(ignored inputs)COMMENT submitted by: Johannes Waldmann
Rewrite Rules:
   [ c(c(?x)) -> b(b(?x)),
     a(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(b(?x)) -> a(c(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ c(c(b(?x_2))) = b(b(a(?x_2))),
     c(b(a(?x_5))) = b(b(b(?x_5))),
     a(b(b(?x))) = b(b(c(?x))),
     a(c(b(?x_2))) = b(b(a(?x_2))),
     a(b(a(?x_5))) = b(b(b(?x_5))),
     c(b(b(?x_1))) = c(b(c(?x_1))),
     b(b(b(?x_4))) = a(c(a(?x_4))),
     b(b(b(?x_1))) = b(b(c(?x_1))),
     c(a(c(?x_3))) = b(a(b(?x_3))),
     c(b(b(?x_4))) = b(a(a(?x_4))),
     c(b(b(?x))) = b(b(c(?x))),
     b(a(c(?x))) = a(c(b(?x))) ]
Outer CPs:
   [  ]
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(c(b(?x_3))) = b(b(a(?x_3))),
     c(b(a(?x_6))) = b(b(b(?x_6))),
     b(b(b(b(?x_1)))) = c(b(b(c(?x_1)))),
     b(b(c(b(?x_3)))) = c(b(b(a(?x_3)))),
     b(b(b(a(?x_6)))) = c(b(b(b(?x_6)))),
     b(b(b(b(?x_1)))) = a(b(b(c(?x_1)))),
     b(b(c(b(?x_3)))) = a(b(b(a(?x_3)))),
     b(b(b(a(?x_6)))) = a(b(b(b(?x_6)))),
     b(b(c(?x))) = c(b(b(?x))),
     b(b(c(?x))) = a(b(b(?x))),
     a(b(b(?x_2))) = b(b(c(?x_2))),
     a(c(b(?x_3))) = b(b(a(?x_3))),
     a(b(a(?x_6))) = b(b(b(?x_6))),
     c(b(b(b(?x_2)))) = c(b(b(c(?x_2)))),
     c(b(c(b(?x_3)))) = c(b(b(a(?x_3)))),
     c(b(b(a(?x_6)))) = c(b(b(b(?x_6)))),
     b(b(b(b(?x_2)))) = b(b(b(c(?x_2)))),
     b(b(c(b(?x_3)))) = b(b(b(a(?x_3)))),
     b(b(b(a(?x_6)))) = b(b(b(b(?x_6)))),
     c(b(c(?x))) = c(b(b(?x))),
     b(b(c(?x))) = b(b(b(?x))),
     c(b(b(?x_3))) = c(b(c(?x_3))),
     b(b(b(b(?x_3)))) = c(c(b(c(?x_3)))),
     b(b(b(b(?x_3)))) = a(c(b(c(?x_3)))),
     b(b(a(?x))) = c(c(b(?x))),
     b(b(a(?x))) = a(c(b(?x))),
     b(a(c(?x_1))) = a(c(b(?x_1))),
     b(b(b(?x_5))) = a(c(a(?x_5))),
     a(c(a(c(?x_1)))) = b(a(c(b(?x_1)))),
     a(c(b(b(?x_5)))) = b(a(c(a(?x_5)))),
     b(a(a(c(?x_1)))) = c(a(c(b(?x_1)))),
     b(a(b(b(?x_5)))) = c(a(c(a(?x_5)))),
     a(c(b(?x))) = b(a(c(?x))),
     b(a(b(?x))) = c(a(c(?x))),
     b(b(b(?x_3))) = b(b(c(?x_3))),
     a(c(b(b(?x_3)))) = b(b(b(c(?x_3)))),
     b(a(b(b(?x_3)))) = c(b(b(c(?x_3)))),
     a(c(a(?x))) = b(b(b(?x))),
     b(a(a(?x))) = c(b(b(?x))),
     c(a(c(?x_5))) = b(a(b(?x_5))),
     c(b(b(?x_6))) = b(a(a(?x_6))),
     b(b(a(c(?x_5)))) = c(b(a(b(?x_5)))),
     b(b(b(b(?x_6)))) = c(b(a(a(?x_6)))),
     b(b(a(c(?x_5)))) = a(b(a(b(?x_5)))),
     b(b(b(b(?x_6)))) = a(b(a(a(?x_6)))),
     b(b(b(?x))) = c(b(a(?x))),
     b(b(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(c(b(?x_2))), b(b(a(?x_2)))> by Rules <2, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([],0)], [([(b,1)],4)]>
Critical Pair <c(b(a(?x_5))), b(b(b(?x_5)))> by Rules <5, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],4),([],5),([],4)], []>
 joinable by a reduction of rules <[([],5),([],4),([(b,1)],4)], []>
Critical Pair <a(b(b(?x))), b(b(c(?x)))> by Rules <0, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],3),([(a,1)],0)]>
