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))
  +(|0|(x),j(y)) -> j(+(x,y))
  +(j(x),|0|(y)) -> j(+(x,y))
  +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
  +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
  +(|1|(x),j(y)) -> |0|(+(x,y))
  +(j(x),|1|(y)) -> |0|(+(x,y))
  +(+(x,y),z) -> +(x,+(y,z))
  opp(|#|()) -> |#|()
  opp(|0|(x)) -> |0|(opp(x))
  opp(|1|(x)) -> j(opp(x))
  opp(j(x)) -> |1|(opp(x))
  -(x,y) -> +(x,opp(y))
  *(|#|(),x) -> |#|()
  *(|0|(x),y) -> |0|(*(x,y))
  *(|1|(x),y) -> +(|0|(*(x,y)),y)
  *(j(x),y) -> -(|0|(*(x,y)),y)
  *(*(x,y),z) -> *(x,*(y,z))


-- 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: +#(|0|(x),j(y)) -> +#(x,y)
p6: +#(j(x),|0|(y)) -> +#(x,y)
p7: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p8: +#(|1|(x),|1|(y)) -> +#(x,y)
p9: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p10: +#(j(x),j(y)) -> +#(x,y)
p11: +#(|1|(x),j(y)) -> |0|#(+(x,y))
p12: +#(|1|(x),j(y)) -> +#(x,y)
p13: +#(j(x),|1|(y)) -> |0|#(+(x,y))
p14: +#(j(x),|1|(y)) -> +#(x,y)
p15: +#(+(x,y),z) -> +#(x,+(y,z))
p16: +#(+(x,y),z) -> +#(y,z)
p17: opp#(|0|(x)) -> |0|#(opp(x))
p18: opp#(|0|(x)) -> opp#(x)
p19: opp#(|1|(x)) -> opp#(x)
p20: opp#(j(x)) -> opp#(x)
p21: -#(x,y) -> +#(x,opp(y))
p22: -#(x,y) -> opp#(y)
p23: *#(|0|(x),y) -> |0|#(*(x,y))
p24: *#(|0|(x),y) -> *#(x,y)
p25: *#(|1|(x),y) -> +#(|0|(*(x,y)),y)
p26: *#(|1|(x),y) -> |0|#(*(x,y))
p27: *#(|1|(x),y) -> *#(x,y)
p28: *#(j(x),y) -> -#(|0|(*(x,y)),y)
p29: *#(j(x),y) -> |0|#(*(x,y))
p30: *#(j(x),y) -> *#(x,y)
p31: *#(*(x,y),z) -> *#(x,*(y,z))
p32: *#(*(x,y),z) -> *#(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

  {p24, p27, p30, p31, p32}
  {p2, p3, p4, p5, p6, p7, p8, p9, p10, p12, p14, p15, p16}
  {p18, p19, p20}


-- Reduction pair.

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: *#(j(x),y) -> *#(x,y)
p4: *#(|1|(x),y) -> *#(x,y)
p5: *#(|0|(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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 + 1
          *_A(x1,x2) = x1 + x2 + 13
          j_A(x1) = x1 + 12
          |1|_A(x1) = x1 + 6
          |0|_A(x1) = x1 + 9
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 4
          opp_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        |1| = |0| = opp > * > j = |#| > *# = +
      
  
    partial status:
  
      pi(*#) = [1]
      pi(*) = [1]
      pi(j) = []
      pi(|1|) = [1]
      pi(|0|) = []
      pi(|#|) = []
      pi(+) = []
      pi(opp) = [1]

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: *#(j(x),y) -> *#(x,y)
p3: *#(|1|(x),y) -> *#(x,y)
p4: *#(|0|(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

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),y) -> *#(x,y)
p3: *#(|1|(x),y) -> *#(x,y)
p4: *#(j(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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 + 1
          *_A(x1,x2) = x1 + x2 + 13
          |0|_A(x1) = x1 + 3
          |1|_A(x1) = x1 + 6
          j_A(x1) = x1 + 12
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 4
          opp_A(x1) = x1 + 2
  
    base order 2:
  
      precedence:
      
