11.20/10.02 YES 11.20/10.02 (ignored inputs)COMMENT full experiments for [35] submitted by: Takahito Aoto and Yoshihito Toyama 11.20/10.02 Rewrite Rules: 11.20/10.02 [ +(0,?y) -> ?y, 11.20/10.02 +(s(?x),?y) -> s(+(?x,?y)), 11.20/10.02 +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.02 +(?x,?y) -> +(?y,?x), 11.20/10.02 s(s(?x)) -> s(?x), 11.20/10.02 s(?x) -> s(s(?x)) ] 11.20/10.02 Apply Direct Methods... 11.20/10.02 Inner CPs: 11.20/10.02 [ +(s(?x_4),?y_1) = s(+(s(?x_4),?y_1)), 11.20/10.02 +(s(s(?x_5)),?y_1) = s(+(?x_5,?y_1)), 11.20/10.02 +(?y,?z_2) = +(0,+(?y,?z_2)), 11.20/10.02 +(s(+(?x_1,?y_1)),?z_2) = +(s(?x_1),+(?y_1,?z_2)), 11.20/10.02 +(+(?y_3,?x_3),?z_2) = +(?x_3,+(?y_3,?z_2)), 11.20/10.02 s(s(s(?x_5))) = s(?x_5), 11.20/10.02 +(+(?x,+(?y,?z)),?z_1) = +(+(?x,?y),+(?z,?z_1)), 11.20/10.02 s(s(?x)) = s(s(?x)) ] 11.20/10.02 Outer CPs: 11.20/10.02 [ ?y = +(?y,0), 11.20/10.02 s(+(?x_1,?y_1)) = +(?y_1,s(?x_1)), 11.20/10.02 +(?x_2,+(?y_2,?z_2)) = +(?z_2,+(?x_2,?y_2)), 11.20/10.02 s(?x_4) = s(s(s(?x_4))) ] 11.20/10.02 not Overlay, check Termination... 11.20/10.02 unknown/not Terminating 11.20/10.02 unknown Knuth & Bendix 11.20/10.02 Linear 11.20/10.02 unknown Development Closed 11.20/10.02 unknown Strongly Closed 11.20/10.02 unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow 11.20/10.02 unknown Upside-Parallel-Closed/Outside-Closed 11.20/10.02 (inner) Parallel CPs: (not computed) 11.20/10.02 unknown Toyama (Parallel CPs) 11.20/10.02 Simultaneous CPs: 11.20/10.02 [ +(?y,0) = ?y, 11.20/10.02 +(0,+(?y,?z_3)) = +(?y,?z_3), 11.20/10.02 +(?y,s(?x_5)) = s(+(s(?x_5),?y)), 11.20/10.02 +(?y,s(s(?x))) = s(+(?x,?y)), 11.20/10.02 +(?y,s(?x)) = s(+(?x,?y)), 11.20/10.02 +(s(?x_5),?y) = s(+(s(?x_5),?y)), 11.20/10.02 +(s(s(?x)),?y) = s(+(?x,?y)), 11.20/10.02 +(s(?x_8),+(?y,?z_3)) = +(s(+(s(?x_8),?y)),?z_3), 11.20/10.02 +(s(s(?x)),+(?y,?z_3)) = +(s(+(?x,?y)),?z_3), 11.20/10.02 +(s(?x),+(?y,?z_3)) = +(s(+(?x,?y)),?z_3), 11.20/10.02 +(?z,+(?x_1,+(?y_1,?y))) = +(+(?x_1,?y_1),+(?y,?z)), 11.20/10.02 +(?z,?y) = +(0,+(?y,?z)), 11.20/10.02 +(?z,s(+(?x_3,?y))) = +(s(?x_3),+(?y,?z)), 11.20/10.02 +(?z,+(?y,?x)) = +(?x,+(?y,?z)), 11.20/10.02 +(?z,+(?x,?y)) = +(?x,+(?y,?z)), 11.20/10.02 +(+(?x_1,+(?y_1,?y)),?z) = +(+(?x_1,?y_1),+(?y,?z)), 11.20/10.02 +(?y,?z) = +(0,+(?y,?z)), 11.20/10.02 +(s(+(?x_3,?y)),?z) = +(s(?x_3),+(?y,?z)), 11.20/10.02 +(+(?y,?x),?z) = +(?x,+(?y,?z)), 11.20/10.02 +(+(?x_2,+(?y_2,?y)),+(?z,?z_1)) = +(+(+(?x_2,?y_2),+(?y,?z)),?z_1), 11.20/10.02 +(?y,+(?z,?z_1)) = +(+(0,+(?y,?z)),?z_1), 11.20/10.02 +(s(+(?x_4,?y)),+(?z,?z_1)) = +(+(s(?x_4),+(?y,?z)),?z_1), 