(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
   [ c(c(?x)) -> b(a(?x)),
     b(b(?x)) -> c(c(?x)),
     b(a(?x)) -> a(a(?x)),
     a(b(?x)) -> a(b(?x)),
     b(c(?x)) -> c(a(?x)),
     b(b(?x)) -> b(b(?x)),
     a(b(?x)) -> b(b(?x)),
     a(b(?x)) -> b(b(?x)),
     a(c(?x)) -> c(b(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ b(a(a(?x_2))) = c(c(a(?x_2))),
     b(c(a(?x_4))) = c(c(c(?x_4))),
     b(b(b(?x_5))) = c(c(b(?x_5))),
     b(a(b(?x_3))) = a(a(b(?x_3))),
     b(b(b(?x_6))) = a(a(b(?x_6))),
     b(b(b(?x_7))) = a(a(b(?x_7))),
     b(c(b(?x_8))) = a(a(c(?x_8))),
     a(c(c(?x_1))) = a(b(b(?x_1))),
     a(a(a(?x_2))) = a(b(a(?x_2))),
     a(c(a(?x_4))) = a(b(c(?x_4))),
     a(b(b(?x_5))) = a(b(b(?x_5))),
     b(b(a(?x))) = c(a(c(?x))),
     b(c(c(?x_1))) = b(b(b(?x_1))),
     b(a(a(?x_2))) = b(b(a(?x_2))),
     b(c(a(?x_4))) = b(b(c(?x_4))),
     a(c(c(?x_1))) = b(b(b(?x_1))),
     a(a(a(?x_2))) = b(b(a(?x_2))),
     a(c(a(?x_4))) = b(b(c(?x_4))),
     a(b(b(?x_5))) = b(b(b(?x_5))),
     a(c(c(?x_1))) = b(b(b(?x_1))),
     a(a(a(?x_2))) = b(b(a(?x_2))),
     a(c(a(?x_4))) = b(b(c(?x_4))),
     a(b(b(?x_5))) = b(b(b(?x_5))),
     a(b(a(?x))) = c(b(c(?x))),
     c(b(a(?x))) = b(a(c(?x))),
     b(c(c(?x))) = c(c(b(?x))),
     b(b(b(?x))) = b(b(b(?x))) ]
Outer CPs:
   [ c(c(?x_1)) = b(b(?x_1)),
     a(b(?x_3)) = b(b(?x_3)),
     a(b(?x_3)) = b(b(?x_3)),
     b(b(?x_6)) = b(b(?x_6)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
inner CP cond (upside-parallel)
innter CP Cond (outside)
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ c(b(a(?x_1))) = b(a(c(?x_1))),
     b(a(b(a(?x_1)))) = c(b(a(c(?x_1)))),
     c(a(b(a(?x_1)))) = b(b(a(c(?x_1)))),
     c(b(b(a(?x_1)))) = a(b(a(c(?x_1)))),
     b(a(c(?x))) = c(b(a(?x))),
     c(a(c(?x))) = b(b(a(?x))),
     c(b(c(?x))) = a(b(a(?x))),
     b(b(?x)) = c(c(?x)),
     b(c(c(?x_1))) = c(c(b(?x_1))),
     b(a(a(?x_3))) = c(c(a(?x_3))),
     b(c(a(?x_5))) = c(c(c(?x_5))),
     b(b(b(?x_6))) = c(c(b(?x_6))),
     c(c(c(c(?x_1)))) = b(c(c(b(?x_1)))),
     c(c(a(a(?x_3)))) = b(c(c(a(?x_3)))),
     c(c(c(a(?x_5)))) = b(c(c(c(?x_5)))),
     c(c(b(b(?x_6)))) = b(c(c(b(?x_6)))),
     a(b(c(c(?x_1)))) = a(c(c(b(?x_1)))),
     a(b(a(a(?x_3)))) = a(c(c(a(?x_3)))),
     a(b(c(a(?x_5)))) = a(c(c(c(?x_5)))),
     a(b(b(b(?x_6)))) = a(c(c(b(?x_6)))),
     b(b(c(c(?x_1)))) = b(c(c(b(?x_1)))),
     b(b(a(a(?x_3)))) = b(c(c(a(?x_3)))),
     b(b(c(a(?x_5)))) = b(c(c(c(?x_5)))),
     b(b(b(b(?x_6)))) = b(c(c(b(?x_6)))),
     b(b(c(c(?x_1)))) = a(c(c(b(?x_1)))),
     b(b(a(a(?x_3)))) = a(c(c(a(?x_3)))),
     b(b(c(a(?x_5)))) = a(c(c(c(?x_5)))),
     b(b(b(b(?x_6)))) = a(c(c(b(?x_6)))),
