/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
+(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
+(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
+(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
+(#,x:S) -> x:S
+(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
+(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
+(x:S,#) -> x:S
-(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
-(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
-(#,x:S) -> #
-(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
-(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
-(x:S,#) -> x:S
0(#) -> #
ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
ge(#,0(x:S)) -> ge(#,x:S)
ge(#,1(x:S)) -> ffalse
ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
ge(x:S,#) -> ttrue
if(ffalse,x:S,y:S) -> y:S
if(ttrue,x:S,y:S) -> x:S
log(x:S) -> -(log'(x:S),1(#))
log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
log'(#) -> #
log'(1(x:S)) -> +(log'(x:S),1(#))
not(ffalse) -> ttrue
not(ttrue) -> ffalse
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(0(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(0(x:S),0(y:S)) -> 0#(+(x:S,y:S))
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> 0#(+(+(x:S,y:S),1(#)))
 -#(0(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(0(x:S),0(y:S)) -> 0#(-(x:S,y:S))
 -#(0(x:S),1(y:S)) -> -#(-(x:S,y:S),1(#))
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> 0#(-(x:S,y:S))
 GE(0(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(0(x:S),1(y:S)) -> GE(y:S,x:S)
 GE(0(x:S),1(y:S)) -> NOT(ge(y:S,x:S))
 GE(#,0(x:S)) -> GE(#,x:S)
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
 LOG(x:S) -> -#(log'(x:S),1(#))
 LOG(x:S) -> LOG'(x:S)
 LOG'(0(x:S)) -> +#(log'(x:S),1(#))
 LOG'(0(x:S)) -> GE(x:S,1(#))
 LOG'(0(x:S)) -> IF(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 LOG'(0(x:S)) -> LOG'(x:S)
 LOG'(1(x:S)) -> +#(log'(x:S),1(#))
 LOG'(1(x:S)) -> LOG'(x:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1: 

SCC Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(0(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(0(x:S),0(y:S)) -> 0#(+(x:S,y:S))
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> 0#(+(+(x:S,y:S),1(#)))
 -#(0(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(0(x:S),0(y:S)) -> 0#(-(x:S,y:S))
 -#(0(x:S),1(y:S)) -> -#(-(x:S,y:S),1(#))
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> 0#(-(x:S,y:S))
 GE(0(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(0(x:S),1(y:S)) -> GE(y:S,x:S)
 GE(0(x:S),1(y:S)) -> NOT(ge(y:S,x:S))
 GE(#,0(x:S)) -> GE(#,x:S)
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
 LOG(x:S) -> -#(log'(x:S),1(#))
 LOG(x:S) -> LOG'(x:S)
 LOG'(0(x:S)) -> +#(log'(x:S),1(#))
 LOG'(0(x:S)) -> GE(x:S,1(#))
 LOG'(0(x:S)) -> IF(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 LOG'(0(x:S)) -> LOG'(x:S)
 LOG'(1(x:S)) -> +#(log'(x:S),1(#))
 LOG'(1(x:S)) -> LOG'(x:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GE(#,0(x:S)) -> GE(#,x:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->->Cycle:
->->-> Pairs:
 GE(0(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(0(x:S),1(y:S)) -> GE(y:S,x:S)
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->->Cycle:
->->-> Pairs:
 -#(0(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(0(x:S),1(y:S)) -> -#(-(x:S,y:S),1(#))
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->->Cycle:
->->-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(0(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->->Cycle:
->->-> Pairs:
 LOG'(0(x:S)) -> LOG'(x:S)
 LOG'(1(x:S)) -> LOG'(x:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 GE(#,0(x:S)) -> GE(#,x:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Projection:
 pi(GE) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 GE(0(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(0(x:S),1(y:S)) -> GE(y:S,x:S)
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[1](X) = X
[GE](X1,X2) = X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 GE(0(x:S),1(y:S)) -> GE(y:S,x:S)
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GE(0(x:S),1(y:S)) -> GE(y:S,x:S)
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 GE(0(x:S),1(y:S)) -> GE(y:S,x:S)
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[1](X) = 2.X
[GE](X1,X2) = 2.X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.2: 

