19.50/5.71	WORST_CASE(Omega(n^1), O(n^1))
19.54/5.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.54/5.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.54/5.73	
19.54/5.73	
19.54/5.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.54/5.73	
19.54/5.73	(0) CpxTRS
19.54/5.73	(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
19.54/5.73	(2) CdtProblem
19.54/5.73	    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
19.54/5.73	    (4) CdtProblem
19.54/5.73	    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
19.54/5.73	    (6) CdtProblem
19.54/5.73	    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 61 ms]
19.54/5.73	    (8) CdtProblem
19.54/5.73	    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 1 ms]
19.54/5.73	    (10) CdtProblem
19.54/5.73	    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
19.54/5.73	    (12) BOUNDS(1, 1)
19.54/5.73	(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
19.54/5.73	(14) CpxTRS
19.54/5.73	    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
19.54/5.73	    (16) CpxTRS
19.54/5.73	    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
19.54/5.73	    (18) typed CpxTrs
19.54/5.73	    (19) OrderProof [LOWER BOUND(ID), 0 ms]
19.54/5.73	    (20) typed CpxTrs
19.54/5.73	    (21) RewriteLemmaProof [LOWER BOUND(ID), 1875 ms]
19.54/5.73	    (22) BEST
19.54/5.73	        (23) proven lower bound
19.54/5.73	            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
19.54/5.73	            (25) BOUNDS(n^1, INF)
19.54/5.73	        (26) typed CpxTrs
19.54/5.73	
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(0)
19.54/5.73	Obligation:
19.54/5.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.54/5.73	
19.54/5.73	
19.54/5.73	The TRS R consists of the following rules:
19.54/5.73	
19.54/5.73	   sum(0) -> 0
19.54/5.73	   sum(s(x)) -> +(sum(x), s(x))
19.54/5.73	   sum1(0) -> 0
19.54/5.73	   sum1(s(x)) -> s(+(sum1(x), +(x, x)))
19.54/5.73	
19.54/5.73	S is empty.
19.54/5.73	Rewrite Strategy: INNERMOST
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(1) CpxTrsToCdtProof (UPPER BOUND(ID))
19.54/5.73	Converted Cpx (relative) TRS to CDT
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(2)
19.54/5.73	Obligation:
19.54/5.73	Complexity Dependency Tuples Problem
19.54/5.73	
19.54/5.73	Rules:
19.54/5.73	   sum(0) -> 0
19.54/5.73	   sum(s(z0)) -> +(sum(z0), s(z0))
19.54/5.73	   sum1(0) -> 0
19.54/5.73	   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
19.54/5.73	Tuples:
19.54/5.73	   SUM(0) -> c
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(0) -> c2
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	S tuples:
19.54/5.73	   SUM(0) -> c
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(0) -> c2
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	K tuples:none
19.54/5.73	Defined Rule Symbols:   sum_1, sum1_1
19.54/5.73	
19.54/5.73	Defined Pair Symbols:   SUM_1, SUM1_1
19.54/5.73	
19.54/5.73	Compound Symbols:   c, c1_1, c2, c3_1
19.54/5.73	
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
19.54/5.73	Removed 2 trailing nodes:
19.54/5.73	   SUM(0) -> c
19.54/5.73	   SUM1(0) -> c2
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(4)
19.54/5.73	Obligation:
19.54/5.73	Complexity Dependency Tuples Problem
19.54/5.73	
19.54/5.73	Rules:
19.54/5.73	   sum(0) -> 0
19.54/5.73	   sum(s(z0)) -> +(sum(z0), s(z0))
19.54/5.73	   sum1(0) -> 0
19.54/5.73	   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
19.54/5.73	Tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	S tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	K tuples:none
19.54/5.73	Defined Rule Symbols:   sum_1, sum1_1
19.54/5.73	
19.54/5.73	Defined Pair Symbols:   SUM_1, SUM1_1
19.54/5.73	
19.54/5.73	Compound Symbols:   c1_1, c3_1
19.54/5.73	
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
19.54/5.73	The following rules are not usable and were removed:
19.54/5.73	   sum(0) -> 0
19.54/5.73	   sum(s(z0)) -> +(sum(z0), s(z0))
19.54/5.73	   sum1(0) -> 0
19.54/5.73	   sum1(s(z0)) -> s(+(sum1(z0), +(z0, z0)))
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(6)
19.54/5.73	Obligation:
19.54/5.73	Complexity Dependency Tuples Problem
19.54/5.73	
19.54/5.73	Rules:none
19.54/5.73	Tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	S tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	K tuples:none
