MAYBE
(ignored inputs)COMMENT doi:10.4230/LIPIcs.RTA.2015.257 [81] Example 1
Rewrite Rules:
   [ f(f(?x)) -> ?x,
     f(?x) -> f(f(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ f(f(f(?x_1))) = ?x_1,
     f(?x) = f(?x) ]
Outer CPs:
   [ ?x = f(f(f(?x))) ]
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:
   [ f(f(?x_1)) = f(?x_1),
     f(f(f(f(?x)))) = ?x,
     f(f(f(?x))) = ?x,
     f(?x_1) = f(?x_1),
     ?x_1 = f(f(?x_1)),
     ?x_1 = f(f(f(?x_1))) ]
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 <f(f(f(?x_1))), ?x_1> by Rules <1, 0> preceded by [(f,1)]
 joinable by a reduction of rules <[([(f,1),(f,1)],1),([(f,1),(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1),(f,1)],1),([(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1),(f,1)],1),([],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],1),([(f,1),(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],1),([(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],1),([],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],0),([],1),([],0)], []>
 joinable by a reduction of rules <[([],1),([(f,1),(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([],1),([(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([],1),([],0),([],0)], []>
 joinable by a reduction of rules <[([],0),([],1),([],0)], []>
Critical Pair <f(?x), f(?x)> by Rules <0, 0> preceded by [(f,1)]
 joinable by a reduction of rules <[], []>
Critical Pair <f(f(f(?x))), ?x> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([(f,1),(f,1)],1),([(f,1),(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1),(f,1)],1),([(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1),(f,1)],1),([],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],1),([(f,1),(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],1),([(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],1),([],0),([],0)], []>
 joinable by a reduction of rules <[([(f,1)],0),([],1),([],0)], []>
 joinable by a reduction of rules <[([],1),([(f,1),(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([],1),([(f,1)],0),([],0)], []>
 joinable by a reduction of rules <[([],1),([],0),([],0)], []>
 joinable by a reduction of rules <[([],0),([],1),([],0)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 4 rules by 3 steps unfolding
obtain 16 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:
   [ f(f(?x)) -> ?x ]
P:
   [ f(?x) -> f(f(?x)) ]
P: not reversible
failure(Step 1)
STEP: 2 (linear)
S:
   [ f(f(?x)) -> ?x ]
P:
   [ f(?x) -> f(f(?x)) ]
P: not reversible
failure(Step 2)
STEP: 3 (relative)
S:
   [ f(f(?x)) -> ?x ]
P:
   [ f(?x) -> f(f(?x)) ]
P: not reversible
failure(Step 3)
failure(no possibility remains)
unknown Reduction-Preserving Completion
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ f(f(?x)) -> ?x,
     f(?x) -> f(f(?x)) ]
Sort Assignment:
 f : 5=>5
maximal types: {5}
Persistent Decomposition failed: Can't judge

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

Try Commutative Decomposition for...
   [ f(f(?x)) -> ?x,
     f(?x) -> f(f(?x)) ]
Outside Critical Pair: <f(f(f(?x))), ?x> by Rules <1, 0>
develop reducts from lhs term...
<{1}, f(f(f(f(f(f(?x))))))>
<{0,1}, f(f(?x))>
<{1}, f(f(f(f(f(?x)))))>
<{0}, f(?x)>
<{1}, f(f(f(f(?x))))>
<{}, f(f(f(?x)))>
develop reducts from rhs term...
<{}, ?x>
Inside Critical Pair: <f(f(f(?x_1))), ?x_1> by Rules <1, 0>
develop reducts from lhs term...
<{1}, f(f(f(f(f(f(?x_1))))))>
<{0,1}, f(f(?x_1))>
<{1}, f(f(f(f(f(?x_1)))))>
<{0}, f(?x_1)>
<{1}, f(f(f(f(?x_1))))>
<{}, f(f(f(?x_1)))>
develop reducts from rhs term...
<{}, ?x_1>
Commutative Decomposition failed: Can't judge
No further decomposition possible


Combined result: Can't judge
442.trs: Failure(unknown CR)
(142 msec.)