/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
+(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,+(y,z)) -> +(+(x,y),z)
+(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(#) -> #
and(x,false) -> false
and(x,true) -> x
bs(l(x)) -> true
bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
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
max(l(x)) -> x
max(n(x,y,z)) -> max(z)
min(l(x)) -> x
min(n(x,y,z)) -> min(y)
not(false) -> true
not(true) -> false
size(l(x)) -> 1(#)
size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
val(l(x)) -> x
val(n(x,y,z)) -> x
wb(l(x)) -> true
wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(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(#)))
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
 -#(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))
 BS(n(x,y,z)) -> AND(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 BS(n(x,y,z)) -> AND(bs(y),bs(z))
 BS(n(x,y,z)) -> AND(ge(x,max(y)),ge(min(z),x))
 BS(n(x,y,z)) -> BS(y)
 BS(n(x,y,z)) -> BS(z)
 BS(n(x,y,z)) -> GE(min(z),x)
 BS(n(x,y,z)) -> GE(x,max(y))
 BS(n(x,y,z)) -> MAX(y)
 BS(n(x,y,z)) -> MIN(z)
 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)
 MAX(n(x,y,z)) -> MAX(z)
 MIN(n(x,y,z)) -> MIN(y)
 SIZE(n(x,y,z)) -> +#(+(size(x),size(y)),1(#))
 SIZE(n(x,y,z)) -> +#(size(x),size(y))
 SIZE(n(x,y,z)) -> SIZE(x)
 SIZE(n(x,y,z)) -> SIZE(y)
 WB(n(x,y,z)) -> -#(size(y),size(z))
 WB(n(x,y,z)) -> -#(size(z),size(y))
 WB(n(x,y,z)) -> AND(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
 WB(n(x,y,z)) -> AND(wb(y),wb(z))
 WB(n(x,y,z)) -> GE(size(y),size(z))
 WB(n(x,y,z)) -> GE(1(#),-(size(y),size(z)))
 WB(n(x,y,z)) -> GE(1(#),-(size(z),size(y)))
 WB(n(x,y,z)) -> IF(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y))))
 WB(n(x,y,z)) -> SIZE(y)
 WB(n(x,y,z)) -> SIZE(z)
 WB(n(x,y,z)) -> WB(y)
 WB(n(x,y,z)) -> WB(z)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1: 

SCC Processor:
-> Pairs:
 +#(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(#)))
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
 -#(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))
 BS(n(x,y,z)) -> AND(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 BS(n(x,y,z)) -> AND(bs(y),bs(z))
 BS(n(x,y,z)) -> AND(ge(x,max(y)),ge(min(z),x))
 BS(n(x,y,z)) -> BS(y)
 BS(n(x,y,z)) -> BS(z)
 BS(n(x,y,z)) -> GE(min(z),x)
 BS(n(x,y,z)) -> GE(x,max(y))
 BS(n(x,y,z)) -> MAX(y)
 BS(n(x,y,z)) -> MIN(z)
 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)
 MAX(n(x,y,z)) -> MAX(z)
 MIN(n(x,y,z)) -> MIN(y)
 SIZE(n(x,y,z)) -> +#(+(size(x),size(y)),1(#))
 SIZE(n(x,y,z)) -> +#(size(x),size(y))
 SIZE(n(x,y,z)) -> SIZE(x)
 SIZE(n(x,y,z)) -> SIZE(y)
 WB(n(x,y,z)) -> -#(size(y),size(z))
 WB(n(x,y,z)) -> -#(size(z),size(y))
 WB(n(x,y,z)) -> AND(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
 WB(n(x,y,z)) -> AND(wb(y),wb(z))
 WB(n(x,y,z)) -> GE(size(y),size(z))
 WB(n(x,y,z)) -> GE(1(#),-(size(y),size(z)))
 WB(n(x,y,z)) -> GE(1(#),-(size(z),size(y)))
 WB(n(x,y,z)) -> IF(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y))))
 WB(n(x,y,z)) -> SIZE(y)
 WB(n(x,y,z)) -> SIZE(z)
 WB(n(x,y,z)) -> WB(y)
 WB(n(x,y,z)) -> WB(z)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MIN(n(x,y,z)) -> MIN(y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->Cycle:
->->-> Pairs:
 MAX(n(x,y,z)) -> MAX(z)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->Cycle:
->->-> Pairs:
 GE(#,0(x)) -> GE(#,x)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->Cycle:
->->-> Pairs:
 BS(n(x,y,z)) -> BS(y)
 BS(n(x,y,z)) -> BS(z)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->Cycle:
->->-> Pairs:
 +#(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)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->Cycle:
->->-> Pairs:
 SIZE(n(x,y,z)) -> SIZE(x)
 SIZE(n(x,y,z)) -> SIZE(y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->->Cycle:
->->-> Pairs:
 WB(n(x,y,z)) -> WB(y)
 WB(n(x,y,z)) -> WB(z)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))


