YES

Problem 1: 

(VAR v_NonEmpty:S b:S m:S n:S x:S y:S)
(RULES
empty(cons(n:S,x:S)) -> ffalse
empty(nil) -> ttrue
head(cons(n:S,x:S)) -> n:S
if(ffalse,b:S,x:S) -> if2(b:S,x:S)
if(ttrue,b:S,x:S) -> weight_undefined_error
if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
if2(ttrue,x:S) -> head(x:S)
sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
sum(nil,y:S) -> y:S
tail(cons(n:S,x:S)) -> x:S
tail(nil) -> nil
weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 IF(ffalse,b:S,x:S) -> IF2(b:S,x:S)
 IF2(ffalse,x:S) -> SUM(x:S,cons(0,tail(tail(x:S))))
 IF2(ffalse,x:S) -> TAIL(tail(x:S))
 IF2(ffalse,x:S) -> TAIL(x:S)
 IF2(ffalse,x:S) -> WEIGHT(sum(x:S,cons(0,tail(tail(x:S)))))
 IF2(ttrue,x:S) -> HEAD(x:S)
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
 WEIGHT(x:S) -> EMPTY(tail(x:S))
 WEIGHT(x:S) -> EMPTY(x:S)
 WEIGHT(x:S) -> IF(empty(x:S),empty(tail(x:S)),x:S)
 WEIGHT(x:S) -> TAIL(x:S)
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)

Problem 1: 

SCC Processor:
-> Pairs:
 IF(ffalse,b:S,x:S) -> IF2(b:S,x:S)
 IF2(ffalse,x:S) -> SUM(x:S,cons(0,tail(tail(x:S))))
 IF2(ffalse,x:S) -> TAIL(tail(x:S))
 IF2(ffalse,x:S) -> TAIL(x:S)
 IF2(ffalse,x:S) -> WEIGHT(sum(x:S,cons(0,tail(tail(x:S)))))
 IF2(ttrue,x:S) -> HEAD(x:S)
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
 WEIGHT(x:S) -> EMPTY(tail(x:S))
 WEIGHT(x:S) -> EMPTY(x:S)
 WEIGHT(x:S) -> IF(empty(x:S),empty(tail(x:S)),x:S)
 WEIGHT(x:S) -> TAIL(x:S)
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
->->-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
->->Cycle:
->->-> Pairs:
 IF(ffalse,b:S,x:S) -> IF2(b:S,x:S)
 IF2(ffalse,x:S) -> WEIGHT(sum(x:S,cons(0,tail(tail(x:S)))))
 WEIGHT(x:S) -> IF(empty(x:S),empty(tail(x:S)),x:S)
->->-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S)
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[empty](X) = 0
[head](X) = 0
[if](X1,X2,X3) = 0
[if2](X1,X2) = 0
[sum](X1,X2) = 0
[tail](X) = 0
[weight](X) = 0
[0] = 2
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[fSNonEmpty] = 0
[false] = 0
[nil] = 0
[s](X) = 2.X
[true] = 0
[weight_undefined_error] = 0
[EMPTY](X) = 0
[HEAD](X) = 0
[IF](X1,X2,X3) = 0
[IF2](X1,X2) = 0
[SUM](X1,X2) = 2.X1
[TAIL](X) = 0
[WEIGHT](X) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
->->-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S))
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[empty](X) = 0
[head](X) = 0
[if](X1,X2,X3) = 0
[if2](X1,X2) = 0
[sum](X1,X2) = 0
[tail](X) = 0
[weight](X) = 0
[0] = 0
[cons](X1,X2) = 2.X1
[fSNonEmpty] = 0
[false] = 0
[nil] = 0
[s](X) = X + 2
[true] = 0
[weight_undefined_error] = 0
[EMPTY](X) = 0
[HEAD](X) = 0
[IF](X1,X2,X3) = 0
[IF2](X1,X2) = 0
[SUM](X1,X2) = 2.X1 + X2
[TAIL](X) = 0
[WEIGHT](X) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 IF(ffalse,b:S,x:S) -> IF2(b:S,x:S)
 IF2(ffalse,x:S) -> WEIGHT(sum(x:S,cons(0,tail(tail(x:S)))))
 WEIGHT(x:S) -> IF(empty(x:S),empty(tail(x:S)),x:S)
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
-> Usable rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[empty](X) = 2.X + 1/4
[head](X) = 0
[if](X1,X2,X3) = 0
[if2](X1,X2) = 0
[sum](X1,X2) = X2
[tail](X) = 1/4.X
[weight](X) = 0
[0] = 1/2
[cons](X1,X2) = 4.X2 + 1/4
[fSNonEmpty] = 0
[false] = 3/4
[nil] = 0
[s](X) = 2/3.X + 1/4
[true] = 1/4
[weight_undefined_error] = 0
[EMPTY](X) = 0
[HEAD](X) = 0
[IF](X1,X2,X3) = 2/3.X2 + 1/4.X3 + 1/3
[IF2](X1,X2) = 2/3.X1 + 1/4.X2 + 1/4
[SUM](X1,X2) = 0
[TAIL](X) = 0
[WEIGHT](X) = X + 1/2

Problem 1.2: 

SCC Processor:
-> Pairs:
 IF2(ffalse,x:S) -> WEIGHT(sum(x:S,cons(0,tail(tail(x:S)))))
 WEIGHT(x:S) -> IF(empty(x:S),empty(tail(x:S)),x:S)
-> Rules:
 empty(cons(n:S,x:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(n:S,x:S)) -> n:S
 if(ffalse,b:S,x:S) -> if2(b:S,x:S)
 if(ttrue,b:S,x:S) -> weight_undefined_error
 if2(ffalse,x:S) -> weight(sum(x:S,cons(0,tail(tail(x:S)))))
 if2(ttrue,x:S) -> head(x:S)
 sum(cons(0,x:S),y:S) -> sum(x:S,y:S)
 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S))
 sum(nil,y:S) -> y:S
 tail(cons(n:S,x:S)) -> x:S
 tail(nil) -> nil
 weight(x:S) -> if(empty(x:S),empty(tail(x:S)),x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.