(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
   [ a(b(?x)) -> c(a(?x)),
     c(a(?x)) -> a(b(?x)),
     b(b(?x)) -> b(a(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(c(?x)) -> b(b(?x)),
     b(b(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ a(b(a(?x_2))) = c(a(b(?x_2))),
     a(b(b(?x_6))) = c(a(b(?x_6))),
     a(a(a(?x_7))) = c(a(c(?x_7))),
     a(a(a(?x_8))) = c(a(c(?x_8))),
     c(c(a(?x))) = a(b(b(?x))),
     c(b(b(?x_4))) = a(b(a(?x_4))),
     b(b(b(?x_6))) = b(a(b(?x_6))),
     b(a(a(?x_7))) = b(a(c(?x_7))),
     b(a(a(?x_8))) = b(a(c(?x_8))),
     c(b(a(?x_2))) = a(c(b(?x_2))),
     c(b(b(?x_6))) = a(c(b(?x_6))),
     c(a(a(?x_7))) = a(c(c(?x_7))),
     c(a(a(?x_8))) = a(c(c(?x_8))),
     a(c(a(?x))) = b(b(b(?x))),
     c(a(b(?x_1))) = b(b(a(?x_1))),
     c(a(c(?x_3))) = b(b(b(?x_3))),
     b(b(a(?x_2))) = b(b(b(?x_2))),
     b(a(a(?x_7))) = b(b(c(?x_7))),
     b(a(a(?x_8))) = b(b(c(?x_8))),
     b(a(b(?x_1))) = a(a(a(?x_1))),
     b(a(c(?x_3))) = a(a(b(?x_3))),
     b(b(b(?x_5))) = a(a(c(?x_5))),
     b(a(b(?x_1))) = a(a(a(?x_1))),
     b(a(c(?x_3))) = a(a(b(?x_3))),
     b(b(b(?x_5))) = a(a(c(?x_5))),
     b(b(a(?x))) = b(a(b(?x))),
     a(b(b(?x))) = b(b(a(?x))),
     c(b(b(?x))) = b(b(c(?x))),
     b(b(b(?x))) = b(b(b(?x))) ]
Outer CPs:
   [ b(a(?x_2)) = b(b(?x_2)),
     a(a(?x_7)) = a(a(?x_7)) ]
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:
   [ a(b(a(?x_3))) = c(a(b(?x_3))),
     a(b(b(?x_7))) = c(a(b(?x_7))),
     a(a(a(?x_8))) = c(a(c(?x_8))),
     a(b(b(a(?x_3)))) = c(c(a(b(?x_3)))),
     a(b(b(b(?x_7)))) = c(c(a(b(?x_7)))),
     a(b(a(a(?x_8)))) = c(c(a(c(?x_8)))),
     b(b(b(a(?x_3)))) = a(c(a(b(?x_3)))),
     b(b(b(b(?x_7)))) = a(c(a(b(?x_7)))),
     b(b(a(a(?x_8)))) = a(c(a(c(?x_8)))),
     a(b(b(?x))) = c(c(a(?x))),
     b(b(b(?x))) = a(c(a(?x))),
     c(c(a(?x_2))) = a(b(b(?x_2))),
     c(b(b(?x_5))) = a(b(a(?x_5))),
     b(b(c(a(?x_2)))) = c(a(b(b(?x_2)))),
     b(b(b(b(?x_5)))) = c(a(b(a(?x_5)))),
     a(a(c(a(?x_2)))) = b(a(b(b(?x_2)))),
     a(a(b(b(?x_5)))) = b(a(b(a(?x_5)))),
     b(b(a(?x))) = c(a(b(?x))),
     a(a(a(?x))) = b(a(b(?x))),
     b(b(?x)) = b(a(?x)),
     b(b(a(?x_1))) = b(a(b(?x_1))),
     b(b(b(?x_7))) = b(a(b(?x_7))),
     b(a(a(?x_8))) = b(a(c(?x_8))),
     b(a(b(a(?x_1)))) = b(b(a(b(?x_1)))),
     b(a(b(b(?x_7)))) = b(b(a(b(?x_7)))),
     b(a(a(a(?x_8)))) = b(b(a(c(?x_8)))),
     c(a(b(a(?x_1)))) = a(b(a(b(?x_1)))),
     c(a(b(b(?x_7)))) = a(b(a(b(?x_7)))),
     c(a(a(a(?x_8)))) = a(b(a(c(?x_8)))),
     a(c(b(a(?x_1)))) = c(b(a(b(?x_1)))),
     a(c(b(b(?x_7)))) = c(b(a(b(?x_7)))),
     a(c(a(a(?x_8)))) = c(b(a(c(?x_8)))),
     b(b(b(a(?x_1)))) = b(b(a(b(?x_1)))),
     b(b(b(b(?x_7)))) = b(b(a(b(?x_7)))),
     b(b(a(a(?x_8)))) = b(b(a(c(?x_8)))),
     b(a(b(?x))) = b(b(a(?x))),
     c(a(b(?x))) = a(b(a(?x))),
     a(c(b(?x))) = c(b(a(?x))),
     b(b(b(?x))) = b(b(a(?x))),
     c(b(a(?x_4))) = a(c(b(?x_4))),
     c(b(b(?x_7))) = a(c(b(?x_7))),
     c(a(a(?x_8))) = a(c(c(?x_8))),
