(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(b(?x)),
     c(b(?x)) -> c(c(?x)),
     b(c(?x)) -> a(c(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ a(c(a(?x_5))) = b(c(b(?x_5))),
     a(a(b(?x_6))) = b(c(c(?x_6))),
     a(c(c(?x_7))) = b(c(b(?x_7))),
     a(a(a(?x_4))) = c(c(b(?x_4))),
     a(a(c(?x_8))) = c(c(c(?x_8))),
     a(b(c(?x))) = c(c(c(?x))),
     a(c(c(?x_1))) = c(c(b(?x_1))),
     a(c(a(?x_3))) = c(c(b(?x_3))),
     a(a(a(?x_4))) = c(a(b(?x_4))),
     a(a(c(?x_8))) = c(a(c(?x_8))),
     b(a(c(?x_8))) = a(a(c(?x_8))),
     c(a(a(?x_4))) = c(a(b(?x_4))),
     c(a(c(?x_8))) = c(a(c(?x_8))),
     c(c(a(?x_5))) = a(b(b(?x_5))),
     c(c(c(?x_7))) = a(b(b(?x_7))),
     c(a(a(?x_4))) = c(c(b(?x_4))),
     c(a(c(?x_8))) = c(c(c(?x_8))),
     b(c(a(?x_5))) = a(c(b(?x_5))),
     b(a(b(?x_6))) = a(c(c(?x_6))),
     b(c(c(?x_7))) = a(c(b(?x_7))),
     a(c(c(?x))) = c(c(a(?x))),
     b(a(a(?x))) = a(a(b(?x))),
     c(a(b(?x))) = a(b(c(?x))) ]
Outer CPs:
   [ c(c(?x_1)) = c(a(?x_1)),
     c(a(?x_5)) = c(c(?x_5)) ]
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(a(?x_6))) = b(c(b(?x_6))),
     a(a(b(?x_7))) = b(c(c(?x_7))),
     a(c(c(?x_8))) = b(c(b(?x_8))),
     c(c(c(a(?x_6)))) = a(b(c(b(?x_6)))),
     c(c(a(b(?x_7)))) = a(b(c(c(?x_7)))),
     c(c(c(c(?x_8)))) = a(b(c(b(?x_8)))),
     c(c(c(?x))) = a(b(c(?x))),
     c(a(?x)) = c(c(?x)),
     a(a(a(?x_5))) = c(c(b(?x_5))),
     a(a(c(?x_9))) = c(c(c(?x_9))),
     c(c(a(a(?x_5)))) = a(c(c(b(?x_5)))),
     c(c(a(c(?x_9)))) = a(c(c(c(?x_9)))),
     c(c(b(?x))) = a(c(c(?x))),
     a(c(c(?x_1))) = c(c(a(?x_1))),
     a(b(c(?x_2))) = c(c(c(?x_2))),
     a(c(c(?x_3))) = c(c(b(?x_3))),
     a(c(a(?x_4))) = c(c(b(?x_4))),
     c(c(c(c(?x_1)))) = a(c(c(a(?x_1)))),
     c(c(b(c(?x_2)))) = a(c(c(c(?x_2)))),
     c(c(c(c(?x_3)))) = a(c(c(b(?x_3)))),
     c(c(c(a(?x_4)))) = a(c(c(b(?x_4)))),
     c(c(a(?x))) = a(c(c(?x))),
     c(c(?x)) = c(a(?x)),
     a(a(a(?x_5))) = c(a(b(?x_5))),
     a(a(c(?x_9))) = c(a(c(?x_9))),
     c(c(a(a(?x_5)))) = a(c(a(b(?x_5)))),
     c(c(a(c(?x_9)))) = a(c(a(c(?x_9)))),
     c(c(b(?x))) = a(c(a(?x))),
     b(a(a(?x_1))) = a(a(b(?x_1))),
     b(a(c(?x_9))) = a(a(c(?x_9))),
     a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))),
     a(a(a(c(?x_9)))) = b(a(a(c(?x_9)))),
     c(c(a(a(?x_1)))) = a(a(a(b(?x_1)))),
     c(c(a(c(?x_9)))) = a(a(a(c(?x_9)))),
     c(a(a(a(?x_1)))) = a(a(a(b(?x_1)))),
     c(a(a(c(?x_9)))) = a(a(a(c(?x_9)))),
     c(a(a(a(?x_1)))) = c(a(a(b(?x_1)))),
     c(a(a(c(?x_9)))) = c(a(a(c(?x_9)))),
     c(c(a(a(?x_1)))) = c(a(a(b(?x_1)))),
     c(c(a(c(?x_9)))) = c(a(a(c(?x_9)))),
     a(a(b(?x))) = b(a(a(?x))),
     c(c(b(?x))) = a(a(a(?x))),
     c(a(b(?x))) = a(a(a(?x))),
     c(a(b(?x))) = c(a(a(?x))),
     c(c(b(?x))) = c(a(a(?x))),
     c(a(a(?x_6))) = c(a(b(?x_6))),
     c(a(c(?x_9))) = c(a(c(?x_9))),
     b(c(a(a(?x_6)))) = a(c(a(b(?x_6)))),
     b(c(a(c(?x_9)))) = a(c(a(c(?x_9)))),
     a(b(a(a(?x_6)))) = c(c(a(b(?x_6)))),
     a(b(a(c(?x_9)))) = c(c(a(c(?x_9)))),
     a(c(a(a(?x_6)))) = b(c(a(b(?x_6)))),
     a(c(a(c(?x_9)))) = b(c(a(c(?x_9)))),
     b(c(b(?x))) = a(c(a(?x))),
     a(b(b(?x))) = c(c(a(?x))),
     a(c(b(?x))) = b(c(a(?x))),
     c(a(b(?x_1))) = a(b(c(?x_1))),
     c(c(a(?x_7))) = a(b(b(?x_7))),
     c(c(c(?x_8))) = a(b(b(?x_8))),
     a(b(a(b(?x_1)))) = c(a(b(c(?x_1)))),
     a(b(c(a(?x_7)))) = c(a(b(b(?x_7)))),
     a(b(c(c(?x_8)))) = c(a(b(b(?x_8)))),
     b(c(a(b(?x_1)))) = a(a(b(c(?x_1)))),
     b(c(c(a(?x_7)))) = a(a(b(b(?x_7)))),
     b(c(c(c(?x_8)))) = a(a(b(b(?x_8)))),
     a(c(a(b(?x_1)))) = b(a(b(c(?x_1)))),
     a(c(c(a(?x_7)))) = b(a(b(b(?x_7)))),
     a(c(c(c(?x_8)))) = b(a(b(b(?x_8)))),
     a(b(c(?x))) = c(a(b(?x))),
     b(c(c(?x))) = a(a(b(?x))),
     a(c(c(?x))) = b(a(b(?x))),
     c(a(a(?x_6))) = c(c(b(?x_6))),
     c(a(c(?x_9))) = c(c(c(?x_9))),
     b(c(a(a(?x_6)))) = a(c(c(b(?x_6)))),
     b(c(a(c(?x_9)))) = a(c(c(c(?x_9)))),
     a(b(a(a(?x_6)))) = c(c(c(b(?x_6)))),
     a(b(a(c(?x_9)))) = c(c(c(c(?x_9)))),
     a(c(a(a(?x_6)))) = b(c(c(b(?x_6)))),
     a(c(a(c(?x_9)))) = b(c(c(c(?x_9)))),
     b(c(b(?x))) = a(c(c(?x))),