        *# = * > |0| = j > + = opp > |1| > |#|
      
  
    partial status:
  
      pi(*#) = [1]
      pi(*) = []
      pi(|0|) = [1]
      pi(|1|) = [1]
      pi(j) = []
      pi(|#|) = []
      pi(+) = []
      pi(opp) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: *#(|0|(x),y) -> *#(x,y)
p2: *#(|1|(x),y) -> *#(x,y)
p3: *#(j(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

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),y) -> *#(x,y)
p2: *#(j(x),y) -> *#(x,y)
p3: *#(|1|(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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
          |0|_A(x1) = x1 + 3
          j_A(x1) = x1 + 12
          |1|_A(x1) = x1 + 6
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 4
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        + > * > |0| > opp > |#| > j = |1| > *#
      
  
    partial status:
  
      pi(*#) = [1]
      pi(|0|) = []
      pi(j) = [1]
      pi(|1|) = [1]
      pi(|#|) = []
      pi(+) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: *#(j(x),y) -> *#(x,y)
p2: *#(|1|(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: *#(j(x),y) -> *#(x,y)
p2: *#(|1|(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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
          j_A(x1) = x1 + 12
          |1|_A(x1) = x1 + 6
          |0|_A(x1) = x1 + 3
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 4
          opp_A(x1) = x1 + 4
          *_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        + > opp = * > *# = |0| > j > |1| > |#|
      
  
    partial status:
  
      pi(*#) = [1]
      pi(j) = [1]
      pi(|1|) = [1]
      pi(|0|) = []
      pi(|#|) = []
      pi(+) = []
      pi(opp) = [1]
      pi(*) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: *#(|1|(x),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

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),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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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 + 3
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 4
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        |0| = + = * > opp > *# > |1| > |#|
      
  
    partial status:
  
      pi(*#) = [1, 2]
      pi(|1|) = []
      pi(|0|) = []
      pi(|#|) = []
      pi(+) = [2]
      pi(opp) = [1]
      pi(*) = [1, 2]

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: +#(j(x),|1|(y)) -> +#(x,y)
p5: +#(|1|(x),j(y)) -> +#(x,y)
p6: +#(j(x),j(y)) -> +#(x,y)
p7: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p8: +#(|0|(x),j(y)) -> +#(x,y)
p9: +#(|1|(x),|1|(y)) -> +#(x,y)
p10: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p11: +#(|0|(x),|1|(y)) -> +#(x,y)
p12: +#(j(x),|0|(y)) -> +#(x,y)
p13: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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 + 4
          |0|_A(x1) = x1 + 1
          +_A(x1,x2) = x1 + x2
          j_A(x1) = x1 + 3
          |1|_A(x1) = x1 + 3
          |#|_A() = 0
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        +# > + > opp = * > |0| = j = |#| > |1|
      
  
    partial status:
  
      pi(+#) = []
      pi(|0|) = []
      pi(+) = [1, 2]
      pi(j) = [1]
      pi(|1|) = [1]
      pi(|#|) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

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: +#(j(x),|1|(y)) -> +#(x,y)
p4: +#(|1|(x),j(y)) -> +#(x,y)
p5: +#(j(x),j(y)) -> +#(x,y)
p6: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p7: +#(|0|(x),j(y)) -> +#(x,y)
p8: +#(|1|(x),|1|(y)) -> +#(x,y)
p9: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p10: +#(|0|(x),|1|(y)) -> +#(x,y)
p11: +#(j(x),|0|(y)) -> +#(x,y)
p12: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: +#(+(x,y),z) -> +#(y,z)
p2: +#(|1|(x),|0|(y)) -> +#(x,y)
p3: +#(j(x),|0|(y)) -> +#(x,y)
p4: +#(|0|(x),|1|(y)) -> +#(x,y)
p5: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p6: +#(|1|(x),|1|(y)) -> +#(x,y)
p7: +#(|0|(x),j(y)) -> +#(x,y)
p8: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p9: +#(j(x),j(y)) -> +#(x,y)
p10: +#(|1|(x),j(y)) -> +#(x,y)
p11: +#(j(x),|1|(y)) -> +#(x,y)
p12: +#(+(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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 + 4
          +_A(x1,x2) = x1 + x2
          |1|_A(x1) = x1 + 3
          |0|_A(x1) = x1 + 1
          j_A(x1) = x1 + 3
          |#|_A() = 0
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        +# = opp > + = |1| = |0| = * > j > |#|
      
