302.41/291.50	WORST_CASE(Omega(n^1), ?)
302.41/291.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
302.41/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
302.41/291.51	
302.41/291.51	
302.41/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.41/291.51	
302.41/291.51	(0) CpxTRS
302.41/291.51	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
302.41/291.51	(2) CpxTRS
302.41/291.51	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
302.41/291.51	(4) typed CpxTrs
302.41/291.51	(5) OrderProof [LOWER BOUND(ID), 0 ms]
302.41/291.51	(6) typed CpxTrs
302.41/291.51	(7) RewriteLemmaProof [LOWER BOUND(ID), 275 ms]
302.41/291.51	(8) BEST
302.41/291.51	    (9) proven lower bound
302.41/291.51	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
302.41/291.51	        (11) BOUNDS(n^1, INF)
302.41/291.51	    (12) typed CpxTrs
302.41/291.51	
302.41/291.51	
302.41/291.51	----------------------------------------
302.41/291.51	
302.41/291.51	(0)
302.41/291.51	Obligation:
302.41/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.41/291.51	
302.41/291.51	
302.41/291.51	The TRS R consists of the following rules:
302.41/291.51	
302.41/291.51	   cond1(true, x, y) -> cond2(gr(x, 0), x, y)
302.41/291.51	   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y)
302.41/291.51	   cond2(false, x, y) -> cond3(gr(y, 0), x, y)
302.41/291.51	   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y))
302.41/291.51	   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y)
302.41/291.51	   gr(0, x) -> false
302.41/291.51	   gr(s(x), 0) -> true
302.41/291.51	   gr(s(x), s(y)) -> gr(x, y)
302.41/291.51	   or(false, false) -> false
302.41/291.51	   or(true, x) -> true
302.41/291.51	   or(x, true) -> true
302.41/291.51	   p(0) -> 0
302.41/291.51	   p(s(x)) -> x
302.41/291.51	
302.41/291.51	S is empty.
302.41/291.51	Rewrite Strategy: FULL
302.41/291.51	----------------------------------------
302.41/291.51	
302.41/291.51	(1) RenamingProof (BOTH BOUNDS(ID, ID))
302.41/291.51	Renamed function symbols to avoid clashes with predefined symbol.
302.41/291.51	----------------------------------------
302.41/291.51	
302.41/291.51	(2)
302.41/291.51	Obligation:
302.41/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.41/291.51	
302.41/291.51	
302.41/291.51	The TRS R consists of the following rules:
302.41/291.51	
302.41/291.51	   cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
302.41/291.51	   cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
302.41/291.51	   cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
302.41/291.51	   cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
302.41/291.51	   cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
302.41/291.51	   gr(0', x) -> false
302.41/291.51	   gr(s(x), 0') -> true
302.41/291.51	   gr(s(x), s(y)) -> gr(x, y)
302.41/291.51	   or(false, false) -> false
302.41/291.51	   or(true, x) -> true
302.41/291.51	   or(x, true) -> true
302.41/291.51	   p(0') -> 0'
302.41/291.51	   p(s(x)) -> x
302.41/291.51	
302.41/291.51	S is empty.
302.41/291.51	Rewrite Strategy: FULL
302.41/291.51	----------------------------------------
302.41/291.51	
302.41/291.51	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
302.41/291.51	Infered types.