     a(b(b(?x))) = c(c(c(?x))),
     a(c(b(?x))) = b(c(c(?x))),
     b(c(a(?x_7))) = a(c(b(?x_7))),
     b(a(b(?x_8))) = a(c(c(?x_8))),
     b(c(c(?x_9))) = a(c(b(?x_9))),
     c(c(c(a(?x_7)))) = a(a(c(b(?x_7)))),
     c(c(a(b(?x_8)))) = a(a(c(c(?x_8)))),
     c(c(c(c(?x_9)))) = a(a(c(b(?x_9)))),
     c(a(c(a(?x_7)))) = a(a(c(b(?x_7)))),
     c(a(a(b(?x_8)))) = a(a(c(c(?x_8)))),
     c(a(c(c(?x_9)))) = a(a(c(b(?x_9)))),
     a(a(c(a(?x_7)))) = b(a(c(b(?x_7)))),
     a(a(a(b(?x_8)))) = b(a(c(c(?x_8)))),
     a(a(c(c(?x_9)))) = b(a(c(b(?x_9)))),
     c(a(c(a(?x_7)))) = c(a(c(b(?x_7)))),
     c(a(a(b(?x_8)))) = c(a(c(c(?x_8)))),
     c(a(c(c(?x_9)))) = c(a(c(b(?x_9)))),
     c(c(c(a(?x_7)))) = c(a(c(b(?x_7)))),
     c(c(a(b(?x_8)))) = c(a(c(c(?x_8)))),
     c(c(c(c(?x_9)))) = c(a(c(b(?x_9)))),
     c(c(c(?x))) = a(a(c(?x))),
     c(a(c(?x))) = a(a(c(?x))),
     a(a(c(?x))) = b(a(c(?x))),
     c(c(c(?x))) = c(a(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 <a(c(a(?x_5))), b(c(b(?x_5)))> by Rules <5, 0> preceded by [(a,1)]
 joinable by a reduction of rules <[([],0)], [([(b,1)],5)]>
Critical Pair <a(a(b(?x_6))), b(c(c(?x_6)))> by Rules <6, 0> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],1)], [([],8)]>
Critical Pair <a(c(c(?x_7))), b(c(b(?x_7)))> by Rules <7, 0> preceded by [(a,1)]
 joinable by a reduction of rules <[([],0)], [([(b,1)],7)]>
Critical Pair <a(a(a(?x_4))), c(c(b(?x_4)))> by Rules <4, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[([],2)], [([(c,1)],5)]>
Critical Pair <a(a(c(?x_8))), c(c(c(?x_8)))> by Rules <8, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[([],2)], []>
 joinable by a reduction of rules <[([(a,1)],0)], [([],6)]>
Critical Pair <a(b(c(?x))), c(c(c(?x)))> by Rules <0, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[([],1)], []>
 joinable by a reduction of rules <[], [([],6)]>
Critical Pair <a(c(c(?x_1))), c(c(b(?x_1)))> by Rules <1, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],6),([],2)], []>
Critical Pair <a(c(a(?x_3))), c(c(b(?x_3)))> by Rules <3, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],6),([(a,1)],4),([(a,1)],2),([(a,1)],6),([(a,1)],3)]>
Critical Pair <a(a(a(?x_4))), c(a(b(?x_4)))> by Rules <4, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[([],2)], [([(c,1)],3)]>
Critical Pair <a(a(c(?x_8))), c(a(c(?x_8)))> by Rules <8, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],0),([],3)], []>
 joinable by a reduction of rules <[([],2)], [([(c,1)],0),([],7)]>
 joinable by a reduction of rules <[([(a,1)],0),([],1)], [([(c,1)],0),([],7)]>
Critical Pair <b(a(c(?x_8))), a(a(c(?x_8)))> by Rules <8, 4> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],0),([],4)], []>
Critical Pair <c(a(a(?x_4))), c(a(b(?x_4)))> by Rules <4, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],2)], [([(c,1)],1)]>
Critical Pair <c(a(c(?x_8))), c(a(c(?x_8)))> by Rules <8, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <c(c(a(?x_5))), a(b(b(?x_5)))> by Rules <5, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[], [([(a,1)],4),([],2)]>
 joinable by a reduction of rules <[], [([],3),([(c,1)],3)]>
 joinable by a reduction of rules <[], [([],1),([(c,1)],5)]>
Critical Pair <c(c(c(?x_7))), a(b(b(?x_7)))> by Rules <7, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],6)], [([],3)]>
Critical Pair <c(a(a(?x_4))), c(c(b(?x_4)))> by Rules <4, 7> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],2)], [([(c,1)],7)]>
Critical Pair <c(a(c(?x_8))), c(c(c(?x_8)))> by Rules <8, 7> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],0),([],7)], []>
 joinable by a reduction of rules <[], [([],6),([],3)]>
Critical Pair <b(c(a(?x_5))), a(c(b(?x_5)))> by Rules <5, 8> preceded by [(b,1)]
 joinable by a reduction of rules <[([],8)], [([(a,1)],5)]>
Critical Pair <b(a(b(?x_6))), a(c(c(?x_6)))> by Rules <6, 8> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],1)], [([],0)]>
Critical Pair <b(c(c(?x_7))), a(c(b(?x_7)))> by Rules <7, 8> preceded by [(b,1)]
 joinable by a reduction of rules <[([],8)], [([(a,1)],7)]>
Critical Pair <a(c(c(?x))), c(c(a(?x)))> by Rules <2, 2> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],6),([],2),([(c,1)],5)], []>
