/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
div_active(0,s(y)) -> 0
div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0)
div_active(x,y) -> div(x,y)
ge_active(0,s(y)) -> false
ge_active(s(x),s(y)) -> ge_active(x,y)
ge_active(x,0) -> true
ge_active(x,y) -> ge(x,y)
if_active(false,x,y) -> mark(y)
if_active(true,x,y) -> mark(x)
if_active(x,y,z) -> if(x,y,z)
mark(0) -> 0
mark(div(x,y)) -> div_active(mark(x),y)
mark(ge(x,y)) -> ge_active(x,y)
mark(if(x,y,z)) -> if_active(mark(x),y,z)
mark(minus(x,y)) -> minus_active(x,y)
mark(s(x)) -> s(mark(x))
minus_active(0,y) -> 0
minus_active(s(x),s(y)) -> minus_active(x,y)
minus_active(x,y) -> minus(x,y)
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.3: 

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

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 IF_ACTIVE(false,x,y) -> MARK(y)
 IF_ACTIVE(true,x,y) -> MARK(x)
 MARK(div(x,y)) -> MARK(x)
 MARK(if(x,y,z)) -> IF_ACTIVE(mark(x),y,z)
 MARK(if(x,y,z)) -> MARK(x)
 MARK(s(x)) -> MARK(x)
-> Rules:
 div_active(0,s(y)) -> 0
 div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0)
 div_active(x,y) -> div(x,y)
 ge_active(0,s(y)) -> false
 ge_active(s(x),s(y)) -> ge_active(x,y)
 ge_active(x,0) -> true
 ge_active(x,y) -> ge(x,y)
 if_active(false,x,y) -> mark(y)
 if_active(true,x,y) -> mark(x)
 if_active(x,y,z) -> if(x,y,z)
 mark(0) -> 0
 mark(div(x,y)) -> div_active(mark(x),y)
 mark(ge(x,y)) -> ge_active(x,y)
 mark(if(x,y,z)) -> if_active(mark(x),y,z)
 mark(minus(x,y)) -> minus_active(x,y)
 mark(s(x)) -> s(mark(x))
 minus_active(0,y) -> 0
 minus_active(s(x),s(y)) -> minus_active(x,y)
 minus_active(x,y) -> minus(x,y)
-> Usable rules:
 div_active(0,s(y)) -> 0
 div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0)
 div_active(x,y) -> div(x,y)
 ge_active(0,s(y)) -> false
 ge_active(s(x),s(y)) -> ge_active(x,y)
 ge_active(x,0) -> true
 ge_active(x,y) -> ge(x,y)
 if_active(false,x,y) -> mark(y)
 if_active(true,x,y) -> mark(x)
 if_active(x,y,z) -> if(x,y,z)
 mark(0) -> 0
 mark(div(x,y)) -> div_active(mark(x),y)
 mark(ge(x,y)) -> ge_active(x,y)
 mark(if(x,y,z)) -> if_active(mark(x),y,z)
 mark(minus(x,y)) -> minus_active(x,y)
 mark(s(x)) -> s(mark(x))
 minus_active(0,y) -> 0
 minus_active(s(x),s(y)) -> minus_active(x,y)
 minus_active(x,y) -> minus(x,y)
->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 + 2.X3 + 1
[mark](X) = X
[minus_active](X1,X2) = 0
[0] = 0
[div](X1,X2) = 2.X1 + 2.X2 + 2
[false] = 1
[ge](X1,X2) = 1
[if](X1,X2,X3) = X1 + X2 + 2.X3 + 1
[minus](X1,X2) = 0
[s](X) = X + 2
[true] = 1
[IF_ACTIVE](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[MARK](X) = 2.X + 2

Problem 1.3: 

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

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.