302.41/291.51	----------------------------------------
302.41/291.51	
302.41/291.51	(4)
302.41/291.51	Obligation:
302.41/291.51	TRS:
302.41/291.51	Rules:
302.41/291.51	cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
302.41/291.51	cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
302.41/291.51	cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
302.41/291.51	cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
302.41/291.51	cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
302.41/291.51	gr(0', x) -> false
302.41/291.51	gr(s(x), 0') -> true
302.41/291.51	gr(s(x), s(y)) -> gr(x, y)
302.41/291.51	or(false, false) -> false
302.41/291.51	or(true, x) -> true
302.41/291.51	or(x, true) -> true
302.41/291.51	p(0') -> 0'
302.41/291.51	p(s(x)) -> x
302.41/291.51	
302.41/291.51	Types:
302.41/291.51	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.41/291.51	true :: true:false
302.41/291.51	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.41/291.51	gr :: 0':s -> 0':s -> true:false
302.41/291.51	0' :: 0':s
302.41/291.51	or :: true:false -> true:false -> true:false
302.41/291.51	p :: 0':s -> 0':s
302.41/291.51	false :: true:false
302.41/291.51	cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	s :: 0':s -> 0':s
302.50/291.51	hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
302.50/291.51	hole_true:false2_0 :: true:false
302.50/291.51	hole_0':s3_0 :: 0':s
302.50/291.51	gen_0':s4_0 :: Nat -> 0':s
302.50/291.51	
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(5) OrderProof (LOWER BOUND(ID))
302.50/291.51	Heuristically decided to analyse the following defined symbols:
302.50/291.51	cond1, cond2, gr, cond3
302.50/291.51	
302.50/291.51	They will be analysed ascendingly in the following order:
302.50/291.51	cond1 = cond2
302.50/291.51	gr < cond1
302.50/291.51	cond1 = cond3
302.50/291.51	gr < cond2
302.50/291.51	cond2 = cond3
302.50/291.51	gr < cond3
302.50/291.51	
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(6)
302.50/291.51	Obligation:
302.50/291.51	TRS:
302.50/291.51	Rules:
302.50/291.51	cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
302.50/291.51	cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
302.50/291.51	cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
302.50/291.51	cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
302.50/291.51	cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
302.50/291.51	gr(0', x) -> false
302.50/291.51	gr(s(x), 0') -> true
302.50/291.51	gr(s(x), s(y)) -> gr(x, y)
302.50/291.51	or(false, false) -> false
302.50/291.51	or(true, x) -> true
302.50/291.51	or(x, true) -> true
302.50/291.51	p(0') -> 0'
302.50/291.51	p(s(x)) -> x
302.50/291.51	
302.50/291.51	Types:
302.50/291.51	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	true :: true:false
302.50/291.51	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	gr :: 0':s -> 0':s -> true:false
302.50/291.51	0' :: 0':s
302.50/291.51	or :: true:false -> true:false -> true:false
302.50/291.51	p :: 0':s -> 0':s
302.50/291.51	false :: true:false
302.50/291.51	cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	s :: 0':s -> 0':s
302.50/291.51	hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
302.50/291.51	hole_true:false2_0 :: true:false
302.50/291.51	hole_0':s3_0 :: 0':s
302.50/291.51	gen_0':s4_0 :: Nat -> 0':s
302.50/291.51	
302.50/291.51	
302.50/291.51	Generator Equations:
302.50/291.51	gen_0':s4_0(0) <=> 0'
302.50/291.51	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
302.50/291.51	
302.50/291.51	
302.50/291.51	The following defined symbols remain to be analysed:
302.50/291.51	gr, cond1, cond2, cond3
302.50/291.51	
302.50/291.51	They will be analysed ascendingly in the following order:
302.50/291.51	cond1 = cond2
302.50/291.51	gr < cond1
302.50/291.51	cond1 = cond3
302.50/291.51	gr < cond2
302.50/291.51	cond2 = cond3
302.50/291.51	gr < cond3
302.50/291.51	
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(7) RewriteLemmaProof (LOWER BOUND(ID))
302.50/291.51	Proved the following rewrite lemma:
302.50/291.51	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
302.50/291.51	
302.50/291.51	Induction Base:
302.50/291.51	gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
302.50/291.51	false
302.50/291.51	
302.50/291.51	Induction Step:
302.50/291.51	gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
302.50/291.51	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
302.50/291.51	false
302.50/291.51	