 joinable by a reduction of rules <[([(a,1)],6),([],2),([(c,1)],7)], [([],6),([],3),([(c,1)],2)]>
Critical Pair <b(a(a(?x))), a(a(b(?x)))> by Rules <4, 4> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],2),([],8)], [([(a,1)],1)]>
 joinable by a reduction of rules <[([(b,1)],2)], [([(a,1)],1),([],0)]>
Critical Pair <c(a(b(?x))), a(b(c(?x)))> by Rules <6, 6> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],1)], [([],1)]>
Critical Pair <c(a(?x_3)), c(c(?x_3))> by Rules <3, 1> preceded by []
 joinable by a reduction of rules <[], [([],6),([],3)]>
Critical Pair <c(c(?x_7)), c(a(?x_7))> by Rules <7, 5> preceded by []
 joinable by a reduction of rules <[([],6),([],3)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 14 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(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<a(c(c(?x))), c(c(a(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<a(c(a(?x_1))), c(c(b(?x_1)))> --> <a(c(a(?x_1))), c(c(a(?x_1)))> => no
<a(a(a(?x_2))), c(a(b(?x_2)))> --> <a(c(c(?x_2))), c(c(a(?x_2)))> => no
<b(a(a(?x))), a(a(b(?x)))> --> <b(c(c(?x))), a(c(a(?x)))> => no
<c(a(a(?x_2))), c(a(b(?x_2)))> --> <c(c(c(?x_2))), c(c(a(?x_2)))> => no
<c(c(?x_4)), c(a(?x_4))> --> <c(c(?x_4)), c(a(?x_4))> => no
<c(a(a(?x_2))), c(c(b(?x_2)))> --> <c(c(c(?x_2))), c(c(a(?x_2)))> => no
<c(a(?x_3)), c(c(?x_3))> --> <c(a(?x_3)), c(c(?x_3))> => no
PCP_in(symP,S):
<a(b(c(?x_4))), c(c(c(?x_4)))> --> <c(a(c(?x_4))), c(c(c(?x_4)))> => no
<a(c(c(?x_3))), c(c(b(?x_3)))> --> <a(c(c(?x_3))), c(c(a(?x_3)))> => no
<a(a(c(?x_1))), c(a(c(?x_1)))> --> <c(c(c(?x_1))), c(a(c(?x_1)))> => no
<b(a(c(?x_1))), a(a(c(?x_1)))> --> <b(a(c(?x_1))), c(c(c(?x_1)))> => no
<c(a(c(?x_1))), c(a(c(?x_1)))> --> <c(a(c(?x_1))), c(a(c(?x_1)))> => yes
<c(a(c(?x_1))), c(c(c(?x_1)))> --> <c(a(c(?x_1))), c(c(c(?x_1)))> => no
CP(S,symP):
<c(a(?x)), c(c(?x))> --> <c(a(?x)), c(c(?x))> => no
<a(a(a(?x))), c(c(b(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<b(c(a(?x))), a(c(b(?x)))> --> <b(c(a(?x))), a(c(a(?x)))> => yes
<c(c(a(?x))), a(b(b(?x)))> --> <c(c(a(?x))), a(c(c(?x)))> => no
<a(c(a(?x))), b(c(b(?x)))> --> <a(c(a(?x))), b(c(a(?x)))> => yes
<b(c(c(?x))), a(c(b(?x)))> --> <b(c(c(?x))), a(c(a(?x)))> => no
<c(c(c(?x))), a(b(b(?x)))> --> <c(c(c(?x))), a(c(c(?x)))> => no
<a(c(c(?x))), b(c(b(?x)))> --> <a(c(c(?x))), b(c(a(?x)))> => no
check joinability condition:
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of a(c(a(?x_1))) and c(c(a(?x_1))): maybe not joinable
check modulo joinability of a(c(c(?x_2))) and c(c(a(?x_2))): maybe not joinable
check modulo joinability of b(c(c(?x))) and a(c(a(?x))): joinable by {1}
check modulo joinability of c(c(c(?x_2))) and c(c(a(?x_2))): joinable by {1}
check modulo joinability of c(c(?x_4)) and c(a(?x_4)): joinable by {1}
check modulo joinability of c(c(c(?x_2))) and c(c(a(?x_2))): joinable by {1}
check modulo joinability of c(a(?x_3)) and c(c(?x_3)): joinable by {1}
check modulo joinability of c(a(c(?x_4))) and c(c(c(?x_4))): joinable by {1}
check modulo reachablity from c(c(a(?x_3))) to a(c(c(?x_3))): maybe not reachable
check modulo joinability of c(c(c(?x_1))) and c(a(c(?x_1))): joinable by {1}
check modulo reachablity from c(c(c(?x_1))) to b(a(c(?x_1))): maybe not reachable
check modulo reachablity from c(c(c(?x_1))) to c(a(c(?x_1))): reachable by {1}
check modulo reachablity from c(a(?x)) to c(c(?x)): maybe not reachable
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of c(c(a(?x))) and a(c(c(?x))): maybe not joinable
check modulo joinability of b(c(c(?x))) and a(c(a(?x))): joinable by {1}
check modulo joinability of c(c(c(?x))) and a(c(c(?x))): maybe not joinable
check modulo joinability of a(c(c(?x))) and b(c(a(?x))): joinable by {0,1}
failed
failure(Step 1)
   [ c(c(?x)) -> c(a(?x)) ]
Added S-Rules:
   [ c(c(?x)) -> c(a(?x)) ]
Added P-Rules:
   [ ]
replace: c(b(?x)) -> c(c(?x)) => c(b(?x)) -> a(b(?x))
STEP: 2 (linear)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<a(c(c(?x))), c(c(a(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<a(c(a(?x_1))), c(c(b(?x_1)))> --> <a(c(a(?x_1))), c(c(a(?x_1)))> => no
