YES
(ignored inputs)COMMENT associative law + commutative law
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 12 rules by 3 steps unfolding
strenghten f(?x_3,?y_3) and f(?y_3,?x_3)
strenghten f(?x,f(?y,?x_4)) and f(?x_4,f(?x,?y))
strenghten f(?x,f(?y,?x_5)) and f(f(?x,?y),?x_5)
strenghten f(?x_1,f(?x_10,?y_1)) and f(?x_10,f(?y_1,?x_1))
strenghten f(?x_1,f(?y_1,?x_12)) and f(?x_12,f(?y_1,?x_1))
strenghten f(?x_1,f(?y_1,?z)) and f(f(?y_1,?x_1),?z)
strenghten f(?x_2,f(?x_10,?y_2)) and f(?x_10,f(?x_2,?y_2))
strenghten f(?x_6,f(?x_1,?y_1)) and f(?x_6,f(?y_1,?x_1))
strenghten f(f(?x_7,?y_7),?x_6) and f(?x_6,f(?x_7,?y_7))
strenghten f(f(?x_9,?y_9),?x_8) and f(?x_8,f(?y_9,?x_9))
strenghten f(f(?x_11,?y_11),?x_10) and f(?x_11,f(?x_10,?y_11))
strenghten f(f(?x_13,?y_13),?x_12) and f(?x_13,f(?y_13,?x_12))
strenghten f(f(?x_15,?y_15),?x_14) and f(?y_15,f(?x_14,?x_15))
strenghten f(f(?x_17,?y_17),?x_16) and f(?y_17,f(?x_17,?x_16))
strenghten f(?x,f(?x_2,f(?x_1,?y_1))) and f(?x_2,f(?x_1,f(?x,?y_1)))
strenghten f(?x,f(?y,f(?x_7,?y_7))) and f(f(?x,?y),f(?x_7,?y_7))
strenghten f(?x,f(?y,f(?x_9,?y_9))) and f(f(?x,?y),f(?y_9,?x_9))
strenghten f(?x,f(?y,f(?x_11,?y_11))) and f(?x_11,f(f(?x,?y),?y_11))
obtain 100 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) (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

Final result: CR
103.trs: Success(CR)
(3456 msec.)