(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
   [ c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)),
     c(a(?x)) -> b(c(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ c(c(c(?x_1))) = a(c(b(?x_1))),
     c(c(a(?x_4))) = a(c(a(?x_4))),
     c(b(c(?x_6))) = a(c(c(?x_6))),
     c(b(c(?x_7))) = a(c(a(?x_7))),
     c(c(b(?x_2))) = c(c(a(?x_2))),
     c(c(c(?x_5))) = c(c(c(?x_5))),
     b(b(a(?x_3))) = c(b(b(?x_3))),
     a(c(b(?x_2))) = b(a(a(?x_2))),
     a(c(c(?x_5))) = b(a(c(?x_5))),
     c(b(a(?x_3))) = c(a(b(?x_3))),
     b(a(c(?x))) = c(c(c(?x))),
     b(c(c(?x_1))) = c(c(b(?x_1))),
     b(c(a(?x_4))) = c(c(a(?x_4))),
     b(b(c(?x_6))) = c(c(c(?x_6))),
     b(b(c(?x_7))) = c(c(a(?x_7))),
     c(a(c(?x))) = b(c(c(?x))),
     c(c(c(?x_1))) = b(c(b(?x_1))),
     c(c(a(?x_4))) = b(c(a(?x_4))),
     c(b(c(?x_7))) = b(c(a(?x_7))),
     c(b(a(?x_3))) = b(c(b(?x_3))),
     c(a(c(?x))) = a(c(c(?x))),
     c(b(c(?x))) = b(c(c(?x))) ]
Outer CPs:
   [ a(c(?x)) = b(c(?x)),
     c(a(?x_4)) = b(c(?x_4)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ b(c(?x)) = a(c(?x)),
     c(a(c(?x_1))) = a(c(c(?x_1))),
     c(c(c(?x_2))) = a(c(b(?x_2))),
     c(c(a(?x_5))) = a(c(a(?x_5))),
     c(b(c(?x_7))) = a(c(c(?x_7))),
     c(b(c(?x_8))) = a(c(a(?x_8))),
     a(c(a(c(?x_1)))) = c(a(c(c(?x_1)))),
     a(c(c(c(?x_2)))) = c(a(c(b(?x_2)))),
     a(c(c(a(?x_5)))) = c(a(c(a(?x_5)))),
     a(c(b(c(?x_7)))) = c(a(c(c(?x_7)))),
     a(c(b(c(?x_8)))) = c(a(c(a(?x_8)))),
     c(c(a(c(?x_1)))) = b(a(c(c(?x_1)))),
     c(c(c(c(?x_2)))) = b(a(c(b(?x_2)))),
     c(c(c(a(?x_5)))) = b(a(c(a(?x_5)))),
     c(c(b(c(?x_7)))) = b(a(c(c(?x_7)))),
     c(c(b(c(?x_8)))) = b(a(c(a(?x_8)))),
     b(c(a(c(?x_1)))) = c(a(c(c(?x_1)))),
     b(c(c(c(?x_2)))) = c(a(c(b(?x_2)))),
     b(c(c(a(?x_5)))) = c(a(c(a(?x_5)))),
     b(c(b(c(?x_7)))) = c(a(c(c(?x_7)))),
     b(c(b(c(?x_8)))) = c(a(c(a(?x_8)))),
     a(c(c(?x))) = c(a(c(?x))),
     c(c(c(?x))) = b(a(c(?x))),
     b(c(c(?x))) = c(a(c(?x))),
     c(c(b(?x_3))) = c(c(a(?x_3))),
     c(c(c(?x_6))) = c(c(c(?x_6))),
     a(c(c(b(?x_3)))) = c(c(c(a(?x_3)))),
     a(c(c(c(?x_6)))) = c(c(c(c(?x_6)))),
     c(c(c(b(?x_3)))) = b(c(c(a(?x_3)))),
     c(c(c(c(?x_6)))) = b(c(c(c(?x_6)))),
     b(c(c(b(?x_3)))) = c(c(c(a(?x_3)))),
     b(c(c(c(?x_6)))) = c(c(c(c(?x_6)))),
     a(c(b(?x))) = c(c(c(?x))),
     c(c(b(?x))) = b(c(c(?x))),
     b(c(b(?x))) = c(c(c(?x))),
     b(b(a(?x_4))) = c(b(b(?x_4))),
     c(c(b(a(?x_4)))) = c(c(b(b(?x_4)))),
     b(a(b(a(?x_4)))) = a(c(b(b(?x_4)))),
     c(c(a(?x))) = c(c(b(?x))),
     b(a(a(?x))) = a(c(b(?x))),
     a(c(b(?x_4))) = b(a(a(?x_4))),
     a(c(c(?x_6))) = b(a(c(?x_6))),
     c(b(c(b(?x_4)))) = b(b(a(a(?x_4)))),
     c(b(c(c(?x_6)))) = b(b(a(c(?x_6)))),
     c(a(c(b(?x_4)))) = c(b(a(a(?x_4)))),
     c(a(c(c(?x_6)))) = c(b(a(c(?x_6)))),
     b(c(c(b(?x_4)))) = c(b(a(a(?x_4)))),
     b(c(c(c(?x_6)))) = c(b(a(c(?x_6)))),
     c(b(b(?x))) = b(b(a(?x))),
     c(a(b(?x))) = c(b(a(?x))),
     b(c(b(?x))) = c(b(a(?x))),
     b(c(?x)) = c(a(?x)),
     c(b(a(?x_5))) = c(a(b(?x_5))),
     a(c(b(a(?x_5)))) = c(c(a(b(?x_5)))),
     c(c(b(a(?x_5)))) = b(c(a(b(?x_5)))),
     b(c(b(a(?x_5)))) = c(c(a(b(?x_5)))),
     a(c(a(?x))) = c(c(a(?x))),
     c(c(a(?x))) = b(c(a(?x))),
     b(c(a(?x))) = c(c(a(?x))),
     b(a(c(?x_2))) = c(c(c(?x_2))),
     b(c(c(?x_3))) = c(c(b(?x_3))),
     b(b(c(?x_7))) = c(c(c(?x_7))),
     b(b(c(?x_8))) = c(c(a(?x_8))),
     c(c(a(c(?x_2)))) = c(c(c(c(?x_2)))),