     c(c(b(?x))) = b(c(c(?x))),
     a(b(b(?x))) = a(c(c(?x))),
     b(b(b(?x))) = b(c(c(?x))),
     b(b(b(?x))) = a(c(c(?x))),
     b(a(b(?x_4))) = a(a(b(?x_4))),
     b(b(b(?x_7))) = a(a(b(?x_7))),
     b(c(b(?x_9))) = a(a(c(?x_9))),
     c(c(a(b(?x_4)))) = b(a(a(b(?x_4)))),
     c(c(b(b(?x_7)))) = b(a(a(b(?x_7)))),
     c(c(c(b(?x_9)))) = b(a(a(c(?x_9)))),
     a(b(a(b(?x_4)))) = a(a(a(b(?x_4)))),
     a(b(b(b(?x_7)))) = a(a(a(b(?x_7)))),
     a(b(c(b(?x_9)))) = a(a(a(c(?x_9)))),
     b(b(a(b(?x_4)))) = b(a(a(b(?x_4)))),
     b(b(b(b(?x_7)))) = b(a(a(b(?x_7)))),
     b(b(c(b(?x_9)))) = b(a(a(c(?x_9)))),
     b(b(a(b(?x_4)))) = a(a(a(b(?x_4)))),
     b(b(b(b(?x_7)))) = a(a(a(b(?x_7)))),
     b(b(c(b(?x_9)))) = a(a(a(c(?x_9)))),
     c(c(a(?x))) = b(a(a(?x))),
     a(b(a(?x))) = a(a(a(?x))),
     b(b(a(?x))) = b(a(a(?x))),
     b(b(a(?x))) = a(a(a(?x))),
     b(b(?x)) = a(b(?x)),
     a(c(c(?x_3))) = a(b(b(?x_3))),
     a(a(a(?x_4))) = a(b(a(?x_4))),
     a(c(a(?x_5))) = a(b(c(?x_5))),
     a(b(b(?x_6))) = a(b(b(?x_6))),
     a(a(c(c(?x_3)))) = b(a(b(b(?x_3)))),
     a(a(a(a(?x_4)))) = b(a(b(a(?x_4)))),
     a(a(c(a(?x_5)))) = b(a(b(c(?x_5)))),
     a(a(b(b(?x_6)))) = b(a(b(b(?x_6)))),
     a(a(b(?x))) = b(a(b(?x))),
     b(b(a(?x_2))) = c(a(c(?x_2))),
     c(c(b(a(?x_2)))) = b(c(a(c(?x_2)))),
     a(b(b(a(?x_2)))) = a(c(a(c(?x_2)))),
     b(b(b(a(?x_2)))) = b(c(a(c(?x_2)))),
     b(b(b(a(?x_2)))) = a(c(a(c(?x_2)))),
     c(c(c(?x))) = b(c(a(?x))),
     a(b(c(?x))) = a(c(a(?x))),
     b(b(c(?x))) = b(c(a(?x))),
     b(b(c(?x))) = a(c(a(?x))),
     c(c(?x)) = b(b(?x)),
     b(b(b(?x_1))) = b(b(b(?x_1))),
     b(c(c(?x_3))) = b(b(b(?x_3))),
     b(a(a(?x_4))) = b(b(a(?x_4))),
     b(c(a(?x_6))) = b(b(c(?x_6))),
     b(b(b(b(?x_1)))) = b(b(b(b(?x_1)))),
     b(b(c(c(?x_3)))) = b(b(b(b(?x_3)))),
     b(b(a(a(?x_4)))) = b(b(b(a(?x_4)))),
     b(b(c(a(?x_6)))) = b(b(b(c(?x_6)))),
     c(c(b(b(?x_1)))) = b(b(b(b(?x_1)))),
     c(c(c(c(?x_3)))) = b(b(b(b(?x_3)))),
     c(c(a(a(?x_4)))) = b(b(b(a(?x_4)))),
     c(c(c(a(?x_6)))) = b(b(b(c(?x_6)))),
     a(b(b(b(?x_1)))) = a(b(b(b(?x_1)))),
     a(b(c(c(?x_3)))) = a(b(b(b(?x_3)))),
     a(b(a(a(?x_4)))) = a(b(b(a(?x_4)))),
     a(b(c(a(?x_6)))) = a(b(b(c(?x_6)))),
     b(b(b(b(?x_1)))) = a(b(b(b(?x_1)))),
     b(b(c(c(?x_3)))) = a(b(b(b(?x_3)))),
     b(b(a(a(?x_4)))) = a(b(b(a(?x_4)))),
     b(b(c(a(?x_6)))) = a(b(b(c(?x_6)))),
     c(c(b(?x))) = b(b(b(?x))),
     b(b(b(?x))) = a(b(b(?x))),
     a(b(?x)) = b(b(?x)),
     b(b(?x)) = b(b(?x)),
     a(c(c(?x_3))) = b(b(b(?x_3))),
     a(a(a(?x_4))) = b(b(a(?x_4))),
     a(c(a(?x_6))) = b(b(c(?x_6))),
     a(b(b(?x_7))) = b(b(b(?x_7))),
     a(a(c(c(?x_3)))) = b(b(b(b(?x_3)))),
