/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
half(0) -> 0
half(s(0)) -> 0
half(s(s(x:S))) -> s(half(x:S))
if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
if(ttrue,x:S,s(y:S)) -> y:S
inc(0) -> 0
inc(s(x:S)) -> s(inc(x:S))
le(0,y:S) -> ttrue
le(s(x:S),0) -> ffalse
le(s(x:S),s(y:S)) -> le(x:S,y:S)
log(x:S) -> log2(x:S,0)
log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
 IF(ffalse,x:S,y:S) -> HALF(x:S)
 IF(ffalse,x:S,y:S) -> LOG2(half(x:S),y:S)
 INC(s(x:S)) -> INC(x:S)
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 LOG(x:S) -> LOG2(x:S,0)
 LOG2(x:S,y:S) -> IF(le(x:S,s(0)),x:S,inc(y:S))
 LOG2(x:S,y:S) -> INC(y:S)
 LOG2(x:S,y:S) -> LE(x:S,s(0))
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))

Problem 1: 

SCC Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
 IF(ffalse,x:S,y:S) -> HALF(x:S)
 IF(ffalse,x:S,y:S) -> LOG2(half(x:S),y:S)
 INC(s(x:S)) -> INC(x:S)
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 LOG(x:S) -> LOG2(x:S,0)
 LOG2(x:S,y:S) -> IF(le(x:S,s(0)),x:S,inc(y:S))
 LOG2(x:S,y:S) -> INC(y:S)
 LOG2(x:S,y:S) -> LE(x:S,s(0))
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
->->-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->->Cycle:
->->-> Pairs:
 INC(s(x:S)) -> INC(x:S)
->->-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->->Cycle:
->->-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
->->-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->->Cycle:
->->-> Pairs:
 IF(ffalse,x:S,y:S) -> LOG2(half(x:S),y:S)
 LOG2(x:S,y:S) -> IF(le(x:S,s(0)),x:S,inc(y:S))
->->-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Projection:
 pi(LE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 INC(s(x:S)) -> INC(x:S)
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Projection:
 pi(INC) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Projection:
 pi(HALF) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 IF(ffalse,x:S,y:S) -> LOG2(half(x:S),y:S)
 LOG2(x:S,y:S) -> IF(le(x:S,s(0)),x:S,inc(y:S))
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
-> Usable rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[half](X) = 1/2.X
[if](X1,X2,X3) = 0
[inc](X) = 2.X + 1
[le](X1,X2) = 1/2.X1
[log](X) = 0
[log2](X1,X2) = 0
[0] = 0
[fSNonEmpty] = 0
[false] = 1/2
[s](X) = 2.X + 2
[true] = 0
[HALF](X) = 0
[IF](X1,X2,X3) = 2.X1 + X2
[INC](X) = 0
[LE](X1,X2) = 0
[LOG](X) = 0
[LOG2](X1,X2) = 2.X1 + 1/2

Problem 1.4: 

SCC Processor:
-> Pairs:
 LOG2(x:S,y:S) -> IF(le(x:S,s(0)),x:S,inc(y:S))
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> log2(half(x:S),y:S)
 if(ttrue,x:S,s(y:S)) -> y:S
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 log(x:S) -> log2(x:S,0)
 log2(x:S,y:S) -> if(le(x:S,s(0)),x:S,inc(y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.