NO
(ignored inputs)COMMENT Example 8 from [HM11]: doi: http://dx.doi.org/10.1007/s10817-011-9238-x
Rewrite Rules:
   [ f(a,a) -> c,
     f(b,?x) -> f(?x,?x),
     f(?x,b) -> f(?x,?x),
     a -> b ]
Apply Direct Methods...
Inner CPs:
   [ f(b,a) = c,
     f(a,b) = c ]
Outer CPs:
   [ f(b,b) = f(b,b) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth & Bendix
Left-Linear, not Right-Linear
unknown Development 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:
   [ f(b,b) = c,
     f(b,a) = c,
     f(a,b) = c,
     f(b,b) = f(b,b),
     c = f(b,a),
     c = f(a,b) ]
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(b,a), c> by Rules <3, 0> preceded by [(f,1)]
 joinable by a reduction of rules <[([],1),([],0)], []>
Critical Pair <f(a,b), c> by Rules <3, 0> preceded by [(f,2)]
 joinable by a reduction of rules <[([],2),([],0)], []>
Critical Pair <f(b,b), f(b,b)> by Rules <2, 1> preceded by []
 joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
   [ f(a,a) -> c,
     f(b,?x) -> f(?x,?x),
     f(?x_1,b) -> f(?x_1,?x_1),
     a -> b ]
Sort Assignment:
 a : =>9
 b : =>9
 c : =>10
 f : 9*9=>10
non-linear variables: {?x,?x_1}
non-linear types: {9}
types leq non-linear types: {9}
rules applicable to terms of non-linear types:
   [ a -> b ]
NLR:
0: {}
1: {3}
2: {3}
3: {}
terms of non-linear types are innermost terminating
Critical Pair <f(b,a), c> by Rules <3, 0>
 joinable by a reduction of rules [->(1),->(0)]
 joinable by a reduction of rules [->(3),(3)<-,->(2),->(0)]
Critical Pair <f(a,b), c> by Rules <3, 0>
 joinable by a reduction of rules [->(2),->(0)]
 joinable by a reduction of rules [->(3),(3)<-,->(1),->(0)]
Critical Pair <f(b,b), f(b,b)> by Rules <2, 1>
 joinable by a reduction of rules []
Not Satisfiable
unknown Quasi-Linear & Linearized-Decreasing
   [ f(a,a) -> c,
     f(b,?x) -> f(?x,?x),
     f(?x_1,b) -> f(?x_1,?x_1),
     a -> b ]
Sort Assignment:
 a : =>9
 b : =>9
 c : =>10
 f : 9*9=>10
non-linear variables: {?x,?x_1}
non-linear types: {9}
types leq non-linear types: {9}
rules applicable to terms of non-linear types:
   [ a -> b ]
terms of non-linear types are terminating
Critical Pair <f(b,a), c> by Rules <3, 0>
 joinable by a reduction of rules [->(1),->(0)]
 joinable by a reduction of rules [->(3),(3)<-,->(2),->(0)]
Critical Pair <f(a,b), c> by Rules <3, 0>
 joinable by a reduction of rules [->(2),->(0)]
 joinable by a reduction of rules [->(3),(3)<-,->(1),->(0)]
Critical Pair <f(b,b), f(b,b)> by Rules <2, 1>
 joinable by a reduction of rules []
Not Satisfiable
unknown Strongly Quasi-Linear & Hierarchically Decreasing
check Non-Confluence...
obtain 14 rules by 3 steps unfolding
strenghten f(a,a) and c
strenghten f(a,b) and c
strenghten f(b,a) and c
strenghten f(b,b) and c
strenghten f(?x_3,?x_3) and f(?x_3,b)
strenghten f(?x_7,?x_7) and f(b,?x_7)
strenghten f(a,a) and f(b,b)
obtain 9 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Root-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) (success)
Witness for Non-Confluence: <f(b,b), c>
Direct Methods: not CR

Final result: not CR
113.trs: Success(not CR)
(189 msec.)