     c(c(c(c(?x_3)))) = c(c(c(b(?x_3)))),
     c(c(c(a(?x_6)))) = c(c(c(a(?x_6)))),
     c(c(b(c(?x_7)))) = c(c(c(c(?x_7)))),
     c(c(b(c(?x_8)))) = c(c(c(a(?x_8)))),
     b(a(a(c(?x_2)))) = a(c(c(c(?x_2)))),
     b(a(c(c(?x_3)))) = a(c(c(b(?x_3)))),
     b(a(c(a(?x_6)))) = a(c(c(a(?x_6)))),
     b(a(b(c(?x_7)))) = a(c(c(c(?x_7)))),
     b(a(b(c(?x_8)))) = a(c(c(a(?x_8)))),
     b(a(c(?x))) = a(c(c(?x))),
     a(c(?x)) = b(c(?x)),
     c(b(c(?x_1))) = b(c(c(?x_1))),
     c(a(c(?x_2))) = b(c(c(?x_2))),
     c(c(c(?x_3))) = b(c(b(?x_3))),
     c(b(c(?x_8))) = b(c(a(?x_8))),
     b(c(b(c(?x_1)))) = c(b(c(c(?x_1)))),
     b(c(a(c(?x_2)))) = c(b(c(c(?x_2)))),
     b(c(c(c(?x_3)))) = c(b(c(b(?x_3)))),
     b(c(c(a(?x_6)))) = c(b(c(a(?x_6)))),
     b(c(b(c(?x_8)))) = c(b(c(a(?x_8)))),
     a(c(b(c(?x_1)))) = c(b(c(c(?x_1)))),
     a(c(a(c(?x_2)))) = c(b(c(c(?x_2)))),
     a(c(c(c(?x_3)))) = c(b(c(b(?x_3)))),
     a(c(c(a(?x_6)))) = c(b(c(a(?x_6)))),
     a(c(b(c(?x_8)))) = c(b(c(a(?x_8)))),
     c(c(b(c(?x_1)))) = b(b(c(c(?x_1)))),
     c(c(a(c(?x_2)))) = b(b(c(c(?x_2)))),
     c(c(c(c(?x_3)))) = b(b(c(b(?x_3)))),
     c(c(c(a(?x_6)))) = b(b(c(a(?x_6)))),
     c(c(b(c(?x_8)))) = b(b(c(a(?x_8)))),
     b(c(c(?x))) = c(b(c(?x))),
     a(c(c(?x))) = c(b(c(?x))),
     c(c(c(?x))) = b(b(c(?x))),
     c(a(?x)) = b(c(?x)),
     c(b(a(?x_5))) = b(c(b(?x_5))),
     a(c(b(a(?x_5)))) = c(b(c(b(?x_5)))),
     c(c(b(a(?x_5)))) = b(b(c(b(?x_5)))),
     b(c(b(a(?x_5)))) = c(b(c(b(?x_5)))),
     a(c(a(?x))) = c(b(c(?x))),
     c(c(a(?x))) = b(b(c(?x))),
     b(c(a(?x))) = c(b(c(?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 <c(c(c(?x_1))), a(c(b(?x_1)))> by Rules <1, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([],0)], [([(a,1)],1)]>
Critical Pair <c(c(a(?x_4))), a(c(a(?x_4)))> by Rules <4, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([],0)], []>
Critical Pair <c(b(c(?x_6))), a(c(c(?x_6)))> by Rules <6, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],5),([],0)], []>
 joinable by a reduction of rules <[([],1),([],0)], []>
Critical Pair <c(b(c(?x_7))), a(c(a(?x_7)))> by Rules <7, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],5),([],0)], [([(a,1)],7),([(a,1)],5)]>
 joinable by a reduction of rules <[([],1),([],0)], [([(a,1)],7),([(a,1)],5)]>
Critical Pair <c(c(b(?x_2))), c(c(a(?x_2)))> by Rules <2, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],1),([(c,1)],6)], [([(c,1)],7)]>
 joinable by a reduction of rules <[([(c,1)],1)], [([(c,1)],7),([(c,1)],5)]>
 joinable by a reduction of rules <[([(c,1)],1)], [([(c,1)],7),([],1)]>
Critical Pair <c(c(c(?x_5))), c(c(c(?x_5)))> by Rules <5, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <b(b(a(?x_3))), c(b(b(?x_3)))> by Rules <3, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],2),([],5)], [([],1)]>
 joinable by a reduction of rules <[([(b,1)],2)], [([],1),([],6)]>
Critical Pair <a(c(b(?x_2))), b(a(a(?x_2)))> by Rules <2, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],2),([(c,1)],2),([],0)]>
Critical Pair <a(c(c(?x_5))), b(a(c(?x_5)))> by Rules <5, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],6),([],3)], []>
Critical Pair <c(b(a(?x_3))), c(a(b(?x_3)))> by Rules <3, 4> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([(c,1)],3)]>
Critical Pair <b(a(c(?x))), c(c(c(?x)))> by Rules <0, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[([],2)], [([(c,1)],6)]>
Critical Pair <b(c(c(?x_1))), c(c(b(?x_1)))> by Rules <1, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[([],5)], [([(c,1)],1)]>