     a(a(a(a(?x_4)))) = b(b(b(a(?x_4)))),
     a(a(c(a(?x_6)))) = b(b(b(c(?x_6)))),
     a(a(b(b(?x_7)))) = b(b(b(b(?x_7)))),
     a(a(b(?x))) = b(b(b(?x))),
     a(b(a(?x_2))) = c(b(c(?x_2))),
     a(a(b(a(?x_2)))) = b(c(b(c(?x_2)))),
     a(a(c(?x))) = b(c(b(?x))) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <b(a(a(?x_2))), c(c(a(?x_2)))> by Rules <2, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([],0)]>
Critical Pair <b(c(a(?x_4))), c(c(c(?x_4)))> by Rules <4, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([],4)], [([(c,1)],0),([(c,1)],2)]>
Critical Pair <b(b(b(?x_5))), c(c(b(?x_5)))> by Rules <5, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([],1)], []>
Critical Pair <b(a(b(?x_3))), a(a(b(?x_3)))> by Rules <3, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[([],2)], []>
Critical Pair <b(b(b(?x_6))), a(a(b(?x_6)))> by Rules <6, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([(a,1)],7),([],7)]>
 joinable by a reduction of rules <[], [([(a,1)],7),([],6)]>
 joinable by a reduction of rules <[], [([(a,1)],6),([],7)]>
 joinable by a reduction of rules <[], [([(a,1)],6),([],6)]>
Critical Pair <b(b(b(?x_7))), a(a(b(?x_7)))> by Rules <7, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([(a,1)],7),([],7)]>
 joinable by a reduction of rules <[], [([(a,1)],7),([],6)]>
 joinable by a reduction of rules <[], [([(a,1)],6),([],7)]>
 joinable by a reduction of rules <[], [([(a,1)],6),([],6)]>
Critical Pair <b(c(b(?x_8))), a(a(c(?x_8)))> by Rules <8, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[([],4),([(c,1)],7)], [([(a,1)],8),([],8)]>
 joinable by a reduction of rules <[([],4),([(c,1)],6)], [([(a,1)],8),([],8)]>
Critical Pair <a(c(c(?x_1))), a(b(b(?x_1)))> by Rules <1, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([(a,1)],1)]>
Critical Pair <a(a(a(?x_2))), a(b(a(?x_2)))> by Rules <2, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([(a,1)],2)]>
Critical Pair <a(c(a(?x_4))), a(b(c(?x_4)))> by Rules <4, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([(a,1)],4)]>
Critical Pair <a(b(b(?x_5))), a(b(b(?x_5)))> by Rules <5, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <b(b(a(?x))), c(a(c(?x)))> by Rules <0, 4> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([(c,1)],8),([],0),([(b,1)],7),([(b,1)],1),([(b,1)],0)]>
 joinable by a reduction of rules <[], [([(c,1)],8),([],0),([(b,1)],6),([(b,1)],1),([(b,1)],0)]>
Critical Pair <b(c(c(?x_1))), b(b(b(?x_1)))> by Rules <1, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([(b,1)],1)]>
Critical Pair <b(a(a(?x_2))), b(b(a(?x_2)))> by Rules <2, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([(b,1)],2)]>
