          |#|_A() = (1,1)
  
    precedence:
    
      +# = |0| = + > |1| > |#|
    
    partial status:
  
      pi(+#) = []
      pi(|1|) = []
      pi(|0|) = []
      pi(+) = []
      pi(|#|) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- 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: +#(+(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}


-- 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: +#(|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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          +#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (7,4)
          |1|_A(x1) = ((1,0),(0,0)) x1 + (6,2)
          |0|_A(x1) = ((1,0),(1,0)) x1 + (1,3)
          +_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,4)
          |#|_A() = (2,1)
  
    precedence:
    
      |1| = |0| = + > +# > |#|
    
    partial status:
  
      pi(+#) = [1]
      pi(|1|) = []
      pi(|0|) = []
      pi(+) = []
      pi(|#|) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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))

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: +#(|1|(x),|0|(y)) -> +#(x,y)
p2: +#(+(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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          +#_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (2,20)
          |1|_A(x1) = ((1,0),(0,0)) x1 + (7,19)
          |0|_A(x1) = ((1,0),(1,0)) x1 + (1,16)
          +_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (4,5)
          |#|_A() = (1,18)
  
    precedence:
    
      |1| = |0| = + > +# = |#|
    
    partial status:
  
      pi(+#) = [1]
      pi(|1|) = []
      pi(|0|) = []
      pi(+) = []
      pi(|#|) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: +#(|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}


-- Reduction pair.

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

p1: +#(|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

  (no rules)

Take the reduction pair:

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

The next rules are strictly ordered:

  p1

We remove them from the problem.  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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          -#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (2,4)
          |0|_A(x1) = x1 + (11,3)
          |1|_A(x1) = ((1,0),(1,1)) x1 + (6,2)
          -_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,7)
          |#|_A() = (1,1)
  
    precedence:
    
      -# = |0| = |1| = - = |#|
    
    partial status:
  
      pi(-#) = []
      pi(|0|) = []
      pi(|1|) = [1]
      pi(-) = [2]
      pi(|#|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: -#(|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, r9, r10, r11, r12, r13, r14

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          -#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,2)
          |1|_A(x1) = ((1,0),(1,1)) x1 + (5,1)
          |0|_A(x1) = ((1,0),(1,1)) x1 + (9,3)
          -_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,4)
          |#|_A() = (1,21)
  
    precedence:
    
      -# = |1| = |0| = - = |#|
    
    partial status:
  
      pi(-#) = []
      pi(|1|) = [1]
      pi(|0|) = []
      pi(-) = [2]
      pi(|#|) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          -#_A(x1,x2) = x1 + x2 + (22,45)
          |1|_A(x1) = ((1,0),(1,1)) x1 + (11,28)
          |0|_A(x1) = ((1,0),(1,0)) x1 + (21,44)
          -_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (8,9)
          |#|_A() = (1,11)
  
    precedence:
    
      -# = |1| = |0| = - > |#|
    
    partial status:
  
      pi(-#) = []
      pi(|1|) = []
      pi(|0|) = []
      pi(-) = []
      pi(|#|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: -#(|1|(x),|0|(y)) -> -#(x,y)
p2: -#(|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}


-- Reduction pair.

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

p1: -#(|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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          -#_A(x1,x2) = ((1,0),(0,0)) x1 + (2,1)
          |1|_A(x1) = x1 + (3,2)
          |0|_A(x1) = (1,2)
  
    precedence:
    
      -# = |1| > |0|
    
    partial status:
  
      pi(-#) = []
      pi(|1|) = []
      pi(|0|) = []

The next rules are strictly ordered:

  p1

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

-- 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, r9, r10, r11, r12, r13, r14

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          -#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,7)
          |0|_A(x1) = ((1,0),(1,0)) x1 + (11,1)
          |1|_A(x1) = ((1,0),(1,0)) x1 + (6,4)
          -_A(x1,x2) = x1 + x2 + (2,12)
          |#|_A() = (2,2)
  
    precedence:
    
      -# = - > |1| > |0| = |#|
    
    partial status:
  
      pi(-#) = []
      pi(|0|) = []
      pi(|1|) = []
      pi(-) = [1, 2]
      pi(|#|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  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

  (no rules)

Take the reduction pair:

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

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: ge#(|1|(x),|1|(y)) -> ge#(x,y)
p2: ge#(|1|(x),|0|(y)) -> ge#(x,y)
p3: 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 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#(|1|(x),|1|(y)) -> ge#(x,y)
p2: ge#(|0|(x),|1|(y)) -> ge#(y,x)
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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          ge#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,2)
          |1|_A(x1) = ((1,0),(1,1)) x1 + (1,3)
          |0|_A(x1) = ((1,0),(1,1)) x1 + (3,1)
  
    precedence:
    
      ge# = |1| > |0|
    
    partial status:
  
      pi(ge#) = []
      pi(|1|) = []
      pi(|0|) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: ge#(|1|(x),|1|(y)) -> ge#(x,y)
p2: 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}


-- Reduction pair.

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

p1: ge#(|1|(x),|1|(y)) -> ge#(x,y)
p2: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          ge#_A(x1,x2) = ((1,0),(1,0)) x1 + (1,2)
          |1|_A(x1) = ((1,0),(0,0)) x1 + (2,1)
          |0|_A(x1) = (2,3)
  
    precedence:
    
      ge# = |1| = |0|
    
    partial status:
  
      pi(ge#) = []
      pi(|1|) = []
      pi(|0|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: 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}


-- Reduction pair.

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

p1: 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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          ge#_A(x1,x2) = ((1,0),(0,0)) x1 + (2,1)
          |1|_A(x1) = x1 + (3,2)
          |0|_A(x1) = (1,2)
  
    precedence:
    
      ge# = |1| > |0|
    
    partial status:
  
      pi(ge#) = []
      pi(|1|) = []
      pi(|0|) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          ge#_A(x1,x2) = ((1,0),(1,1)) x2 + (2,2)
          |#|_A() = (1,1)
          |0|_A(x1) = x1 + (3,3)
  
    precedence:
    
      ge# > |#| = |0|
    
    partial status:
  
      pi(ge#) = [2]
      pi(|#|) = []
      pi(|0|) = [1]

The next rules are strictly ordered:

  p1

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