Critical Pair <b(c(a(?x_4))), c(c(a(?x_4)))> by Rules <4, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[([],5)], []>
 joinable by a reduction of rules <[], [([],6)]>
Critical Pair <b(b(c(?x_6))), c(c(c(?x_6)))> by Rules <6, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],5)], [([],6)]>
Critical Pair <b(b(c(?x_7))), c(c(a(?x_7)))> by Rules <7, 5> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([],6),([(b,1)],7)]>
 joinable by a reduction of rules <[([(b,1)],5),([],5)], [([(c,1)],7),([(c,1)],5)]>
 joinable by a reduction of rules <[([(b,1)],5),([],5)], [([(c,1)],7),([],1)]>
Critical Pair <c(a(c(?x))), b(c(c(?x)))> by Rules <0, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([],7)], []>
Critical Pair <c(c(c(?x_1))), b(c(b(?x_1)))> by Rules <1, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([],6)], [([(b,1)],1)]>
Critical Pair <c(c(a(?x_4))), b(c(a(?x_4)))> by Rules <4, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([],6)], []>
 joinable by a reduction of rules <[], [([],5)]>
Critical Pair <c(b(c(?x_7))), b(c(a(?x_7)))> by Rules <7, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([],5),([(c,1)],7)]>
 joinable by a reduction of rules <[([(c,1)],5),([],6)], [([(b,1)],7),([(b,1)],5)]>
 joinable by a reduction of rules <[([],1),([],6)], [([(b,1)],7),([(b,1)],5)]>
Critical Pair <c(b(a(?x_3))), b(c(b(?x_3)))> by Rules <3, 7> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],2)], [([],5)]>
Critical Pair <c(a(c(?x))), a(c(c(?x)))> by Rules <0, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([],7),([(b,1)],0)], [([(a,1)],6),([],3)]>
Critical Pair <c(b(c(?x))), b(c(c(?x)))> by Rules <6, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],5)], [([],5)]>
 joinable by a reduction of rules <[([],1)], [([],5)]>
Critical Pair <b(c(?x_6)), a(c(?x_6))> by Rules <6, 0> preceded by []
 joinable by a reduction of rules <[([],5),([],0)], []>
Critical Pair <b(c(?x_7)), c(a(?x_7))> by Rules <7, 4> preceded by []
 joinable by a reduction of rules <[], [([],7)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 16 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)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> b(c(?x)) ]
P:
   [ c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> b(c(?x)) ]
P:
   [ c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> b(c(?x)) ]
P:
   [ c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)) ]
Check relative termination:
   [ c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> b(c(?x)) ]
   [ c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)*x1
  c:= (1)+(1)*x1
retract c(c(?x)) -> a(c(?x))
retract c(c(?x)) -> b(c(?x))
Polynomial Interpretation:
  a:= (2)+(2)*x1
  b:= (2)*x1
  c:= (1)*x1
retract c(c(?x)) -> a(c(?x))
retract b(a(?x)) -> c(b(?x))
retract c(a(?x)) -> b(c(?x))
retract c(c(?x)) -> b(c(?x))
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)+(2)*x1*x1
  c:= (1)*x1*x1
retract c(c(?x)) -> a(c(?x))
retract c(b(?x)) -> c(c(?x))
retract b(a(?x)) -> c(b(?x))
retract c(a(?x)) -> b(c(?x))
retract b(c(?x)) -> c(c(?x))
retract c(c(?x)) -> b(c(?x))
Polynomial Interpretation:
  a:= (2)*x1
  b:= (1)+(1)*x1
  c:= (1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)),
     c(a(?x)) -> b(c(?x)) ]
Sort Assignment:
 a : 15=>15
 b : 15=>15
 c : 15=>15
maximal types: {15}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)),
     c(a(?x)) -> b(c(?x)) ]
Layer Preserving Decomposition failed: Can't judge

Try Commutative Decomposition for...
