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


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


YES

Problem 1: 

(VAR x y)
(RULES
bitIter(x,y) -> if(zero(x),x,inc(y))
bits(x) -> bitIter(x,0)
half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
if(false,x,y) -> bitIter(half(x),y)
if(true,x,y) -> p(y)
inc(0) -> 0
inc(s(x)) -> s(inc(x))
p(0) -> 0
p(s(x)) -> x
zero(0) -> true
zero(s(x)) -> false
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 bitIter(x,y) -> if(zero(x),x,inc(y))
 bits(x) -> bitIter(x,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> bitIter(half(x),y)
 if(true,x,y) -> p(y)
 inc(0) -> 0
 inc(s(x)) -> s(inc(x))
 p(0) -> 0
 p(s(x)) -> x
 zero(0) -> true
 zero(s(x)) -> false
-> 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,y) -> IF(zero(x),x,inc(y))
 BITITER(x,y) -> INC(y)
 BITITER(x,y) -> ZERO(x)
 BITS(x) -> BITITER(x,0)
 HALF(s(s(x))) -> HALF(x)
 IF(false,x,y) -> BITITER(half(x),y)
 IF(false,x,y) -> HALF(x)
 IF(true,x,y) -> P(y)
 INC(s(x)) -> INC(x)
-> Rules:
 bitIter(x,y) -> if(zero(x),x,inc(y))
 bits(x) -> bitIter(x,0)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> bitIter(half(x),y)
 if(true,x,y) -> p(y)
 inc(0) -> 0
 inc(s(x)) -> s(inc(x))
 p(0) -> 0
 p(s(x)) -> x
 zero(0) -> true
 zero(s(x)) -> false

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.