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

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

Try Commutative Decomposition for...
   [ b(b(?x)) -> a(a(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(b(?x)) -> a(b(?x)),
     b(c(?x)) -> a(a(?x)),
     a(b(?x)) -> b(c(?x)) ]
Outside Critical Pair: <b(c(?x_6)), a(b(?x_6))> by Rules <6, 4>
develop reducts from lhs term...
<{5}, a(a(?x_6))>
<{}, b(c(?x_6))>
develop reducts from rhs term...
<{6}, b(c(?x_6))>
<{4}, a(b(?x_6))>
<{}, a(b(?x_6))>
Inside Critical Pair: <b(a(c(?x_3))), a(a(a(?x_3)))> by Rules <3, 0>
develop reducts from lhs term...
<{3}, a(c(c(?x_3)))>
<{2}, b(b(a(?x_3)))>
<{}, b(a(c(?x_3)))>
develop reducts from rhs term...
<{1}, b(c(a(?x_3)))>
<{1}, a(b(c(?x_3)))>
<{}, a(a(a(?x_3)))>
Inside Critical Pair: <b(a(a(?x_5))), a(a(c(?x_5)))> by Rules <5, 0>
develop reducts from lhs term...
<{3}, a(c(a(?x_5)))>
<{1}, b(b(c(?x_5)))>
<{}, b(a(a(?x_5)))>
develop reducts from rhs term...
<{1}, b(c(c(?x_5)))>
<{2}, a(b(a(?x_5)))>
<{}, a(a(c(?x_5)))>
Inside Critical Pair: <a(b(a(?x_2))), b(c(c(?x_2)))> by Rules <2, 1>
develop reducts from lhs term...
<{6}, b(c(a(?x_2)))>
<{4}, a(b(a(?x_2)))>
<{3}, a(a(c(?x_2)))>
<{}, a(b(a(?x_2)))>
develop reducts from rhs term...
<{5}, a(a(c(?x_2)))>
<{}, b(c(c(?x_2)))>
Inside Critical Pair: <a(a(b(?x_4))), b(c(b(?x_4)))> by Rules <4, 1>
develop reducts from lhs term...
<{1}, b(c(b(?x_4)))>
<{6}, a(b(c(?x_4)))>
<{4}, a(a(b(?x_4)))>
<{}, a(a(b(?x_4)))>
develop reducts from rhs term...
<{5}, a(a(b(?x_4)))>
<{}, b(c(b(?x_4)))>
Inside Critical Pair: <a(b(c(?x_6))), b(c(b(?x_6)))> by Rules <6, 1>
develop reducts from lhs term...
<{6}, b(c(c(?x_6)))>
<{4}, a(b(c(?x_6)))>
<{5}, a(a(a(?x_6)))>
<{}, a(b(c(?x_6)))>
develop reducts from rhs term...
<{5}, a(a(b(?x_6)))>
<{}, b(c(b(?x_6)))>
Inside Critical Pair: <b(b(c(?x_1))), a(c(a(?x_1)))> by Rules <1, 3>
develop reducts from lhs term...
<{0}, a(a(c(?x_1)))>
<{5}, b(a(a(?x_1)))>
<{}, b(b(c(?x_1)))>
develop reducts from rhs term...
<{2}, b(a(a(?x_1)))>
<{}, a(c(a(?x_1)))>
Inside Critical Pair: <b(b(a(?x_2))), a(c(c(?x_2)))> by Rules <2, 3>
develop reducts from lhs term...
<{0}, a(a(a(?x_2)))>
<{3}, b(a(c(?x_2)))>
<{}, b(b(a(?x_2)))>
develop reducts from rhs term...
<{2}, b(a(c(?x_2)))>
<{}, a(c(c(?x_2)))>
Inside Critical Pair: <b(a(b(?x_4))), a(c(b(?x_4)))> by Rules <4, 3>
develop reducts from lhs term...
