MAYBE
(ignored inputs)COMMENT the following rules are removed from the original TRS a ( c ( x ) ) -> a ( c ( x ) ) a ( c ( x ) ) -> a ( c ( x ) )
Rewrite Rules:
[ a(c(?x)) -> b(a(?x)),
a(a(?x)) -> a(b(?x)),
b(b(?x)) -> a(c(?x)),
c(c(?x)) -> c(a(?x)),
c(b(?x)) -> a(c(?x)),
a(b(?x)) -> a(c(?x)) ]
Apply Direct Methods...
Inner CPs:
[ a(c(a(?x_3))) = b(a(c(?x_3))),
a(a(c(?x_4))) = b(a(b(?x_4))),
a(b(a(?x))) = a(b(c(?x))),
a(a(c(?x_5))) = a(b(b(?x_5))),
c(a(c(?x_4))) = c(a(b(?x_4))),
c(a(c(?x_2))) = a(c(b(?x_2))),
a(a(c(?x_2))) = a(c(b(?x_2))),
a(a(b(?x))) = a(b(a(?x))),
b(a(c(?x))) = a(c(b(?x))),
c(c(a(?x))) = c(a(c(?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(c(a(?x_4))) = b(a(c(?x_4))),
a(a(c(?x_5))) = b(a(b(?x_5))),
a(b(c(a(?x_4)))) = a(b(a(c(?x_4)))),
a(b(a(c(?x_5)))) = a(b(a(b(?x_5)))),
a(b(c(?x))) = a(b(a(?x))),
a(a(b(?x_1))) = a(b(a(?x_1))),
a(b(a(?x_2))) = a(b(c(?x_2))),
a(a(c(?x_6))) = a(b(b(?x_6))),
a(b(a(b(?x_1)))) = a(a(b(a(?x_1)))),
a(b(b(a(?x_2)))) = a(a(b(c(?x_2)))),
a(b(a(c(?x_6)))) = a(a(b(b(?x_6)))),
a(b(a(?x))) = a(a(b(?x))),
b(a(c(?x_1))) = a(c(b(?x_1))),
a(c(a(c(?x_1)))) = b(a(c(b(?x_1)))),
a(c(a(c(?x_1)))) = c(a(c(b(?x_1)))),
a(c(a(c(?x_1)))) = a(a(c(b(?x_1)))),
a(c(b(?x))) = b(a(c(?x))),
a(c(b(?x))) = c(a(c(?x))),
a(c(b(?x))) = a(a(c(?x))),
c(c(a(?x_1))) = c(a(c(?x_1))),
c(a(c(?x_5))) = c(a(b(?x_5))),
c(a(c(a(?x_1)))) = c(c(a(c(?x_1)))),
c(a(a(c(?x_5)))) = c(c(a(b(?x_5)))),
b(a(c(a(?x_1)))) = a(c(a(c(?x_1)))),
b(a(a(c(?x_5)))) = a(c(a(b(?x_5)))),
c(a(c(?x))) = c(c(a(?x))),
b(a(c(?x))) = a(c(a(?x))),
c(a(c(?x_4))) = a(c(b(?x_4))),
b(a(a(c(?x_4)))) = a(a(c(b(?x_4)))),
c(a(a(c(?x_4)))) = c(a(c(b(?x_4)))),
b(a(b(?x))) = a(a(c(?x))),
c(a(b(?x))) = c(a(c(?x))),
a(a(c(?x_4))) = a(c(b(?x_4))),
a(b(a(c(?x_4)))) = a(a(c(b(?x_4)))),
a(b(b(?x))) = a(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 by Rules <3, 0> preceded by [(a,1)]
joinable by a reduction of rules <[], [([(b,1)],0),([],2)]>
Critical Pair by Rules <4, 0> preceded by [(a,1)]
joinable by a reduction of rules <[([],1),([],5),([],0)], [([(b,1)],5)]>
joinable by a reduction of rules <[([(a,1)],0),([],5)], [([(b,1)],5),([(b,1)],0),([],2)]>
joinable by a reduction of rules <[([],1),([],5),([(a,1)],3)], [([(b,1)],5),([(b,1)],0),([],2)]>
Critical Pair by Rules <0, 1> preceded by [(a,1)]
joinable by a reduction of rules <[([],5)], [([],5),([(a,1)],3)]>
Critical Pair by Rules <5, 1> preceded by [(a,1)]
joinable by a reduction of rules <[], [([(a,1)],2)]>
Critical Pair by Rules <4, 3> preceded by [(c,1)]
joinable by a reduction of rules <[], [([(c,1)],5)]>
Critical Pair by Rules <2, 4> preceded by [(c,1)]
joinable by a reduction of rules <[([(c,1)],0),([],4)], [([(a,1)],4),([(a,1)],0),([],5)]>
Critical Pair by Rules <2, 5> preceded by [(a,1)]
joinable by a reduction of rules <[], [([(a,1)],4)]>
Critical Pair by Rules <1, 1> preceded by [(a,1)]
joinable by a reduction of rules <[([(a,1)],5),([(a,1)],0)], []>
Critical Pair by Rules <2, 2> preceded by [(b,1)]
joinable by a reduction of rules <[], [([],0),([(b,1)],5)]>
Critical Pair by Rules <3, 3> preceded by [(c,1)]
joinable by a reduction of rules <[([],3),([(c,1)],1),([(c,1)],5)], []>
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...
