/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 v_NonEmpty:S x:S y:S z:S)
(RULES
div_active(0,s(y:S)) -> 0
div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
div_active(x:S,y:S) -> div(x:S,y:S)
ge_active(0,s(y:S)) -> ffalse
ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
ge_active(x:S,0) -> ttrue
ge_active(x:S,y:S) -> ge(x:S,y:S)
if_active(ffalse,x:S,y:S) -> mark(y:S)
if_active(ttrue,x:S,y:S) -> mark(x:S)
if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
mark(0) -> 0
mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
mark(s(x:S)) -> s(mark(x:S))
minus_active(0,y:S) -> 0
minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
minus_active(x:S,y:S) -> minus(x:S,y:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 DIV_ACTIVE(s(x:S),s(y:S)) -> GE_ACTIVE(x:S,y:S)
 DIV_ACTIVE(s(x:S),s(y:S)) -> IF_ACTIVE(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 GE_ACTIVE(s(x:S),s(y:S)) -> GE_ACTIVE(x:S,y:S)
 IF_ACTIVE(ffalse,x:S,y:S) -> MARK(y:S)
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> DIV_ACTIVE(mark(x:S),y:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(ge(x:S,y:S)) -> GE_ACTIVE(x:S,y:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(minus(x:S,y:S)) -> MINUS_ACTIVE(x:S,y:S)
 MARK(s(x:S)) -> MARK(x:S)
 MINUS_ACTIVE(s(x:S),s(y:S)) -> MINUS_ACTIVE(x:S,y:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)

Problem 1: 

SCC Processor:
-> Pairs:
 DIV_ACTIVE(s(x:S),s(y:S)) -> GE_ACTIVE(x:S,y:S)
 DIV_ACTIVE(s(x:S),s(y:S)) -> IF_ACTIVE(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 GE_ACTIVE(s(x:S),s(y:S)) -> GE_ACTIVE(x:S,y:S)
 IF_ACTIVE(ffalse,x:S,y:S) -> MARK(y:S)
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> DIV_ACTIVE(mark(x:S),y:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(ge(x:S,y:S)) -> GE_ACTIVE(x:S,y:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(minus(x:S,y:S)) -> MINUS_ACTIVE(x:S,y:S)
 MARK(s(x:S)) -> MARK(x:S)
 MINUS_ACTIVE(s(x:S),s(y:S)) -> MINUS_ACTIVE(x:S,y:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS_ACTIVE(s(x:S),s(y:S)) -> MINUS_ACTIVE(x:S,y:S)
->->-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->->Cycle:
->->-> Pairs:
 GE_ACTIVE(s(x:S),s(y:S)) -> GE_ACTIVE(x:S,y:S)
->->-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->->Cycle:
->->-> Pairs:
 DIV_ACTIVE(s(x:S),s(y:S)) -> IF_ACTIVE(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 IF_ACTIVE(ffalse,x:S,y:S) -> MARK(y:S)
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> DIV_ACTIVE(mark(x:S),y:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
->->-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS_ACTIVE(s(x:S),s(y:S)) -> MINUS_ACTIVE(x:S,y:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Projection:
 pi(MINUS_ACTIVE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 GE_ACTIVE(s(x:S),s(y:S)) -> GE_ACTIVE(x:S,y:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Projection:
 pi(GE_ACTIVE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 DIV_ACTIVE(s(x:S),s(y:S)) -> IF_ACTIVE(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 IF_ACTIVE(ffalse,x:S,y:S) -> MARK(y:S)
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> DIV_ACTIVE(mark(x:S),y:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
-> Usable rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div_active](X1,X2) = 2.X1 + 2.X2 + 2
[ge_active](X1,X2) = 2.X1
[if_active](X1,X2,X3) = X1 + X2 + 2.X3 + 1
[mark](X) = X
[minus_active](X1,X2) = 0
[0] = 0
[div](X1,X2) = 2.X1 + 2.X2 + 2
[false] = 0
[ge](X1,X2) = 2.X1
[if](X1,X2,X3) = X1 + X2 + 2.X3 + 1
[minus](X1,X2) = 0
[s](X) = X + 1
[true] = 0
[DIV_ACTIVE](X1,X2) = 2.X1 + 2.X2 + 2
[IF_ACTIVE](X1,X2,X3) = X1 + X2 + 2.X3
[MARK](X) = X

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_ACTIVE(ffalse,x:S,y:S) -> MARK(y:S)
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> DIV_ACTIVE(mark(x:S),y:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_ACTIVE(ffalse,x:S,y:S) -> MARK(y:S)
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
->->-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 IF_ACTIVE(ffalse,x:S,y:S) -> MARK(y:S)
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
-> Usable rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[div_active](X1,X2) = 2.X1 + 2.X2 + 2
[ge_active](X1,X2) = 1
[if_active](X1,X2,X3) = X1 + X2 + X3
[mark](X) = X
[minus_active](X1,X2) = X1
[0] = 1
[div](X1,X2) = 2.X1 + 2.X2 + 2
[false] = 1
[ge](X1,X2) = 1
[if](X1,X2,X3) = X1 + X2 + X3
[minus](X1,X2) = X1
[s](X) = X + 2
[true] = 1
[IF_ACTIVE](X1,X2,X3) = X1 + X2 + X3
[MARK](X) = X

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
->->-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)

Problem 1.3: 

Subterm Processor:
-> Pairs:
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
 MARK(div(x:S,y:S)) -> MARK(x:S)
 MARK(if(x:S,y:S,z:S)) -> IF_ACTIVE(mark(x:S),y:S,z:S)
 MARK(if(x:S,y:S,z:S)) -> MARK(x:S)
 MARK(s(x:S)) -> MARK(x:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Projection:
 pi(IF_ACTIVE) = 2
 pi(MARK) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_ACTIVE(ttrue,x:S,y:S) -> MARK(x:S)
-> Rules:
 div_active(0,s(y:S)) -> 0
 div_active(s(x:S),s(y:S)) -> if_active(ge_active(x:S,y:S),s(div(minus(x:S,y:S),s(y:S))),0)
 div_active(x:S,y:S) -> div(x:S,y:S)
 ge_active(0,s(y:S)) -> ffalse
 ge_active(s(x:S),s(y:S)) -> ge_active(x:S,y:S)
 ge_active(x:S,0) -> ttrue
 ge_active(x:S,y:S) -> ge(x:S,y:S)
 if_active(ffalse,x:S,y:S) -> mark(y:S)
 if_active(ttrue,x:S,y:S) -> mark(x:S)
 if_active(x:S,y:S,z:S) -> if(x:S,y:S,z:S)
 mark(0) -> 0
 mark(div(x:S,y:S)) -> div_active(mark(x:S),y:S)
 mark(ge(x:S,y:S)) -> ge_active(x:S,y:S)
 mark(if(x:S,y:S,z:S)) -> if_active(mark(x:S),y:S,z:S)
 mark(minus(x:S,y:S)) -> minus_active(x:S,y:S)
 mark(s(x:S)) -> s(mark(x:S))
 minus_active(0,y:S) -> 0
 minus_active(s(x:S),s(y:S)) -> minus_active(x:S,y:S)
 minus_active(x:S,y:S) -> minus(x:S,y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.