11.20/10.02 +(+(?y,?x),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1), 11.20/10.02 +(+(?x,?y),+(?z,?z_1)) = +(+(?x,+(?y,?z)),?z_1), 11.20/10.02 ?y = +(?y,0), 11.20/10.02 s(+(?x_2,?y)) = +(?y,s(?x_2)), 11.20/10.02 +(?x_3,+(?y_3,?y)) = +(?y,+(?x_3,?y_3)), 11.20/10.02 +(?x,+(?y,?z_4)) = +(+(?y,?x),?z_4), 11.20/10.02 s(s(s(?x_1))) = s(s(?x_1)), 11.20/10.02 s(s(s(s(?x)))) = s(?x), 11.20/10.02 s(s(s(?x))) = s(?x), 11.20/10.02 s(s(?x_1)) = s(s(?x_1)), 11.20/10.02 s(s(?x_1)) = s(s(s(?x_1))), 11.20/10.02 s(+(s(?x_4),?y_3)) = +(s(s(?x_4)),?y_3), 11.20/10.02 s(+(s(s(?x)),?y_3)) = +(s(?x),?y_3), 11.20/10.02 s(+(s(?x),?y_3)) = +(s(?x),?y_3), 11.20/10.02 s(?x_5) = s(s(s(?x_5))), 11.20/10.02 s(+(?x,?y_3)) = +(s(s(?x)),?y_3) ] 11.20/10.02 unknown Okui (Simultaneous CPs) 11.20/10.02 unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping 11.20/10.02 unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping 11.20/10.02 check Locally Decreasing Diagrams by Rule Labelling... 11.20/10.02 Critical Pair <+(s(?x_4),?y_1), s(+(s(?x_4),?y_1))> by Rules <4, 1> preceded by [(+,1)] 11.20/10.02 joinable by a reduction of rules <[([(+,1)],5),([],1)], []> 11.20/10.02 joinable by a reduction of rules <[([],1),([],5)], [([(s,1)],1)]> 11.20/10.02 joinable by a reduction of rules <[([],1)], [([(s,1)],1),([],4)]> 11.20/10.02 Critical Pair <+(s(s(?x_5)),?y_1), s(+(?x_5,?y_1))> by Rules <5, 1> preceded by [(+,1)] 11.20/10.02 joinable by a reduction of rules <[([(+,1)],4),([],1)], []> 11.20/10.02 joinable by a reduction of rules <[([],1),([(s,1)],1)], [([],5)]> 11.20/10.02 Critical Pair <+(?y,?z_2), +(0,+(?y,?z_2))> by Rules <0, 2> preceded by [(+,1)] 11.20/10.02 joinable by a reduction of rules <[], [([],0)]> 11.20/10.02 Critical Pair <+(s(+(?x_1,?y_1)),?z_2), +(s(?x_1),+(?y_1,?z_2))> by Rules <1, 2> preceded by [(+,1)] 11.20/10.02 joinable by a reduction of rules <[([],1),([(s,1)],2)], [([],1)]> 11.20/10.02 Critical Pair <+(+(?y_3,?x_3),?z_2), +(?x_3,+(?y_3,?z_2))> by Rules <3, 2> preceded by [(+,1)] 11.20/10.02 joinable by a reduction of rules <[([(+,1)],3),([],2)], []> 11.20/10.02 joinable by a reduction of rules <[([],2),([(+,2)],3)], [([],3),([],2)]> 11.20/10.02 Critical Pair by Rules <5, 4> preceded by [(s,1)] 11.20/10.02 joinable by a reduction of rules <[([(s,1)],4)], [([],5)]> 11.20/10.02 joinable by a reduction of rules <[([],4)], [([],5)]> 11.20/10.02 Critical Pair <+(+(?x,+(?y,?z)),?z_1), +(+(?x,?y),+(?z,?z_1))> by Rules <2, 2> preceded by [(+,1)] 11.20/10.02 joinable by a reduction of rules <[([],2),([(+,2)],2)], [([],2)]> 11.20/10.02 