YES (ignored inputs)COMMENT the following rules are removed from the original TRS b ( b ( x ) ) -> b ( b ( x ) ) Rewrite Rules: [ c(b(?x)) -> c(a(?x)), b(b(?x)) -> c(a(?x)), a(a(?x)) -> c(a(?x)), c(c(?x)) -> a(c(?x)), b(a(?x)) -> c(b(?x)), c(a(?x)) -> a(c(?x)), c(c(?x)) -> c(a(?x)) ] Apply Direct Methods... Inner CPs: [ c(c(a(?x_1))) = c(a(b(?x_1))), c(c(b(?x_4))) = c(a(a(?x_4))), b(c(b(?x_4))) = c(a(a(?x_4))), c(c(a(?x))) = a(c(b(?x))), c(a(c(?x_5))) = a(c(a(?x_5))), c(c(a(?x_6))) = a(c(c(?x_6))), b(c(a(?x_2))) = c(b(a(?x_2))), c(c(a(?x_2))) = a(c(a(?x_2))), c(c(a(?x))) = c(a(b(?x))), c(a(c(?x_3))) = c(a(c(?x_3))), c(a(c(?x_5))) = c(a(a(?x_5))), b(c(a(?x))) = c(a(b(?x))), a(c(a(?x))) = c(a(a(?x))), c(a(c(?x))) = a(c(c(?x))), c(c(a(?x))) = c(a(c(?x))) ] Outer CPs: [ a(c(?x_3)) = c(a(?x_3)) ] 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: [ c(c(a(?x_2))) = c(a(b(?x_2))), c(c(b(?x_5))) = c(a(a(?x_5))), a(c(c(a(?x_2)))) = c(c(a(b(?x_2)))), a(c(c(b(?x_5)))) = c(c(a(a(?x_5)))), c(a(c(a(?x_2)))) = c(c(a(b(?x_2)))), c(a(c(b(?x_5)))) = c(c(a(a(?x_5)))), a(c(b(?x))) = c(c(a(?x))), c(a(b(?x))) = c(c(a(?x))), b(c(a(?x_1))) = c(a(b(?x_1))), b(c(b(?x_5))) = c(a(a(?x_5))), c(a(c(a(?x_1)))) = b(c(a(b(?x_1)))), c(a(c(b(?x_5)))) = b(c(a(a(?x_5)))), c(a(b(?x))) = b(c(a(?x))), a(c(a(?x_1))) = c(a(a(?x_1))), c(a(c(a(?x_1)))) = a(c(a(a(?x_1)))), c(b(c(a(?x_1)))) = b(c(a(a(?x_1)))), a(c(c(a(?x_1)))) = c(c(a(a(?x_1)))), c(a(a(?x))) = a(c(a(?x))), c(b(a(?x))) = b(c(a(?x))), a(c(a(?x))) = c(c(a(?x))), c(a(?x)) = a(c(?x)), c(a(c(?x_1))) = a(c(c(?x_1))), c(c(a(?x_2))) = a(c(b(?x_2))), c(a(c(?x_6))) = a(c(a(?x_6))), c(c(a(?x_7))) = a(c(c(?x_7))), a(c(a(c(?x_1)))) = c(a(c(c(?x_1)))), a(c(c(a(?x_2)))) = c(a(c(b(?x_2)))), a(c(a(c(?x_6)))) = c(a(c(a(?x_6)))), a(c(c(a(?x_7)))) = c(a(c(c(?x_7)))), c(a(a(c(?x_1)))) = c(a(c(c(?x_1)))), c(a(c(a(?x_2)))) = c(a(c(b(?x_2)))), c(a(a(c(?x_6)))) = c(a(c(a(?x_6)))), c(a(c(a(?x_7)))) = c(a(c(c(?x_7)))), a(c(c(?x))) = c(a(c(?x))), c(a(c(?x))) = c(a(c(?x))), b(c(a(?x_4))) = c(b(a(?x_4))), c(a(c(a(?x_4)))) = c(c(b(a(?x_4)))), c(a(c(a(?x_4)))) = b(c(b(a(?x_4)))), c(a(a(?x))) = c(c(b(?x))), c(a(a(?x))) = b(c(b(?x))), c(c(a(?x_4))) = a(c(a(?x_4))), a(c(c(a(?x_4)))) = c(a(c(a(?x_4)))), c(a(c(a(?x_4)))) = c(a(c(a(?x_4)))), a(c(a(?x))) = c(a(c(?x))), c(a(a(?x))) = c(a(c(?x))), a(c(?x)) = c(a(?x)), c(c(a(?x_1))) = c(a(c(?x_1))), c(a(c(?x_7))) = c(a(a(?x_7))), c(a(c(a(?x_1)))) = c(c(a(c(?x_1)))), c(a(a(c(?x_5)))) = c(c(a(c(?x_5)))), c(a(a(c(?x_7)))) = c(c(a(a(?x_7)))), a(c(c(a(?x_1)))) = c(c(a(c(?x_1)))), a(c(a(c(?x_5)))) = c(c(a(c(?x_5)))), a(c(a(c(?x_7)))) = c(c(a(a(?x_7)))), c(a(c(?x))) = c(c(a(?x))), a(c(c(?x))) = c(c(a(?