/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 x y)
(RULES
half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
help(x,y) -> ifb(lt(y,x),x,y)
ifa(false,x) -> logZeroError
ifa(true,x) -> help(x,1)
ifb(false,x,y) -> y
ifb(true,x,y) -> help(half(x),s(y))
logarithm(x) -> ifa(lt(0,x),x)
lt(0,s(x)) -> true
lt(s(x),s(y)) -> lt(x,y)
lt(x,0) -> false
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
-> 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))) -> HALF(x)
 HELP(x,y) -> IFB(lt(y,x),x,y)
 HELP(x,y) -> LT(y,x)
 IFA(true,x) -> HELP(x,1)
 IFB(true,x,y) -> HALF(x)
 IFB(true,x,y) -> HELP(half(x),s(y))
 LOGARITHM(x) -> IFA(lt(0,x),x)
 LOGARITHM(x) -> LT(0,x)
 LT(s(x),s(y)) -> LT(x,y)
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false

Problem 1: 

SCC Processor:
-> Pairs:
 HALF(s(s(x))) -> HALF(x)
 HELP(x,y) -> IFB(lt(y,x),x,y)
 HELP(x,y) -> LT(y,x)
 IFA(true,x) -> HELP(x,1)
 IFB(true,x,y) -> HALF(x)
 IFB(true,x,y) -> HELP(half(x),s(y))
 LOGARITHM(x) -> IFA(lt(0,x),x)
 LOGARITHM(x) -> LT(0,x)
 LT(s(x),s(y)) -> LT(x,y)
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LT(s(x),s(y)) -> LT(x,y)
->->-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->->Cycle:
->->-> Pairs:
 HALF(s(s(x))) -> HALF(x)
->->-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->->Cycle:
->->-> Pairs:
 HELP(x,y) -> IFB(lt(y,x),x,y)
 IFB(true,x,y) -> HELP(half(x),s(y))
->->-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LT(s(x),s(y)) -> LT(x,y)
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Projection:
 pi(LT) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> 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:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Projection:
 pi(HALF) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 HELP(x,y) -> IFB(lt(y,x),x,y)
 IFB(true,x,y) -> HELP(half(x),s(y))
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
-> Usable rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[half](X) = 1/2.X
[lt](X1,X2) = 2.X2 + 1/2
[0] = 0
[false] = 0
[s](X) = X + 2
[true] = 2
[HELP](X1,X2) = 2.X1 + 1
[IFB](X1,X2,X3) = 1/2.X1 + X2 + 1/2

Problem 1.3: 

SCC Processor:
-> Pairs:
 IFB(true,x,y) -> HELP(half(x),s(y))
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 help(x,y) -> ifb(lt(y,x),x,y)
 ifa(false,x) -> logZeroError
 ifa(true,x) -> help(x,1)
 ifb(false,x,y) -> y
 ifb(true,x,y) -> help(half(x),s(y))
 logarithm(x) -> ifa(lt(0,x),x)
 lt(0,s(x)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.