   [ c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)),
     b(a(?x)) -> c(b(?x)),
     a(b(?x)) -> b(a(?x)),
     c(a(?x)) -> c(a(?x)),
     b(c(?x)) -> c(c(?x)),
     c(c(?x)) -> b(c(?x)),
     c(a(?x)) -> b(c(?x)) ]
Outside Critical Pair: <b(c(?x_6)), a(c(?x_6))> by Rules <6, 0>
develop reducts from lhs term...
<{5}, c(c(?x_6))>
<{}, b(c(?x_6))>
develop reducts from rhs term...
<{}, a(c(?x_6))>
Outside Critical Pair: <b(c(?x_7)), c(a(?x_7))> by Rules <7, 4>
develop reducts from lhs term...
<{5}, c(c(?x_7))>
<{}, b(c(?x_7))>
develop reducts from rhs term...
<{7}, b(c(?x_7))>
<{4}, c(a(?x_7))>
<{}, c(a(?x_7))>
Inside Critical Pair: <c(c(c(?x_1))), a(c(b(?x_1)))> by Rules <1, 0>
develop reducts from lhs term...
<{6}, b(c(c(?x_1)))>
<{0}, a(c(c(?x_1)))>
<{6}, c(b(c(?x_1)))>
<{0}, c(a(c(?x_1)))>
<{}, c(c(c(?x_1)))>
develop reducts from rhs term...
<{1}, a(c(c(?x_1)))>
<{}, a(c(b(?x_1)))>
Inside Critical Pair: <c(c(a(?x_4))), a(c(a(?x_4)))> by Rules <4, 0>
develop reducts from lhs term...
<{6}, b(c(a(?x_4)))>
<{0}, a(c(a(?x_4)))>
<{7}, c(b(c(?x_4)))>
<{4}, c(c(a(?x_4)))>
<{}, c(c(a(?x_4)))>
develop reducts from rhs term...
<{7}, a(b(c(?x_4)))>
<{4}, a(c(a(?x_4)))>
<{}, a(c(a(?x_4)))>
Inside Critical Pair: <c(b(c(?x_6))), a(c(c(?x_6)))> by Rules <6, 0>
develop reducts from lhs term...
<{1}, c(c(c(?x_6)))>
<{5}, c(c(c(?x_6)))>
<{}, c(b(c(?x_6)))>
develop reducts from rhs term...
<{6}, a(b(c(?x_6)))>
<{0}, a(a(c(?x_6)))>
<{}, a(c(c(?x_6)))>
Inside Critical Pair: <c(b(c(?x_7))), a(c(a(?x_7)))> by Rules <7, 0>
develop reducts from lhs term...
<{1}, c(c(c(?x_7)))>
<{5}, c(c(c(?x_7)))>
<{}, c(b(c(?x_7)))>
develop reducts from rhs term...
<{7}, a(b(c(?x_7)))>
<{4}, a(c(a(?x_7)))>
<{}, a(c(a(?x_7)))>
Inside Critical Pair: <c(c(b(?x_2))), c(c(a(?x_2)))> by Rules <2, 1>
develop reducts from lhs term...
<{6}, b(c(b(?x_2)))>
<{0}, a(c(b(?x_2)))>
<{1}, c(c(c(?x_2)))>
<{}, c(c(b(?x_2)))>
develop reducts from rhs term...
<{6}, b(c(a(?x_2)))>
<{0}, a(c(a(?x_2)))>
<{7}, c(b(c(?x_2)))>
<{4}, c(c(a(?x_2)))>
<{}, c(c(a(?x_2)))>
Inside Critical Pair: <c(c(c(?x_5))), c(c(c(?x_5)))> by Rules <5, 1>
develop reducts from lhs term...
<{6}, b(c(c(?x_5)))>
<{0}, a(c(c(?x_5)))>
<{6}, c(b(c(?x_5)))>
<{0}, c(a(c(?x_5)))>
<{}, c(c(c(?x_5)))>
develop reducts from rhs term...
<{6}, b(c(c(?x_5)))>
<{0}, a(c(c(?x_5)))>
<{6}, c(b(c(?x_5)))>
<{0}, c(a(c(?x_5)))>
<{}, c(c(c(?x_5)))>
Inside Critical Pair: <b(b(a(?x_3))), c(b(b(?x_3)))> by Rules <3, 2>
develop reducts from lhs term...
<{2}, b(c(b(?x_3)))>
<{}, b(b(a(?x_3)))>
develop reducts from rhs term...
<{1}, c(c(b(?x_3)))>
<{}, c(b(b(?x_3)))>
Inside Critical Pair: <a(c(b(?x_2))), b(a(a(?x_2)))> by Rules <2, 3>
develop reducts from lhs term...
<{1}, a(c(c(?x_2)))>
<{}, a(c(b(?x_2)))>
develop reducts from rhs term...
<{2}, c(b(a(?x_2)))>
<{}, b(a(a(?x_2)))>
Inside Critical Pair: <a(c(c(?x_5))), b(a(c(?x_5)))> by Rules <5, 3>
develop reducts from lhs term...