     b(b(b(a(?x_4)))) = c(a(c(b(?x_4)))),
     b(b(b(b(?x_7)))) = c(a(c(b(?x_7)))),
     b(b(a(a(?x_8)))) = c(a(c(c(?x_8)))),
     a(a(b(a(?x_4)))) = b(a(c(b(?x_4)))),
     a(a(b(b(?x_7)))) = b(a(c(b(?x_7)))),
     a(a(a(a(?x_8)))) = b(a(c(c(?x_8)))),
     b(b(b(?x))) = c(a(c(?x))),
     a(a(b(?x))) = b(a(c(?x))),
     a(b(b(?x_1))) = b(b(a(?x_1))),
     a(c(a(?x_2))) = b(b(b(?x_2))),
     b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))),
     b(b(c(a(?x_2)))) = a(b(b(b(?x_2)))),
     a(b(b(b(?x_1)))) = c(b(b(a(?x_1)))),
     a(b(c(a(?x_2)))) = c(b(b(b(?x_2)))),
     b(b(a(?x))) = a(b(b(?x))),
     a(b(a(?x))) = c(b(b(?x))),
     c(b(b(?x_1))) = b(b(c(?x_1))),
     c(a(b(?x_3))) = b(b(a(?x_3))),
     c(a(c(?x_5))) = b(b(b(?x_5))),
     b(b(b(b(?x_1)))) = c(b(b(c(?x_1)))),
     b(b(a(b(?x_3)))) = c(b(b(a(?x_3)))),
     b(b(a(c(?x_5)))) = c(b(b(b(?x_5)))),
     a(a(b(b(?x_1)))) = b(b(b(c(?x_1)))),
     a(a(a(b(?x_3)))) = b(b(b(a(?x_3)))),
     a(a(a(c(?x_5)))) = b(b(b(b(?x_5)))),
     b(b(c(?x))) = c(b(b(?x))),
     a(a(c(?x))) = b(b(b(?x))),
     b(a(?x)) = b(b(?x)),
     b(b(b(?x_1))) = b(b(b(?x_1))),
     b(b(a(?x_4))) = b(b(b(?x_4))),
     b(a(a(?x_8))) = b(b(c(?x_8))),
     b(b(b(b(?x_1)))) = b(b(b(b(?x_1)))),
     b(b(b(a(?x_4)))) = b(b(b(b(?x_4)))),
     b(b(a(a(?x_8)))) = b(b(b(c(?x_8)))),
     c(a(b(b(?x_1)))) = a(b(b(b(?x_1)))),
     c(a(b(a(?x_4)))) = a(b(b(b(?x_4)))),
     c(a(a(a(?x_8)))) = a(b(b(c(?x_8)))),
     b(a(b(b(?x_1)))) = b(b(b(b(?x_1)))),
     b(a(b(a(?x_4)))) = b(b(b(b(?x_4)))),
     b(a(a(a(?x_8)))) = b(b(b(c(?x_8)))),
     a(c(b(b(?x_1)))) = c(b(b(b(?x_1)))),
     a(c(b(a(?x_4)))) = c(b(b(b(?x_4)))),
     a(c(a(a(?x_8)))) = c(b(b(c(?x_8)))),
     c(a(b(?x))) = a(b(b(?x))),
     b(a(b(?x))) = b(b(b(?x))),
     a(c(b(?x))) = c(b(b(?x))),
     a(a(?x)) = a(a(?x)),
     b(a(b(?x_3))) = a(a(a(?x_3))),
     b(a(c(?x_5))) = a(a(b(?x_5))),
     b(b(b(?x_7))) = a(a(c(?x_7))),
     c(a(a(b(?x_3)))) = a(a(a(a(?x_3)))),
     c(a(a(c(?x_5)))) = a(a(a(b(?x_5)))),
     c(a(b(b(?x_7)))) = a(a(a(c(?x_7)))),
     b(a(a(b(?x_3)))) = b(a(a(a(?x_3)))),
     b(a(a(c(?x_5)))) = b(a(a(b(?x_5)))),
     b(a(b(b(?x_7)))) = b(a(a(c(?x_7)))),
     a(c(a(b(?x_3)))) = c(a(a(a(?x_3)))),
     a(c(a(c(?x_5)))) = c(a(a(b(?x_5)))),
     a(c(b(b(?x_7)))) = c(a(a(c(?x_7)))),
     b(b(a(b(?x_3)))) = b(a(a(a(?x_3)))),
     b(b(a(c(?x_5)))) = b(a(a(b(?x_5)))),
     b(b(b(b(?x_7)))) = b(a(a(c(?x_7)))),
     c(a(c(?x))) = a(a(a(?x))),
     b(a(c(?x))) = b(a(a(?x))),
     a(c(c(?x))) = c(a(a(?x))),
     b(b(c(?x))) = b(a(a(?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 <a(b(a(?x_2))), c(a(b(?x_2)))> by Rules <2, 0> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],1),([(a,1)],2)]>
Critical Pair <a(b(b(?x_6))), c(a(b(?x_6)))> by Rules <6, 0> preceded by [(a,1)]
 joinable by a reduction of rules <[([],0)], []>
 joinable by a reduction of rules <[], [([],1)]>
Critical Pair <a(a(a(?x_7))), c(a(c(?x_7)))> by Rules <7, 0> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],1),([(a,1)],8)]>