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 <1, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],3)], [([],5),([(a,1)],0)]> joinable by a reduction of rules <[([],6),([],5)], [([],5),([(a,1)],0)]> Critical Pair by Rules <4, 0> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0)], [([(c,1)],2)]> Critical Pair by Rules <4, 1> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0),([(b,1)],5),([],4),([],0)], [([(c,1)],2),([(c,1)],5)]> joinable by a reduction of rules <[([(b,1)],0),([(b,1)],5),([],4),([],0)], [([],5),([(a,1)],5),([],2)]> joinable by a reduction of rules <[([(b,1)],0),([(b,1)],5),([],4),([],0)], [([(c,1)],2),([],3),([(a,1)],5),([],2)]> Critical Pair by Rules <0, 3> preceded by [(c,1)] joinable by a reduction of rules <[([],3)], [([(a,1)],0)]> Critical Pair by Rules <5, 3> preceded by [(c,1)] joinable by a reduction of rules <[([],5),([(a,1)],6)], []> joinable by a reduction of rules <[([],5),([(a,1)],3)], [([(a,1)],5)]> joinable by a reduction of rules <[], [([(a,1)],5),([],2)]> Critical Pair by Rules <6, 3> preceded by [(c,1)] joinable by a reduction of rules <[([],3)], [([(a,1)],6)]> Critical Pair by Rules <2, 4> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],5),([],4),([],0)], [([(c,1)],4),([(c,1)],0),([(c,1)],5)]> joinable by a reduction of rules <[([(b,1)],5),([],4),([],0)], [([],0),([(c,1)],2),([(c,1)],5)]> Critical Pair by Rules <2, 5> preceded by [(c,1)] joinable by a reduction of rules <[([],3)], []> Critical Pair by Rules <0, 6> preceded by [(c,1)] joinable by a reduction of rules <[([],3)], [([],5),([(a,1)],0)]> joinable by a reduction of rules <[([],6),([],5)], [([],5),([(a,1)],0)]> Critical Pair by Rules <3, 6> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <5, 6> preceded by [(c,1)] joinable by a reduction of rules <[([],5),([(a,1)],6)], [([],5)]> joinable by a reduction of rules <[], [([(c,1)],2),([(c,1)],5)]> joinable by a reduction of rules <[([],5),([(a,1)],6)], [([(c,1)],2),([],3)]> joinable by a reduction of rules <[([],5),([(a,1)],3)], [([],5),([(a,1)],5)]> Critical Pair by Rules <1, 1> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],5),([],4),([],0)], [([],5),([(a,1)],0),([(a,1)],5),([],2)]> Critical Pair by Rules <2, 2> preceded by [(a,1)] joinable by a reduction of rules <[], [([],5)]> Critical Pair by Rules <3, 3> preceded by [(c,1)] joinable by a reduction of rules <[([],5)], []> Critical Pair by Rules <6, 6> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],5)], []> Critical Pair by Rules <6, 3> preceded by [] joinable by a reduction of rules <[([],5)], []> Satisfiable by 2>5>1>7>3>6,4; a(1)b(1)c(1); 2>1>5>3>6>4>7 Diagram Decreasing Direct Methods: CR Combined result: CR /tmp/fileH3hZJC.trs: Success(CR) (42 msec.)