<{6}, a(b(c(?x_5)))>
<{0}, a(a(c(?x_5)))>
<{}, a(c(c(?x_5)))>
develop reducts from rhs term...
<{2}, c(b(c(?x_5)))>
<{}, b(a(c(?x_5)))>
Inside Critical Pair: <c(b(a(?x_3))), c(a(b(?x_3)))> by Rules <3, 4>
develop reducts from lhs term...
<{1}, c(c(a(?x_3)))>
<{2}, c(c(b(?x_3)))>
<{}, c(b(a(?x_3)))>
develop reducts from rhs term...
<{7}, b(c(b(?x_3)))>
<{4}, c(a(b(?x_3)))>
<{3}, c(b(a(?x_3)))>
<{}, c(a(b(?x_3)))>
Inside Critical Pair: <b(a(c(?x))), c(c(c(?x)))> by Rules <0, 5>
develop reducts from lhs term...
<{2}, c(b(c(?x)))>
<{}, b(a(c(?x)))>
develop reducts from rhs term...
<{6}, b(c(c(?x)))>
<{0}, a(c(c(?x)))>
<{6}, c(b(c(?x)))>
<{0}, c(a(c(?x)))>
<{}, c(c(c(?x)))>
Inside Critical Pair: <b(c(c(?x_1))), c(c(b(?x_1)))> by Rules <1, 5>
develop reducts from lhs term...
<{5}, c(c(c(?x_1)))>
<{6}, b(b(c(?x_1)))>
<{0}, b(a(c(?x_1)))>
<{}, b(c(c(?x_1)))>
develop reducts from rhs term...
<{6}, b(c(b(?x_1)))>
<{0}, a(c(b(?x_1)))>
<{1}, c(c(c(?x_1)))>
<{}, c(c(b(?x_1)))>
Inside Critical Pair: <b(c(a(?x_4))), c(c(a(?x_4)))> by Rules <4, 5>
develop reducts from lhs term...
<{5}, c(c(a(?x_4)))>
<{7}, b(b(c(?x_4)))>
<{4}, b(c(a(?x_4)))>
<{}, b(c(a(?x_4)))>
develop reducts from rhs term...
<{6}, b(c(a(?x_4)))>
<{0}, a(c(a(?x_4)))>
<{7}, c(b(c(?x_4)))>
<{4}, c(c(a(?x_4)))>
<{}, c(c(a(?x_4)))>
Inside Critical Pair: <b(b(c(?x_6))), c(c(c(?x_6)))> by Rules <6, 5>
develop reducts from lhs term...
<{5}, b(c(c(?x_6)))>
<{}, b(b(c(?x_6)))>
develop reducts from rhs term...
<{6}, b(c(c(?x_6)))>
<{0}, a(c(c(?x_6)))>
<{6}, c(b(c(?x_6)))>
<{0}, c(a(c(?x_6)))>
<{}, c(c(c(?x_6)))>
Inside Critical Pair: <b(b(c(?x_7))), c(c(a(?x_7)))> by Rules <7, 5>
develop reducts from lhs term...
<{5}, b(c(c(?x_7)))>
<{}, b(b(c(?x_7)))>
develop reducts from rhs term...
<{6}, b(c(a(?x_7)))>
<{0}, a(c(a(?x_7)))>
<{7}, c(b(c(?x_7)))>
<{4}, c(c(a(?x_7)))>
<{}, c(c(a(?x_7)))>
Inside Critical Pair: <c(a(c(?x))), b(c(c(?x)))> by Rules <0, 6>
develop reducts from lhs term...
<{7}, b(c(c(?x)))>
<{4}, c(a(c(?x)))>
<{}, c(a(c(?x)))>
develop reducts from rhs term...
<{5}, c(c(c(?x)))>
<{6}, b(b(c(?x)))>
<{0}, b(a(c(?x)))>
<{}, b(c(c(?x)))>
Inside Critical Pair: <c(c(c(?x_1))), b(c(b(?x_1)))> by Rules <1, 6>
develop reducts from lhs term...
<{6}, b(c(c(?x_1)))>
<{0}, a(c(c(?x_1)))>
<{6}, c(b(c(?x_1)))>
<{0}, c(a(c(?x_1)))>
<{}, c(c(c(?x_1)))>
develop reducts from rhs term...
<{5}, c(c(b(?x_1)))>
<{1}, b(c(c(?x_1)))>
<{}, b(c(b(?x_1)))>
Inside Critical Pair: <c(c(a(?x_4))), b(c(a(?x_4)))> by Rules <4, 6>
develop reducts from lhs term...
<{6}, b(c(a(?x_4)))>
<{0}, a(c(a(?x_4)))>
<{7}, c(b(c(?x_4)))>
<{4}, c(c(a(?x_4)))>
<{}, c(c(a(?x_4)))>
develop reducts from rhs term...
<{5}, c(c(a(?x_4)))>
<{7}, b(b(c(?x_4)))>
<{4}, b(c(a(?x_4)))>
<{}, b(c(a(?x_4)))>
Inside Critical Pair: <c(b(c(?x_7))), b(c(a(?x_7)))> by Rules <7, 6>
develop reducts from lhs term...