 joinable by a reduction of rules <[], [([],1),([(a,1)],7)]>
Critical Pair <a(a(a(?x_8))), c(a(c(?x_8)))> by Rules <8, 0> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],1),([(a,1)],8)]>
 joinable by a reduction of rules <[], [([],1),([(a,1)],7)]>
Critical Pair <c(c(a(?x))), a(b(b(?x)))> by Rules <0, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],1)], [([],0)]>
Critical Pair <c(b(b(?x_4))), a(b(a(?x_4)))> by Rules <4, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([],0),([(c,1)],4)]>
Critical Pair <b(b(b(?x_6))), b(a(b(?x_6)))> by Rules <6, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[([],2)], []>
Critical Pair <b(a(a(?x_7))), b(a(c(?x_7)))> by Rules <7, 2> preceded by [(b,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 12 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:
   [ b(b(?x)) -> b(a(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(c(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)) ]
P:
   [ a(b(?x)) -> c(a(?x)),
     c(a(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
   [ b(b(?x)) -> b(a(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(c(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)) ]
P:
   [ a(b(?x)) -> c(a(?x)),
     c(a(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
   [ b(b(?x)) -> b(a(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(c(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)) ]
P:
   [ a(b(?x)) -> c(a(?x)),
     c(a(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
Check relative termination:
   [ b(b(?x)) -> b(a(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(c(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)) ]
   [ a(b(?x)) -> c(a(?x)),
     c(a(?x)) -> a(b(?x)),
     b(b(?x)) -> b(b(?x)) ]
Polynomial Interpretation:
  a:= (2)*x1
  b:= (2)*x1
  c:= (4)+(2)*x1
retract c(c(?x)) -> b(b(?x))
retract b(c(?x)) -> a(a(?x))
retract b(c(?x)) -> a(a(?x))
retract c(a(?x)) -> a(b(?x))
Polynomial Interpretation:
  a:= (1)+(2)*x1*x1
  b:= (1)+(2)*x1*x1
  c:= (2)*x1*x1
retract c(b(?x)) -> a(c(?x))
retract c(c(?x)) -> b(b(?x))
retract b(c(?x)) -> a(a(?x))
retract b(c(?x)) -> a(a(?x))
retract a(b(?x)) -> c(a(?x))
retract c(a(?x)) -> a(b(?x))
Polynomial Interpretation:
  a:= (1)+(1)*x1
  b:= (1)*x1
  c:= (7)+(1)*x1+(7)*x1*x1
retract c(b(?x)) -> a(c(?x))
retract a(a(?x)) -> b(b(?x))
retract c(c(?x)) -> b(b(?x))
retract b(c(?x)) -> a(a(?x))
retract b(c(?x)) -> a(a(?x))
retract a(b(?x)) -> c(a(?x))
retract c(a(?x)) -> a(b(?x))
Polynomial Interpretation:
  a:= (1)+(1)*x1
  b:= (2)+(1)*x1
  c:= (9)*x1+(2)*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...
   [ a(b(?x)) -> c(a(?x)),
     c(a(?x)) -> a(b(?x)),
     b(b(?x)) -> b(a(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(c(?x)) -> b(b(?x)),
     b(b(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)) ]
Sort Assignment:
 a : 16=>16
 b : 16=>16
 c : 16=>16
maximal types: {16}
Persistent Decomposition failed: Can't judge

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

Try Commutative Decomposition for...