failure(empty P)
unknown Reduction-Preserving Completion
check by Ordered Rewriting...
remove redundants rules and split
R-part:
[ a(c(?x)) -> b(a(?x)),
a(a(?x)) -> a(b(?x)),
b(b(?x)) -> a(c(?x)),
c(c(?x)) -> c(a(?x)),
c(b(?x)) -> a(c(?x)),
a(b(?x)) -> a(c(?x)) ]
E-part:
[ ]
...failed to find a suitable LPO.
unknown Confluence by Ordered Rewriting
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ a(c(?x)) -> b(a(?x)),
a(a(?x)) -> a(b(?x)),
b(b(?x)) -> a(c(?x)),
c(c(?x)) -> c(a(?x)),
c(b(?x)) -> a(c(?x)),
a(b(?x)) -> a(c(?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...
[ a(c(?x)) -> b(a(?x)),
a(a(?x)) -> a(b(?x)),
b(b(?x)) -> a(c(?x)),
c(c(?x)) -> c(a(?x)),
c(b(?x)) -> a(c(?x)),
a(b(?x)) -> a(c(?x)) ]
Layer Preserving Decomposition failed: Can't judge
Try Commutative Decomposition for...
[ a(c(?x)) -> b(a(?x)),
a(a(?x)) -> a(b(?x)),
b(b(?x)) -> a(c(?x)),
c(c(?x)) -> c(a(?x)),
c(b(?x)) -> a(c(?x)),
a(b(?x)) -> a(c(?x)) ]
Inside Critical Pair: by Rules <3, 0>
develop reducts from lhs term...
<{0}, b(a(a(?x_3)))>
<{}, a(c(a(?x_3)))>
develop reducts from rhs term...
<{0}, b(b(a(?x_3)))>
<{}, b(a(c(?x_3)))>
Inside Critical Pair: by Rules <4, 0>
develop reducts from lhs term...
<{1}, a(b(c(?x_4)))>
<{0}, a(b(a(?x_4)))>
<{}, a(a(c(?x_4)))>
develop reducts from rhs term...
<{5}, b(a(c(?x_4)))>
<{}, b(a(b(?x_4)))>
Inside Critical Pair: by Rules <0, 1>
develop reducts from lhs term...
<{5}, a(c(a(?x)))>
<{}, a(b(a(?x)))>
develop reducts from rhs term...
<{5}, a(c(c(?x)))>
<{}, a(b(c(?x)))>
Inside Critical Pair: by Rules <5, 1>
develop reducts from lhs term...
<{1}, a(b(c(?x_5)))>
<{0}, a(b(a(?x_5)))>
<{}, a(a(c(?x_5)))>
develop reducts from rhs term...
<{5}, a(c(b(?x_5)))>
<{2}, a(a(c(?x_5)))>
<{}, a(b(b(?x_5)))>
Inside Critical Pair: by Rules <4, 3>
develop reducts from lhs term...
<{0}, c(b(a(?x_4)))>
<{}, c(a(c(?x_4)))>
develop reducts from rhs term...
<{5}, c(a(c(?x_4)))>
<{}, c(a(b(?x_4)))>
Inside Critical Pair: by Rules <2, 4>
develop reducts from lhs term...
<{0}, c(b(a(?x_2)))>
<{}, c(a(c(?x_2)))>
develop reducts from rhs term...
<{0}, b(a(b(?x_2)))>
<{4}, a(a(c(?x_2)))>
<{}, a(c(b(?x_2)))>
Inside Critical Pair: by Rules <2, 5>
develop reducts from lhs term...
<{1}, a(b(c(?x_2)))>
<{0}, a(b(a(?x_2)))>
<{}, a(a(c(?x_2)))>
develop reducts from rhs term...
<{0}, b(a(b(?x_2)))>
<{4}, a(a(c(?x_2)))>
<{}, a(c(b(?x_2)))>
Commutative Decomposition failed: Can't judge
No further decomposition possible
Combined result: Can't judge
/tmp/fileov1Fc0.trs: Failure(unknown CR)
(2301 msec.)