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


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YES

Problem 1: 

(VAR x y z)
(RULES
+(+(x,y),z) -> +(x,+(y,z))
+(0(x),0(y)) -> 0(+(x,y))
+(0(x),1(y)) -> 1(+(x,y))
+(#,x) -> x
+(1(x),0(y)) -> 1(+(x,y))
+(1(x),1(y)) -> 0(+(+(x,y),1(#)))
+(x,#) -> x
-(0(x),0(y)) -> 0(-(x,y))
-(0(x),1(y)) -> 1(-(-(x,y),1(#)))
-(#,x) -> #
-(1(x),0(y)) -> 1(-(x,y))
-(1(x),1(y)) -> 0(-(x,y))
-(x,#) -> x
0(#) -> #
ge(0(x),0(y)) -> ge(x,y)
ge(0(x),1(y)) -> not(ge(y,x))
ge(#,0(x)) -> ge(#,x)
ge(#,1(x)) -> false
ge(1(x),0(y)) -> ge(x,y)
ge(1(x),1(y)) -> ge(x,y)
ge(x,#) -> true
if(false,x,y) -> y
if(true,x,y) -> x
log(x) -> -(log'(x),1(#))
log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
log'(#) -> #
log'(1(x)) -> +(log'(x),1(#))
not(false) -> true
not(true) -> false
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 5 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 -#(0(x),1(y)) -> -#(-(x,y),1(#))
 -#(0(x),1(y)) -> -#(x,y)
 -#(1(x),0(y)) -> -#(x,y)
 -#(1(x),1(y)) -> -#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
-> Usable rules:
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 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),1(y)) -> -#(x,y)
 -#(1(x),0(y)) -> -#(x,y)
 -#(1(x),1(y)) -> -#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(0(x),1(y)) -> -#(x,y)
 -#(1(x),0(y)) -> -#(x,y)
 -#(1(x),1(y)) -> -#(x,y)
->->-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(+(x,y),z) -> +#(y,z)
 +#(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(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 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,y),z) -> +#(x,+(y,z))
 +#(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(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(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(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(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(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 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,y),z) -> +#(x,+(y,z))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 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,y),z) -> +#(x,+(y,z))
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#)))
 -(#,x) -> #
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 -(x,#) -> x
 0(#) -> #
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(#,0(x)) -> ge(#,x)
 ge(#,1(x)) -> false
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#) -> true
 if(false,x,y) -> y
 if(true,x,y) -> x
 log(x) -> -(log'(x),1(#))
 log'(0(x)) -> if(ge(x,1(#)),+(log'(x),1(#)),#)
 log'(#) -> #
 log'(1(x)) -> +(log'(x),1(#))
 not(false) -> true
 not(true) -> false

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.4: 

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

The problem is finite.

Problem 1.5: 

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

Problem 1.5: 

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

The problem is finite.