/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 v_NonEmpty:S l:S x:S y:S)
(RULES
empty(cons(x:S,l:S)) -> ffalse
empty(nil) -> ttrue
head(cons(x:S,l:S)) -> x:S
if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
if(ttrue,x:S,l:S) -> x:S
last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
rev(nil) -> nil
rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
rev2(x:S,nil) -> nil
tail(cons(x:S,l:S)) -> l:S
tail(nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
-> 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,x:S,l:S) -> HEAD(l:S)
 IF(ffalse,x:S,l:S) -> LAST(head(l:S),tail(l:S))
 IF(ffalse,x:S,l:S) -> TAIL(l:S)
 LAST(x:S,l:S) -> EMPTY(l:S)
 LAST(x:S,l:S) -> IF(empty(l:S),x:S,l:S)
 REV(cons(x:S,l:S)) -> REV2(x:S,l:S)
 REV2(x:S,cons(y:S,l:S)) -> REV(cons(x:S,rev2(y:S,l:S)))
 REV2(x:S,cons(y:S,l:S)) -> REV2(y:S,l:S)
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 IF(ffalse,x:S,l:S) -> HEAD(l:S)
 IF(ffalse,x:S,l:S) -> LAST(head(l:S),tail(l:S))
 IF(ffalse,x:S,l:S) -> TAIL(l:S)
 LAST(x:S,l:S) -> EMPTY(l:S)
 LAST(x:S,l:S) -> IF(empty(l:S),x:S,l:S)
 REV(cons(x:S,l:S)) -> REV2(x:S,l:S)
 REV2(x:S,cons(y:S,l:S)) -> REV(cons(x:S,rev2(y:S,l:S)))
 REV2(x:S,cons(y:S,l:S)) -> REV2(y:S,l:S)
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REV(cons(x:S,l:S)) -> REV2(x:S,l:S)
 REV2(x:S,cons(y:S,l:S)) -> REV(cons(x:S,rev2(y:S,l:S)))
 REV2(x:S,cons(y:S,l:S)) -> REV2(y:S,l:S)
->->-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
->->Cycle:
->->-> Pairs:
 IF(ffalse,x:S,l:S) -> LAST(head(l:S),tail(l:S))
 LAST(x:S,l:S) -> IF(empty(l:S),x:S,l:S)
->->-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 REV(cons(x:S,l:S)) -> REV2(x:S,l:S)
 REV2(x:S,cons(y:S,l:S)) -> REV(cons(x:S,rev2(y:S,l:S)))
 REV2(x:S,cons(y:S,l:S)) -> REV2(y:S,l:S)
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
-> Usable rules:
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[empty](X) = 0
[head](X) = 0
[if](X1,X2,X3) = 0
[last](X1,X2) = 0
[rev](X) = X
[rev2](X1,X2) = X2
[tail](X) = 0
[cons](X1,X2) = 2.X2 + 2
[fSNonEmpty] = 0
[false] = 0
[nil] = 0
[rev1](X1,X2) = X2 + 2
[true] = 0
[EMPTY](X) = 0
[HEAD](X) = 0
[IF](X1,X2,X3) = 0
[LAST](X1,X2) = 0
[REV](X) = X + 2
[REV2](X1,X2) = 2.X2 + 1
[TAIL](X) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 REV2(x:S,cons(y:S,l:S)) -> REV(cons(x:S,rev2(y:S,l:S)))
 REV2(x:S,cons(y:S,l:S)) -> REV2(y:S,l:S)
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REV2(x:S,cons(y:S,l:S)) -> REV2(y:S,l:S)
->->-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil

Problem 1.1: 

Subterm Processor:
-> Pairs:
 REV2(x:S,cons(y:S,l:S)) -> REV2(y:S,l:S)
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
->Projection:
 pi(REV2) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 IF(ffalse,x:S,l:S) -> LAST(head(l:S),tail(l:S))
 LAST(x:S,l:S) -> IF(empty(l:S),x:S,l:S)
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
-> Usable rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[empty](X) = 2.X
[head](X) = X
[if](X1,X2,X3) = 0
[last](X1,X2) = 0
[rev](X) = 0
[rev2](X1,X2) = 0
[tail](X) = 1/2.X
[cons](X1,X2) = X1 + 2.X2 + 1/2
[fSNonEmpty] = 0
[false] = 1/2
[nil] = 0
[rev1](X1,X2) = 0
[true] = 0
[EMPTY](X) = 0
[HEAD](X) = 0
[IF](X1,X2,X3) = 1/2.X1 + X3 + 2
[LAST](X1,X2) = 2.X2 + 2
[REV](X) = 0
[REV2](X1,X2) = 0
[TAIL](X) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 LAST(x:S,l:S) -> IF(empty(l:S),x:S,l:S)
-> Rules:
 empty(cons(x:S,l:S)) -> ffalse
 empty(nil) -> ttrue
 head(cons(x:S,l:S)) -> x:S
 if(ffalse,x:S,l:S) -> last(head(l:S),tail(l:S))
 if(ttrue,x:S,l:S) -> x:S
 last(x:S,l:S) -> if(empty(l:S),x:S,l:S)
 rev(cons(x:S,l:S)) -> cons(rev1(x:S,l:S),rev2(x:S,l:S))
 rev(nil) -> nil
 rev2(x:S,cons(y:S,l:S)) -> rev(cons(x:S,rev2(y:S,l:S)))
 rev2(x:S,nil) -> nil
 tail(cons(x:S,l:S)) -> l:S
 tail(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.