Critical Pair <a(c(b(?x_2))), b(b(a(?x_2)))> by Rules <2, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[([],1)], [([(b,1)],4)]>
Critical Pair <a(b(a(?x_5))), b(b(b(?x_5)))> by Rules <5, 1> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],3),([(a,1)],5)]>
Critical Pair <c(b(b(?x_1))), c(b(c(?x_1)))> by Rules <1, 2> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],3),([],2)], []>
 joinable by a reduction of rules <[([],5),([],4)], [([],5),([(b,1)],1)]>
Critical Pair <b(b(b(?x_4))), a(c(a(?x_4)))> by Rules <4, 3> preceded by [(b,1)]
 joinable by a reduction of rules <[([],3)], [([(a,1)],2)]>
Critical Pair <b(b(b(?x_1))), b(b(c(?x_1)))> by Rules <1, 4> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],3),([],4)], []>
Critical Pair <c(a(c(?x_3))), b(a(b(?x_3)))> by Rules <3, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],1),([],5)], []>
 joinable by a reduction of rules <[([],2),([],5)], [([],4),([(b,1)],3)]>
Critical Pair <c(b(b(?x_4))), b(a(a(?x_4)))> by Rules <4, 5> preceded by [(c,1)]
 joinable by a reduction of rules <[([],5),([],4)], [([],4),([(b,1)],4)]>
Critical Pair <c(b(b(?x))), b(b(c(?x)))> by Rules <0, 0> preceded by [(c,1)]
 joinable by a reduction of rules <[([(c,1)],3),([],2),([],5),([],4)], []>
 joinable by a reduction of rules <[([],5),([],4),([(b,1)],3),([],4)], []>
Critical Pair <b(a(c(?x))), a(c(b(?x)))> by Rules <3, 3> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],1)], [([],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(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<c(b(b(?x))), b(b(c(?x)))> --> <b(b(b(?x))), b(b(c(?x)))> => no
<c(c(b(?x_1))), b(b(a(?x_1)))> --> <b(b(b(?x_1))), b(b(b(?x_1)))> => yes
<c(b(a(?x_3))), b(b(b(?x_3)))> --> <b(b(b(?x_3))), b(b(b(?x_3)))> => yes
<c(b(b(?x_2))), b(a(a(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
PCP_in(symP,S):
<c(b(b(?x_2))), c(b(c(?x_2)))> --> <b(b(b(?x_2))), b(b(c(?x_2)))> => no
<b(b(b(?x_2))), b(b(c(?x_2)))> --> <b(b(b(?x_2))), b(b(c(?x_2)))> => no
<c(a(c(?x_1))), b(a(b(?x_1)))> --> <b(b(c(?x_1))), b(b(b(?x_1)))> => no
CP(S,symP):
<a(b(b(?x))), b(b(c(?x)))> --> <a(b(b(?x))), b(b(c(?x)))> => no
<a(c(b(?x))), b(b(a(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
<b(b(b(?x))), a(c(a(?x)))> --> <b(b(b(?x))), a(b(b(?x)))> => no
<a(b(a(?x))), b(b(b(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
check joinability condition:
check modulo joinability of b(b(b(?x))) and b(b(c(?x))): joinable by {0}
check modulo joinability of b(b(b(?x_2))) and b(b(c(?x_2))): joinable by {0}
check modulo reachablity from b(b(c(?x_2))) to b(b(b(?x_2))): maybe not reachable
check modulo joinability of b(b(c(?x_1))) and b(b(b(?x_1))): joinable by {0}
check modulo reachablity from a(b(b(?x))) to b(b(c(?x))): maybe not reachable
check modulo joinability of a(b(b(?x))) and b(b(b(?x))): joinable by {0}
check modulo joinability of b(b(b(?x))) and a(b(b(?x))): joinable by {0}
check modulo reachablity from a(b(b(?x))) to b(b(b(?x))): maybe not reachable
failed
