/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Proof: DP Processor: DPs: +#(0(x),0(y)) -> +#(x,y) +#(0(x),0(y)) -> 0#(+(x,y)) +#(0(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) *#(0(x),y) -> *#(x,y) *#(0(x),y) -> 0#(*(x,y)) *#(1(x),y) -> *#(x,y) *#(1(x),y) -> 0#(*(x,y)) *#(1(x),y) -> +#(0(*(x,y)),y) sum#(nil()) -> 0#(#()) sum#(cons(x,l)) -> sum#(l) sum#(cons(x,l)) -> +#(x,sum(l)) prod#(cons(x,l)) -> prod#(l) prod#(cons(x,l)) -> *#(x,prod(l)) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) TDG Processor: DPs: +#(0(x),0(y)) -> +#(x,y) +#(0(x),0(y)) -> 0#(+(x,y)) +#(0(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) *#(0(x),y) -> *#(x,y) *#(0(x),y) -> 0#(*(x,y)) *#(1(x),y) -> *#(x,y) *#(1(x),y) -> 0#(*(x,y)) *#(1(x),y) -> +#(0(*(x,y)),y) sum#(nil()) -> 0#(#()) sum#(cons(x,l)) -> sum#(l) sum#(cons(x,l)) -> +#(x,sum(l)) prod#(cons(x,l)) -> prod#(l) prod#(cons(x,l)) -> *#(x,prod(l)) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) graph: prod#(cons(x,l)) -> prod#(l) -> prod#(cons(x,l)) -> *#(x,prod(l)) prod#(cons(x,l)) -> prod#(l) -> prod#(cons(x,l)) -> prod#(l) prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(1(x),y) -> +#(0(*(x,y)),y) prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(1(x),y) -> 0#(*(x,y)) prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(1(x),y) -> *#(x,y) prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(0(x),y) -> 0#(*(x,y)) prod#(cons(x,l)) -> *#(x,prod(l)) -> *#(0(x),y) -> *#(x,y) sum#(cons(x,l)) -> sum#(l) -> sum#(cons(x,l)) -> +#(x,sum(l)) sum#(cons(x,l)) -> sum#(l) -> sum#(cons(x,l)) -> sum#(l) sum#(cons(x,l)) -> sum#(l) -> sum#(nil()) -> 0#(#()) sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(1(x),1(y)) -> +#(x,y) sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(1(x),0(y)) -> +#(x,y) sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(0(x),1(y)) -> +#(x,y) sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(0(x),0(y)) -> 0#(+(x,y)) sum#(cons(x,l)) -> +#(x,sum(l)) -> +#(0(x),0(y)) -> +#(x,y) *#(1(x),y) -> *#(x,y) -> *#(1(x),y) -> +#(0(*(x,y)),y) *#(1(x),y) -> *#(x,y) -> *#(1(x),y) -> 0#(*(x,y)) *#(1(x),y) -> *#(x,y) -> *#(1(x),y) -> *#(x,y) *#(1(x),y) -> *#(x,y) -> *#(0(x),y) -> 0#(*(x,y)) *#(1(x),y) -> *#(x,y) -> *#(0(x),y) -> *#(x,y) *#(1(x),y) -> +#(0(*(x,y)),y) -> +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) *#(1(x),y) -> +#(0(*(x,y)),y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) *#(1(x),y) -> +#(0(*(x,y)),y) -> +#(1(x),1(y)) -> +#(x,y) *#(1(x),y) -> +#(0(*(x,y)),y) -> +#(1(x),0(y)) -> +#(x,y) *#(1(x),y) -> +#(0(*(x,y)),y) -> +#(0(x),1(y)) -> +#(x,y) *#(1(x),y) -> +#(0(*(x,y)),y) -> +#(0(x),0(y)) -> 0#(+(x,y)) *#(1(x),y) -> +#(0(*(x,y)),y) -> +#(0(x),0(y)) -> +#(x,y) *#(0(x),y) -> *#(x,y) -> *#(1(x),y) -> +#(0(*(x,y)),y) *#(0(x),y) -> *#(x,y) -> *#(1(x),y) -> 0#(*(x,y)) *#(0(x),y) -> *#(x,y) -> *#(1(x),y) -> *#(x,y) *#(0(x),y) -> *#(x,y) -> *#(0(x),y) -> 0#(*(x,y)) *#(0(x),y) -> *#(x,y) -> *#(0(x),y) -> *#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(0(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(0(x),0(y)) -> 0#(+(x,y)) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(0(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) +#(1(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(1(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(0(x),0(y)) -> 0#(+(x,y)) +#(1(x),1(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) +#(1(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(1(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(0(x),0(y)) -> 0#(+(x,y)) +#(1(x),0(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) +#(0(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(0(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(0(x),0(y)) -> 0#(+(x,y)) +#(0(x),1(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> 0#(+(+(x,y),1(#()))) +#(0(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(0(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(0(x),0(y)) -> 0#(+(x,y)) +#(0(x),0(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) SCC Processor: #sccs: 4 #rules: 9 #arcs: 69/289 DPs: sum#(cons(x,l)) -> sum#(l) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Subterm Criterion Processor: simple projection: pi(sum#) = 0 problem: DPs: TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Qed DPs: prod#(cons(x,l)) -> prod#(l) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Subterm Criterion Processor: simple projection: pi(prod#) = 0 problem: DPs: TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Qed DPs: *#(0(x),y) -> *#(x,y) *#(1(x),y) -> *#(x,y) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Subterm Criterion Processor: simple projection: pi(*#) = 0 problem: DPs: TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Qed DPs: +#(0(x),0(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) EDG Processor: DPs: +#(0(x),0(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) graph: +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(0(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(1(x),1(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(1(x),0(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(0(x),1(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) +#(0(x),0(y)) -> +#(x,y) -> +#(0(x),0(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(0(x),1(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(1(x),0(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(x,y) +#(0(x),0(y)) -> +#(x,y) -> +#(1(x),1(y)) -> +#(+(x,y),1(#())) Usable Rule Processor: DPs: +#(0(x),0(y)) -> +#(x,y) +#(0(x),1(y)) -> +#(x,y) +#(1(x),0(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(x,y) +#(1(x),1(y)) -> +#(+(x,y),1(#())) TRS: +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() Semantic Labeling Processor: dimension: 1 usable rules: interpretation: [1](x0) = x0 + 1, [+](x0, x1) = x0 + x1, [0](x0) = x0, [#] = 0 labeled: +# 1 + usable (for model): +# 0 1 + # argument filtering: pi(#) = [] pi(0) = 0 pi(+) = 1 pi(1) = 0 pi(+#) = [] precedence: +# ~ 1 ~ + ~ 0 ~ # problem: DPs: +#(0(x),0(y)) -> +#(x,y) TRS: +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() Restore Modifier: DPs: +#(0(x),0(y)) -> +#(x,y) TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) Size-Change Termination Processor: DPs: TRS: 0(#()) -> #() +(x,#()) -> x +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) *(#(),x) -> #() *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) sum(nil()) -> 0(#()) sum(cons(x,l)) -> +(x,sum(l)) prod(nil()) -> 1(#()) prod(cons(x,l)) -> *(x,prod(l)) The DP: +#(0(x),0(y)) -> +#(x,y) has the edges: 0 > 0 1 > 1 Qed