<a(a(a(?x_2))), c(a(b(?x_2)))> --> <a(c(c(?x_2))), c(c(a(?x_2)))> => no
<b(a(a(?x))), a(a(b(?x)))> --> <b(c(c(?x))), a(c(a(?x)))> => no
<c(a(a(?x_2))), c(a(b(?x_2)))> --> <c(c(c(?x_2))), c(c(a(?x_2)))> => no
<c(c(?x_4)), c(a(?x_4))> --> <c(c(?x_4)), c(a(?x_4))> => no
<c(a(a(?x_2))), c(c(b(?x_2)))> --> <c(c(c(?x_2))), c(c(a(?x_2)))> => no
<c(a(?x_3)), c(c(?x_3))> --> <c(a(?x_3)), c(c(?x_3))> => no
CP_in(symP,S):
<a(a(c(?x))), c(a(c(?x)))> --> <c(c(c(?x))), c(a(c(?x)))> => no
<b(a(c(?x))), a(a(c(?x)))> --> <b(a(c(?x))), c(c(c(?x)))> => no
<c(a(c(?x))), c(a(c(?x)))> --> <c(a(c(?x))), c(a(c(?x)))> => yes
<c(a(c(?x))), c(c(c(?x)))> --> <c(a(c(?x))), c(c(c(?x)))> => no
<a(c(c(?x))), c(c(b(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<a(b(c(?x))), c(c(c(?x)))> --> <c(a(c(?x))), c(c(c(?x)))> => no
CP(S,symP):
<c(a(?x)), c(c(?x))> --> <c(a(?x)), c(c(?x))> => no
<a(a(a(?x))), c(c(b(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<b(c(a(?x))), a(c(b(?x)))> --> <b(c(a(?x))), a(c(a(?x)))> => yes
<c(c(a(?x))), a(b(b(?x)))> --> <c(c(a(?x))), a(c(c(?x)))> => no
<a(c(a(?x))), b(c(b(?x)))> --> <a(c(a(?x))), b(c(a(?x)))> => yes
<b(c(c(?x))), a(c(b(?x)))> --> <b(c(c(?x))), a(c(a(?x)))> => no
<c(c(c(?x))), a(b(b(?x)))> --> <c(c(c(?x))), a(c(c(?x)))> => no
<a(c(c(?x))), b(c(b(?x)))> --> <a(c(c(?x))), b(c(a(?x)))> => no
check joinability condition:
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of a(c(a(?x_1))) and c(c(a(?x_1))): maybe not joinable
check modulo joinability of a(c(c(?x_2))) and c(c(a(?x_2))): maybe not joinable
check modulo joinability of b(c(c(?x))) and a(c(a(?x))): joinable by {1}
check modulo joinability of c(c(c(?x_2))) and c(c(a(?x_2))): joinable by {1}
check modulo joinability of c(c(?x_4)) and c(a(?x_4)): maybe not joinable
check modulo joinability of c(c(c(?x_2))) and c(c(a(?x_2))): joinable by {1}
check modulo joinability of c(a(?x_3)) and c(c(?x_3)): maybe not joinable
check modulo joinability of c(c(c(?x))) and c(a(c(?x))): joinable by {1,3}
check modulo reachablity from c(c(c(?x))) to b(a(c(?x))): maybe not reachable
check modulo reachablity from c(c(c(?x))) to c(a(c(?x))): reachable by {1,3}
check modulo reachablity from c(c(a(?x))) to a(c(c(?x))): maybe not reachable
check modulo joinability of c(a(c(?x))) and c(c(c(?x))): joinable by {1,3}
check modulo reachablity from c(a(?x)) to c(c(?x)): maybe not reachable
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of c(c(a(?x))) and a(c(c(?x))): maybe not joinable
check modulo joinability of b(c(c(?x))) and a(c(a(?x))): joinable by {1}
check modulo joinability of c(c(c(?x))) and a(c(c(?x))): maybe not joinable
check modulo joinability of a(c(c(?x))) and b(c(a(?x))): joinable by {1}
failed
failure(Step 2)
   [ c(c(?x)) -> c(a(?x)) ]
Added S-Rules:
   [ c(c(?x)) -> c(a(?x)) ]
Added P-Rules:
   [ ]
replace: c(b(?x)) -> c(c(?x)) => c(b(?x)) -> a(b(?x))
STEP: 3 (relative)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Check relative termination:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)) ]
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (4)+(1)*x1
  c:= (1)*x1
retract a(b(?x)) -> c(a(?x))
retract b(b(?x)) -> a(a(?x))
retract c(b(?x)) -> c(a(?x))
retract c(b(?x)) -> c(c(?x))
retract a(b(?x)) -> c(c(?x))
retract b(c(?x)) -> a(c(?x))
Polynomial Interpretation:
  a:= (2)+(2)*x1+(1)*x1*x1
  b:= (1)+(1)*x1*x1
  c:= (2)+(1)*x1+(1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(a(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 4)
STEP: 5 (linear)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(a(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 5)
STEP: 6 (relative)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(a(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Check relative termination:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> c(c(?x)),
     c(c(?x)) -> c(a(?x)) ]
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (3)*x1
  b:= (3)+(3)*x1
  c:= (3)*x1
retract a(b(?x)) -> c(a(?x))
retract b(b(?x)) -> a(a(?x))
retract c(b(?x)) -> c(a(?x))
retract c(b(?x)) -> c(c(?x))
retract a(b(?x)) -> c(c(?x))
retract b(c(?x)) -> a(c(?x))
Polynomial Interpretation:
  a:= (1)+(3)*x1