  
    partial status:
  
      pi(+#) = []
      pi(+) = [1, 2]
      pi(|1|) = []
      pi(|0|) = []
      pi(j) = [1]
      pi(|#|) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: +#(+(x,y),z) -> +#(y,z)
p2: +#(j(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: +#(|0|(x),j(y)) -> +#(x,y)
p7: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p8: +#(j(x),j(y)) -> +#(x,y)
p9: +#(|1|(x),j(y)) -> +#(x,y)
p10: +#(j(x),|1|(y)) -> +#(x,y)
p11: +#(+(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: +#(+(x,y),z) -> +#(y,z)
p2: +#(+(x,y),z) -> +#(x,+(y,z))
p3: +#(j(x),|1|(y)) -> +#(x,y)
p4: +#(|1|(x),j(y)) -> +#(x,y)
p5: +#(j(x),j(y)) -> +#(x,y)
p6: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p7: +#(|0|(x),j(y)) -> +#(x,y)
p8: +#(|1|(x),|1|(y)) -> +#(x,y)
p9: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p10: +#(|0|(x),|1|(y)) -> +#(x,y)
p11: +#(j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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 + 4
          +_A(x1,x2) = x1 + x2
          j_A(x1) = x1 + 5
          |1|_A(x1) = x1 + 5
          |#|_A() = 0
          |0|_A(x1) = x1 + 3
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        +# > + > * > j = |1| = |0| > opp > |#|
      
  
    partial status:
  
      pi(+#) = [1]
      pi(+) = [1, 2]
      pi(j) = []
      pi(|1|) = []
      pi(|#|) = []
      pi(|0|) = []
      pi(opp) = []
      pi(*) = [1, 2]

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: +#(j(x),|1|(y)) -> +#(x,y)
p3: +#(|1|(x),j(y)) -> +#(x,y)
p4: +#(j(x),j(y)) -> +#(x,y)
p5: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p6: +#(|0|(x),j(y)) -> +#(x,y)
p7: +#(|1|(x),|1|(y)) -> +#(x,y)
p8: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p9: +#(|0|(x),|1|(y)) -> +#(x,y)
p10: +#(j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: +#(+(x,y),z) -> +#(x,+(y,z))
p2: +#(j(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: +#(|0|(x),j(y)) -> +#(x,y)
p7: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p8: +#(j(x),j(y)) -> +#(x,y)
p9: +#(|1|(x),j(y)) -> +#(x,y)
p10: +#(j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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 + 6
          +_A(x1,x2) = x1 + x2
          j_A(x1) = x1 + 5
          |0|_A(x1) = x1 + 3
          |1|_A(x1) = x1 + 5
          |#|_A() = 0
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        +# > + > * > |0| > opp > j = |1| > |#|
      
  
    partial status:
  
