/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 l x)
(RULES
conv(x) -> conviter(x,cons(0,nil))
conviter(x,l) -> if(zero(x),x,l)
half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
if(true,x,l) -> l
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
zero(0) -> true
zero(s(x)) -> false
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(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:
 CONV(x) -> CONVITER(x,cons(0,nil))
 CONVITER(x,l) -> IF(zero(x),x,l)
 CONVITER(x,l) -> ZERO(x)
 HALF(s(s(x))) -> HALF(x)
 IF(false,x,l) -> CONVITER(half(x),cons(lastbit(x),l))
 IF(false,x,l) -> HALF(x)
 IF(false,x,l) -> LASTBIT(x)
 LASTBIT(s(s(x))) -> LASTBIT(x)
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false

Problem 1: 

SCC Processor:
-> Pairs:
 CONV(x) -> CONVITER(x,cons(0,nil))
 CONVITER(x,l) -> IF(zero(x),x,l)
 CONVITER(x,l) -> ZERO(x)
 HALF(s(s(x))) -> HALF(x)
 IF(false,x,l) -> CONVITER(half(x),cons(lastbit(x),l))
 IF(false,x,l) -> HALF(x)
 IF(false,x,l) -> LASTBIT(x)
 LASTBIT(s(s(x))) -> LASTBIT(x)
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LASTBIT(s(s(x))) -> LASTBIT(x)
->->-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
->->Cycle:
->->-> Pairs:
 HALF(s(s(x))) -> HALF(x)
->->-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
->->Cycle:
->->-> Pairs:
 CONVITER(x,l) -> IF(zero(x),x,l)
 IF(false,x,l) -> CONVITER(half(x),cons(lastbit(x),l))
->->-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LASTBIT(s(s(x))) -> LASTBIT(x)
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
->Projection:
 pi(LASTBIT) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(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:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
->Projection:
 pi(HALF) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(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:
 CONVITER(x,l) -> IF(zero(x),x,l)
 IF(false,x,l) -> CONVITER(half(x),cons(lastbit(x),l))
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
-> Usable rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[half](X) = 1/2.X
[lastbit](X) = 2.X
[zero](X) = 1/2.X + 1/2
[0] = 0
[cons](X1,X2) = 1/2.X2 + 2
[false] = 1
[s](X) = 2.X + 1
[true] = 1/2
[CONVITER](X1,X2) = 2.X1 + 2
[IF](X1,X2,X3) = X1 + X2 + 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF(false,x,l) -> CONVITER(half(x),cons(lastbit(x),l))
-> Rules:
 conv(x) -> conviter(x,cons(0,nil))
 conviter(x,l) -> if(zero(x),x,l)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,l) -> conviter(half(x),cons(lastbit(x),l))
 if(true,x,l) -> l
 lastbit(0) -> 0
 lastbit(s(0)) -> s(0)
 lastbit(s(s(x))) -> lastbit(x)
 zero(0) -> true
 zero(s(x)) -> false
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.