failure(Step 1)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: b(a(?x)) -> b(b(?x)) => b(a(?x)) -> a(c(?x))
replace: c(c(?x)) -> b(b(?x)) => c(c(?x)) -> a(c(?x))
STEP: 2 (linear)
S:
   [ c(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<c(b(b(?x))), b(b(c(?x)))> --> <b(b(b(?x))), b(b(c(?x)))> => no
<c(c(b(?x_1))), b(b(a(?x_1)))> --> <b(b(b(?x_1))), b(b(b(?x_1)))> => yes
<c(b(a(?x_3))), b(b(b(?x_3)))> --> <b(b(b(?x_3))), b(b(b(?x_3)))> => yes
<c(b(b(?x_2))), b(a(a(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
CP_in(symP,S):
<c(a(c(?x))), b(a(b(?x)))> --> <b(b(c(?x))), b(b(b(?x)))> => no
<c(b(b(?x))), c(b(c(?x)))> --> <b(b(b(?x))), b(b(c(?x)))> => no
<b(b(b(?x))), b(b(c(?x)))> --> <b(b(b(?x))), b(b(c(?x)))> => no
CP(S,symP):
<a(b(b(?x))), b(b(c(?x)))> --> <a(b(b(?x))), b(b(c(?x)))> => no
<a(c(b(?x))), b(b(a(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
<b(b(b(?x))), a(c(a(?x)))> --> <b(b(b(?x))), a(b(b(?x)))> => no
<a(b(a(?x))), b(b(b(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
check joinability condition:
check modulo joinability of b(b(b(?x))) and b(b(c(?x))): joinable by {0}
check modulo joinability of b(b(c(?x))) and b(b(b(?x))): joinable by {0}
check modulo joinability of b(b(b(?x))) and b(b(c(?x))): joinable by {0}
check modulo reachablity from b(b(c(?x))) to b(b(b(?x))): maybe not reachable
check modulo reachablity from a(b(b(?x))) to b(b(c(?x))): maybe not reachable
check modulo joinability of a(b(b(?x))) and b(b(b(?x))): joinable by {0}
check modulo joinability of b(b(b(?x))) and a(b(b(?x))): joinable by {0}
check modulo reachablity from a(b(b(?x))) to b(b(b(?x))): maybe not reachable
failed
failure(Step 2)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: b(a(?x)) -> b(b(?x)) => b(a(?x)) -> a(c(?x))
replace: c(c(?x)) -> b(b(?x)) => c(c(?x)) -> a(c(?x))
STEP: 3 (relative)
S:
   [ c(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Check relative termination:
   [ c(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (2)+(1)*x1
  b:= (1)*x1
  c:= (1)*x1
retract c(a(?x)) -> c(b(?x))
retract b(a(?x)) -> b(b(?x))
retract a(c(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)*x1
  c:= (8)+(1)*x1
retract c(a(?x)) -> c(b(?x))
retract b(a(?x)) -> b(b(?x))
retract c(b(?x)) -> b(a(?x))
retract a(c(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)*x1
  c:= (2)+(1)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<c(a(c(?x))), a(c(c(?x)))> --> <b(b(c(?x))), a(a(c(?x)))> => no
<c(c(b(?x_1))), a(c(a(?x_1)))> --> <b(b(b(?x_1))), a(b(b(?x_1)))> => no
<c(b(a(?x_3))), a(c(b(?x_3)))> --> <b(b(b(?x_3))), a(b(b(?x_3)))> => no
<c(b(b(?x_2))), b(a(a(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
PCP_in(symP,S):
<c(b(b(?x_2))), c(b(c(?x_2)))> --> <b(b(b(?x_2))), b(b(c(?x_2)))> => no
<b(b(b(?x_2))), b(b(c(?x_2)))> --> <b(b(b(?x_2))), b(b(c(?x_2)))> => no
<c(a(c(?x_1))), b(a(b(?x_1)))> --> <b(b(c(?x_1))), b(b(b(?x_1)))> => no
CP(S,symP):
<a(a(c(?x))), b(b(c(?x)))> --> <a(a(c(?x))), b(b(c(?x)))> => no