      pi(+#) = [1, 2]
      pi(+) = [1, 2]
      pi(j) = [1]
      pi(|0|) = []
      pi(|1|) = [1]
      pi(|#|) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: +#(j(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: +#(|0|(x),j(y)) -> +#(x,y)
p6: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p7: +#(j(x),j(y)) -> +#(x,y)
p8: +#(|1|(x),j(y)) -> +#(x,y)
p9: +#(j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: +#(j(x),|0|(y)) -> +#(x,y)
p2: +#(j(x),|1|(y)) -> +#(x,y)
p3: +#(|1|(x),j(y)) -> +#(x,y)
p4: +#(j(x),j(y)) -> +#(x,y)
p5: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p6: +#(|0|(x),j(y)) -> +#(x,y)
p7: +#(|1|(x),|1|(y)) -> +#(x,y)
p8: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p9: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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 + 6
          j_A(x1) = x1 + 5
          |0|_A(x1) = x1 + 3
          |1|_A(x1) = x1 + 5
          +_A(x1,x2) = x1 + x2
          |#|_A() = 0
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        +# > + > * > |0| > opp > j = |1| > |#|
      
  
    partial status:
  
      pi(+#) = [1]
      pi(j) = [1]
      pi(|0|) = []
      pi(|1|) = [1]
      pi(+) = [1, 2]
      pi(|#|) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: +#(j(x),|1|(y)) -> +#(x,y)
p2: +#(|1|(x),j(y)) -> +#(x,y)
p3: +#(j(x),j(y)) -> +#(x,y)
p4: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p5: +#(|0|(x),j(y)) -> +#(x,y)
p6: +#(|1|(x),|1|(y)) -> +#(x,y)
p7: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p8: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: +#(j(x),|1|(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: +#(|0|(x),j(y)) -> +#(x,y)
p6: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p7: +#(j(x),j(y)) -> +#(x,y)
p8: +#(|1|(x),j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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 + 6
          j_A(x1) = x1 + 5
          |1|_A(x1) = x1 + 5
          |0|_A(x1) = x1 + 3
          +_A(x1,x2) = x1 + x2
          |#|_A() = 0
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        +# > + > * > |0| > opp > j = |1| > |#|
      
  
    partial status:
  
      pi(+#) = [2]
      pi(j) = [1]
      pi(|1|) = [1]
      pi(|0|) = []
      pi(+) = [1, 2]
      pi(|#|) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

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)
p2: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p3: +#(|1|(x),|1|(y)) -> +#(x,y)
p4: +#(|0|(x),j(y)) -> +#(x,y)
p5: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p6: +#(j(x),j(y)) -> +#(x,y)
p7: +#(|1|(x),j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: +#(|0|(x),|1|(y)) -> +#(x,y)
p2: +#(|1|(x),j(y)) -> +#(x,y)
p3: +#(j(x),j(y)) -> +#(x,y)
p4: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p5: +#(|0|(x),j(y)) -> +#(x,y)
p6: +#(|1|(x),|1|(y)) -> +#(x,y)
p7: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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
          |0|_A(x1) = x1 + 1
          |1|_A(x1) = x1 + 3
          j_A(x1) = x1 + 3
          +_A(x1,x2) = x1 + x2
          |#|_A() = 0
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        +# > + > |0| = |1| = opp = * > j = |#|
      
  
    partial status:
  
      pi(+#) = []
      pi(|0|) = []
      pi(|1|) = [1]
      pi(j) = [1]
      pi(+) = [1, 2]
      pi(|#|) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: +#(|1|(x),j(y)) -> +#(x,y)
p2: +#(j(x),j(y)) -> +#(x,y)
p3: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p4: +#(|0|(x),j(y)) -> +#(x,y)
p5: +#(|1|(x),|1|(y)) -> +#(x,y)
p6: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

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: +#(|1|(x),j(y)) -> +#(x,y)
p2: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p3: +#(|1|(x),|1|(y)) -> +#(x,y)
p4: +#(|0|(x),j(y)) -> +#(x,y)
p5: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p6: +#(j(x),j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r3, r14, r19