   [ a(b(?x)) -> c(a(?x)),
     c(a(?x)) -> a(b(?x)),
     b(b(?x)) -> b(a(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(c(?x)) -> b(b(?x)),
     b(b(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)) ]
Outside Critical Pair: <b(b(?x_6)), b(a(?x_6))> by Rules <6, 2>
develop reducts from lhs term...
<{6}, b(b(?x_6))>
<{2}, b(a(?x_6))>
<{}, b(b(?x_6))>
develop reducts from rhs term...
<{}, b(a(?x_6))>
Outside Critical Pair: <a(a(?x_8)), a(a(?x_8))> by Rules <8, 7>
develop reducts from lhs term...
<{4}, b(b(?x_8))>
<{}, a(a(?x_8))>
develop reducts from rhs term...
<{4}, b(b(?x_8))>
<{}, a(a(?x_8))>
Inside Critical Pair: <a(b(a(?x_2))), c(a(b(?x_2)))> by Rules <2, 0>
develop reducts from lhs term...
<{0}, c(a(a(?x_2)))>
<{}, a(b(a(?x_2)))>
develop reducts from rhs term...
<{1}, a(b(b(?x_2)))>
<{0}, c(c(a(?x_2)))>
<{}, c(a(b(?x_2)))>
Inside Critical Pair: <a(b(b(?x_6))), c(a(b(?x_6)))> by Rules <6, 0>
develop reducts from lhs term...
<{0}, c(a(b(?x_6)))>
<{6}, a(b(b(?x_6)))>
<{2}, a(b(a(?x_6)))>
<{}, a(b(b(?x_6)))>
develop reducts from rhs term...
<{1}, a(b(b(?x_6)))>
<{0}, c(c(a(?x_6)))>
<{}, c(a(b(?x_6)))>
Inside Critical Pair: <a(a(a(?x_7))), c(a(c(?x_7)))> by Rules <7, 0>
develop reducts from lhs term...
<{4}, b(b(a(?x_7)))>
<{4}, a(b(b(?x_7)))>
<{}, a(a(a(?x_7)))>
develop reducts from rhs term...
<{1}, a(b(c(?x_7)))>
<{}, c(a(c(?x_7)))>
Inside Critical Pair: <a(a(a(?x_8))), c(a(c(?x_8)))> by Rules <8, 0>
develop reducts from lhs term...
<{4}, b(b(a(?x_8)))>
<{4}, a(b(b(?x_8)))>
<{}, a(a(a(?x_8)))>
develop reducts from rhs term...
<{1}, a(b(c(?x_8)))>
<{}, c(a(c(?x_8)))>
Inside Critical Pair: <c(c(a(?x))), a(b(b(?x)))> by Rules <0, 1>
develop reducts from lhs term...
<{5}, b(b(a(?x)))>
<{1}, c(a(b(?x)))>
<{}, c(c(a(?x)))>
develop reducts from rhs term...
<{0}, c(a(b(?x)))>
<{6}, a(b(b(?x)))>
<{2}, a(b(a(?x)))>
<{}, a(b(b(?x)))>
Inside Critical Pair: <c(b(b(?x_4))), a(b(a(?x_4)))> by Rules <4, 1>
develop reducts from lhs term...
<{3}, a(c(b(?x_4)))>
<{6}, c(b(b(?x_4)))>
<{2}, c(b(a(?x_4)))>
<{}, c(b(b(?x_4)))>
develop reducts from rhs term...
<{0}, c(a(a(?x_4)))>
<{}, a(b(a(?x_4)))>
Inside Critical Pair: <b(b(b(?x_6))), b(a(b(?x_6)))> by Rules <6, 2>
develop reducts from lhs term...
<{2}, b(a(b(?x_6)))>
<{6}, b(b(b(?x_6)))>
<{2}, b(b(a(?x_6)))>
<{}, b(b(b(?x_6)))>
develop reducts from rhs term...
<{0}, b(c(a(?x_6)))>
<{}, b(a(b(?x_6)))>
Inside Critical Pair: <b(a(a(?x_7))), b(a(c(?x_7)))> by Rules <7, 2>
develop reducts from lhs term...
<{4}, b(b(b(?x_7)))>
<{}, b(a(a(?x_7)))>
develop reducts from rhs term...
<{}, b(a(c(?x_7)))>
Inside Critical Pair: <b(a(a(?x_8))), b(a(c(?x_8)))> by Rules <8, 2>
develop reducts from lhs term...
<{4}, b(b(b(?x_8)))>
<{}, b(a(a(?x_8)))>
develop reducts from rhs term...