<{3}, a(c(b(?x_4)))>
<{6}, b(b(c(?x_4)))>
<{4}, b(a(b(?x_4)))>
<{}, b(a(b(?x_4)))>
develop reducts from rhs term...
<{2}, b(a(b(?x_4)))>
<{}, a(c(b(?x_4)))>
Inside Critical Pair: <b(b(c(?x_6))), a(c(b(?x_6)))> by Rules <6, 3>
develop reducts from lhs term...
<{0}, a(a(c(?x_6)))>
<{5}, b(a(a(?x_6)))>
<{}, b(b(c(?x_6)))>
develop reducts from rhs term...
<{2}, b(a(b(?x_6)))>
<{}, a(c(b(?x_6)))>
Inside Critical Pair: <a(a(a(?x))), a(b(b(?x)))> by Rules <0, 4>
develop reducts from lhs term...
<{1}, b(c(a(?x)))>
<{1}, a(b(c(?x)))>
<{}, a(a(a(?x)))>
develop reducts from rhs term...
<{6}, b(c(b(?x)))>
<{4}, a(b(b(?x)))>
<{0}, a(a(a(?x)))>
<{}, a(b(b(?x)))>
Inside Critical Pair: <a(a(c(?x_3))), a(b(a(?x_3)))> by Rules <3, 4>
develop reducts from lhs term...
<{1}, b(c(c(?x_3)))>
<{2}, a(b(a(?x_3)))>
<{}, a(a(c(?x_3)))>
develop reducts from rhs term...
<{6}, b(c(a(?x_3)))>
<{4}, a(b(a(?x_3)))>
<{3}, a(a(c(?x_3)))>
<{}, a(b(a(?x_3)))>
Inside Critical Pair: <a(a(a(?x_5))), a(b(c(?x_5)))> by Rules <5, 4>
develop reducts from lhs term...
<{1}, b(c(a(?x_5)))>
<{1}, a(b(c(?x_5)))>
<{}, a(a(a(?x_5)))>
develop reducts from rhs term...
<{6}, b(c(c(?x_5)))>
<{4}, a(b(c(?x_5)))>
<{5}, a(a(a(?x_5)))>
<{}, a(b(c(?x_5)))>
Inside Critical Pair: <a(a(a(?x))), b(c(b(?x)))> by Rules <0, 6>
develop reducts from lhs term...
<{1}, b(c(a(?x)))>
<{1}, a(b(c(?x)))>
<{}, a(a(a(?x)))>
develop reducts from rhs term...
<{5}, a(a(b(?x)))>
<{}, b(c(b(?x)))>
Inside Critical Pair: <a(a(c(?x_3))), b(c(a(?x_3)))> by Rules <3, 6>
develop reducts from lhs term...
<{1}, b(c(c(?x_3)))>
<{2}, a(b(a(?x_3)))>
<{}, a(a(c(?x_3)))>
develop reducts from rhs term...
<{5}, a(a(a(?x_3)))>
<{}, b(c(a(?x_3)))>
Inside Critical Pair: <a(a(a(?x_5))), b(c(c(?x_5)))> by Rules <5, 6>
develop reducts from lhs term...
<{1}, b(c(a(?x_5)))>
<{1}, a(b(c(?x_5)))>
<{}, a(a(a(?x_5)))>
develop reducts from rhs term...