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
          |1|_A(x1) = x1 + 9
          j_A(x1) = x1 + 12
          +_A(x1,x2) = x1 + 4
          |#|_A() = 0
          |0|_A(x1) = 10
          opp_A(x1) = 0
          *_A(x1,x2) = x1 + 11
  
    base order 2:
  
      precedence:
      
        j > * > |0| > |1| = + > +# = opp > |#|
      
  
    partial status:
  
      pi(+#) = []
      pi(|1|) = [1]
      pi(j) = [1]
      pi(+) = []
      pi(|#|) = []
      pi(|0|) = []
      pi(opp) = []
      pi(*) = [1]

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: +#(|1|(x),j(y)) -> +#(x,y)
p2: +#(|1|(x),|1|(y)) -> +#(+(x,y),|1|(|#|()))
p3: +#(|1|(x),|1|(y)) -> +#(x,y)
p4: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p5: +#(j(x),j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

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),j(y)) -> +#(x,y)
p2: +#(j(x),j(y)) -> +#(x,y)
p3: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p4: +#(|1|(x),|1|(y)) -> +#(x,y)
p5: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r3, r14, r19

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 + 1
          |1|_A(x1) = x1 + 11
          j_A(x1) = x1 + 9
          +_A(x1,x2) = x1 + 3
          |#|_A() = 0
          |0|_A(x1) = x1 + 4
          opp_A(x1) = x1 + 1
          *_A(x1,x2) = x1 + x2 + 12
  
    base order 2:
  
      precedence:
      
        +# = * > |1| = j > |0| > + = opp > |#|
      
  
    partial status:
  
      pi(+#) = [2]
      pi(|1|) = []
      pi(j) = []
      pi(+) = [1]
      pi(|#|) = []
      pi(|0|) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p4



-- SCC decomposition.

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

p1: +#(|1|(x),j(y)) -> +#(x,y)
p2: +#(j(x),j(y)) -> +#(x,y)
p3: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p4: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

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),j(y)) -> +#(x,y)
p2: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))
p3: +#(j(x),j(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r4, r5, r6, r10, r11, r12, r14, r15, r17, r19, r20

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 + 3
          |1|_A(x1) = 2
          j_A(x1) = x1 + 4
          +_A(x1,x2) = 5
          |#|_A() = 0
          |0|_A(x1) = 1
          opp_A(x1) = 3
          *_A(x1,x2) = x1 + x2 + 5
  
    base order 2:
  
      precedence:
      
        +# > j > opp > |1| > * > + = |0| > |#|
      
  
    partial status:
  
      pi(+#) = []
      pi(|1|) = []
      pi(j) = []
      pi(+) = []
      pi(|#|) = []
      pi(|0|) = []
      pi(opp) = []
      pi(*) = [1]

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: +#(|1|(x),j(y)) -> +#(x,y)
p2: +#(j(x),j(y)) -> +#(+(x,y),j(|#|()))

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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

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

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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r5, r6, r10, r14, r17, r19

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
          |1|_A(x1) = 2
          j_A(x1) = x1 + 7
          +_A(x1,x2) = 5
          |#|_A() = 0
          |0|_A(x1) = 1
          opp_A(x1) = 3
          *_A(x1,x2) = 6
  
    base order 2:
  
      precedence:
      
        +# = j > * > + > opp > |1| = |0| > |#|
      
  
    partial status:
  
      pi(+#) = []
      pi(|1|) = []
      pi(j) = []
      pi(+) = []
      pi(|#|) = []
      pi(|0|) = []
      pi(opp) = []
      pi(*) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

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

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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

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

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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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
          j_A(x1) = x1 + 3
          +_A(x1,x2) = x1 + x2
          |#|_A() = 0
          |0|_A(x1) = x1 + 1
          |1|_A(x1) = x1 + 3
          opp_A(x1) = x1
          *_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        +# > + > * > j = |0| = |1| = opp > |#|
      
  
    partial status:
  
      pi(+#) = [1, 2]
      pi(j) = []
      pi(+) = [1, 2]
      pi(|#|) = []
      pi(|0|) = [1]
      pi(|1|) = [1]
      pi(opp) = []
      pi(*) = [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: +#(|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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r19

