315.85/291.48	WORST_CASE(Omega(n^1), ?)
315.85/291.49	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
315.85/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
315.85/291.49	
315.85/291.49	
315.85/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.85/291.49	
315.85/291.49	(0) CpxTRS
315.85/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
315.85/291.49	(2) TRS for Loop Detection
315.85/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
315.85/291.49	(4) BEST
315.85/291.49	    (5) proven lower bound
315.85/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
315.85/291.49	        (7) BOUNDS(n^1, INF)
315.85/291.49	    (8) TRS for Loop Detection
315.85/291.49	
315.85/291.49	
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(0)
315.85/291.49	Obligation:
315.85/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.85/291.49	
315.85/291.49	
315.85/291.49	The TRS R consists of the following rules:
315.85/291.49	
315.85/291.49	   fstsplit(0, x) -> nil
315.85/291.49	   fstsplit(s(n), nil) -> nil
315.85/291.49	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
315.85/291.49	   sndsplit(0, x) -> x
315.85/291.49	   sndsplit(s(n), nil) -> nil
315.85/291.49	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
315.85/291.49	   empty(nil) -> true
315.85/291.49	   empty(cons(h, t)) -> false
315.85/291.49	   leq(0, m) -> true
315.85/291.49	   leq(s(n), 0) -> false
315.85/291.49	   leq(s(n), s(m)) -> leq(n, m)
315.85/291.49	   length(nil) -> 0
315.85/291.49	   length(cons(h, t)) -> s(length(t))
315.85/291.49	   app(nil, x) -> x
315.85/291.49	   app(cons(h, t), x) -> cons(h, app(t, x))
315.85/291.49	   map_f(pid, nil) -> nil
315.85/291.49	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
315.85/291.49	   process(store, m) -> if1(store, m, leq(m, length(store)))
315.85/291.49	   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
315.85/291.49	   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
315.85/291.49	   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m)
315.85/291.49	   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m)
315.85/291.49	
315.85/291.49	S is empty.
315.85/291.49	Rewrite Strategy: FULL
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
315.85/291.49	Transformed a relative TRS into a decreasing-loop problem.
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(2)
315.85/291.49	Obligation:
315.85/291.49	Analyzing the following TRS for decreasing loops:
315.85/291.49	
315.85/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.85/291.49	
315.85/291.49	
315.85/291.49	The TRS R consists of the following rules:
315.85/291.49	
315.85/291.49	   fstsplit(0, x) -> nil
315.85/291.49	   fstsplit(s(n), nil) -> nil
315.85/291.49	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
315.85/291.49	   sndsplit(0, x) -> x
315.85/291.49	   sndsplit(s(n), nil) -> nil
315.85/291.49	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
315.85/291.49	   empty(nil) -> true
315.85/291.49	   empty(cons(h, t)) -> false
315.85/291.49	   leq(0, m) -> true
315.85/291.49	   leq(s(n), 0) -> false
315.85/291.49	   leq(s(n), s(m)) -> leq(n, m)
315.85/291.49	   length(nil) -> 0
315.85/291.49	   length(cons(h, t)) -> s(length(t))
315.85/291.49	   app(nil, x) -> x
315.85/291.49	   app(cons(h, t), x) -> cons(h, app(t, x))
315.85/291.49	   map_f(pid, nil) -> nil
315.85/291.49	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
315.85/291.49	   process(store, m) -> if1(store, m, leq(m, length(store)))
315.85/291.49	   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
315.85/291.49	   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
315.85/291.49	   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m)
315.85/291.49	   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m)
315.85/291.49	
315.85/291.49	S is empty.
315.85/291.49	Rewrite Strategy: FULL
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
315.85/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
315.85/291.49	
315.85/291.49	The rewrite sequence
315.85/291.49	
315.85/291.49	map_f(pid, cons(h, t)) ->^+ app(f(pid, h), map_f(pid, t))
315.85/291.49	
315.85/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
315.85/291.49	
315.85/291.49	The pumping substitution is [t / cons(h, t)].