<{1}, c(c(c(?x_7)))>
<{5}, c(c(c(?x_7)))>
<{}, c(b(c(?x_7)))>
develop reducts from rhs term...
<{5}, c(c(a(?x_7)))>
<{7}, b(b(c(?x_7)))>
<{4}, b(c(a(?x_7)))>
<{}, b(c(a(?x_7)))>
Inside Critical Pair: <c(b(a(?x_3))), b(c(b(?x_3)))> by Rules <3, 7>
develop reducts from lhs term...
<{1}, c(c(a(?x_3)))>
<{2}, c(c(b(?x_3)))>
<{}, c(b(a(?x_3)))>
develop reducts from rhs term...
<{5}, c(c(b(?x_3)))>
<{1}, b(c(c(?x_3)))>
<{}, b(c(b(?x_3)))>
Try A Minimal Decomposition {6,5,3,2,7,0,1}{4}
{6,5,3,2,7,0,1}
(cm)Rewrite Rules:
   [ c(c(?x)) -> b(c(?x)),
     b(c(?x)) -> c(c(?x)),
     a(b(?x)) -> b(a(?x)),
     b(a(?x)) -> c(b(?x)),
     c(a(?x)) -> b(c(?x)),
     c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ c(b(c(?x_4))) = b(c(a(?x_4))),
     c(a(c(?x_5))) = b(c(c(?x_5))),
     c(c(c(?x_6))) = b(c(b(?x_6))),
     b(b(c(?x))) = c(c(c(?x))),
     b(b(c(?x_4))) = c(c(a(?x_4))),
     b(a(c(?x_5))) = c(c(c(?x_5))),
     b(c(c(?x_6))) = c(c(b(?x_6))),
     a(c(c(?x_1))) = b(a(c(?x_1))),
     a(c(b(?x_3))) = b(a(a(?x_3))),
     b(b(a(?x_2))) = c(b(b(?x_2))),
     c(b(a(?x_2))) = b(c(b(?x_2))),
     c(b(c(?x))) = a(c(c(?x))),
     c(b(c(?x_4))) = a(c(a(?x_4))),
     c(c(c(?x_6))) = a(c(b(?x_6))),
     c(c(c(?x_1))) = c(c(c(?x_1))),
     c(c(b(?x_3))) = c(c(a(?x_3))),
     c(b(c(?x))) = b(c(c(?x))),
     c(a(c(?x))) = a(c(c(?x))) ]
Outer CPs:
   [ b(c(?x)) = a(c(?x)) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ a(c(?x)) = b(c(?x)),
     c(b(c(?x_1))) = b(c(c(?x_1))),
     c(b(c(?x_5))) = b(c(a(?x_5))),
     c(a(c(?x_6))) = b(c(c(?x_6))),
     c(c(c(?x_7))) = b(c(b(?x_7))),
     b(c(b(c(?x_1)))) = c(b(c(c(?x_1)))),
     b(c(b(c(?x_5)))) = c(b(c(a(?x_5)))),
     b(c(a(c(?x_6)))) = c(b(c(c(?x_6)))),
     b(c(c(c(?x_7)))) = c(b(c(b(?x_7)))),
     c(c(b(c(?x_1)))) = b(b(c(c(?x_1)))),
     c(c(b(c(?x_5)))) = b(b(c(a(?x_5)))),
     c(c(a(c(?x_6)))) = b(b(c(c(?x_6)))),
     c(c(c(c(?x_7)))) = b(b(c(b(?x_7)))),
     a(c(b(c(?x_1)))) = c(b(c(c(?x_1)))),
     a(c(b(c(?x_5)))) = c(b(c(a(?x_5)))),
     a(c(a(c(?x_6)))) = c(b(c(c(?x_6)))),
     a(c(c(c(?x_7)))) = c(b(c(b(?x_7)))),
     b(c(c(?x))) = c(b(c(?x))),
     c(c(c(?x))) = b(b(c(?x))),
     a(c(c(?x))) = c(b(c(?x))),
     b(b(c(?x_2))) = c(c(c(?x_2))),
     b(b(c(?x_5))) = c(c(a(?x_5))),
     b(a(c(?x_6))) = c(c(c(?x_6))),
     b(c(c(?x_7))) = c(c(b(?x_7))),
     b(a(b(c(?x_2)))) = a(c(c(c(?x_2)))),
     b(a(b(c(?x_5)))) = a(c(c(a(?x_5)))),
     b(a(a(c(?x_6)))) = a(c(c(c(?x_6)))),
     b(a(c(c(?x_7)))) = a(c(c(b(?x_7)))),
     c(c(b(c(?x_2)))) = c(c(c(c(?x_2)))),
     c(c(b(c(?x_5)))) = c(c(c(a(?x_5)))),
     c(c(a(c(?x_6)))) = c(c(c(c(?x_6)))),
     c(c(c(c(?x_7)))) = c(c(c(b(?x_7)))),
     b(a(c(?x))) = a(c(c(?x))),
     c(c(c(?x))) = c(c(c(?x))),
     a(c(c(?x_3))) = b(a(c(?x_3))),
     a(c(b(?x_4))) = b(a(a(?x_4))),
     c(b(c(c(?x_3)))) = b(b(a(c(?x_3)))),
     c(b(c(b(?x_4)))) = b(b(a(a(?x_4)))),
     b(c(c(c(?x_3)))) = c(b(a(c(?x_3)))),
     b(c(c(b(?x_4)))) = c(b(a(a(?x_4)))),
     c(b(b(?x))) = b(b(a(?x))),
     b(c(b(?x))) = c(b(a(?x))),
     b(b(a(?x_4))) = c(b(b(?x_4))),
     b(a(b(a(?x_4)))) = a(c(b(b(?x_4)))),
     c(c(b(a(?x_4)))) = c(c(b(b(?x_4)))),
     b(a(a(?x))) = a(c(b(?x))),
     c(c(a(?x))) = c(c(b(?x))),
     c(b(a(?x_4))) = b(c(b(?x_4))),
     b(c(b(a(?x_4)))) = c(b(c(b(?x_4)))),
     c(c(b(a(?x_4)))) = b(b(c(b(?x_4)))),
     a(c(b(a(?x_4)))) = c(b(c(b(?x_4)))),
     b(c(a(?x))) = c(b(c(?x))),
