/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem 1: 

(VAR v_NonEmpty:S w:S x:S y:S z:S)
(RULES
app(app(eq,app(s,x:S)),0) -> ffalse
app(app(eq,0),app(s,x:S)) -> ffalse
app(app(eq,x:S),x:S) -> ttrue
app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
app(app(lt,0),app(s,y:S)) -> ttrue
app(app(lt,y:S),0) -> ffalse
app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
app(app(member,w:S),null) -> ffalse
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(app(lt,app(s,x:S)),app(s,y:S)) -> APP(app(lt,x:S),y:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(eq,w:S),y:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(if,app(app(eq,w:S),y:S)),ttrue)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(lt,w:S),y:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),x:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),z:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(if,app(app(eq,w:S),y:S))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(if,app(app(lt,w:S),y:S))
-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(lt,app(s,x:S)),app(s,y:S)) -> APP(app(lt,x:S),y:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(eq,w:S),y:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(if,app(app(eq,w:S),y:S)),ttrue)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(lt,w:S),y:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),x:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),z:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(if,app(app(eq,w:S),y:S))
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(if,app(app(lt,w:S),y:S))
-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(lt,app(s,x:S)),app(s,y:S)) -> APP(app(lt,x:S),y:S)
->->-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse
->->Cycle:
->->-> Pairs:
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),x:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),z:S)
->->-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APP(app(lt,app(s,x:S)),app(s,y:S)) -> APP(app(lt,x:S),y:S)
-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse
->Projection:
 pi(APP) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),x:S)
 APP(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> APP(app(member,w:S),z:S)
-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse
->Projection:
 pi(APP) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(eq,app(s,x:S)),0) -> ffalse
 app(app(eq,0),app(s,x:S)) -> ffalse
 app(app(eq,x:S),x:S) -> ttrue
 app(app(lt,app(s,x:S)),app(s,y:S)) -> app(app(lt,x:S),y:S)
 app(app(lt,0),app(s,y:S)) -> ttrue
 app(app(lt,y:S),0) -> ffalse
 app(app(member,w:S),app(app(app(fork,x:S),y:S),z:S)) -> app(app(app(if,app(app(lt,w:S),y:S)),app(app(member,w:S),x:S)),app(app(app(if,app(app(eq,w:S),y:S)),ttrue),app(app(member,w:S),z:S)))
 app(app(member,w:S),null) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.