YES Problem: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) +(minus(+(x,1())),1()) -> minus(x) Proof: DP Processor: DPs: +#(x,minus(y)) -> minus#(x) +#(x,minus(y)) -> +#(minus(x),y) +#(x,minus(y)) -> minus#(+(minus(x),y)) +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) +#(minus(+(x,1())),1()) -> minus#(x) TRS: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) +(minus(+(x,1())),1()) -> minus(x) TDG Processor: DPs: +#(x,minus(y)) -> minus#(x) +#(x,minus(y)) -> +#(minus(x),y) +#(x,minus(y)) -> minus#(+(minus(x),y)) +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) +#(minus(+(x,1())),1()) -> minus#(x) TRS: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) +(minus(+(x,1())),1()) -> minus(x) graph: +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(minus(+(x,1())),1()) -> minus#(x) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,minus(y)) -> minus#(+(minus(x),y)) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,minus(y)) -> +#(minus(x),y) +#(x,+(y,z)) -> +#(+(x,y),z) -> +#(x,minus(y)) -> minus#(x) +#(x,+(y,z)) -> +#(x,y) -> +#(minus(+(x,1())),1()) -> minus#(x) +#(x,+(y,z)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,+(y,z)) -> +#(x,y) -> +#(x,+(y,z)) -> +#(x,y) +#(x,+(y,z)) -> +#(x,y) -> +#(x,minus(y)) -> minus#(+(minus(x),y)) +#(x,+(y,z)) -> +#(x,y) -> +#(x,minus(y)) -> +#(minus(x),y) +#(x,+(y,z)) -> +#(x,y) -> +#(x,minus(y)) -> minus#(x) +#(x,minus(y)) -> +#(minus(x),y) -> +#(minus(+(x,1())),1()) -> minus#(x) +#(x,minus(y)) -> +#(minus(x),y) -> +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,minus(y)) -> +#(minus(x),y) -> +#(x,+(y,z)) -> +#(x,y) +#(x,minus(y)) -> +#(minus(x),y) -> +#(x,minus(y)) -> minus#(+(minus(x),y)) +#(x,minus(y)) -> +#(minus(x),y) -> +#(x,minus(y)) -> +#(minus(x),y) +#(x,minus(y)) -> +#(minus(x),y) -> +#(x,minus(y)) -> minus#(x) SCC Processor: #sccs: 1 #rules: 3 #arcs: 18/36 DPs: +#(x,+(y,z)) -> +#(+(x,y),z) +#(x,minus(y)) -> +#(minus(x),y) +#(x,+(y,z)) -> +#(x,y) TRS: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) +(minus(+(x,1())),1()) -> minus(x) Subterm Criterion Processor: simple projection: pi(+#) = 1 problem: DPs: TRS: minus(0()) -> 0() +(x,0()) -> x +(0(),y) -> y +(minus(1()),1()) -> 0() minus(minus(x)) -> x +(x,minus(y)) -> minus(+(minus(x),y)) +(x,+(y,z)) -> +(+(x,y),z) +(minus(+(x,1())),1()) -> minus(x) Qed