/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)
(RULES
bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
bits(x:S) -> bitIter(x:S,0)
half(0) -> 0
half(s(0)) -> 0
half(s(s(x:S))) -> s(half(x:S))
if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
if(ttrue,x:S,y:S) -> p(y:S)
inc(0) -> 0
inc(s(x:S)) -> s(inc(x:S))
p(0) -> 0
p(s(x:S)) -> x:S
zero(0) -> ttrue
zero(s(x:S)) -> ffalse
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 BITITER(x:S,y:S) -> IF(zero(x:S),x:S,inc(y:S))
 BITITER(x:S,y:S) -> INC(y:S)
 BITITER(x:S,y:S) -> ZERO(x:S)
 BITS(x:S) -> BITITER(x:S,0)
 HALF(s(s(x:S))) -> HALF(x:S)
 IF(ffalse,x:S,y:S) -> BITITER(half(x:S),y:S)
 IF(ffalse,x:S,y:S) -> HALF(x:S)
 IF(ttrue,x:S,y:S) -> P(y:S)
 INC(s(x:S)) -> INC(x:S)
-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse

Problem 1: 

SCC Processor:
-> Pairs:
 BITITER(x:S,y:S) -> IF(zero(x:S),x:S,inc(y:S))
 BITITER(x:S,y:S) -> INC(y:S)
 BITITER(x:S,y:S) -> ZERO(x:S)
 BITS(x:S) -> BITITER(x:S,0)
 HALF(s(s(x:S))) -> HALF(x:S)
 IF(ffalse,x:S,y:S) -> BITITER(half(x:S),y:S)
 IF(ffalse,x:S,y:S) -> HALF(x:S)
 IF(ttrue,x:S,y:S) -> P(y:S)
 INC(s(x:S)) -> INC(x:S)
-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 INC(s(x:S)) -> INC(x:S)
->->-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
->->Cycle:
->->-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
->->-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
->->Cycle:
->->-> Pairs:
 BITITER(x:S,y:S) -> IF(zero(x:S),x:S,inc(y:S))
 IF(ffalse,x:S,y:S) -> BITITER(half(x:S),y:S)
->->-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 INC(s(x:S)) -> INC(x:S)
-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
->Projection:
 pi(INC) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
->Projection:
 pi(HALF) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 BITITER(x:S,y:S) -> IF(zero(x:S),x:S,inc(y:S))
 IF(ffalse,x:S,y:S) -> BITITER(half(x:S),y:S)
-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
-> 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))
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[bitIter](X1,X2) = 0
[bits](X) = 0
[half](X) = 1/2.X
[if](X1,X2,X3) = 0
[inc](X) = 2.X + 1
[p](X) = 0
[zero](X) = X + 1
[0] = 0
[fSNonEmpty] = 0
[false] = 2
[s](X) = 2.X + 2
[true] = 1
[BITITER](X1,X2) = 2.X1 + 2
[BITS](X) = 0
[HALF](X) = 0
[IF](X1,X2,X3) = X1 + X2 + 1/2
[INC](X) = 0
[P](X) = 0
[ZERO](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF(ffalse,x:S,y:S) -> BITITER(half(x:S),y:S)
-> Rules:
 bitIter(x:S,y:S) -> if(zero(x:S),x:S,inc(y:S))
 bits(x:S) -> bitIter(x:S,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(ffalse,x:S,y:S) -> bitIter(half(x:S),y:S)
 if(ttrue,x:S,y:S) -> p(y:S)
 inc(0) -> 0
 inc(s(x:S)) -> s(inc(x:S))
 p(0) -> 0
 p(s(x:S)) -> x:S
 zero(0) -> ttrue
 zero(s(x:S)) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.