  b:= (3)+(1)*x1
  c:= (3)*x1
retract a(a(?x)) -> c(c(?x))
retract a(b(?x)) -> c(a(?x))
retract b(b(?x)) -> a(a(?x))
retract c(b(?x)) -> c(a(?x))
retract c(b(?x)) -> c(c(?x))
retract a(b(?x)) -> c(c(?x))
retract b(c(?x)) -> a(c(?x))
Polynomial Interpretation:
  a:= (3)*x1
  b:= (3)+(1)*x1
  c:= (1)+(3)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 6)
STEP: 7 (parallel)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<a(c(c(?x))), c(c(a(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<a(c(a(?x_1))), c(c(b(?x_1)))> --> <a(c(a(?x_1))), c(c(a(?x_1)))> => no
<a(a(a(?x_2))), c(a(b(?x_2)))> --> <a(c(c(?x_2))), c(c(a(?x_2)))> => no
<b(a(a(?x))), a(a(b(?x)))> --> <b(c(c(?x))), a(c(a(?x)))> => no
<c(a(a(?x_2))), c(a(b(?x_2)))> --> <c(c(c(?x_2))), c(c(a(?x_2)))> => no
<a(b(?x_4)), c(a(?x_4))> --> <c(a(?x_4)), c(a(?x_4))> => yes
<c(a(a(?x_2))), a(b(b(?x_2)))> --> <c(c(c(?x_2))), a(c(c(?x_2)))> => no
<c(a(?x_3)), a(b(?x_3))> --> <c(a(?x_3)), c(a(?x_3))> => yes
PCP_in(symP,S):
<a(b(c(?x_4))), c(c(c(?x_4)))> --> <c(a(c(?x_4))), c(c(c(?x_4)))> => no
<a(c(c(?x_3))), c(c(b(?x_3)))> --> <a(c(c(?x_3))), c(c(a(?x_3)))> => no
<a(a(c(?x_1))), c(a(c(?x_1)))> --> <c(c(c(?x_1))), c(a(c(?x_1)))> => no
<b(a(c(?x_1))), a(a(c(?x_1)))> --> <b(a(c(?x_1))), c(c(c(?x_1)))> => no
<c(a(c(?x_1))), c(a(c(?x_1)))> --> <c(a(c(?x_1))), c(a(c(?x_1)))> => yes
<c(a(c(?x_1))), a(b(c(?x_1)))> --> <c(a(c(?x_1))), c(a(c(?x_1)))> => yes
CP(S,symP):
<c(a(?x)), c(c(?x))> --> <c(a(?x)), c(c(?x))> => no
<a(a(a(?x))), c(c(b(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<b(c(a(?x))), a(c(b(?x)))> --> <b(c(a(?x))), a(c(a(?x)))> => yes
<c(c(a(?x))), a(b(b(?x)))> --> <c(c(a(?x))), a(c(c(?x)))> => no
<a(c(a(?x))), b(c(b(?x)))> --> <a(c(a(?x))), b(c(a(?x)))> => yes
<b(a(b(?x))), a(c(b(?x)))> --> <b(c(a(?x))), a(c(a(?x)))> => yes
<c(a(b(?x))), a(b(b(?x)))> --> <c(c(a(?x))), a(c(c(?x)))> => no
<a(a(b(?x))), b(c(b(?x)))> --> <a(c(a(?x))), b(c(a(?x)))> => yes
check joinability condition:
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of a(c(a(?x_1))) and c(c(a(?x_1))): maybe not joinable
check modulo joinability of a(c(c(?x_2))) and c(c(a(?x_2))): maybe not joinable
check modulo joinability of b(c(c(?x))) and a(c(a(?x))): joinable by {1}
check modulo joinability of c(c(c(?x_2))) and c(c(a(?x_2))): joinable by {1}
check modulo joinability of c(c(c(?x_2))) and a(c(c(?x_2))): maybe not joinable
check modulo joinability of c(a(c(?x_4))) and c(c(c(?x_4))): joinable by {1}
check modulo reachablity from c(c(a(?x_3))) to a(c(c(?x_3))): maybe not reachable
check modulo joinability of c(c(c(?x_1))) and c(a(c(?x_1))): joinable by {1}
check modulo reachablity from c(c(c(?x_1))) to b(a(c(?x_1))): maybe not reachable
check modulo reachablity from c(a(?x)) to c(c(?x)): maybe not reachable
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of c(c(a(?x))) and a(c(c(?x))): maybe not joinable
check modulo joinability of c(c(a(?x))) and a(c(c(?x))): maybe not joinable
failed
failure(Step 7)
   [ c(c(?x)) -> c(a(?x)) ]
Added S-Rules:
   [ c(c(?x)) -> c(a(?x)) ]
Added P-Rules:
   [ ]
replace: c(b(?x)) -> a(b(?x)) => c(b(?x)) -> c(c(?x))
STEP: 8 (linear)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<a(c(c(?x))), c(c(a(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<a(c(a(?x_1))), c(c(b(?x_1)))> --> <a(c(a(?x_1))), c(c(a(?x_1)))> => no
<a(a(a(?x_2))), c(a(b(?x_2)))> --> <a(c(c(?x_2))), c(c(a(?x_2)))> => no
<b(a(a(?x))), a(a(b(?x)))> --> <b(c(c(?x))), a(c(a(?x)))> => no
<c(a(a(?x_2))), c(a(b(?x_2)))> --> <c(c(c(?x_2))), c(c(a(?x_2)))> => no
<a(b(?x_4)), c(a(?x_4))> --> <c(a(?x_4)), c(a(?x_4))> => yes
<c(a(a(?x_2))), a(b(b(?x_2)))> --> <c(c(c(?x_2))), a(c(c(?x_2)))> => no
<c(a(?x_3)), a(b(?x_3))> --> <c(a(?x_3)), c(a(?x_3))> => yes
CP_in(symP,S):
<a(a(c(?x))), c(a(c(?x)))> --> <c(c(c(?x))), c(a(c(?x)))> => no
<b(a(c(?x))), a(a(c(?x)))> --> <b(a(c(?x))), c(c(c(?x)))> => no
<c(a(c(?x))), c(a(c(?x)))> --> <c(a(c(?x))), c(a(c(?x)))> => yes
<c(a(c(?x))), a(b(c(?x)))> --> <c(a(c(?x))), c(a(c(?x)))> => yes
<a(c(c(?x))), c(c(b(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<a(b(c(?x))), c(c(c(?x)))> --> <c(a(c(?x))), c(c(c(?x)))> => no