<a(c(b(?x))), b(b(a(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
<b(b(b(?x))), a(c(a(?x)))> --> <b(b(b(?x))), a(b(b(?x)))> => no
<a(b(a(?x))), b(b(b(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
check joinability condition:
check modulo joinability of b(b(c(?x))) and a(a(c(?x))): joinable by {0}
check modulo joinability of b(b(b(?x_1))) and a(b(b(?x_1))): joinable by {0}
check modulo joinability of b(b(b(?x_3))) and a(b(b(?x_3))): joinable by {0}
check modulo joinability of b(b(b(?x_2))) and b(b(c(?x_2))): joinable by {0}
check modulo reachablity from b(b(c(?x_2))) to b(b(b(?x_2))): maybe not reachable
check modulo joinability of b(b(c(?x_1))) and b(b(b(?x_1))): joinable by {0}
check modulo reachablity from a(a(c(?x))) to b(b(c(?x))): maybe not reachable
check modulo joinability of a(b(b(?x))) and b(b(b(?x))): joinable by {0}
check modulo joinability of b(b(b(?x))) and a(b(b(?x))): joinable by {0}
check modulo reachablity from a(b(b(?x))) to b(b(b(?x))): maybe not reachable
failed
failure(Step 4)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: b(a(?x)) -> b(b(?x)) => b(a(?x)) -> a(c(?x))
replace: c(c(?x)) -> a(c(?x)) => c(c(?x)) -> b(b(?x))
STEP: 5 (linear)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: terminating
CP(S,S):
<c(a(c(?x))), a(c(c(?x)))> --> <b(b(c(?x))), a(a(c(?x)))> => no
<c(c(b(?x_1))), a(c(a(?x_1)))> --> <b(b(b(?x_1))), a(b(b(?x_1)))> => no
<c(b(a(?x_3))), a(c(b(?x_3)))> --> <b(b(b(?x_3))), a(b(b(?x_3)))> => no
<c(b(b(?x_2))), b(a(a(?x_2)))> --> <b(b(b(?x_2))), b(b(b(?x_2)))> => yes
CP_in(symP,S):
<c(a(c(?x))), b(a(b(?x)))> --> <b(b(c(?x))), b(b(b(?x)))> => no
<c(b(b(?x))), c(b(c(?x)))> --> <b(b(b(?x))), b(b(c(?x)))> => no
<b(b(b(?x))), b(b(c(?x)))> --> <b(b(b(?x))), b(b(c(?x)))> => no
CP(S,symP):
<a(a(c(?x))), b(b(c(?x)))> --> <a(a(c(?x))), b(b(c(?x)))> => no
<a(c(b(?x))), b(b(a(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
<b(b(b(?x))), a(c(a(?x)))> --> <b(b(b(?x))), a(b(b(?x)))> => no
<a(b(a(?x))), b(b(b(?x)))> --> <a(b(b(?x))), b(b(b(?x)))> => no
check joinability condition:
check modulo joinability of b(b(c(?x))) and a(a(c(?x))): joinable by {0,1}
check modulo joinability of b(b(b(?x_1))) and a(b(b(?x_1))): joinable by {0}
check modulo joinability of b(b(b(?x_3))) and a(b(b(?x_3))): joinable by {0}
check modulo joinability of b(b(c(?x))) and b(b(b(?x))): joinable by {0}
check modulo joinability of b(b(b(?x))) and b(b(c(?x))): joinable by {0}
check modulo reachablity from b(b(c(?x))) to b(b(b(?x))): maybe not reachable
check modulo reachablity from a(a(c(?x))) to b(b(c(?x))): maybe not reachable
check modulo joinability of a(b(b(?x))) and b(b(b(?x))): joinable by {0}
check modulo joinability of b(b(b(?x))) and a(b(b(?x))): joinable by {0}
check modulo reachablity from a(b(b(?x))) to b(b(b(?x))): maybe not reachable
failed
failure(Step 5)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: b(a(?x)) -> b(b(?x)) => b(a(?x)) -> a(c(?x))
replace: c(c(?x)) -> a(c(?x)) => c(c(?x)) -> b(b(?x))