Critical Pair by Rules <4, 4> preceded by [(s,1)] 11.20/10.02 joinable by a reduction of rules <[], []> 11.20/10.02 Critical Pair <+(?y_3,0), ?y_3> by Rules <3, 0> preceded by [] 11.20/10.02 joinable by a reduction of rules <[([],3),([],0)], []> 11.20/10.02 Critical Pair <+(?y_3,s(?x_1)), s(+(?x_1,?y_3))> by Rules <3, 1> preceded by [] 11.20/10.02 joinable by a reduction of rules <[([],3),([],1)], []> 11.20/10.02 Critical Pair <+(?y_3,+(?x_2,?y_2)), +(?x_2,+(?y_2,?y_3))> by Rules <3, 2> preceded by [] 11.20/10.02 joinable by a reduction of rules <[([],3),([],2)], []> 11.20/10.02 Critical Pair by Rules <5, 4> preceded by [] 11.20/10.02 joinable by a reduction of rules <[([(s,1)],4)], [([],5)]> 11.20/10.02 joinable by a reduction of rules <[([],4)], [([],5)]> 11.20/10.02 unknown Diagram Decreasing 11.20/10.02 check Non-Confluence... 11.20/10.02 obtain 8 rules by 3 steps unfolding 11.20/10.02 obtain 100 candidates for checking non-joinability 11.20/10.02 check by TCAP-Approximation (failure) 11.20/10.02 check by Ordering(rpo), check by Tree-Automata Approximation (failure) 11.20/10.02 check by Interpretation(mod2) (failure) 11.20/10.02 check by Descendants-Approximation, check by Ordering(poly) (failure) 11.20/10.02 unknown Non-Confluence 11.20/10.02 Check relative termination: 11.20/10.02 [ +(0,?y) -> ?y, 11.20/10.02 +(s(?x),?y) -> s(+(?x,?y)), 11.20/10.02 s(s(?x)) -> s(?x), 11.20/10.02 s(?x) -> s(s(?x)) ] 11.20/10.02 [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.02 +(?x,?y) -> +(?y,?x) ] 11.20/10.02 not relatively terminatiing 11.20/10.02 unknown Huet (modulo AC) 11.20/10.02 check by Reduction-Preserving Completion... 11.20/10.02 STEP: 1 (parallel) 11.20/10.02 S: 11.20/10.02 [ +(0,?y) -> ?y, 11.20/10.02 +(s(?x),?y) -> s(+(?x,?y)) ] 11.20/10.02 P: 11.20/10.02 [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.02 +(?x,?y) -> +(?y,?x), 11.20/10.02 s(s(?x)) -> s(?x), 11.20/10.02 s(?x) -> s(s(?x)) ] 11.20/10.02 S: terminating 11.20/10.02 CP(S,S): 11.20/10.02 PCP_in(symP,S): 11.20/10.02 <+(s(?x_5),?y), s(+(s(?x_5),?y))> --> => no 11.20/10.02 <+(s(s(?x_4)),?y), s(+(?x_4,?y))> --> => no 11.20/10.02 CP(S,symP): 11.20/10.02 <+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes 11.20/10.02 <+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes 11.20/10.02 <+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(+(?x_1,0),?y)> => no 11.20/10.02 --> => no 11.20/10.02 <+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> => yes 11.20/10.02 --> => yes 11.20/10.02 <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> => no 