CP(S,symP):
<c(a(?x)), c(c(?x))> --> <c(a(?x)), c(c(?x))> => no
<a(a(a(?x))), c(c(b(?x)))> --> <a(c(c(?x))), c(c(a(?x)))> => no
<b(c(a(?x))), a(c(b(?x)))> --> <b(c(a(?x))), a(c(a(?x)))> => yes
<c(c(a(?x))), a(b(b(?x)))> --> <c(c(a(?x))), a(c(c(?x)))> => no
<a(c(a(?x))), b(c(b(?x)))> --> <a(c(a(?x))), b(c(a(?x)))> => yes
<b(a(b(?x))), a(c(b(?x)))> --> <b(c(a(?x))), a(c(a(?x)))> => yes
<c(a(b(?x))), a(b(b(?x)))> --> <c(c(a(?x))), a(c(c(?x)))> => no
<a(a(b(?x))), b(c(b(?x)))> --> <a(c(a(?x))), b(c(a(?x)))> => yes
check joinability condition:
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of a(c(a(?x_1))) and c(c(a(?x_1))): maybe not joinable
check modulo joinability of a(c(c(?x_2))) and c(c(a(?x_2))): maybe not joinable
check modulo joinability of b(c(c(?x))) and a(c(a(?x))): joinable by {1}
check modulo joinability of c(c(c(?x_2))) and c(c(a(?x_2))): joinable by {1}
check modulo joinability of c(c(c(?x_2))) and a(c(c(?x_2))): maybe not joinable
check modulo joinability of c(c(c(?x))) and c(a(c(?x))): joinable by {1,3}
check modulo reachablity from c(c(c(?x))) to b(a(c(?x))): maybe not reachable
check modulo reachablity from c(c(a(?x))) to a(c(c(?x))): maybe not reachable
check modulo joinability of c(a(c(?x))) and c(c(c(?x))): joinable by {1,3}
check modulo reachablity from c(a(?x)) to c(c(?x)): maybe not reachable
check modulo joinability of a(c(c(?x))) and c(c(a(?x))): maybe not joinable
check modulo joinability of c(c(a(?x))) and a(c(c(?x))): maybe not joinable
check modulo joinability of c(c(a(?x))) and a(c(c(?x))): maybe not joinable
failed
failure(Step 8)
   [ c(c(?x)) -> c(a(?x)) ]
Added S-Rules:
   [ c(c(?x)) -> c(a(?x)) ]
Added P-Rules:
   [ ]
replace: c(b(?x)) -> a(b(?x)) => c(b(?x)) -> c(c(?x))
STEP: 9 (relative)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Check relative termination:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)) ]
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)*x1
  c:= (2)+(1)*x1
retract c(b(?x)) -> a(b(?x))
retract c(c(?x)) -> a(b(?x))
Polynomial Interpretation:
  a:= (1)*x1
  b:= (8)+(1)*x1
  c:= (1)*x1
retract a(b(?x)) -> c(a(?x))
retract c(b(?x)) -> c(a(?x))
retract c(b(?x)) -> a(b(?x))
retract a(b(?x)) -> c(c(?x))
retract c(c(?x)) -> a(b(?x))
retract b(c(?x)) -> a(c(?x))
Polynomial Interpretation:
  a:= (1)*x1*x1
  b:= (1)+(1)*x1*x1
  c:= (1)*x1*x1
retract a(b(?x)) -> c(a(?x))
retract b(b(?x)) -> a(a(?x))
retract c(b(?x)) -> c(a(?x))
retract c(b(?x)) -> a(b(?x))
retract a(b(?x)) -> c(c(?x))
retract c(c(?x)) -> a(b(?x))
retract b(c(?x)) -> a(c(?x))
Polynomial Interpretation:
  a:= (3)+(1)*x1+(1)*x1*x1
  b:= (1)+(2)*x1
  c:= (2)+(1)*x1+(1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 9)
STEP: 10 (parallel)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)),
     c(c(?x)) -> c(a(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 10)
STEP: 11 (linear)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)),
     c(c(?x)) -> c(a(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 11)
STEP: 12 (relative)
S:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)),
     c(c(?x)) -> c(a(?x)) ]
P:
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Check relative termination:
   [ a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(b(?x)) -> a(b(?x)),
     c(c(?x)) -> c(a(?x)) ]
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     c(c(?x)) -> a(b(?x)),
     b(c(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (2)+(1)*x1
  b:= (1)*x1
  c:= (1)*x1
retract a(a(?x)) -> c(c(?x))
retract a(c(?x)) -> b(c(?x))
retract a(b(?x)) -> c(c(?x))
Polynomial Interpretation:
  a:= (1)*x1*x1
  b:= (2)+(1)*x1*x1
  c:= (2)*x1+(1)*x1*x1
retract a(a(?x)) -> c(c(?x))
retract a(b(?x)) -> c(a(?x))
retract b(b(?x)) -> a(a(?x))
retract c(b(?x)) -> c(a(?x))
retract c(b(?x)) -> a(b(?x))
retract a(c(?x)) -> b(c(?x))
retract a(b(?x)) -> c(c(?x))
retract b(c(?x)) -> a(c(?x))
Polynomial Interpretation:
  a:= (1)*x1
  b:= (2)*x1*x1
  c:= (1)+(1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 12)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

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

Try Commutative Decomposition for...