<{5}, a(a(c(?x_5)))>
<{}, b(c(c(?x_5)))>
Try A Minimal Decomposition {5,0,3,6,1,2}{4}
{5,0,3,6,1,2}
(cm)Rewrite Rules:
   [ b(c(?x)) -> a(a(?x)),
     b(b(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(b(?x)) -> b(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ b(a(a(?x))) = a(a(c(?x))),
     b(a(c(?x_2))) = a(a(a(?x_2))),
     b(b(c(?x_3))) = a(c(b(?x_3))),
     b(b(c(?x_4))) = a(c(a(?x_4))),
     b(b(a(?x_5))) = a(c(c(?x_5))),
     a(a(a(?x))) = b(c(c(?x))),
     a(a(a(?x_1))) = b(c(b(?x_1))),
     a(a(c(?x_2))) = b(c(a(?x_2))),
     a(b(c(?x_3))) = b(c(b(?x_3))),
     a(b(a(?x_5))) = b(c(c(?x_5))),
     b(a(a(?x))) = a(a(b(?x))),
     a(b(c(?x))) = b(c(a(?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:
   [ a(a(c(?x))) = b(a(a(?x))),
     b(c(c(?x))) = a(a(a(?x))),
     b(a(a(?x_1))) = a(a(b(?x_1))),
     b(a(a(?x_2))) = a(a(c(?x_2))),
     b(a(c(?x_3))) = a(a(a(?x_3))),
     a(a(a(a(?x_1)))) = b(a(a(b(?x_1)))),
     a(a(a(a(?x_2)))) = b(a(a(c(?x_2)))),
     a(a(a(c(?x_3)))) = b(a(a(a(?x_3)))),
     b(c(a(a(?x_1)))) = a(a(a(b(?x_1)))),
     b(c(a(a(?x_2)))) = a(a(a(c(?x_2)))),
     b(c(a(c(?x_3)))) = a(a(a(a(?x_3)))),
     a(a(b(?x))) = b(a(a(?x))),
     b(c(b(?x))) = a(a(a(?x))),
     b(b(c(?x_4))) = a(c(b(?x_4))),
     b(b(c(?x_5))) = a(c(a(?x_5))),
     b(b(a(?x_6))) = a(c(c(?x_6))),
     a(a(b(c(?x_4)))) = b(a(c(b(?x_4)))),
     a(a(b(c(?x_5)))) = b(a(c(a(?x_5)))),
     a(a(b(a(?x_6)))) = b(a(c(c(?x_6)))),
     b(c(b(c(?x_4)))) = a(a(c(b(?x_4)))),
     b(c(b(c(?x_5)))) = a(a(c(a(?x_5)))),
     b(c(b(a(?x_6)))) = a(a(c(c(?x_6)))),
     a(a(a(?x))) = b(a(c(?x))),
     b(c(a(?x))) = a(a(c(?x))),
     a(a(a(?x_2))) = b(c(c(?x_2))),
     a(a(a(?x_3))) = b(c(b(?x_3))),
     a(a(c(?x_4))) = b(c(a(?x_4))),
     a(c(a(a(?x_2)))) = b(b(c(c(?x_2)))),
     a(c(a(a(?x_3)))) = b(b(c(b(?x_3)))),
     a(c(a(c(?x_4)))) = b(b(c(a(?x_4)))),
     b(c(a(a(?x_2)))) = a(b(c(c(?x_2)))),
     b(c(a(a(?x_3)))) = a(b(c(b(?x_3)))),
     b(c(a(c(?x_4)))) = a(b(c(a(?x_4)))),
     a(c(b(?x))) = b(b(c(?x))),
     b(c(b(?x))) = a(b(c(?x))),
     a(b(c(?x_1))) = b(c(a(?x_1))),
     a(b(c(?x_5))) = b(c(b(?x_5))),
     a(b(a(?x_6))) = b(c(c(?x_6))),
     b(c(b(c(?x_1)))) = a(b(c(a(?x_1)))),
     b(c(b(c(?x_5)))) = a(b(c(b(?x_5)))),
     b(c(b(a(?x_6)))) = a(b(c(c(?x_6)))),
     a(c(b(c(?x_1)))) = b(b(c(a(?x_1)))),
     a(c(b(c(?x_5)))) = b(b(c(b(?x_5)))),
     a(c(b(a(?x_6)))) = b(b(c(c(?x_6)))),
     b(c(a(?x))) = a(b(c(?x))),
     a(c(a(?x))) = b(b(c(?x))),
     a(c(c(?x))) = b(b(a(?x))),
     b(c(c(?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 <b(a(a(?x))), a(a(c(?x)))> by Rules <0, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],4),([],1)], []>
Critical Pair <b(a(c(?x_2))), a(a(a(?x_2)))> by Rules <2, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],5),([],1)], []>
