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))
help(x:S,y:S) -> ifb(lt(y:S,x:S),x:S,y:S)
ifa(ffalse,x:S) -> logZeroError
ifa(ttrue,x:S) -> help(x:S,1)
ifb(ffalse,x:S,y:S) -> y:S
ifb(ttrue,x:S,y:S) -> help(half(x:S),s(y:S))
logarithm(x:S) -> ifa(lt(0,x:S),x:S)
lt(0,s(x:S)) -> ttrue
lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
lt(x:S,0) -> ffalse
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LT(s(x:S),s(y:S)) -> LT(x:S,y:S)
-> Rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 help(x:S,y:S) -> ifb(lt(y:S,x:S),x:S,y:S)
 ifa(ffalse,x:S) -> logZeroError
 ifa(ttrue,x:S) -> help(x:S,1)
 ifb(ffalse,x:S,y:S) -> y:S
 ifb(ttrue,x:S,y:S) -> help(half(x:S),s(y:S))
 logarithm(x:S) -> ifa(lt(0,x:S),x:S)
 lt(0,s(x:S)) -> ttrue
 lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
 lt(x:S,0) -> ffalse
->Projection:
 pi(LT) = 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))
 help(x:S,y:S) -> ifb(lt(y:S,x:S),x:S,y:S)
 ifa(ffalse,x:S) -> logZeroError
 ifa(ttrue,x:S) -> help(x:S,1)
 ifb(ffalse,x:S,y:S) -> y:S
 ifb(ttrue,x:S,y:S) -> help(half(x:S),s(y:S))
 logarithm(x:S) -> ifa(lt(0,x:S),x:S)
 lt(0,s(x:S)) -> ttrue
 lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
 lt(x:S,0) -> 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:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 help(x:S,y:S) -> ifb(lt(y:S,x:S),x:S,y:S)
 ifa(ffalse,x:S) -> logZeroError
 ifa(ttrue,x:S) -> help(x:S,1)
 ifb(ffalse,x:S,y:S) -> y:S
 ifb(ttrue,x:S,y:S) -> help(half(x:S),s(y:S))
 logarithm(x:S) -> ifa(lt(0,x:S),x:S)
 lt(0,s(x:S)) -> ttrue
 lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
 lt(x:S,0) -> ffalse
->Projection:
 pi(HALF) = 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))
 help(x:S,y:S) -> ifb(lt(y:S,x:S),x:S,y:S)
 ifa(ffalse,x:S) -> logZeroError
 ifa(ttrue,x:S) -> help(x:S,1)
 ifb(ffalse,x:S,y:S) -> y:S
 ifb(ttrue,x:S,y:S) -> help(half(x:S),s(y:S))
 logarithm(x:S) -> ifa(lt(0,x:S),x:S)
 lt(0,s(x:S)) -> ttrue
 lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
 lt(x:S,0) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.