<{}, b(a(c(?x_8)))>
Inside Critical Pair: <c(b(a(?x_2))), a(c(b(?x_2)))> by Rules <2, 3>
develop reducts from lhs term...
<{3}, a(c(a(?x_2)))>
<{}, c(b(a(?x_2)))>
develop reducts from rhs term...
<{3}, a(a(c(?x_2)))>
<{}, a(c(b(?x_2)))>
Inside Critical Pair: <c(b(b(?x_6))), a(c(b(?x_6)))> by Rules <6, 3>
develop reducts from lhs term...
<{3}, a(c(b(?x_6)))>
<{6}, c(b(b(?x_6)))>
<{2}, c(b(a(?x_6)))>
<{}, c(b(b(?x_6)))>
develop reducts from rhs term...
<{3}, a(a(c(?x_6)))>
<{}, a(c(b(?x_6)))>
Inside Critical Pair: <c(a(a(?x_7))), a(c(c(?x_7)))> by Rules <7, 3>
develop reducts from lhs term...
<{1}, a(b(a(?x_7)))>
<{4}, c(b(b(?x_7)))>
<{}, c(a(a(?x_7)))>
develop reducts from rhs term...
<{5}, a(b(b(?x_7)))>
<{}, a(c(c(?x_7)))>
Inside Critical Pair: <c(a(a(?x_8))), a(c(c(?x_8)))> by Rules <8, 3>
develop reducts from lhs term...
<{1}, a(b(a(?x_8)))>
<{4}, c(b(b(?x_8)))>
<{}, c(a(a(?x_8)))>
develop reducts from rhs term...
<{5}, a(b(b(?x_8)))>
<{}, a(c(c(?x_8)))>
Inside Critical Pair: <a(c(a(?x))), b(b(b(?x)))> by Rules <0, 4>
develop reducts from lhs term...
<{1}, a(a(b(?x)))>
<{}, a(c(a(?x)))>
develop reducts from rhs term...
<{2}, b(a(b(?x)))>
<{6}, b(b(b(?x)))>
<{2}, b(b(a(?x)))>
<{}, b(b(b(?x)))>
Inside Critical Pair: <c(a(b(?x_1))), b(b(a(?x_1)))> by Rules <1, 5>
develop reducts from lhs term...
<{1}, a(b(b(?x_1)))>
<{0}, c(c(a(?x_1)))>
<{}, c(a(b(?x_1)))>
develop reducts from rhs term...
<{6}, b(b(a(?x_1)))>
<{2}, b(a(a(?x_1)))>
<{}, b(b(a(?x_1)))>
Inside Critical Pair: <c(a(c(?x_3))), b(b(b(?x_3)))> by Rules <3, 5>
develop reducts from lhs term...
<{1}, a(b(c(?x_3)))>
<{}, c(a(c(?x_3)))>
develop reducts from rhs term...
<{2}, b(a(b(?x_3)))>
<{6}, b(b(b(?x_3)))>
<{2}, b(b(a(?x_3)))>
<{}, b(b(b(?x_3)))>
Inside Critical Pair: <b(b(a(?x_2))), b(b(b(?x_2)))> by Rules <2, 6>
develop reducts from lhs term...
<{6}, b(b(a(?x_2)))>
<{2}, b(a(a(?x_2)))>
<{}, b(b(a(?x_2)))>
develop reducts from rhs term...
<{2}, b(a(b(?x_2)))>
<{6}, b(b(b(?x_2)))>
<{2}, b(b(a(?x_2)))>
<{}, b(b(b(?x_2)))>
Inside Critical Pair: <b(a(a(?x_7))), b(b(c(?x_7)))> by Rules <7, 6>
develop reducts from lhs term...
<{4}, b(b(b(?x_7)))>
<{}, b(a(a(?x_7)))>
develop reducts from rhs term...
<{6}, b(b(c(?x_7)))>
<{2}, b(a(c(?x_7)))>
<{8}, b(a(a(?x_7)))>
<{7}, b(a(a(?x_7)))>
<{}, b(b(c(?x_7)))>
Inside Critical Pair: <b(a(a(?x_8))), b(b(c(?x_8)))> by Rules <8, 6>
develop reducts from lhs term...
<{4}, b(b(b(?x_8)))>
<{}, b(a(a(?x_8)))>
develop reducts from rhs term...
<{6}, b(b(c(?x_8)))>
<{2}, b(a(c(?x_8)))>
<{8}, b(a(a(?x_8)))>
<{7}, b(a(a(?x_8)))>
<{}, b(b(c(?x_8)))>
Inside Critical Pair: <b(a(b(?x_1))), a(a(a(?x_1)))> by Rules <1, 7>
develop reducts from lhs term...
