YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
even(0) -> ttrue
even(s(0)) -> ffalse
even(s(s(x:S))) -> even(x:S)
half(0) -> 0
half(s(s(x:S))) -> s(half(x:S))
if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
plus(0,y:S) -> y:S
plus(s(x:S),y:S) -> s(plus(x:S,y:S))
times(0,y:S) -> 0
times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 EVEN(s(s(x:S))) -> EVEN(x:S)
 HALF(s(s(x:S))) -> HALF(x:S)
 IF_TIMES(ffalse,s(x:S),y:S) -> PLUS(y:S,times(x:S,y:S))
 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
 IF_TIMES(ttrue,s(x:S),y:S) -> HALF(s(x:S))
 IF_TIMES(ttrue,s(x:S),y:S) -> PLUS(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 TIMES(s(x:S),y:S) -> EVEN(s(x:S))
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)

Problem 1: 

SCC Processor:
-> Pairs:
 EVEN(s(s(x:S))) -> EVEN(x:S)
 HALF(s(s(x:S))) -> HALF(x:S)
 IF_TIMES(ffalse,s(x:S),y:S) -> PLUS(y:S,times(x:S,y:S))
 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
 IF_TIMES(ttrue,s(x:S),y:S) -> HALF(s(x:S))
 IF_TIMES(ttrue,s(x:S),y:S) -> PLUS(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 TIMES(s(x:S),y:S) -> EVEN(s(x:S))
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
->->-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->->Cycle:
->->-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
->->-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->->Cycle:
->->-> Pairs:
 EVEN(s(s(x:S))) -> EVEN(x:S)
->->-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->->Cycle:
->->-> Pairs:
 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
->->-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->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:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Projection:
 pi(HALF) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 EVEN(s(s(x:S))) -> EVEN(x:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Projection:
 pi(EVEN) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 IF_TIMES(ffalse,s(x:S),y:S) -> TIMES(x:S,y:S)
 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
-> Usable rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[even](X) = 1
[half](X) = X
[if_times](X1,X2,X3) = 0
[plus](X1,X2) = 0
[times](X1,X2) = 0
[0] = 0
[fSNonEmpty] = 0
[false] = 1
[s](X) = 2.X + 2
[true] = 1
[EVEN](X) = 0
[HALF](X) = 0
[IF_TIMES](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
->->-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 IF_TIMES(ttrue,s(x:S),y:S) -> TIMES(half(s(x:S)),y:S)
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
-> Usable rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[even](X) = 1/2.X + 1/2
[half](X) = 1/2.X + 1/2
[if_times](X1,X2,X3) = 0
[plus](X1,X2) = 0
[times](X1,X2) = 0
[0] = 1
[fSNonEmpty] = 0
[false] = 1/2
[s](X) = 2.X + 2
[true] = 1
[EVEN](X) = 0
[HALF](X) = 0
[IF_TIMES](X1,X2,X3) = X1 + X2 + 1/2.X3 + 1/2
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 1/2.X2

Problem 1.4: 

SCC Processor:
-> Pairs:
 TIMES(s(x:S),y:S) -> IF_TIMES(even(s(x:S)),s(x:S),y:S)
-> Rules:
 even(0) -> ttrue
 even(s(0)) -> ffalse
 even(s(s(x:S))) -> even(x:S)
 half(0) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if_times(ffalse,s(x:S),y:S) -> plus(y:S,times(x:S,y:S))
 if_times(ttrue,s(x:S),y:S) -> plus(times(half(s(x:S)),y:S),times(half(s(x:S)),y:S))
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 times(0,y:S) -> 0
 times(s(x:S),y:S) -> if_times(even(s(x:S)),s(x:S),y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.