Critical Pair <b(b(c(?x_3))), a(c(b(?x_3)))> by Rules <3, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[], [([],5),([(b,1)],3)]>
Critical Pair <b(b(c(?x_4))), a(c(a(?x_4)))> by Rules <4, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],0)], [([],5)]>
Critical Pair <b(b(a(?x_5))), a(c(c(?x_5)))> by Rules <5, 2> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],2)], [([],5)]>
Critical Pair <a(a(a(?x))), b(c(c(?x)))> by Rules <0, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],4),([],3)], []>
Critical Pair <a(a(a(?x_1))), b(c(b(?x_1)))> by Rules <1, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],4)], [([],0),([(a,1)],3)]>
Critical Pair <a(a(c(?x_2))), b(c(a(?x_2)))> by Rules <2, 3> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],5),([],3)], []>
Critical Pair <a(b(c(?x_3))), b(c(b(?x_3)))> by Rules <3, 4> preceded by [(a,1)]
 joinable by a reduction of rules <[], [([],0),([(a,1)],3)]>
Critical Pair <a(b(a(?x_5))), b(c(c(?x_5)))> by Rules <5, 4> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],2)], [([],0)]>
Critical Pair <b(a(a(?x))), a(a(b(?x)))> by Rules <1, 1> preceded by [(b,1)]
 joinable by a reduction of rules <[([(b,1)],4),([],1),([],4)], [([(a,1)],3),([],3)]>
 joinable by a reduction of rules <[([(b,1)],4),([],1)], [([(a,1)],3),([],3),([],0)]>
Critical Pair <a(b(c(?x))), b(c(a(?x)))> by Rules <4, 4> preceded by [(a,1)]
 joinable by a reduction of rules <[([(a,1)],0)], [([],0)]>
unknown Diagram Decreasing
check Non-Confluence...
obtain 11 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)) -> a(a(?x)),
     a(b(?x)) -> b(c(?x)) ]
P:
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
S: terminating
CP(S,S):
<b(a(a(?x))), a(a(b(?x)))> --> <b(a(a(?x))), b(c(c(?x)))> => no
<a(a(a(?x))), b(c(b(?x)))> --> <a(a(a(?x))), b(c(b(?x)))> => no
PCP_in(symP,S):
<b(a(c(?x_4))), a(a(a(?x_4)))> --> <b(a(c(?x_4))), a(a(a(?x_4)))> => no
<b(a(a(?x_3))), a(a(c(?x_3)))> --> <b(a(a(?x_3))), a(a(c(?x_3)))> => no
<a(a(c(?x_4))), b(c(a(?x_4)))> --> <a(a(c(?x_4))), b(c(a(?x_4)))> => no
<a(a(a(?x_3))), b(c(c(?x_3)))> --> <a(a(a(?x_3))), b(c(c(?x_3)))> => no
CP(S,symP):
<a(b(c(?x))), b(c(b(?x)))> --> <b(c(c(?x))), b(c(b(?x)))> => no
<b(b(c(?x))), a(c(b(?x)))> --> <a(a(c(?x))), a(c(b(?x)))> => no
check joinability condition:
check modulo joinability of b(a(a(?x))) and b(c(c(?x))): joinable by {0,2}
check modulo joinability of a(a(a(?x))) and b(c(b(?x))): joinable by {0,2}
check modulo reachablity from a(a(a(?x_4))) to b(a(c(?x_4))): maybe not reachable
check modulo reachablity from a(a(c(?x_3))) to b(a(a(?x_3))): maybe not reachable
check modulo reachablity from b(c(a(?x_4))) to a(a(c(?x_4))): reachable by {0,2}
check modulo reachablity from b(c(c(?x_3))) to a(a(a(?x_3))): reachable by {1,2}
check modulo reachablity from b(c(c(?x))) to b(c(b(?x))): maybe not reachable