   [ a(c(?x)) -> b(c(?x)),
     a(b(?x)) -> c(c(?x)),
     a(a(?x)) -> c(c(?x)),
     a(b(?x)) -> c(a(?x)),
     b(b(?x)) -> a(a(?x)),
     c(b(?x)) -> c(a(?x)),
     c(c(?x)) -> a(b(?x)),
     c(b(?x)) -> c(c(?x)),
     b(c(?x)) -> a(c(?x)) ]
Outside Critical Pair: <c(a(?x_3)), c(c(?x_3))> by Rules <3, 1>
develop reducts from lhs term...
<{}, c(a(?x_3))>
develop reducts from rhs term...
<{6}, a(b(?x_3))>
<{}, c(c(?x_3))>
Outside Critical Pair: <c(c(?x_7)), c(a(?x_7))> by Rules <7, 5>
develop reducts from lhs term...
<{6}, a(b(?x_7))>
<{}, c(c(?x_7))>
develop reducts from rhs term...
<{}, c(a(?x_7))>
Inside Critical Pair: <a(c(a(?x_5))), b(c(b(?x_5)))> by Rules <5, 0>
develop reducts from lhs term...
<{0}, b(c(a(?x_5)))>
<{}, a(c(a(?x_5)))>
develop reducts from rhs term...
<{8}, a(c(b(?x_5)))>
<{7}, b(c(c(?x_5)))>
<{5}, b(c(a(?x_5)))>
<{}, b(c(b(?x_5)))>
Inside Critical Pair: <a(a(b(?x_6))), b(c(c(?x_6)))> by Rules <6, 0>
develop reducts from lhs term...
<{2}, c(c(b(?x_6)))>
<{3}, a(c(a(?x_6)))>
<{1}, a(c(c(?x_6)))>
<{}, a(a(b(?x_6)))>
develop reducts from rhs term...
<{8}, a(c(c(?x_6)))>
<{6}, b(a(b(?x_6)))>
<{}, b(c(c(?x_6)))>
Inside Critical Pair: <a(c(c(?x_7))), b(c(b(?x_7)))> by Rules <7, 0>
develop reducts from lhs term...
<{0}, b(c(c(?x_7)))>
<{6}, a(a(b(?x_7)))>
<{}, a(c(c(?x_7)))>
develop reducts from rhs term...
<{8}, a(c(b(?x_7)))>
<{7}, b(c(c(?x_7)))>
<{5}, b(c(a(?x_7)))>
<{}, b(c(b(?x_7)))>
Inside Critical Pair: <a(a(a(?x_4))), c(c(b(?x_4)))> by Rules <4, 1>
develop reducts from lhs term...
<{2}, c(c(a(?x_4)))>
<{2}, a(c(c(?x_4)))>
<{}, a(a(a(?x_4)))>
develop reducts from rhs term...
<{6}, a(b(b(?x_4)))>
<{7}, c(c(c(?x_4)))>
<{5}, c(c(a(?x_4)))>
<{}, c(c(b(?x_4)))>
Inside Critical Pair: <a(a(c(?x_8))), c(c(c(?x_8)))> by Rules <8, 1>
develop reducts from lhs term...
<{2}, c(c(c(?x_8)))>
<{0}, a(b(c(?x_8)))>
<{}, a(a(c(?x_8)))>
develop reducts from rhs term...
<{6}, a(b(c(?x_8)))>
<{6}, c(a(b(?x_8)))>
<{}, c(c(c(?x_8)))>
Inside Critical Pair: <a(b(c(?x))), c(c(c(?x)))> by Rules <0, 2>
develop reducts from lhs term...
<{3}, c(a(c(?x)))>
<{1}, c(c(c(?x)))>
<{8}, a(a(c(?x)))>
<{}, a(b(c(?x)))>
develop reducts from rhs term...
<{6}, a(b(c(?x)))>
<{6}, c(a(b(?x)))>
<{}, c(c(c(?x)))>
Inside Critical Pair: <a(c(c(?x_1))), c(c(b(?x_1)))> by Rules <1, 2>
develop reducts from lhs term...
<{0}, b(c(c(?x_1)))>
<{6}, a(a(b(?x_1)))>
<{}, a(c(c(?x_1)))>
develop reducts from rhs term...
<{6}, a(b(b(?x_1)))>
<{7}, c(c(c(?x_1)))>
<{5}, c(c(a(?x_1)))>
<{}, c(c(b(?x_1)))>
Inside Critical Pair: <a(c(a(?x_3))), c(c(b(?x_3)))> by Rules <3, 2>
develop reducts from lhs term...
