YES
(ignored inputs)COMMENT associative law + commutative law submitted by: Takahito Aoto , Junichi Yoshida , and Yoshihito Toyama
Rewrite Rules:
   [ f(f(?x,?y),?z) -> f(?x,f(?y,?z)),
     f(?x,?y) -> f(?y,?x) ]
Apply Direct Methods...
Inner CPs:
   [ f(f(?y_1,?x_1),?z) = f(?x_1,f(?y_1,?z)),
     f(f(?x,f(?y,?z)),?z_1) = f(f(?x,?y),f(?z,?z_1)) ]
Outer CPs:
   [ f(?x,f(?y,?z)) = f(?z,f(?x,?y)) ]
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(?z,f(?x_1,f(?y_1,?y))) = f(f(?x_1,?y_1),f(?y,?z)),
     f(?z,f(?y,?x)) = f(?x,f(?y,?z)),
     f(?z,f(?x,?y)) = f(?x,f(?y,?z)),
     f(f(?x_1,f(?y_1,?y)),?z) = f(f(?x_1,?y_1),f(?y,?z)),
     f(f(?y,?x),?z) = f(?x,f(?y,?z)),
     f(f(?x_2,f(?y_2,?y)),f(?z,?z_1)) = f(f(f(?x_2,?y_2),f(?y,?z)),?z_1),
     f(f(?y,?x),f(?z,?z_1)) = f(f(?x,f(?y,?z)),?z_1),
     f(f(?x,?y),f(?z,?z_1)) = f(f(?x,f(?y,?z)),?z_1),
     f(?x_1,f(?y_1,?y)) = f(?y,f(?x_1,?y_1)),
     f(?x,f(?y,?z_2)) = f(f(?y,?x),?z_2) ]
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(?y_1,?x_1),?z), f(?x_1,f(?y_1,?z))> by Rules <1, 0> preceded by [(f,1)]
 joinable by a reduction of rules <[([(f,1)],1),([],0)], []>
 joinable by a reduction of rules <[([],0),([(f,2)],1)], [([],1),([],0)]>
Critical Pair <f(f(?x,f(?y,?z)),?z_1), f(f(?x,?y),f(?z,?z_1))> by Rules <0, 0> preceded by [(f,1)]
 joinable by a reduction of rules <[([],0),([(f,2)],0)], [([],0)]>
Critical Pair <f(?y_1,f(?x,?y)), f(?x,f(?y,?y_1))> by Rules <1, 0> preceded by []
 joinable by a reduction of rules <[([],1),([],0)], []>
unknown Diagram Decreasing
check Non-Confluence...
obtain 9 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
Check relative termination:
   [ ]
   [ f(f(?x,?y),?z) -> f(?x,f(?y,?z)),
     f(?x,?y) -> f(?y,?x) ]
relatively terminating
Huet (modulo AC)
Direct Methods: CR

Combined result: CR
103.trs: Success(CR)
(2026 msec.)