check modulo reachablity from a(a(c(?x))) to a(c(b(?x))): maybe not reachable
failed
failure(Step 1)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: a(b(?x)) -> b(c(?x)) => a(b(?x)) -> a(a(?x))
STEP: 2 (linear)
S:
   [ b(b(?x)) -> a(a(?x)),
     a(b(?x)) -> b(c(?x)) ]
P:
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
S: terminating
CP(S,S):
<b(a(a(?x))), a(a(b(?x)))> --> <b(a(a(?x))), b(c(c(?x)))> => no
<a(a(a(?x))), b(c(b(?x)))> --> <a(a(a(?x))), b(c(b(?x)))> => no
CP_in(symP,S):
<b(a(a(?x))), a(a(c(?x)))> --> <b(a(a(?x))), a(a(c(?x)))> => no
<a(a(a(?x))), b(c(c(?x)))> --> <a(a(a(?x))), b(c(c(?x)))> => no
<b(a(c(?x))), a(a(a(?x)))> --> <b(a(c(?x))), a(a(a(?x)))> => no
<a(a(c(?x))), b(c(a(?x)))> --> <a(a(c(?x))), b(c(a(?x)))> => no
CP(S,symP):
<a(b(c(?x))), b(c(b(?x)))> --> <b(c(c(?x))), b(c(b(?x)))> => no
<b(b(c(?x))), a(c(b(?x)))> --> <a(a(c(?x))), a(c(b(?x)))> => no
check joinability condition:
check modulo joinability of b(a(a(?x))) and b(c(c(?x))): joinable by {0,1,2}
check modulo joinability of a(a(a(?x))) and b(c(b(?x))): joinable by {0,2}
check modulo reachablity from a(a(c(?x))) to b(a(a(?x))): maybe not reachable
check modulo reachablity from b(c(c(?x))) to a(a(a(?x))): reachable by {1,2}
check modulo reachablity from a(a(a(?x))) to b(a(c(?x))): maybe not reachable
check modulo reachablity from b(c(a(?x))) to a(a(c(?x))): reachable by {0,2}
check modulo reachablity from b(c(c(?x))) to b(c(b(?x))): maybe not reachable
check modulo reachablity from a(a(c(?x))) to a(c(b(?x))): maybe not reachable
failed
failure(Step 2)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: a(b(?x)) -> b(c(?x)) => a(b(?x)) -> a(a(?x))
STEP: 3 (relative)
S:
   [ b(b(?x)) -> a(a(?x)),
     a(b(?x)) -> b(c(?x)) ]
P:
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
Check relative termination:
   [ b(b(?x)) -> a(a(?x)),
     a(b(?x)) -> b(c(?x)) ]
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
Polynomial Interpretation:
  a:= (2)*x1
  b:= (1)+(2)*x1
  c:= (2)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 3)
STEP: 4 (parallel)
S:
   [ b(b(?x)) -> a(a(?x)),
     a(b(?x)) -> a(a(?x)) ]
P:
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
S: terminating
CP(S,S):
<b(a(a(?x))), a(a(b(?x)))> --> <b(a(a(?x))), a(a(a(?x)))> => no
<a(a(a(?x))), a(a(b(?x)))> --> <a(a(a(?x))), a(a(a(?x)))> => yes
PCP_in(symP,S):
<b(a(c(?x_4))), a(a(a(?x_4)))> --> <b(a(c(?x_4))), a(a(a(?x_4)))> => no
<b(a(a(?x_3))), a(a(c(?x_3)))> --> <b(a(a(?x_3))), a(a(c(?x_3)))> => no
<a(a(c(?x_4))), a(a(a(?x_4)))> --> <a(a(c(?x_4))), a(a(a(?x_4)))> => no
<a(a(a(?x_3))), a(a(c(?x_3)))> --> <a(a(a(?x_3))), a(a(c(?x_3)))> => no
CP(S,symP):
<a(a(a(?x))), b(c(b(?x)))> --> <a(a(a(?x))), b(c(b(?x)))> => no
<b(a(a(?x))), a(c(b(?x)))> --> <b(a(a(?x))), a(c(b(?x)))> => no
check joinability condition:
