/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 n x y z)
(RULES
app(add(n,x),y) -> add(n,app(x,y))
app(nil,y) -> y
head(add(n,x)) -> n
if(false,x,y,z) -> shuff(reverse(tail(x)),z)
if(true,x,y,z) -> y
null(add(n,x)) -> false
null(nil) -> true
reverse(add(n,x)) -> app(reverse(x),add(n,nil))
reverse(nil) -> nil
shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
shuffle(x) -> shuff(x,nil)
tail(add(n,x)) -> x
tail(nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 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:
 APP(add(n,x),y) -> APP(x,y)
 IF(false,x,y,z) -> REVERSE(tail(x))
 IF(false,x,y,z) -> SHUFF(reverse(tail(x)),z)
 IF(false,x,y,z) -> TAIL(x)
 REVERSE(add(n,x)) -> APP(reverse(x),add(n,nil))
 REVERSE(add(n,x)) -> REVERSE(x)
 SHUFF(x,y) -> APP(y,add(head(x),nil))
 SHUFF(x,y) -> HEAD(x)
 SHUFF(x,y) -> IF(null(x),x,y,app(y,add(head(x),nil)))
 SHUFF(x,y) -> NULL(x)
 SHUFFLE(x) -> SHUFF(x,nil)
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 APP(add(n,x),y) -> APP(x,y)
 IF(false,x,y,z) -> REVERSE(tail(x))
 IF(false,x,y,z) -> SHUFF(reverse(tail(x)),z)
 IF(false,x,y,z) -> TAIL(x)
 REVERSE(add(n,x)) -> APP(reverse(x),add(n,nil))
 REVERSE(add(n,x)) -> REVERSE(x)
 SHUFF(x,y) -> APP(y,add(head(x),nil))
 SHUFF(x,y) -> HEAD(x)
 SHUFF(x,y) -> IF(null(x),x,y,app(y,add(head(x),nil)))
 SHUFF(x,y) -> NULL(x)
 SHUFFLE(x) -> SHUFF(x,nil)
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(add(n,x),y) -> APP(x,y)
->->-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->->Cycle:
->->-> Pairs:
 REVERSE(add(n,x)) -> REVERSE(x)
->->-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->->Cycle:
->->-> Pairs:
 IF(false,x,y,z) -> SHUFF(reverse(tail(x)),z)
 SHUFF(x,y) -> IF(null(x),x,y,app(y,add(head(x),nil)))
->->-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APP(add(n,x),y) -> APP(x,y)
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->Projection:
 pi(APP) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 REVERSE(add(n,x)) -> REVERSE(x)
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->Projection:
 pi(REVERSE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 IF(false,x,y,z) -> SHUFF(reverse(tail(x)),z)
 SHUFF(x,y) -> IF(null(x),x,y,app(y,add(head(x),nil)))
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
-> Usable rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 tail(add(n,x)) -> x
 tail(nil) -> nil
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 1/2.X1.X2 + X1 + X2
[head](X) = 2.X.X + 1/2
[null](X) = X
[reverse](X) = X
[tail](X) = 1/2.X + 1/2
[add](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2
[false] = 2
[nil] = 0
[true] = 0
[IF](X1,X2,X3,X4) = 1/2.X1 + 1/2.X2 + 1
[SHUFF](X1,X2) = X1 + 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 SHUFF(x,y) -> IF(null(x),x,y,app(y,add(head(x),nil)))
-> Rules:
 app(add(n,x),y) -> add(n,app(x,y))
 app(nil,y) -> y
 head(add(n,x)) -> n
 if(false,x,y,z) -> shuff(reverse(tail(x)),z)
 if(true,x,y,z) -> y
 null(add(n,x)) -> false
 null(nil) -> true
 reverse(add(n,x)) -> app(reverse(x),add(n,nil))
 reverse(nil) -> nil
 shuff(x,y) -> if(null(x),x,y,app(y,add(head(x),nil)))
 shuffle(x) -> shuff(x,nil)
 tail(add(n,x)) -> x
 tail(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.