     c(c(a(?x))) = b(b(c(?x))),
     a(c(a(?x))) = c(b(c(?x))),
     b(c(?x)) = a(c(?x)),
     c(a(c(?x_1))) = a(c(c(?x_1))),
     c(b(c(?x_2))) = a(c(c(?x_2))),
     c(b(c(?x_6))) = a(c(a(?x_6))),
     c(c(c(?x_7))) = a(c(b(?x_7))),
     a(c(a(c(?x_1)))) = c(a(c(c(?x_1)))),
     a(c(b(c(?x_2)))) = c(a(c(c(?x_2)))),
     a(c(b(c(?x_6)))) = c(a(c(a(?x_6)))),
     a(c(c(c(?x_7)))) = c(a(c(b(?x_7)))),
     b(c(a(c(?x_1)))) = c(a(c(c(?x_1)))),
     b(c(b(c(?x_2)))) = c(a(c(c(?x_2)))),
     b(c(b(c(?x_6)))) = c(a(c(a(?x_6)))),
     b(c(c(c(?x_7)))) = c(a(c(b(?x_7)))),
     c(c(a(c(?x_1)))) = b(a(c(c(?x_1)))),
     c(c(b(c(?x_2)))) = b(a(c(c(?x_2)))),
     c(c(b(c(?x_6)))) = b(a(c(a(?x_6)))),
     c(c(c(c(?x_7)))) = b(a(c(b(?x_7)))),
     a(c(c(?x))) = c(a(c(?x))),
     b(c(c(?x))) = c(a(c(?x))),
     c(c(c(?x))) = b(a(c(?x))),
     c(c(b(?x_5))) = c(c(a(?x_5))),
     b(c(c(c(?x_3)))) = c(c(c(c(?x_3)))),
     b(c(c(b(?x_5)))) = c(c(c(a(?x_5)))),
     c(c(c(c(?x_3)))) = b(c(c(c(?x_3)))),
     c(c(c(b(?x_5)))) = b(c(c(a(?x_5)))),
     a(c(c(c(?x_3)))) = c(c(c(c(?x_3)))),
     a(c(c(b(?x_5)))) = c(c(c(a(?x_5)))),
     b(c(b(?x))) = c(c(c(?x))),
     c(c(b(?x))) = b(c(c(?x))),
     a(c(b(?x))) = c(c(c(?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 <c(b(c(?x_4))), b(c(a(?x_4)))> by Rules <4, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([],1),([(c,1)],4)]>
 joinable by a reduction of rules <[([(c,1)],1),([],0)], [([(b,1)],4),([(b,1)],1)]>
 joinable by a reduction of rules <[([],6),([],0)], [([(b,1)],4),([(b,1)],1)]>
Critical Pair <c(a(c(?x_5))), b(c(c(?x_5)))> by Rules <5, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([],4)], []>
Critical Pair <c(c(c(?x_6))), b(c(b(?x_6)))> by Rules <6, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([],0)], [([(b,1)],6)]>
Critical Pair <b(b(c(?x))), c(c(c(?x)))> by Rules <0, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],1)], [([],0)]>
Critical Pair <b(b(c(?x_4))), c(c(a(?x_4)))> by Rules <4, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([],0),([(b,1)],4)]>
 joinable by a reduction of rules <[([(b,1)],1),([],1)], [([(c,1)],4),([(c,1)],1)]>
 joinable by a reduction of rules <[([(b,1)],1),([],1)], [([(c,1)],4),([],6)]>
Critical Pair <b(a(c(?x_5))), c(c(c(?x_5)))> by Rules <5, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([],3)], [([(c,1)],0)]>
Critical Pair <b(c(c(?x_6))), c(c(b(?x_6)))> by Rules <6, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([],1)], [([(c,1)],6)]>
Critical Pair <a(c(c(?x_1))), b(a(c(?x_1)))> by Rules <1, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],0),([],2)], []>
Critical Pair <a(c(b(?x_3))), b(a(a(?x_3)))> by Rules <3, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],3),([(c,1)],3),([],5)]>
Critical Pair <b(b(a(?x_2))), c(b(b(?x_2)))> by Rules <2, 3> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],3),([],1)], [([],6)]>
 joinable by a reduction of rules <[([(b,1)],3)], [([],6),([],0)]>
Critical Pair <c(b(a(?x_2))), b(c(b(?x_2)))> by Rules <2, 4> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],3)], [([],1)]>
Critical Pair <c(b(c(?x))), a(c(c(?x)))> by Rules <0, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],1),([],5)], []>
 joinable by a reduction of rules <[([],6),([],5)], []>
Critical Pair <c(b(c(?x_4))), a(c(a(?x_4)))> by Rules <4, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],1),([],5)], [([(a,1)],4),([(a,1)],1)]>