STEP: 6 (relative)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Check relative termination:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)*x1
  c:= (8)+(1)*x1
retract c(b(?x)) -> b(a(?x))
retract a(c(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)+(1)*x1+(2)*x1*x1
  b:= (1)*x1+(2)*x1*x1
  c:= (1)*x1+(2)*x1*x1
retract c(a(?x)) -> c(b(?x))
retract b(a(?x)) -> b(b(?x))
retract c(b(?x)) -> b(a(?x))
retract a(c(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)*x1
  b:= (2)+(1)*x1*x1
  c:= (1)+(1)*x1*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 6)
STEP: 7 (parallel)
S:
   [ c(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 7)
STEP: 8 (linear)
S:
   [ c(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 8)
STEP: 9 (relative)
S:
   [ c(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Check relative termination:
   [ c(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (1)*x1
  b:= (1)*x1
  c:= (1)+(1)*x1
retract c(c(?x)) -> b(b(?x))
retract c(b(?x)) -> b(a(?x))
retract a(c(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)+(2)*x1*x1
  b:= (1)+(1)*x1*x1
  c:= (1)*x1
retract c(c(?x)) -> b(b(?x))
retract b(a(?x)) -> a(c(?x))
retract c(b(?x)) -> b(a(?x))
retract a(c(?x)) -> b(b(?x))
retract b(b(?x)) -> a(c(?x))
Polynomial Interpretation:
  a:= (1)+(2)*x1*x1
  b:= (1)*x1*x1
  c:= (1)+(2)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 9)
STEP: 10 (parallel)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 10)
STEP: 11 (linear)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
S: unknown termination
failure(Step 11)
STEP: 12 (relative)
S:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
P:
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Check relative termination:
   [ c(c(?x)) -> a(c(?x)),
     c(a(?x)) -> c(b(?x)),
     b(a(?x)) -> a(c(?x)),
     c(b(?x)) -> b(a(?x)) ]
   [ a(c(?x)) -> b(b(?x)),
     b(b(?x)) -> a(c(?x)) ]
Polynomial Interpretation:
  a:= (1)+(2)*x1
  b:= (2)*x1
  c:= (2)*x1
retract c(a(?x)) -> c(b(?x))
retract b(a(?x)) -> a(c(?x))
retract a(c(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (2)+(1)*x1
  b:= (2)*x1
  c:= (2)+(2)*x1
retract c(c(?x)) -> a(c(?x))
retract c(a(?x)) -> c(b(?x))
retract b(a(?x)) -> a(c(?x))
retract a(c(?x)) -> b(b(?x))
Polynomial Interpretation:
  a:= (1)+(1)*x1
  b:= (1)+(1)*x1+(2)*x1*x1
  c:= (5)*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...
   [ c(c(?x)) -> b(b(?x)),
     a(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(b(?x)) -> a(c(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
Sort Assignment:
 a : 13=>13
 b : 13=>13
 c : 13=>13
maximal types: {13}
Persistent Decomposition failed: Can't judge

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

Try Commutative Decomposition for...
   [ c(c(?x)) -> b(b(?x)),
     a(c(?x)) -> b(b(?x)),
     c(a(?x)) -> c(b(?x)),
     b(b(?x)) -> a(c(?x)),
     b(a(?x)) -> b(b(?x)),
     c(b(?x)) -> b(a(?x)) ]
Inside Critical Pair: <c(c(b(?x_2))), b(b(a(?x_2)))> by Rules <2, 0>
develop reducts from lhs term...