302.50/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(8)
302.50/291.51	Complex Obligation (BEST)
302.50/291.51	
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(9)
302.50/291.51	Obligation:
302.50/291.51	Proved the lower bound n^1 for the following obligation:
302.50/291.51	
302.50/291.51	TRS:
302.50/291.51	Rules:
302.50/291.51	cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
302.50/291.51	cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
302.50/291.51	cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
302.50/291.51	cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
302.50/291.51	cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
302.50/291.51	gr(0', x) -> false
302.50/291.51	gr(s(x), 0') -> true
302.50/291.51	gr(s(x), s(y)) -> gr(x, y)
302.50/291.51	or(false, false) -> false
302.50/291.51	or(true, x) -> true
302.50/291.51	or(x, true) -> true
302.50/291.51	p(0') -> 0'
302.50/291.51	p(s(x)) -> x
302.50/291.51	
302.50/291.51	Types:
302.50/291.51	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	true :: true:false
302.50/291.51	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	gr :: 0':s -> 0':s -> true:false
302.50/291.51	0' :: 0':s
302.50/291.51	or :: true:false -> true:false -> true:false
302.50/291.51	p :: 0':s -> 0':s
302.50/291.51	false :: true:false
302.50/291.51	cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	s :: 0':s -> 0':s
302.50/291.51	hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
302.50/291.51	hole_true:false2_0 :: true:false
302.50/291.51	hole_0':s3_0 :: 0':s
302.50/291.51	gen_0':s4_0 :: Nat -> 0':s
302.50/291.51	
302.50/291.51	
302.50/291.51	Generator Equations:
302.50/291.51	gen_0':s4_0(0) <=> 0'
302.50/291.51	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
302.50/291.51	
302.50/291.51	
302.50/291.51	The following defined symbols remain to be analysed:
302.50/291.51	gr, cond1, cond2, cond3
302.50/291.51	
302.50/291.51	They will be analysed ascendingly in the following order:
302.50/291.51	cond1 = cond2
302.50/291.51	gr < cond1
302.50/291.51	cond1 = cond3
302.50/291.51	gr < cond2
302.50/291.51	cond2 = cond3
302.50/291.51	gr < cond3
302.50/291.51	
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(10) LowerBoundPropagationProof (FINISHED)
302.50/291.51	Propagated lower bound.
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(11)
302.50/291.51	BOUNDS(n^1, INF)
302.50/291.51	
302.50/291.51	----------------------------------------
302.50/291.51	
302.50/291.51	(12)
302.50/291.51	Obligation:
302.50/291.51	TRS:
302.50/291.51	Rules:
302.50/291.51	cond1(true, x, y) -> cond2(gr(x, 0'), x, y)
302.50/291.51	cond2(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
302.50/291.51	cond2(false, x, y) -> cond3(gr(y, 0'), x, y)
302.50/291.51	cond3(true, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
302.50/291.51	cond3(false, x, y) -> cond1(or(gr(x, 0'), gr(y, 0')), x, y)
302.50/291.51	gr(0', x) -> false
302.50/291.51	gr(s(x), 0') -> true
302.50/291.51	gr(s(x), s(y)) -> gr(x, y)
302.50/291.51	or(false, false) -> false
302.50/291.51	or(true, x) -> true
302.50/291.51	or(x, true) -> true
302.50/291.51	p(0') -> 0'
302.50/291.51	p(s(x)) -> x
302.50/291.51	
302.50/291.51	Types:
302.50/291.51	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	true :: true:false
302.50/291.51	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	gr :: 0':s -> 0':s -> true:false
302.50/291.51	0' :: 0':s
302.50/291.51	or :: true:false -> true:false -> true:false
302.50/291.51	p :: 0':s -> 0':s
302.50/291.51	false :: true:false
302.50/291.51	cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
302.50/291.51	s :: 0':s -> 0':s
302.50/291.51	hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
302.50/291.51	hole_true:false2_0 :: true:false
302.50/291.51	hole_0':s3_0 :: 0':s
302.50/291.51	gen_0':s4_0 :: Nat -> 0':s
302.50/291.51	
302.50/291.51	
302.50/291.51	Lemmas:
302.50/291.51	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
302.50/291.51	
302.50/291.51	
302.50/291.51	Generator Equations:
302.50/291.51	gen_0':s4_0(0) <=> 0'
302.50/291.51	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
302.50/291.51	
302.50/291.51	
302.50/291.51	The following defined symbols remain to be analysed:
302.50/291.51	cond2, cond1, cond3
302.50/291.51	
302.50/291.51	They will be analysed ascendingly in the following order:
302.50/291.51	cond1 = cond2
302.50/291.51	cond1 = cond3
302.50/291.51	cond2 = cond3
302.50/291.53	EOF