(ignored inputs)COMMENT submitted by: Johannes Waldmann Rewrite Rules: [ c(b(?x)) -> c(a(?x)), b(b(?x)) -> b(b(?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(b(b(?x_1))) = c(a(b(?x_1))), c(c(a(?x_2))) = c(a(b(?x_2))), c(c(b(?x_5))) = c(a(a(?x_5))), b(c(a(?x_2))) = b(b(b(?x_2))), b(c(b(?x_5))) = b(b(a(?x_5))), b(b(b(?x_1))) = c(a(b(?x_1))), b(c(b(?x_5))) = c(a(a(?x_5))), c(c(a(?x))) = a(c(b(?x))), c(a(c(?x_6))) = a(c(a(?x_6))), c(c(a(?x_7))) = a(c(c(?x_7))), b(c(a(?x_3))) = c(b(a(?x_3))), c(c(a(?x_3))) = a(c(a(?x_3))), c(c(a(?x))) = c(a(b(?x))), c(a(c(?x_4))) = c(a(c(?x_4))), c(a(c(?x_6))) = c(a(a(?x_6))), b(b(b(?x))) = b(b(b(?x))), 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: [ b(b(?x_1)) = c(a(?x_1)), a(c(?x_4)) = c(a(?x_4)) ] 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(b(b(?x_2))) = c(a(b(?x_2))), c(c(a(?x_3))) = c(a(b(?x_3))), c(c(b(?x_6))) = c(a(a(?x_6))), a(c(b(b(?x_2)))) = c(c(a(b(?x_2)))), a(c(c(a(?x_3)))) = c(c(a(b(?x_3)))), a(c(c(b(?x_6)))) = c(c(a(a(?x_6)))), c(a(b(b(?x_2)))) = c(c(a(b(?x_2)))), c(a(c(a(?x_3)))) = c(c(a(b(?x_3)))), c(a(c(b(?x_6)))) = c(c(a(a(?x_6)))), a(c(b(?x))) = c(c(a(?x))), c(a(b(?x))) = c(c(a(?x))), c(a(?x)) = b(b(?x)), b(b(b(?x_1))) = b(b(b(?x_1))), b(c(a(?x_3))) = b(b(b(?x_3))), b(c(b(?x_6))) = b(b(a(?x_6))), b(b(b(b(?x_1)))) = b(b(b(b(?x_1)))), b(b(c(a(?x_3)))) = b(b(b(b(?x_3)))), b(b(c(b(?x_6)))) = b(b(b(a(?x_6)))), c(a(b(b(?x_1)))) = c(b(b(b(?x_1)))), c(a(c(a(?x_3)))) = c(b(b(b(?x_3)))), c(a(c(b(?x_6)))) = c(b(b(a(?x_6)))), c(a(b(b(?x_1)))) = b(b(b(b(?x_1)))), c(a(c(a(?x_3)))) = b(b(b(b(?x_3)))), c(a(c(b(?x_6)))) = b(b(b(a(?x_6)))), c(a(b(?x))) = c(b(b(?x))), c(a(b(?x))) = b(b(b(?x))), b(b(?x)) = c(a(?x)), b(c(a(?x_1))) = c(a(b(?x_1))), b(b(b(?x_3))) = c(a(b(?x_3))), b(c(b(?x_6))) = c(a(a(?x_6))), c(a(c(a(?x_1)))) = b(c(a(b(?x_1)))), c(a(b(b(?x_3)))) = b(c(a(b(?x_3)))), c(a(c(b(?x_6)))) = b(c(a(a(?x_6)))), b(b(c(a(?x_1)))) = b(c(a(b(?x_1)))), b(b(b(b(?x_3)))) = b(c(a(b(?x_3)))), b(b(c(b(?x_6)))) = b(c(a(a(?x_6)))), c(a(b(?x))) = b(c(a(?x))), b(b(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_7))) = a(c(a(?x_7))), c(c(a(?x_8))) = a(c(c(?x_8))), 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_7)))) = c(a(c(a(?x_7)))), a(c(c(a(?x_8)))) = c(a(c(c(?x_8)))), 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_7)))) = c(a(c(a(?x_7)))), c(a(c(a(?x_8)))) = c(a(c(c(?x_8)))), a(c(c(?x))) = c(a(c(?x))), c(a(c(?x))) = c(a(c(?x))), b(c(a(?x_5))) = c(b(a(?x_5))), c(a(c(a(?x_5)))) = c(c(b(a(?x_5)))), b(b(c(a(?x_5)))) = b(c(b(a(?x_5)))), c(a(c(a(?x_5)))) = b(c(b(a(?x_5)))), c(a(a(?x))) = c(c(b(?x))), b(b(a(?x))) = b(c(b(?x))), c(a(a(?x))) = b(c(b(?x))), c(c(a(?x_5))) = a(c(a(?x_5))), a(c(c(a(?x_5)))) = c(a(c(a(?x_5)))), c(a(c(a(?x_5)))) = c(a(c(a(?x_5)))), 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_8))) = c(a(a(?x_8))), c(a(c(a(?x_1)))) = c(c(a(c(?x_1)))), c(a(a(c(?x_6)))) = c(c(a(c(?x_6)))), c(a(a(c(?x_8)))) = c(c(a(a(?x_8)))), a(c(c(a(?x_1)))) = c(c(a(c(?x_1)))), a(c(a(c(?x_6)))) = c(c(a(c(?x_6)))), a(c(a(c(?x_8)))) = c(c(a(a(?x_8)))), 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 <[([],0)], []> Critical Pair by Rules <2, 0> preceded by [(c,1)] joinable by a reduction of rules <[([],4)], [([],6),([(a,1)],0)]> joinable by a reduction of rules <[([],7),([],6)], [([],6),([(a,1)],0)]> Critical Pair by Rules <5, 0> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],0)], [([(c,1)],3)]> Critical Pair by Rules <2, 1> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],2)]> Critical Pair by Rules <5, 1> preceded by [(b,1)] joinable by a reduction of rules <[], [([(b,1)],5)]> Critical Pair by Rules <1, 2> preceded by [(b,1)] joinable by a reduction of rules <[([],2)], []> Critical Pair by Rules <5, 2> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],0),([(b,1)],6),([],5),([],0)], [([(c,1)],3),([(c,1)],6)]> joinable by a reduction of rules <[([(b,1)],0),([(b,1)],6),([],5),([],0)], [([],6),([(a,1)],6),([],3)]> joinable by a reduction of rules <[([(b,1)],0),([(b,1)],6),([],5),([],0)], [([(c,1)],3),([],4),([(a,1)],6),([],3)]> Critical Pair by Rules <0, 4> preceded by [(c,1)] joinable by a reduction of rules <[([],4)], [([(a,1)],0)]> Critical Pair by Rules <6, 4> preceded by [(c,1)] joinable by a reduction of rules <[([],6),([(a,1)],7)], []> joinable by a reduction of rules <[([],6),([(a,1)],4)], [([(a,1)],6)]> joinable by a reduction of rules <[], [([(a,1)],6),([],3)]> Critical Pair by Rules <7, 4> preceded by [(c,1)] joinable by a reduction of rules <[([],4)], [([(a,1)],7)]> Critical Pair by Rules <3, 5> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6),([],5),([],0)], [([(c,1)],5),([(c,1)],0),([(c,1)],6)]> joinable by a reduction of rules <[([(b,1)],6),([],5),([],0)], [([],0),([(c,1)],3),([(c,1)],6)]> Critical Pair by Rules <3, 6> preceded by [(c,1)] joinable by a reduction of rules <[([],4)], []> Critical Pair by Rules <0, 7> preceded by [(c,1)] joinable by a reduction of rules <[([],4)], [([],6),([(a,1)],0)]> joinable by a reduction of rules <[([],7),([],6)], [([],6),([(a,1)],0)]> Critical Pair by Rules <4, 7> preceded by [(c,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <6, 7> preceded by [(c,1)] joinable by a reduction of rules <[([],6),([(a,1)],7)], [([],6)]> joinable by a reduction of rules <[], [([(c,1)],3),([(c,1)],6)]> joinable by a reduction of rules <[([],6),([(a,1)],7)], [([(c,1)],3),([],4)]> joinable by a reduction of rules <[([],6),([(a,1)],4)], [([],6),([(a,1)],6)]> Critical Pair by Rules <1, 1> preceded by [(b,1)] joinable by a reduction of rules <[], []> Critical Pair by Rules <2, 2> preceded by [(b,1)] joinable by a reduction of rules <[([(b,1)],6),([],5),([],0)], [([],6),([(a,1)],0),([(a,1)],6),([],3)]> Critical Pair by Rules <3, 3> preceded by [(a,1)] joinable by a reduction of rules <[], [([],6)]> Critical Pair by Rules <4, 4> preceded by [(c,1)] joinable by a reduction of rules <[([],6)], []> Critical Pair by Rules <7, 7> preceded by [(c,1)] joinable by a reduction of rules <[([(c,1)],6)], []> Critical Pair by Rules <2, 1> preceded by [] joinable by a reduction of rules <[], [([],2)]> Critical Pair by Rules <7, 4> preceded by [] joinable by a reduction of rules <[([],6)], []> Satisfiable by 8>6,3>1>4,5>7,2; a(1)b(2)c(1); 6,3>1>7>5>8>2,4 Diagram Decreasing Direct Methods: CR Combined result: CR 1016.trs: Success(CR) YES (11 msec.)