check modulo joinability of b(a(a(?x))) and a(a(a(?x))): joinable by {0}
check modulo reachablity from a(a(a(?x_4))) to b(a(c(?x_4))): maybe not reachable
check modulo reachablity from a(a(c(?x_3))) to b(a(a(?x_3))): maybe not reachable
check modulo reachablity from a(a(a(?x_4))) to a(a(c(?x_4))): reachable by {0}
check modulo reachablity from a(a(c(?x_3))) to a(a(a(?x_3))): reachable by {1}
check modulo reachablity from a(a(a(?x))) to b(c(b(?x))): maybe not reachable
check modulo reachablity from b(a(a(?x))) to a(c(b(?x))): maybe not reachable
failed
failure(Step 4)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: a(b(?x)) -> a(a(?x)) => a(b(?x)) -> b(c(?x))
STEP: 5 (linear)
S:
   [ b(b(?x)) -> a(a(?x)),
     a(b(?x)) -> a(a(?x)) ]
P:
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
S: terminating
CP(S,S):
<b(a(a(?x))), a(a(b(?x)))> --> <b(a(a(?x))), a(a(a(?x)))> => no
<a(a(a(?x))), a(a(b(?x)))> --> <a(a(a(?x))), a(a(a(?x)))> => yes
CP_in(symP,S):
<b(a(a(?x))), a(a(c(?x)))> --> <b(a(a(?x))), a(a(c(?x)))> => no
<a(a(a(?x))), a(a(c(?x)))> --> <a(a(a(?x))), a(a(c(?x)))> => no
<b(a(c(?x))), a(a(a(?x)))> --> <b(a(c(?x))), a(a(a(?x)))> => no
<a(a(c(?x))), a(a(a(?x)))> --> <a(a(c(?x))), a(a(a(?x)))> => no
CP(S,symP):
<a(a(a(?x))), b(c(b(?x)))> --> <a(a(a(?x))), b(c(b(?x)))> => no
<b(a(a(?x))), a(c(b(?x)))> --> <b(a(a(?x))), a(c(b(?x)))> => no
check joinability condition:
check modulo joinability of b(a(a(?x))) and a(a(a(?x))): joinable by {0}
check modulo reachablity from a(a(c(?x))) to b(a(a(?x))): maybe not reachable
check modulo reachablity from a(a(c(?x))) to a(a(a(?x))): reachable by {0,1}
check modulo reachablity from a(a(a(?x))) to b(a(c(?x))): maybe not reachable
check modulo reachablity from a(a(a(?x))) to a(a(c(?x))): reachable by {0}
check modulo reachablity from a(a(a(?x))) to b(c(b(?x))): maybe not reachable
check modulo reachablity from b(a(a(?x))) to a(c(b(?x))): maybe not reachable
failed
failure(Step 5)
   [ ]
Added S-Rules:
   [ ]
Added P-Rules:
   [ ]
replace: a(b(?x)) -> a(a(?x)) => a(b(?x)) -> b(c(?x))
STEP: 6 (relative)
S:
   [ b(b(?x)) -> a(a(?x)),
     a(b(?x)) -> a(a(?x)) ]
P:
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
Check relative termination:
   [ b(b(?x)) -> a(a(?x)),
     a(b(?x)) -> a(a(?x)) ]
   [ b(c(?x)) -> a(a(?x)),
     b(a(?x)) -> a(c(?x)),
     a(a(?x)) -> b(c(?x)),
     a(c(?x)) -> b(a(?x)) ]
Polynomial Interpretation:
  a:= (2)*x1
  b:= (4)+(2)*x1
  c:= (3)+(2)*x1
relatively terminating
S/P: relatively terminating
check CP condition:
failed
failure(Step 6)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

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

{4}
(cm)Rewrite Rules:
   [ a(b(?x)) -> a(b(?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
996.trs: Failure(unknown CR)
MAYBE
(10843 msec.)