11.20/10.02 --> => no 11.20/10.02 check joinability condition: 11.20/10.02 check modulo joinability of s(+(?x_5,?y)) and s(s(+(?x_5,?y))): joinable by {0} 11.20/10.02 check modulo joinability of s(s(+(?x_4,?y))) and s(+(?x_4,?y)): joinable by {0} 11.20/10.02 check modulo reachablity from +(?x_1,?y) to +(+(?x_1,0),?y): maybe not reachable 11.20/10.02 check modulo reachablity from ?y to +(?y,0): maybe not reachable 11.20/10.02 check modulo reachablity from +(?x_1,s(+(?x,?y))) to +(+(?x_1,s(?x)),?y): maybe not reachable 11.20/10.02 check modulo reachablity from s(+(?x,?y)) to +(?y,s(?x)): maybe not reachable 11.20/10.02 failed 11.20/10.02 failure(Step 1) 11.20/10.02 [ +(?y,0) -> ?y, 11.20/10.02 +(?y,s(?x)) -> s(+(?x,?y)) ] 11.20/10.02 Added S-Rules: 11.20/10.02 [ +(?y,0) -> ?y, 11.20/10.02 +(?y,s(?x)) -> s(+(?x,?y)) ] 11.20/10.02 Added P-Rules: 11.20/10.02 [ ] 11.20/10.02 replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(+(?y,?x)) 11.20/10.02 replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(s(+(?x,?y))) 11.20/10.02 STEP: 2 (linear) 11.20/10.02 S: 11.20/10.02 [ +(0,?y) -> ?y, 11.20/10.02 +(s(?x),?y) -> s(+(?x,?y)) ] 11.20/10.02 P: 11.20/10.02 [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.02 +(?x,?y) -> +(?y,?x), 11.20/10.02 s(s(?x)) -> s(?x), 11.20/10.02 s(?x) -> s(s(?x)) ] 11.20/10.02 S: terminating 11.20/10.02 CP(S,S): 11.20/10.02 CP_in(symP,S): 11.20/10.02 <+(s(s(?x)),?y_1), s(+(?x,?y_1))> --> => yes 11.20/10.02 <+(s(?x),?y_1), s(+(s(?x),?y_1))> --> => yes 11.20/10.02 CP(S,symP): 11.20/10.02 <+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes 11.20/10.02 <+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes 11.20/10.02 <+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(+(?x_1,0),?y)> => no 11.20/10.02 --> => no 11.20/10.02 <+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> => yes 11.20/10.02 --> => yes 11.20/10.02 <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> => no 11.20/10.02 --> => no 11.20/10.02 check joinability condition: 11.20/10.02 check modulo reachablity from +(?x_1,?y) to +(+(?x_1,0),?y): maybe not reachable 11.20/10.02 check modulo reachablity from ?y to +(?y,0): maybe not reachable 11.20/10.02 check modulo reachablity from +(?x_1,s(+(?x,?y))) to +(+(?x_1,s(?x)),?y): maybe not reachable 11.20/10.02 check modulo reachablity from s(+(?x,?y)) to +(?y,s(?x)): maybe not reachable 11.20/10.02 failed 11.20/10.02 failure(Step 2) 11.20/10.02 [ +(?y,0) -> ?y, 11.20/10.02 +(?y,s(?x)) -> s(+(?x,?y)) ] 11.20/10.02 Added S-Rules: 11.20/10.02 [ +(?y,0) -> ?y, 11.20/10.02 +(?y,s(?x)) -> s(+(?x,?y)) ] 