The problem is decomposed in 9 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MIN(n(x,y,z)) -> MIN(y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(MIN) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 MAX(n(x,y,z)) -> MAX(z)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(MAX) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 GE(#,0(x)) -> GE(#,x)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(GE) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> 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 1.4: 

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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.4: 

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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> 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 1.4: 

SCC Processor:
-> Pairs:
 GE(1(x),0(y)) -> GE(x,y)
 GE(1(x),1(y)) -> GE(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GE(1(x),0(y)) -> GE(x,y)
 GE(1(x),1(y)) -> GE(x,y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.4: 

Subterm Processor:
-> Pairs:
 GE(1(x),0(y)) -> GE(x,y)
 GE(1(x),1(y)) -> GE(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(GE) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 BS(n(x,y,z)) -> BS(y)
 BS(n(x,y,z)) -> BS(z)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(BS) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> 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 1.6: 

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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.6: 

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:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> 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 1.6: 

SCC Processor:
-> Pairs:
 -#(0(x),1(y)) -> -#(x,y)
 -#(1(x),0(y)) -> -#(x,y)
 -#(1(x),1(y)) -> -#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(0(x),1(y)) -> -#(x,y)
 -#(1(x),0(y)) -> -#(x,y)
 -#(1(x),1(y)) -> -#(x,y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.6: 

Subterm Processor:
-> Pairs:
 -#(0(x),1(y)) -> -#(x,y)
 -#(1(x),0(y)) -> -#(x,y)
 -#(1(x),1(y)) -> -#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(-#) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Reduction Pair Processor:
-> Pairs:
 +#(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)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> Usable rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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) = X1 + 2.X2

Problem 1.7: 

SCC Processor:
-> Pairs:
 +#(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)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(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)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.7: 

Reduction Pair Processor:
-> Pairs:
 +#(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)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> Usable rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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 1.7: 

SCC Processor:
-> Pairs:
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.7: 

Reduction Pair Processor:
-> Pairs:
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> Usable rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(x,#) -> x
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + 2.X2
[0](X) = X + 1
[#] = 0
[1](X) = X + 1
[+#](X1,X2) = X1 + 2.X2

Problem 1.7: 

SCC Processor:
-> Pairs:
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.7: 

Reduction Pair Processor:
-> Pairs:
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
-> Usable rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(x,#) -> x
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[0](X) = 2.X
[#] = 0
[1](X) = 2.X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.7: 

SCC Processor:
-> Pairs:
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
->->-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))

Problem 1.7: 

Subterm Processor:
-> Pairs:
 +#(1(x),1(y)) -> +#(x,y)
 +#(x,+(y,z)) -> +#(+(x,y),z)
 +#(x,+(y,z)) -> +#(x,y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(+#) = 2

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.8: 

Subterm Processor:
-> Pairs:
 SIZE(n(x,y,z)) -> SIZE(x)
 SIZE(n(x,y,z)) -> SIZE(y)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(SIZE) = 1

Problem 1.8: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.9: 

Subterm Processor:
-> Pairs:
 WB(n(x,y,z)) -> WB(y)
 WB(n(x,y,z)) -> WB(z)
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Projection:
 pi(WB) = 1

Problem 1.9: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(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,+(y,z)) -> +(+(x,y),z)
 +(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(#) -> #
 and(x,false) -> false
 and(x,true) -> x
 bs(l(x)) -> true
 bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z)))
 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
 max(l(x)) -> x
 max(n(x,y,z)) -> max(z)
 min(l(x)) -> x
 min(n(x,y,z)) -> min(y)
 not(false) -> true
 not(true) -> false
 size(l(x)) -> 1(#)
 size(n(x,y,z)) -> +(+(size(x),size(y)),1(#))
 val(l(x)) -> x
 val(n(x,y,z)) -> x
 wb(l(x)) -> true
 wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#),-(size(y),size(z))),ge(1(#),-(size(z),size(y)))),and(wb(y),wb(z)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.