<{0}, b(c(a(?x_1)))>
<{}, b(a(b(?x_1)))>
develop reducts from rhs term...
<{4}, b(b(a(?x_1)))>
<{4}, a(b(b(?x_1)))>
<{}, a(a(a(?x_1)))>
Inside Critical Pair: <b(a(c(?x_3))), a(a(b(?x_3)))> by Rules <3, 7>
develop reducts from lhs term...
<{}, b(a(c(?x_3)))>
develop reducts from rhs term...
<{4}, b(b(b(?x_3)))>
<{0}, a(c(a(?x_3)))>
<{}, a(a(b(?x_3)))>
Inside Critical Pair: <b(b(b(?x_5))), a(a(c(?x_5)))> by Rules <5, 7>
develop reducts from lhs term...
<{2}, b(a(b(?x_5)))>
<{6}, b(b(b(?x_5)))>
<{2}, b(b(a(?x_5)))>
<{}, b(b(b(?x_5)))>
develop reducts from rhs term...
<{4}, b(b(c(?x_5)))>
<{}, a(a(c(?x_5)))>
Inside Critical Pair: <b(a(b(?x_1))), a(a(a(?x_1)))> by Rules <1, 8>
develop reducts from lhs term...
<{0}, b(c(a(?x_1)))>
<{}, b(a(b(?x_1)))>
develop reducts from rhs term...
<{4}, b(b(a(?x_1)))>
<{4}, a(b(b(?x_1)))>
<{}, a(a(a(?x_1)))>
Inside Critical Pair: <b(a(c(?x_3))), a(a(b(?x_3)))> by Rules <3, 8>
develop reducts from lhs term...
<{}, b(a(c(?x_3)))>
develop reducts from rhs term...
<{4}, b(b(b(?x_3)))>
<{0}, a(c(a(?x_3)))>
<{}, a(a(b(?x_3)))>
Inside Critical Pair: <b(b(b(?x_5))), a(a(c(?x_5)))> by Rules <5, 8>
develop reducts from lhs term...
<{2}, b(a(b(?x_5)))>
<{6}, b(b(b(?x_5)))>
<{2}, b(b(a(?x_5)))>
<{}, b(b(b(?x_5)))>
develop reducts from rhs term...
<{4}, b(b(c(?x_5)))>
<{}, a(a(c(?x_5)))>
Try A Minimal Decomposition {5,3,4,1,8,7,0,2}{6}
{5,3,4,1,8,7,0,2}
(cm)Rewrite Rules:
   [ c(c(?x)) -> b(b(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(a(?x)) -> a(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> b(a(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ c(a(c(?x_1))) = b(b(b(?x_1))),
     c(a(b(?x_3))) = b(b(a(?x_3))),
     c(a(a(?x_4))) = a(c(c(?x_4))),
     c(a(a(?x_5))) = a(c(c(?x_5))),
     c(b(a(?x_7))) = a(c(b(?x_7))),
     a(c(a(?x_6))) = b(b(b(?x_6))),
     c(b(b(?x_2))) = a(b(a(?x_2))),
     c(c(a(?x_6))) = a(b(b(?x_6))),
     b(b(b(?x))) = a(a(c(?x))),
     b(a(c(?x_1))) = a(a(b(?x_1))),
     b(a(b(?x_3))) = a(a(a(?x_3))),
     b(b(b(?x))) = a(a(c(?x))),
     b(a(c(?x_1))) = a(a(b(?x_1))),
     b(a(b(?x_3))) = a(a(a(?x_3))),
     a(a(a(?x_4))) = c(a(c(?x_4))),
     a(a(a(?x_5))) = c(a(c(?x_5))),
     a(b(a(?x_7))) = c(a(b(?x_7))),
     b(a(a(?x_4))) = b(a(c(?x_4))),
     b(a(a(?x_5))) = b(a(c(?x_5))),
     c(b(b(?x))) = b(b(c(?x))),
     a(b(b(?x))) = b(b(a(?x))),
     b(b(a(?x))) = b(a(b(?x))) ]
Outer CPs:
   [ a(a(?x_4)) = a(a(?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
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(b(?x_1))) = b(b(c(?x_1))),
     c(a(c(?x_2))) = b(b(b(?x_2))),
     c(a(b(?x_4))) = b(b(a(?x_4))),
     b(b(b(b(?x_1)))) = c(b(b(c(?x_1)))),
     b(b(a(c(?x_2)))) = c(b(b(b(?x_2)))),
     b(b(a(b(?x_4)))) = c(b(b(a(?x_4)))),
     a(a(b(b(?x_1)))) = b(b(b(c(?x_1)))),
     a(a(a(c(?x_2)))) = b(b(b(b(?x_2)))),
     a(a(a(b(?x_4)))) = b(b(b(a(?x_4)))),
     b(b(c(?x))) = c(b(b(?x))),
     a(a(c(?x))) = b(b(b(?x))),
     c(a(a(?x_5))) = a(c(c(?x_5))),
     c(b(a(?x_8))) = a(c(b(?x_8))),
     b(b(a(a(?x_5)))) = c(a(c(c(?x_5)))),
     b(b(b(a(?x_8)))) = c(a(c(b(?x_8)))),