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
          |1|_A(x1) = x1 + 3
          +_A(x1,x2) = x1 + x2
          |#|_A() = 0
          |0|_A(x1) = x1 + 1
          j_A(x1) = x1 + 3
          opp_A(x1) = 0
          *_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        * > +# > + > |0| > opp > |1| > j > |#|
      
  
    partial status:
  
      pi(+#) = [1, 2]
      pi(|1|) = []
      pi(+) = [1, 2]
      pi(|#|) = []
      pi(|0|) = [1]
      pi(j) = [1]
      pi(opp) = []
      pi(*) = [1, 2]

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: opp#(|0|(x)) -> opp#(x)
p2: opp#(j(x)) -> opp#(x)
p3: opp#(|1|(x)) -> opp#(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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:
          opp#_A(x1) = x1 + 7
          |0|_A(x1) = x1 + 3
          j_A(x1) = x1 + 12
          |1|_A(x1) = x1 + 6
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 4
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        + > * > |0| > opp > opp# > j = |1| > |#|
      
  
    partial status:
  
      pi(opp#) = [1]
      pi(|0|) = []
      pi(j) = [1]
      pi(|1|) = []
      pi(|#|) = []
      pi(+) = []
      pi(opp) = [1]
      pi(*) = [1]

The following rules are removed:

  p3



-- SCC decomposition.

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

p1: opp#(|0|(x)) -> opp#(x)
p2: opp#(j(x)) -> opp#(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: opp#(|0|(x)) -> opp#(x)
p2: opp#(j(x)) -> opp#(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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:
          opp#_A(x1) = x1
          |0|_A(x1) = x1 + 3
          j_A(x1) = x1 + 11
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 5
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 12
  
    base order 2:
  
      precedence:
      
        + > * > |0| > j > opp# = opp > |#|
      
  
    partial status:
  
      pi(opp#) = [1]
      pi(|0|) = []
      pi(j) = []
      pi(|#|) = []
      pi(+) = [1, 2]
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: opp#(j(x)) -> opp#(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: opp#(j(x)) -> opp#(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: +(|0|(x),j(y)) -> j(+(x,y))
r8: +(j(x),|0|(y)) -> j(+(x,y))
r9: +(|1|(x),|1|(y)) -> j(+(+(x,y),|1|(|#|())))
r10: +(j(x),j(y)) -> |1|(+(+(x,y),j(|#|())))
r11: +(|1|(x),j(y)) -> |0|(+(x,y))
r12: +(j(x),|1|(y)) -> |0|(+(x,y))
r13: +(+(x,y),z) -> +(x,+(y,z))
r14: opp(|#|()) -> |#|()
r15: opp(|0|(x)) -> |0|(opp(x))
r16: opp(|1|(x)) -> j(opp(x))
r17: opp(j(x)) -> |1|(opp(x))
r18: -(x,y) -> +(x,opp(y))
r19: *(|#|(),x) -> |#|()
r20: *(|0|(x),y) -> |0|(*(x,y))
r21: *(|1|(x),y) -> +(|0|(*(x,y)),y)
r22: *(j(x),y) -> -(|0|(*(x,y)),y)
r23: *(*(x,y),z) -> *(x,*(y,z))

The set of usable rules consists of

  r1, r2, r3, r14, r19

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:
          opp#_A(x1) = x1
          j_A(x1) = x1 + 12
          |0|_A(x1) = x1 + 3
          |#|_A() = 1
          +_A(x1,x2) = x1 + x2 + 4
          opp_A(x1) = x1 + 2
          *_A(x1,x2) = x1 + x2 + 13
  
    base order 2:
  
      precedence:
      
        + > * > |0| > opp > j > opp# > |#|
      
  
    partial status:
  
      pi(opp#) = [1]
      pi(j) = [1]
      pi(|0|) = []
      pi(|#|) = []
      pi(+) = []
      pi(opp) = [1]
      pi(*) = [1, 2]

The following rules are removed:

  p1

Then no dependency pair remains.