Critical Pair <b(c(a(?x_4))), b(b(c(?x_4)))> by Rules <4, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([(b,1)],4)]>
Critical Pair <a(c(c(?x_1))), b(b(b(?x_1)))> by Rules <1, 6> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],0),([],7)], [([(b,1)],1),([(b,1)],0)]>
 joinable by a reduction of rules <[([(a,1)],0),([],6)], [([(b,1)],1),([(b,1)],0)]>
Critical Pair <a(a(a(?x_2))), b(b(a(?x_2)))> by Rules <2, 6> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([(b,1)],2),([],2)]>
Critical Pair <a(c(a(?x_4))), b(b(c(?x_4)))> by Rules <4, 6> preceded by [(a,1)]
 joinable by a reduction of rules <[([],8)], [([],1),([(c,1)],0)]>
 joinable by a reduction of rules <[([],8),([(c,1)],2)], [([(b,1)],4),([],4)]>
Critical Pair <a(b(b(?x_5))), b(b(b(?x_5)))> by Rules <5, 6> preceded by [(a,1)]
 joinable by a reduction of rules <[([],7)], []>
 joinable by a reduction of rules <[([],6)], []>
Critical Pair <a(c(c(?x_1))), b(b(b(?x_1)))> by Rules <1, 7> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],0),([],7)], [([(b,1)],1),([(b,1)],0)]>
 joinable by a reduction of rules <[([(a,1)],0),([],6)], [([(b,1)],1),([(b,1)],0)]>
Critical Pair <a(a(a(?x_2))), b(b(a(?x_2)))> by Rules <2, 7> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([(b,1)],2),([],2)]>
Critical Pair <a(c(a(?x_4))), b(b(c(?x_4)))> by Rules <4, 7> preceded by [(a,1)]
 joinable by a reduction of rules <[([],8)], [([],1),([(c,1)],0)]>
 joinable by a reduction of rules <[([],8),([(c,1)],2)], [([(b,1)],4),([],4)]>
Critical Pair <a(b(b(?x_5))), b(b(b(?x_5)))> by Rules <5, 7> preceded by [(a,1)]
 joinable by a reduction of rules <[([],7)], []>
 joinable by a reduction of rules <[([],6)], []>
Critical Pair <a(b(a(?x))), c(b(c(?x)))> by Rules <0, 8> preceded by [(a,1)]
 joinable by a reduction of rules <[([],7),([],1)], [([(c,1)],4)]>
 joinable by a reduction of rules <[([],6),([],1)], [([(c,1)],4)]>
 joinable by a reduction of rules <[([],7),([(b,1)],2)], [([(c,1)],4),([],0)]>
 joinable by a reduction of rules <[([],6),([(b,1)],2)], [([(c,1)],4),([],0)]>
Critical Pair <c(b(a(?x))), b(a(c(?x)))> by Rules <0, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([(b,1)],8),([],4),([(c,1)],7),([(c,1)],1),([(c,1)],0)]>
 joinable by a reduction of rules <[], [([(b,1)],8),([],4),([(c,1)],6),([(c,1)],1),([(c,1)],0)]>
 joinable by a reduction of rules <[], [([],2),([(a,1)],8),([],8),([(c,1)],1),([(c,1)],0)]>
Critical Pair <b(c(c(?x))), c(c(b(?x)))> by Rules <1, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([],4),([(c,1)],8)], []>
Critical Pair <b(b(b(?x))), b(b(b(?x)))> by Rules <5, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <b(b(?x_5)), c(c(?x_5))> by Rules <5, 1> preceded by []
 joinable by a reduction of rules <[([],1)], []>