 joinable by a reduction of rules <[([],6),([],5)], [([(a,1)],4),([(a,1)],1)]>
Critical Pair <c(c(c(?x_6))), a(c(b(?x_6)))> by Rules <6, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[([],5)], [([(a,1)],6)]>
Critical Pair <c(c(c(?x_1))), c(c(c(?x_1)))> by Rules <1, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <c(c(b(?x_3))), c(c(a(?x_3)))> by Rules <3, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],6),([(c,1)],0)], [([(c,1)],4)]>
 joinable by a reduction of rules <[([(c,1)],6)], [([(c,1)],4),([(c,1)],1)]>
 joinable by a reduction of rules <[([(c,1)],6)], [([(c,1)],4),([],6)]>
Critical Pair <c(b(c(?x))), b(c(c(?x)))> by Rules <0, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],1)], [([],1)]>
 joinable by a reduction of rules <[([],6)], [([],1)]>
Critical Pair <c(a(c(?x))), a(c(c(?x)))> by Rules <5, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[([],4),([(b,1)],5)], [([(a,1)],0),([],2)]>
Critical Pair <a(c(?x_5)), b(c(?x_5))> by Rules <5, 0> preceded by []
 joinable by a reduction of rules <[], [([],1),([],5)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 15 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:
   [ a(b(?x)) -> b(a(?x)),
     b(a(?x)) -> c(b(?x)),
     c(a(?x)) -> b(c(?x)),
     c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)) ]
P:
   [ c(c(?x)) -> b(c(?x)),
     b(c(?x)) -> c(c(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
   [ a(b(?x)) -> b(a(?x)),
     b(a(?x)) -> c(b(?x)),
     c(a(?x)) -> b(c(?x)),
     c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)) ]
P:
   [ c(c(?x)) -> b(c(?x)),
     b(c(?x)) -> c(c(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
   [ a(b(?x)) -> b(a(?x)),
     b(a(?x)) -> c(b(?x)),
     c(a(?x)) -> b(c(?x)),
     c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)) ]
P:
   [ c(c(?x)) -> b(c(?x)),
     b(c(?x)) -> c(c(?x)) ]
Check relative termination:
   [ a(b(?x)) -> b(a(?x)),
     b(a(?x)) -> c(b(?x)),
     c(a(?x)) -> b(c(?x)),
     c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)) ]
   [ c(c(?x)) -> b(c(?x)),
     b(c(?x)) -> c(c(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)*x1
  c:= (1)+(1)*x1
retract c(c(?x)) -> a(c(?x))
retract c(c(?x)) -> b(c(?x))
Polynomial Interpretation:
  a:= (1)+(2)*x1*x1
  b:= (1)+(2)*x1*x1
  c:= (2)*x1*x1
retract b(a(?x)) -> c(b(?x))
retract c(a(?x)) -> b(c(?x))
retract c(c(?x)) -> a(c(?x))
retract c(b(?x)) -> c(c(?x))
retract c(c(?x)) -> b(c(?x))
retract b(c(?x)) -> c(c(?x))
Polynomial Interpretation:
  a:= (1)*x1+(1)*x1*x1
  b:= (1)+(1)*x1
  c:= (1)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ c(c(?x)) -> b(c(?x)),
     b(c(?x)) -> c(c(?x)),
     a(b(?x)) -> b(a(?x)),
     b(a(?x)) -> c(b(?x)),
     c(a(?x)) -> b(c(?x)),
     c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)) ]
Sort Assignment:
 a : 14=>14
 b : 14=>14
 c : 14=>14
maximal types: {14}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ c(c(?x)) -> b(c(?x)),
     b(c(?x)) -> c(c(?x)),
     a(b(?x)) -> b(a(?x)),
     b(a(?x)) -> c(b(?x)),
     c(a(?x)) -> b(c(?x)),
     c(c(?x)) -> a(c(?x)),
     c(b(?x)) -> c(c(?x)) ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible

{4}
(cm)Rewrite Rules:
   [ c(a(?x)) -> c(a(?x)) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
Development Closed
Direct Methods: CR
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
1012.trs: Failure(unknown CR)
MAYBE
(9015 msec.)