/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
fib(0) -> pair(s(0),0)
fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
plus(0,X:S) -> X:S
plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))

Conditional Termination Problem 2:
-> Pairs:
 FIB(s(X:S)) -> FIB(X:S)
 FIB(s(X:S)) -> PLUS(Y:S,Z:S) | fib(X:S) ->* pair(Y:S,Z:S)
-> QPairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
-> QPairs:
 Empty
->->-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))

Problem 1.1: 

Conditional Subterm Processor:
-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 FIB(s(X:S)) -> FIB(X:S)
 FIB(s(X:S)) -> PLUS(Y:S,Z:S) | fib(X:S) ->* pair(Y:S,Z:S)
-> QPairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,s(Y:S))
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FIB(s(X:S)) -> FIB(X:S)
-> QPairs:
 Empty
->->-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))

Problem 1.2: 

Conditional Subterm Processor:
-> Pairs:
 FIB(s(X:S)) -> FIB(X:S)
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
->Projection:
 pi(FIB) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(s(0),0)
 fib(s(X:S)) -> pair(W:S,Y:S) | fib(X:S) ->* pair(Y:S,Z:S), plus(Y:S,Z:S) ->* W:S
 plus(0,X:S) -> X:S
 plus(s(X:S),Y:S) -> plus(X:S,s(Y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.