     a(a(a(a(?x_5)))) = b(a(c(c(?x_5)))),
     a(a(b(a(?x_8)))) = b(a(c(b(?x_8)))),
     b(b(b(?x))) = c(a(c(?x))),
     a(a(b(?x))) = b(a(c(?x))),
     a(b(b(?x_1))) = b(b(a(?x_1))),
     a(c(a(?x_7))) = b(b(b(?x_7))),
     b(b(b(b(?x_1)))) = a(b(b(a(?x_1)))),
     b(b(c(a(?x_7)))) = a(b(b(b(?x_7)))),
     a(b(b(b(?x_1)))) = c(b(b(a(?x_1)))),
     a(b(c(a(?x_7)))) = c(b(b(b(?x_7)))),
     b(b(a(?x))) = a(b(b(?x))),
     a(b(a(?x))) = c(b(b(?x))),
     c(b(b(?x_4))) = a(b(a(?x_4))),
     c(c(a(?x_7))) = a(b(b(?x_7))),
     b(b(b(b(?x_4)))) = c(a(b(a(?x_4)))),
     b(b(c(a(?x_7)))) = c(a(b(b(?x_7)))),
     a(a(b(b(?x_4)))) = b(a(b(a(?x_4)))),
     a(a(c(a(?x_7)))) = b(a(b(b(?x_7)))),
     b(b(a(?x))) = c(a(b(?x))),
     a(a(a(?x))) = b(a(b(?x))),
     a(a(?x)) = a(a(?x)),
     b(b(b(?x_2))) = a(a(c(?x_2))),
     b(a(c(?x_3))) = a(a(b(?x_3))),
     b(a(b(?x_5))) = a(a(a(?x_5))),
     a(c(b(b(?x_2)))) = c(a(a(c(?x_2)))),
     a(c(a(c(?x_3)))) = c(a(a(b(?x_3)))),
     a(c(a(b(?x_5)))) = c(a(a(a(?x_5)))),
     c(a(b(b(?x_2)))) = a(a(a(c(?x_2)))),
     c(a(a(c(?x_3)))) = a(a(a(b(?x_3)))),
     c(a(a(b(?x_5)))) = a(a(a(a(?x_5)))),
     b(a(b(b(?x_2)))) = b(a(a(c(?x_2)))),
     b(a(a(c(?x_3)))) = b(a(a(b(?x_3)))),
     b(a(a(b(?x_5)))) = b(a(a(a(?x_5)))),
     a(c(c(?x))) = c(a(a(?x))),
     c(a(c(?x))) = a(a(a(?x))),
     b(a(c(?x))) = b(a(a(?x))),
     a(a(a(?x_6))) = c(a(c(?x_6))),
     a(b(a(?x_8))) = c(a(b(?x_8))),
     b(b(a(a(?x_6)))) = a(c(a(c(?x_6)))),
     b(b(b(a(?x_8)))) = a(c(a(b(?x_8)))),
     a(b(a(a(?x_6)))) = c(c(a(c(?x_6)))),
     a(b(b(a(?x_8)))) = c(c(a(b(?x_8)))),
     b(b(b(?x))) = a(c(a(?x))),
     a(b(b(?x))) = c(c(a(?x))),
     b(b(a(?x_1))) = b(a(b(?x_1))),
     b(a(a(?x_6))) = b(a(c(?x_6))),
     b(a(b(a(?x_1)))) = b(b(a(b(?x_1)))),
     b(a(a(a(?x_6)))) = b(b(a(c(?x_6)))),
     a(c(b(a(?x_1)))) = c(b(a(b(?x_1)))),
     a(c(a(a(?x_6)))) = c(b(a(c(?x_6)))),
     c(a(b(a(?x_1)))) = a(b(a(b(?x_1)))),
     c(a(a(a(?x_6)))) = a(b(a(c(?x_6)))),
     b(a(b(?x))) = b(b(a(?x))),
     a(c(b(?x))) = c(b(a(?x))),
     c(a(b(?x))) = a(b(a(?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(a(c(?x_1))), b(b(b(?x_1)))> by Rules <1, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([],3),([(a,1)],5),([],2)], [([(b,1)],7)]>
 joinable by a reduction of rules <[([],3),([(a,1)],4),([],2)], [([(b,1)],7)]>
 joinable by a reduction of rules <[([],3),([(a,1)],5)], [([],7),([(b,1)],6),([],5)]>
 joinable by a reduction of rules <[([],3),([(a,1)],5)], [([],7),([(b,1)],6),([],4)]>
 joinable by a reduction of rules <[([],3),([(a,1)],4)], [([],7),([(b,1)],6),([],5)]>
 joinable by a reduction of rules <[([],3),([(a,1)],4)], [([],7),([(b,1)],6),([],4)]>
Critical Pair <c(a(b(?x_3))), b(b(a(?x_3)))> by Rules <3, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],6),([],0)], []>
Critical Pair <c(a(a(?x_4))), a(c(c(?x_4)))> by Rules <4, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[([],3)], [([(a,1)],0),([(a,1)],7)]>
Critical Pair <c(a(a(?x_5))), a(c(c(?x_5)))> by Rules <5, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[([],3)], [([(a,1)],0),([(a,1)],7)]>