Critical Pair <b(b(?x_6)), a(b(?x_6))> by Rules <6, 3> preceded by []
 joinable by a reduction of rules <[], [([],7)]>
 joinable by a reduction of rules <[], [([],6)]>
Critical Pair <b(b(?x_7)), a(b(?x_7))> by Rules <7, 3> preceded by []
 joinable by a reduction of rules <[], [([],7)]>
 joinable by a reduction of rules <[], [([],6)]>
Critical Pair <b(b(?x_7)), b(b(?x_7))> by Rules <7, 6> preceded by []
 joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 13 rules by 3 steps unfolding
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Ordering(rpo), check by Tree-Automata Approximation (failure)
check by Interpretation(mod2) (failure)
check by Descendants-Approximation, check by Ordering(poly) (failure)
unknown Non-Confluence
unknown Huet (modulo AC)
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
   [ c(c(?x)) -> b(a(?x)),
     b(b(?x)) -> c(c(?x)),
     b(a(?x)) -> a(a(?x)),
     b(c(?x)) -> c(a(?x)),
     a(b(?x)) -> b(b(?x)),
     a(b(?x)) -> b(b(?x)),
     a(c(?x)) -> c(b(?x)) ]
P:
   [ a(b(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
   [ c(c(?x)) -> b(a(?x)),
     b(b(?x)) -> c(c(?x)),
     b(a(?x)) -> a(a(?x)),
     b(c(?x)) -> c(a(?x)),
     a(b(?x)) -> b(b(?x)),
     a(b(?x)) -> b(b(?x)),
     a(c(?x)) -> c(b(?x)) ]
P:
   [ a(b(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
   [ c(c(?x)) -> b(a(?x)),
     b(b(?x)) -> c(c(?x)),
     b(a(?x)) -> a(a(?x)),
     b(c(?x)) -> c(a(?x)),
     a(b(?x)) -> b(b(?x)),
     a(b(?x)) -> b(b(?x)),
     a(c(?x)) -> c(b(?x)) ]
P:
   [ a(b(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
Check relative termination:
   [ c(c(?x)) -> b(a(?x)),
     b(b(?x)) -> c(c(?x)),
     b(a(?x)) -> a(a(?x)),
     b(c(?x)) -> c(a(?x)),
     a(b(?x)) -> b(b(?x)),
     a(b(?x)) -> b(b(?x)),
     a(c(?x)) -> c(b(?x)) ]
   [ a(b(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)+(1)*x1
  c:= (1)*x1
retract b(b(?x)) -> c(c(?x))
retract b(a(?x)) -> a(a(?x))
retract b(c(?x)) -> c(a(?x))
Polynomial Interpretation:
  a:= (2)*x1*x1
  b:= (1)+(2)*x1*x1
  c:= (1)+(2)*x1*x1
retract c(c(?x)) -> b(a(?x))
retract b(b(?x)) -> c(c(?x))
retract b(a(?x)) -> a(a(?x))
retract b(c(?x)) -> c(a(?x))
Polynomial Interpretation:
  a:= (2)+(2)*x1
  b:= (2)*x1
  c:= (3)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
success
S:
   [ c(c(?x)) -> b(a(?x)),
     b(b(?x)) -> c(c(?x)),
     b(a(?x)) -> a(a(?x)),
     b(c(?x)) -> c(a(?x)),
     a(b(?x)) -> b(b(?x)),
     a(b(?x)) -> b(b(?x)),
     a(c(?x)) -> c(b(?x)) ]
P:
   [ a(b(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
Success
Reduction-Preserving Completion
Direct Methods: CR

Combined result: CR
1025.trs: Success(CR)
YES
(5189 msec.)