YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
id_inc(x:S) -> s(x:S)
id_inc(x:S) -> x:S
if(ffalse,x:S,y:S) -> y:S
if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
nonZero(0) -> ffalse
nonZero(s(x:S)) -> ttrue
p(s(0)) -> 0
p(s(s(x:S))) -> s(p(s(x:S)))
rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
random(x:S) -> rand(x:S,0)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 IF(ttrue,x:S,y:S) -> ID_INC(y:S)
 IF(ttrue,x:S,y:S) -> P(x:S)
 IF(ttrue,x:S,y:S) -> RAND(p(x:S),id_inc(y:S))
 P(s(s(x:S))) -> P(s(x:S))
 RAND(x:S,y:S) -> IF(nonZero(x:S),x:S,y:S)
 RAND(x:S,y:S) -> NONZERO(x:S)
 RANDOM(x:S) -> RAND(x:S,0)
-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)

Problem 1: 

SCC Processor:
-> Pairs:
 IF(ttrue,x:S,y:S) -> ID_INC(y:S)
 IF(ttrue,x:S,y:S) -> P(x:S)
 IF(ttrue,x:S,y:S) -> RAND(p(x:S),id_inc(y:S))
 P(s(s(x:S))) -> P(s(x:S))
 RAND(x:S,y:S) -> IF(nonZero(x:S),x:S,y:S)
 RAND(x:S,y:S) -> NONZERO(x:S)
 RANDOM(x:S) -> RAND(x:S,0)
-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 P(s(s(x:S))) -> P(s(x:S))
->->-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)
->->Cycle:
->->-> Pairs:
 IF(ttrue,x:S,y:S) -> RAND(p(x:S),id_inc(y:S))
 RAND(x:S,y:S) -> IF(nonZero(x:S),x:S,y:S)
->->-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 P(s(s(x:S))) -> P(s(x:S))
-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)
->Projection:
 pi(P) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 IF(ttrue,x:S,y:S) -> RAND(p(x:S),id_inc(y:S))
 RAND(x:S,y:S) -> IF(nonZero(x:S),x:S,y:S)
-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)
-> Usable rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[id_inc](X) = X.X + 2.X + 2
[nonZero](X) = 2.X
[p](X) = 1/2.X
[0] = 2
[false] = 1/2
[s](X) = X.X + 2.X + 2
[true] = 1/2
[IF](X1,X2,X3) = 1/2.X1 + X2 + 1
[RAND](X1,X2) = 2.X1 + 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 RAND(x:S,y:S) -> IF(nonZero(x:S),x:S,y:S)
-> Rules:
 id_inc(x:S) -> s(x:S)
 id_inc(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> rand(p(x:S),id_inc(y:S))
 nonZero(0) -> ffalse
 nonZero(s(x:S)) -> ttrue
 p(s(0)) -> 0
 p(s(s(x:S))) -> s(p(s(x:S)))
 rand(x:S,y:S) -> if(nonZero(x:S),x:S,y:S)
 random(x:S) -> rand(x:S,0)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.