Critical Pair <c(b(a(?x_7))), a(c(b(?x_7)))> by Rules <7, 1> preceded by [(c,1)]
 joinable by a reduction of rules <[([],1),([(a,1)],3),([],2)], [([(a,1)],1),([],2),([(b,1)],5),([(b,1)],2)]>
 joinable by a reduction of rules <[([],1),([(a,1)],3),([],2)], [([(a,1)],1),([],2),([(b,1)],4),([(b,1)],2)]>
Critical Pair <a(c(a(?x_6))), b(b(b(?x_6)))> by Rules <6, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],3),([],2)], []>
Critical Pair <c(b(b(?x_2))), a(b(a(?x_2)))> by Rules <2, 3> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([],6),([(c,1)],2)]>
Critical Pair <c(c(a(?x_6))), a(b(b(?x_6)))> by Rules <6, 3> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],3)], [([],6)]>
Critical Pair <b(b(b(?x))), a(a(c(?x)))> by Rules <0, 4> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],7),([],7)], [([],2),([(b,1)],5)]>
 joinable by a reduction of rules <[([(b,1)],7),([],7)], [([],2),([(b,1)],4)]>
Critical Pair <b(a(c(?x_1))), a(a(b(?x_1)))> by Rules <1, 4> preceded by [(b,1)]
unknown Diagram Decreasing
check Non-Confluence...
obtain 12 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(b(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)),
     b(b(?x)) -> b(a(?x)) ]
P:
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> c(a(?x)) ]
S: unknown termination
failure(Step 1)
STEP: 2 (linear)
S:
   [ c(c(?x)) -> b(b(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)),
     b(b(?x)) -> b(a(?x)) ]
P:
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> c(a(?x)) ]
S: unknown termination
failure(Step 2)
STEP: 3 (relative)
S:
   [ c(c(?x)) -> b(b(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)),
     b(b(?x)) -> b(a(?x)) ]
P:
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> c(a(?x)) ]
Check relative termination:
   [ c(c(?x)) -> b(b(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)),
     b(b(?x)) -> b(a(?x)) ]
   [ c(a(?x)) -> a(b(?x)),
     a(b(?x)) -> c(a(?x)) ]
Polynomial Interpretation:
  a:= (1)+(1)*x1
  b:= (1)*x1
  c:= (1)*x1
retract a(a(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)+(2)*x1
  c:= (2)*x1
retract c(b(?x)) -> a(c(?x))
retract a(a(?x)) -> b(b(?x))
retract b(c(?x)) -> a(a(?x))
retract b(c(?x)) -> a(a(?x))
retract b(b(?x)) -> b(a(?x))
retract a(b(?x)) -> c(a(?x))
Polynomial Interpretation:
  a:= (3)*x1
  b:= (1)*x1*x1
  c:= (2)+(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)) -> b(b(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(a(?x)) -> a(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> b(a(?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)) -> b(b(?x)),
     c(b(?x)) -> a(c(?x)),
     a(a(?x)) -> b(b(?x)),
     c(a(?x)) -> a(b(?x)),
     b(c(?x)) -> a(a(?x)),
     b(c(?x)) -> a(a(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> b(a(?x)) ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible

{6}
(cm)Rewrite Rules:
   [ b(b(?x)) -> b(b(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ b(b(b(?x))) = b(b(b(?x))) ]
Outer CPs:
   [  ]
not Overlay, check Termination...
unknown/not 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
1022.trs: Failure(unknown CR)
MAYBE
(13533 msec.)