<{0}, b(b(b(?x_2)))>
<{5}, c(b(a(?x_2)))>
<{}, c(c(b(?x_2)))>
develop reducts from rhs term...
<{3}, a(c(a(?x_2)))>
<{4}, b(b(b(?x_2)))>
<{}, b(b(a(?x_2)))>
Inside Critical Pair: <c(b(a(?x_5))), b(b(b(?x_5)))> by Rules <5, 0>
develop reducts from lhs term...
<{5}, b(a(a(?x_5)))>
<{4}, c(b(b(?x_5)))>
<{}, c(b(a(?x_5)))>
develop reducts from rhs term...
<{3}, a(c(b(?x_5)))>
<{3}, b(a(c(?x_5)))>
<{}, b(b(b(?x_5)))>
Inside Critical Pair: <a(b(b(?x))), b(b(c(?x)))> by Rules <0, 1>
develop reducts from lhs term...
<{3}, a(a(c(?x)))>
<{}, a(b(b(?x)))>
develop reducts from rhs term...
<{3}, a(c(c(?x)))>
<{}, b(b(c(?x)))>
Inside Critical Pair: <a(c(b(?x_2))), b(b(a(?x_2)))> by Rules <2, 1>
develop reducts from lhs term...
<{1}, b(b(b(?x_2)))>
<{5}, a(b(a(?x_2)))>
<{}, a(c(b(?x_2)))>
develop reducts from rhs term...
<{3}, a(c(a(?x_2)))>
<{4}, b(b(b(?x_2)))>
<{}, b(b(a(?x_2)))>
Inside Critical Pair: <a(b(a(?x_5))), b(b(b(?x_5)))> by Rules <5, 1>
develop reducts from lhs term...
<{4}, a(b(b(?x_5)))>
<{}, a(b(a(?x_5)))>
develop reducts from rhs term...
<{3}, a(c(b(?x_5)))>
<{3}, b(a(c(?x_5)))>
<{}, b(b(b(?x_5)))>
Inside Critical Pair: <c(b(b(?x_1))), c(b(c(?x_1)))> by Rules <1, 2>
develop reducts from lhs term...
<{5}, b(a(b(?x_1)))>
<{3}, c(a(c(?x_1)))>
<{}, c(b(b(?x_1)))>
develop reducts from rhs term...
<{5}, b(a(c(?x_1)))>
<{}, c(b(c(?x_1)))>
Inside Critical Pair: <b(b(b(?x_4))), a(c(a(?x_4)))> by Rules <4, 3>
develop reducts from lhs term...
<{3}, a(c(b(?x_4)))>
<{3}, b(a(c(?x_4)))>
<{}, b(b(b(?x_4)))>
develop reducts from rhs term...
<{1}, b(b(a(?x_4)))>
<{2}, a(c(b(?x_4)))>
<{}, a(c(a(?x_4)))>
Inside Critical Pair: <b(b(b(?x_1))), b(b(c(?x_1)))> by Rules <1, 4>
develop reducts from lhs term...
<{3}, a(c(b(?x_1)))>
<{3}, b(a(c(?x_1)))>
<{}, b(b(b(?x_1)))>
develop reducts from rhs term...
<{3}, a(c(c(?x_1)))>
<{}, b(b(c(?x_1)))>
Inside Critical Pair: <c(a(c(?x_3))), b(a(b(?x_3)))> by Rules <3, 5>
develop reducts from lhs term...
<{2}, c(b(c(?x_3)))>
<{1}, c(b(b(?x_3)))>
<{}, c(a(c(?x_3)))>
develop reducts from rhs term...
<{4}, b(b(b(?x_3)))>
<{}, b(a(b(?x_3)))>
Inside Critical Pair: <c(b(b(?x_4))), b(a(a(?x_4)))> by Rules <4, 5>
develop reducts from lhs term...
<{5}, b(a(b(?x_4)))>
<{3}, c(a(c(?x_4)))>
<{}, c(b(b(?x_4)))>
develop reducts from rhs term...
<{4}, b(b(a(?x_4)))>
<{}, b(a(a(?x_4)))>
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
989.trs: Failure(unknown CR)
MAYBE
(5527 msec.)