315.85/291.49	
315.85/291.49	The result substitution is [ ].
315.85/291.49	
315.85/291.49	
315.85/291.49	
315.85/291.49	
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(4)
315.85/291.49	Complex Obligation (BEST)
315.85/291.49	
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(5)
315.85/291.49	Obligation:
315.85/291.49	Proved the lower bound n^1 for the following obligation:
315.85/291.49	
315.85/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.85/291.49	
315.85/291.49	
315.85/291.49	The TRS R consists of the following rules:
315.85/291.49	
315.85/291.49	   fstsplit(0, x) -> nil
315.85/291.49	   fstsplit(s(n), nil) -> nil
315.85/291.49	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
315.85/291.49	   sndsplit(0, x) -> x
315.85/291.49	   sndsplit(s(n), nil) -> nil
315.85/291.49	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
315.85/291.49	   empty(nil) -> true
315.85/291.49	   empty(cons(h, t)) -> false
315.85/291.49	   leq(0, m) -> true
315.85/291.49	   leq(s(n), 0) -> false
315.85/291.49	   leq(s(n), s(m)) -> leq(n, m)
315.85/291.49	   length(nil) -> 0
315.85/291.49	   length(cons(h, t)) -> s(length(t))
315.85/291.49	   app(nil, x) -> x
315.85/291.49	   app(cons(h, t), x) -> cons(h, app(t, x))
315.85/291.49	   map_f(pid, nil) -> nil
315.85/291.49	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
315.85/291.49	   process(store, m) -> if1(store, m, leq(m, length(store)))
315.85/291.49	   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
315.85/291.49	   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
315.85/291.49	   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m)
315.85/291.49	   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m)
315.85/291.49	
315.85/291.49	S is empty.
315.85/291.49	Rewrite Strategy: FULL
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(6) LowerBoundPropagationProof (FINISHED)
315.85/291.49	Propagated lower bound.
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(7)
315.85/291.49	BOUNDS(n^1, INF)
315.85/291.49	
315.85/291.49	----------------------------------------
315.85/291.49	
315.85/291.49	(8)
315.85/291.49	Obligation:
315.85/291.49	Analyzing the following TRS for decreasing loops:
315.85/291.49	
315.85/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.85/291.49	
315.85/291.49	
315.85/291.49	The TRS R consists of the following rules:
315.85/291.49	
315.85/291.49	   fstsplit(0, x) -> nil
315.85/291.49	   fstsplit(s(n), nil) -> nil
315.85/291.49	   fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t))
315.85/291.49	   sndsplit(0, x) -> x
315.85/291.49	   sndsplit(s(n), nil) -> nil
315.85/291.49	   sndsplit(s(n), cons(h, t)) -> sndsplit(n, t)
315.85/291.49	   empty(nil) -> true
315.85/291.49	   empty(cons(h, t)) -> false
315.85/291.49	   leq(0, m) -> true
315.85/291.49	   leq(s(n), 0) -> false
315.85/291.49	   leq(s(n), s(m)) -> leq(n, m)
315.85/291.49	   length(nil) -> 0
315.85/291.49	   length(cons(h, t)) -> s(length(t))
315.85/291.49	   app(nil, x) -> x
315.85/291.49	   app(cons(h, t), x) -> cons(h, app(t, x))
315.85/291.49	   map_f(pid, nil) -> nil
315.85/291.49	   map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t))
315.85/291.49	   process(store, m) -> if1(store, m, leq(m, length(store)))
315.85/291.49	   if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store)))
315.85/291.49	   if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
315.85/291.49	   if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m)
315.85/291.49	   if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m)
315.85/291.49	
315.85/291.49	S is empty.
315.85/291.49	Rewrite Strategy: FULL
315.93/291.51	EOF