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


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


YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
*(x:S,0) -> 0
*(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
+(x:S,0) -> x:S
+(x:S,s(y:S)) -> s(+(x:S,y:S))
-(s(x:S),s(y:S)) -> -(x:S,y:S)
-(x:S,0) -> x:S
fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
ge(0,s(y:S)) -> ffalse
ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
ge(x:S,0) -> ttrue
iffact(x:S,ffalse) -> s(0)
iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 *#(x:S,s(y:S)) -> *#(x:S,y:S)
 *#(x:S,s(y:S)) -> +#(*(x:S,y:S),x:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
 FACT(x:S) -> GE(x:S,s(s(0)))
 FACT(x:S) -> IFFACT(x:S,ge(x:S,s(s(0))))
 GE(s(x:S),s(y:S)) -> GE(x:S,y:S)
 IFFACT(x:S,ttrue) -> *#(x:S,fact(-(x:S,s(0))))
 IFFACT(x:S,ttrue) -> -#(x:S,s(0))
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))

Problem 1: 

SCC Processor:
-> Pairs:
 *#(x:S,s(y:S)) -> *#(x:S,y:S)
 *#(x:S,s(y:S)) -> +#(*(x:S,y:S),x:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
 FACT(x:S) -> GE(x:S,s(s(0)))
 FACT(x:S) -> IFFACT(x:S,ge(x:S,s(s(0))))
 GE(s(x:S),s(y:S)) -> GE(x:S,y:S)
 IFFACT(x:S,ttrue) -> *#(x:S,fact(-(x:S,s(0))))
 IFFACT(x:S,ttrue) -> -#(x:S,s(0))
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GE(s(x:S),s(y:S)) -> GE(x:S,y:S)
->->-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->->Cycle:
->->-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
->->-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->->Cycle:
->->-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->->Cycle:
->->-> Pairs:
 *#(x:S,s(y:S)) -> *#(x:S,y:S)
->->-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->->Cycle:
->->-> Pairs:
 FACT(x:S) -> IFFACT(x:S,ge(x:S,s(s(0))))
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
->->-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 GE(s(x:S),s(y:S)) -> GE(x:S,y:S)
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Projection:
 pi(GE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Projection:
 pi(-#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Projection:
 pi(+#) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 *#(x:S,s(y:S)) -> *#(x:S,y:S)
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Projection:
 pi(*#) = 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Narrowing Processor:
-> Pairs:
 FACT(x:S) -> IFFACT(x:S,ge(x:S,s(s(0))))
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Narrowed Pairs:
->->Original Pair:
 FACT(x:S) -> IFFACT(x:S,ge(x:S,s(s(0))))
->-> Narrowed pairs:
 FACT(0) -> IFFACT(0,ffalse)
 FACT(s(x:S)) -> IFFACT(s(x:S),ge(x:S,s(0)))

Problem 1.5: 

SCC Processor:
-> Pairs:
 FACT(0) -> IFFACT(0,ffalse)
 FACT(s(x:S)) -> IFFACT(s(x:S),ge(x:S,s(0)))
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FACT(s(x:S)) -> IFFACT(s(x:S),ge(x:S,s(0)))
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
->->-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))

Problem 1.5: 

Narrowing Processor:
-> Pairs:
 FACT(s(x:S)) -> IFFACT(s(x:S),ge(x:S,s(0)))
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Narrowed Pairs:
->->Original Pair:
 FACT(s(x:S)) -> IFFACT(s(x:S),ge(x:S,s(0)))
->-> Narrowed pairs:
 FACT(s(0)) -> IFFACT(s(0),ffalse)
 FACT(s(s(x:S))) -> IFFACT(s(s(x:S)),ge(x:S,0))
->->Original Pair:
 IFFACT(x:S,ttrue) -> FACT(-(x:S,s(0)))
->-> Narrowed pairs:
 IFFACT(s(x:S),ttrue) -> FACT(-(x:S,0))

Problem 1.5: 

SCC Processor:
-> Pairs:
 FACT(s(0)) -> IFFACT(s(0),ffalse)
 FACT(s(s(x:S))) -> IFFACT(s(s(x:S)),ge(x:S,0))
 IFFACT(s(x:S),ttrue) -> FACT(-(x:S,0))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FACT(s(s(x:S))) -> IFFACT(s(s(x:S)),ge(x:S,0))
 IFFACT(s(x:S),ttrue) -> FACT(-(x:S,0))
->->-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))

Problem 1.5: 

Reduction Pairs Processor:
-> Pairs:
 FACT(s(s(x:S))) -> IFFACT(s(s(x:S)),ge(x:S,0))
 IFFACT(s(x:S),ttrue) -> FACT(-(x:S,0))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
-> Usable rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = 0
[-](X1,X2) = 2.X1 + 1
[fact](X) = 0
[ge](X1,X2) = 2.X1 + X2
[iffact](X1,X2) = 0
[0] = 2
[fSNonEmpty] = 0
[false] = 2
[s](X) = 2.X + 2
[true] = 0
[*#](X1,X2) = 0
[+#](X1,X2) = 0
[-#](X1,X2) = 0
[FACT](X) = X + 2
[GE](X1,X2) = 0
[IFFACT](X1,X2) = X1 + 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 IFFACT(s(x:S),ttrue) -> FACT(-(x:S,0))
-> Rules:
 *(x:S,0) -> 0
 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 fact(x:S) -> iffact(x:S,ge(x:S,s(s(0))))
 ge(0,s(y:S)) -> ffalse
 ge(s(x:S),s(y:S)) -> ge(x:S,y:S)
 ge(x:S,0) -> ttrue
 iffact(x:S,ffalse) -> s(0)
 iffact(x:S,ttrue) -> *(x:S,fact(-(x:S,s(0))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.