Subterm Processor:
-> Pairs:
 GE(1(x:S),0(y:S)) -> GE(x:S,y:S)
 GE(1(x:S),1(y:S)) -> GE(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Projection:
 pi(GE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 -#(0(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(0(x:S),1(y:S)) -> -#(-(x:S,y:S),1(#))
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = X1
[0](X) = 2.X + 2
[#] = 1
[1](X) = 2.X + 2
[-#](X1,X2) = 2.X1 + 2.X2

Problem 1.3: 

SCC Processor:
-> Pairs:
 -#(0(x:S),1(y:S)) -> -#(-(x:S,y:S),1(#))
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(0(x:S),1(y:S)) -> -#(-(x:S,y:S),1(#))
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 -#(0(x:S),1(y:S)) -> -#(-(x:S,y:S),1(#))
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = X1 + X2 + 1
[0](X) = X + 2
[#] = 0
[1](X) = X + 1
[-#](X1,X2) = 2.X1 + 2.X2

Problem 1.3: 

SCC Processor:
-> Pairs:
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.3: 

Subterm Processor:
-> Pairs:
 -#(0(x:S),1(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),0(y:S)) -> -#(x:S,y:S)
 -#(1(x:S),1(y:S)) -> -#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Projection:
 pi(-#) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(0(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 1
[0](X) = X
[#] = 0
[1](X) = X + 1
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.4: 

SCC Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(0(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(0(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(0(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 1
[0](X) = X + 1
[#] = 0
[1](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.4: 

SCC Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(0(x:S),1(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 2
[0](X) = X
[#] = 0
[1](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.4: 

SCC Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),0(y:S)) -> +#(x:S,y:S)
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 2
[0](X) = X
[#] = 0
[1](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.4: 

SCC Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),1(y:S)) -> +#(+(x:S,y:S),1(#))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
-> Usable rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[0](X) = X + 1
[#] = 1
[1](X) = X + 2
[+#](X1,X2) = X1 + X2

Problem 1.4: 

SCC Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse

Problem 1.4: 

Subterm Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(1(x:S),1(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Projection:
 pi(+#) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 LOG'(0(x:S)) -> LOG'(x:S)
 LOG'(1(x:S)) -> LOG'(x:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Projection:
 pi(LOG') = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(0(x:S),0(y:S)) -> 0(+(x:S,y:S))
 +(0(x:S),1(y:S)) -> 1(+(x:S,y:S))
 +(#,x:S) -> x:S
 +(1(x:S),0(y:S)) -> 1(+(x:S,y:S))
 +(1(x:S),1(y:S)) -> 0(+(+(x:S,y:S),1(#)))
 +(x:S,#) -> x:S
 -(0(x:S),0(y:S)) -> 0(-(x:S,y:S))
 -(0(x:S),1(y:S)) -> 1(-(-(x:S,y:S),1(#)))
 -(#,x:S) -> #
 -(1(x:S),0(y:S)) -> 1(-(x:S,y:S))
 -(1(x:S),1(y:S)) -> 0(-(x:S,y:S))
 -(x:S,#) -> x:S
 0(#) -> #
 ge(0(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(0(x:S),1(y:S)) -> not(ge(y:S,x:S))
 ge(#,0(x:S)) -> ge(#,x:S)
 ge(#,1(x:S)) -> ffalse
 ge(1(x:S),0(y:S)) -> ge(x:S,y:S)
 ge(1(x:S),1(y:S)) -> ge(x:S,y:S)
 ge(x:S,#) -> ttrue
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 log(x:S) -> -(log'(x:S),1(#))
 log'(0(x:S)) -> if(ge(x:S,1(#)),+(log'(x:S),1(#)),#)
 log'(#) -> #
 log'(1(x:S)) -> +(log'(x:S),1(#))
 not(ffalse) -> ttrue
 not(ttrue) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.