<{0}, b(c(a(?x_3)))>
<{}, a(c(a(?x_3)))>
develop reducts from rhs term...
<{6}, a(b(b(?x_3)))>
<{7}, c(c(c(?x_3)))>
<{5}, c(c(a(?x_3)))>
<{}, c(c(b(?x_3)))>
Inside Critical Pair: <a(a(a(?x_4))), c(a(b(?x_4)))> by Rules <4, 3>
develop reducts from lhs term...
<{2}, c(c(a(?x_4)))>
<{2}, a(c(c(?x_4)))>
<{}, a(a(a(?x_4)))>
develop reducts from rhs term...
<{3}, c(c(a(?x_4)))>
<{1}, c(c(c(?x_4)))>
<{}, c(a(b(?x_4)))>
Inside Critical Pair: <a(a(c(?x_8))), c(a(c(?x_8)))> by Rules <8, 3>
develop reducts from lhs term...
<{2}, c(c(c(?x_8)))>
<{0}, a(b(c(?x_8)))>
<{}, a(a(c(?x_8)))>
develop reducts from rhs term...
<{0}, c(b(c(?x_8)))>
<{}, c(a(c(?x_8)))>
Inside Critical Pair: <b(a(c(?x_8))), a(a(c(?x_8)))> by Rules <8, 4>
develop reducts from lhs term...
<{0}, b(b(c(?x_8)))>
<{}, b(a(c(?x_8)))>
develop reducts from rhs term...
<{2}, c(c(c(?x_8)))>
<{0}, a(b(c(?x_8)))>
<{}, a(a(c(?x_8)))>
Inside Critical Pair: <c(a(a(?x_4))), c(a(b(?x_4)))> by Rules <4, 5>
develop reducts from lhs term...
<{2}, c(c(c(?x_4)))>
<{}, c(a(a(?x_4)))>
develop reducts from rhs term...
<{3}, c(c(a(?x_4)))>
<{1}, c(c(c(?x_4)))>
<{}, c(a(b(?x_4)))>
Inside Critical Pair: <c(a(c(?x_8))), c(a(c(?x_8)))> by Rules <8, 5>
develop reducts from lhs term...
<{0}, c(b(c(?x_8)))>
<{}, c(a(c(?x_8)))>
develop reducts from rhs term...
<{0}, c(b(c(?x_8)))>
<{}, c(a(c(?x_8)))>
Inside Critical Pair: <c(c(a(?x_5))), a(b(b(?x_5)))> by Rules <5, 6>
develop reducts from lhs term...
<{6}, a(b(a(?x_5)))>
<{}, c(c(a(?x_5)))>
develop reducts from rhs term...
<{3}, c(a(b(?x_5)))>
<{1}, c(c(b(?x_5)))>
<{4}, a(a(a(?x_5)))>
<{}, a(b(b(?x_5)))>
Inside Critical Pair: <c(c(c(?x_7))), a(b(b(?x_7)))> by Rules <7, 6>
develop reducts from lhs term...
<{6}, a(b(c(?x_7)))>
<{6}, c(a(b(?x_7)))>
<{}, c(c(c(?x_7)))>
develop reducts from rhs term...
<{3}, c(a(b(?x_7)))>
<{1}, c(c(b(?x_7)))>
<{4}, a(a(a(?x_7)))>
<{}, a(b(b(?x_7)))>
Inside Critical Pair: <c(a(a(?x_4))), c(c(b(?x_4)))> by Rules <4, 7>
develop reducts from lhs term...
<{2}, c(c(c(?x_4)))>
<{}, c(a(a(?x_4)))>
develop reducts from rhs term...
<{6}, a(b(b(?x_4)))>
<{7}, c(c(c(?x_4)))>
<{5}, c(c(a(?x_4)))>
<{}, c(c(b(?x_4)))>
Inside Critical Pair: <c(a(c(?x_8))), c(c(c(?x_8)))> by Rules <8, 7>
develop reducts from lhs term...
<{0}, c(b(c(?x_8)))>
<{}, c(a(c(?x_8)))>
develop reducts from rhs term...
<{6}, a(b(c(?x_8)))>
<{6}, c(a(b(?x_8)))>
<{}, c(c(c(?x_8)))>
Inside Critical Pair: <b(c(a(?x_5))), a(c(b(?x_5)))> by Rules <5, 8>
develop reducts from lhs term...
<{8}, a(c(a(?x_5)))>
<{}, b(c(a(?x_5)))>
develop reducts from rhs term...
<{0}, b(c(b(?x_5)))>
<{7}, a(c(c(?x_5)))>
<{5}, a(c(a(?x_5)))>
<{}, a(c(b(?x_5)))>
Inside Critical Pair: <b(a(b(?x_6))), a(c(c(?x_6)))> by Rules <6, 8>
develop reducts from lhs term...
<{3}, b(c(a(?x_6)))>
<{1}, b(c(c(?x_6)))>
<{}, b(a(b(?x_6)))>
develop reducts from rhs term...
<{0}, b(c(c(?x_6)))>
<{6}, a(a(b(?x_6)))>
<{}, a(c(c(?x_6)))>
Inside Critical Pair: <b(c(c(?x_7))), a(c(b(?x_7)))> by Rules <7, 8>
develop reducts from lhs term...
<{8}, a(c(c(?x_7)))>
<{6}, b(a(b(?x_7)))>
<{}, b(c(c(?x_7)))>
develop reducts from rhs term...
<{0}, b(c(b(?x_7)))>
<{7}, a(c(c(?x_7)))>
<{5}, a(c(a(?x_7)))>
<{}, a(c(b(?x_7)))>
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
1028.trs: Failure(unknown CR)
MAYBE
(7954 msec.)