11.20/10.02 Added P-Rules: 11.20/10.02 [ ] 11.20/10.02 replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(+(?y,?x)) 11.20/10.02 replace: +(s(?x),?y) -> s(+(?x,?y)) => +(s(?x),?y) -> s(s(+(?x,?y))) 11.20/10.02 STEP: 3 (relative) 11.20/10.02 S: 11.20/10.02 [ +(0,?y) -> ?y, 11.20/10.02 +(s(?x),?y) -> s(+(?x,?y)) ] 11.20/10.02 P: 11.20/10.02 [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.02 +(?x,?y) -> +(?y,?x), 11.20/10.02 s(s(?x)) -> s(?x), 11.20/10.02 s(?x) -> s(s(?x)) ] 11.20/10.02 Check relative termination: 11.20/10.02 [ +(0,?y) -> ?y, 11.20/10.02 +(s(?x),?y) -> s(+(?x,?y)) ] 11.20/10.02 [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.02 +(?x,?y) -> +(?y,?x), 11.20/10.02 s(s(?x)) -> s(?x), 11.20/10.02 s(?x) -> s(s(?x)) ] 11.20/10.02 Polynomial Interpretation: 11.20/10.02 +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2 11.20/10.02 0:= (12) 11.20/10.02 s:= (1)*x1 11.20/10.02 retract +(0,?y) -> ?y 11.20/10.02 Polynomial Interpretation: 11.20/10.02 +:= (2)+(2)*x1+(1)*x1*x2+(2)*x2 11.20/10.02 0:= (10) 11.20/10.02 s:= (1)+(1)*x1 11.20/10.02 relatively terminating 11.20/10.02 S/P: relatively terminating 11.20/10.02 check CP condition: 11.20/10.02 failed 11.20/10.02 failure(Step 3) 11.20/10.02 STEP: 4 (parallel) 11.20/10.02 S: 11.20/10.02 [ +(0,?y) -> ?y, 11.20/10.02 +(s(?x),?y) -> s(+(?x,?y)), 11.20/10.02 +(?y,0) -> ?y, 11.20/10.02 +(?y,s(?x)) -> s(+(?x,?y)) ] 11.20/10.02 P: 11.20/10.02 [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.02 +(?x,?y) -> +(?y,?x), 11.20/10.02 s(s(?x)) -> s(?x), 11.20/10.02 s(?x) -> s(s(?x)) ] 11.20/10.02 S: terminating 11.20/10.02 CP(S,S): 11.20/10.02 <0, 0> --> <0, 0> => yes 11.20/10.02 --> => yes 11.20/10.02 --> => yes 11.20/10.02 --> => yes 11.20/10.02 <0, 0> --> <0, 0> => yes 11.20/10.02 --> => yes 11.20/10.02 --> => yes 11.20/10.02 --> => yes 11.20/10.02 PCP_in(symP,S): 11.20/10.02 <+(s(?x_5),?y), s(+(s(?x_5),?y))> --> => no 11.20/10.02 <+(s(s(?x_4)),?y), s(+(?x_4,?y))> --> => no 11.20/10.02 <+(?y,s(?x_5)), s(+(s(?x_5),?y))> --> => no 11.20/10.02 <+(?y,s(s(?x_4))), s(+(?x_4,?y))> --> => no 11.20/10.02 CP(S,symP): 11.20/10.02 <+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes 11.20/10.02 <+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes 11.20/10.02 <+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(?x_1,?y)> => yes 11.20/10.02 --> => yes 11.20/10.02 <+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> => yes 11.20/10.02 --> => yes 11.20/10.02 <+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> => no 11.20/10.02 --> => yes 11.20/10.02 <+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes 11.20/10.02 <+(?y,?z_1), +(?y,+(0,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes 11.20/10.02 <+(?x_1,?y), +(+(?x_1,?y),0)> --> <+(?x_1,?y), +(?x_1,?y)> => yes 11.20/10.02 --> => yes 11.20/10.02 --> => no 11.20/10.02 <+(s(+(?x,?y)),?z_1), +(?y,+(s(?x),?z_1))> --> => no 11.20/10.02 <+(?x_1,s(+(?x,?y))), +(+(?x_1,?y),s(?x))> --> => no 11.20/10.02 --> => yes 11.20/10.02 check joinability condition: 11.20/10.02 check modulo joinability of s(+(?x_5,?y)) and s(s(+(?x_5,?y))): joinable by {0} 11.20/10.02 check modulo joinability of s(s(+(?x_4,?y))) and s(+(?x_4,?y)): joinable by {0} 11.20/10.03 check modulo joinability of s(+(?x_5,?y)) and s(s(+(?x_5,?y))): joinable by {0} 11.20/10.03 check modulo joinability of s(s(+(?x_4,?y))) and s(+(?x_4,?y)): joinable by {0} 11.20/10.03 check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(+(?x,?x_1),?y)): joinable by {1} 11.20/10.03 check modulo joinability of s(+(?x,+(?x_1,?y_1))) and s(+(+(?x,?y_1),?x_1)): joinable by {1} 11.20/10.03 check modulo joinability of s(+(+(?x,?y),?z_1)) and s(+(+(?x,?z_1),?y)): joinable by {1} 11.20/10.03 check modulo joinability of s(+(+(?x,?y),?x_1)) and s(+(?x,+(?x_1,?y))): joinable by {1} 11.20/10.03 success 11.20/10.03 P': 11.20/10.03 [ s(s(?x)) -> s(?x), 11.20/10.03 +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ] 11.20/10.03 Check relative termination: 11.20/10.03 [ +(0,?y) -> ?y, 11.20/10.03 +(s(?x),?y) -> s(+(?x,?y)), 11.20/10.03 +(?y,0) -> ?y, 11.20/10.03 +(?y,s(?x)) -> s(+(?x,?y)) ] 11.20/10.03 [ s(s(?x)) -> s(?x), 11.20/10.03 +(+(?x,?y),?z) -> +(?x,+(?y,?z)) ] 11.20/10.03 Polynomial Interpretation: 11.20/10.03 +:= (1)*x1+(1)*x2 11.20/10.03 0:= (8) 11.20/10.03 s:= (1)*x1 11.20/10.03 retract +(0,?y) -> ?y 11.20/10.03 retract +(?y,0) -> ?y 11.20/10.03 Polynomial Interpretation: 11.20/10.03 +:= (1)+(2)*x1+(2)*x1*x2+(2)*x2 11.20/10.03 0:= (15) 11.20/10.03 s:= (4)+(1)*x1 11.20/10.03 relatively terminating 11.20/10.03 S/P': relatively terminating 11.20/10.03 S: 11.20/10.03 [ +(0,?y) -> ?y, 11.20/10.03 +(s(?x),?y) -> s(+(?x,?y)), 11.20/10.03 +(?y,0) -> ?y, 11.20/10.03 +(?y,s(?x)) -> s(+(?x,?y)) ] 11.20/10.03 P: 11.20/10.03 [ +(+(?x,?y),?z) -> +(?x,+(?y,?z)), 11.20/10.03 +(?x,?y) -> +(?y,?x), 11.20/10.03 s(s(?x)) -> s(?x), 11.20/10.03 s(?x) -> s(s(?x)) ] 11.20/10.03 Success 11.20/10.03 Reduction-Preserving Completion 11.20/10.03 Direct Methods: CR 11.20/10.03 11.20/10.03 Combined result: CR 11.20/10.03 /export/starexec/sandbox/benchmark/theBenchmark.trs: Success(CR) 11.20/10.03 (3256 msec.) 11.20/10.03 EOF