19.54/5.73	Defined Rule Symbols:none
19.54/5.73	
19.54/5.73	Defined Pair Symbols:   SUM_1, SUM1_1
19.54/5.73	
19.54/5.73	Compound Symbols:   c1_1, c3_1
19.54/5.73	
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
19.54/5.73	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	We considered the (Usable) Rules:none
19.54/5.73	And the Tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	The order we found is given by the following interpretation:
19.54/5.73	
19.54/5.73	Polynomial interpretation :
19.54/5.73	
19.54/5.73	   POL(SUM(x_1)) = 0
19.54/5.73	   POL(SUM1(x_1)) = x_1
19.54/5.73	   POL(c1(x_1)) = x_1
19.54/5.73	   POL(c3(x_1)) = x_1
19.54/5.73	   POL(s(x_1)) = [1] + x_1
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(8)
19.54/5.73	Obligation:
19.54/5.73	Complexity Dependency Tuples Problem
19.54/5.73	
19.54/5.73	Rules:none
19.54/5.73	Tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	S tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	K tuples:
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	Defined Rule Symbols:none
19.54/5.73	
19.54/5.73	Defined Pair Symbols:   SUM_1, SUM1_1
19.54/5.73	
19.54/5.73	Compound Symbols:   c1_1, c3_1
19.54/5.73	
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
19.54/5.73	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	We considered the (Usable) Rules:none
19.54/5.73	And the Tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	The order we found is given by the following interpretation:
19.54/5.73	
19.54/5.73	Polynomial interpretation :
19.54/5.73	
19.54/5.73	   POL(SUM(x_1)) = x_1
19.54/5.73	   POL(SUM1(x_1)) = 0
19.54/5.73	   POL(c1(x_1)) = x_1
19.54/5.73	   POL(c3(x_1)) = x_1
19.54/5.73	   POL(s(x_1)) = [1] + x_1
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(10)
19.54/5.73	Obligation:
19.54/5.73	Complexity Dependency Tuples Problem
19.54/5.73	
19.54/5.73	Rules:none
19.54/5.73	Tuples:
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	S tuples:none
19.54/5.73	K tuples:
19.54/5.73	   SUM1(s(z0)) -> c3(SUM1(z0))
19.54/5.73	   SUM(s(z0)) -> c1(SUM(z0))
19.54/5.73	Defined Rule Symbols:none
19.54/5.73	
19.54/5.73	Defined Pair Symbols:   SUM_1, SUM1_1
19.54/5.73	
19.54/5.73	Compound Symbols:   c1_1, c3_1
19.54/5.73	
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
19.54/5.73	The set S is empty
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(12)
19.54/5.73	BOUNDS(1, 1)
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(13) RenamingProof (BOTH BOUNDS(ID, ID))
19.54/5.73	Renamed function symbols to avoid clashes with predefined symbol.
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(14)
19.54/5.73	Obligation:
19.54/5.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
19.54/5.73	
19.54/5.73	
19.54/5.73	The TRS R consists of the following rules:
19.54/5.73	
19.54/5.73	   sum(0') -> 0'
19.54/5.73	   sum(s(x)) -> +'(sum(x), s(x))
19.54/5.73	   sum1(0') -> 0'
19.54/5.73	   sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))
19.54/5.73	
19.54/5.73	S is empty.
19.54/5.73	Rewrite Strategy: INNERMOST
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(15) SlicingProof (LOWER BOUND(ID))
19.54/5.73	Sliced the following arguments:
19.54/5.73	+'/1
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(16)
19.54/5.73	Obligation:
19.54/5.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
19.54/5.73	
19.54/5.73	
19.54/5.73	The TRS R consists of the following rules:
19.54/5.73	
19.54/5.73	   sum(0') -> 0'
19.54/5.73	   sum(s(x)) -> +'(sum(x))
19.54/5.73	   sum1(0') -> 0'
19.54/5.73	   sum1(s(x)) -> s(+'(sum1(x)))
19.54/5.73	
19.54/5.73	S is empty.
19.54/5.73	Rewrite Strategy: INNERMOST
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
19.54/5.73	Infered types.
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(18)
19.54/5.73	Obligation:
19.54/5.73	Innermost TRS:
19.54/5.73	Rules:
19.54/5.73	sum(0') -> 0'
19.54/5.73	sum(s(x)) -> +'(sum(x))
19.54/5.73	sum1(0') -> 0'
19.54/5.73	sum1(s(x)) -> s(+'(sum1(x)))
19.54/5.73	
19.54/5.73	Types:
19.54/5.73	sum :: 0':s:+' -> 0':s:+'
19.54/5.73	0' :: 0':s:+'
19.54/5.73	s :: 0':s:+' -> 0':s:+'
19.54/5.73	+' :: 0':s:+' -> 0':s:+'
19.54/5.73	sum1 :: 0':s:+' -> 0':s:+'
19.54/5.73	hole_0':s:+'1_0 :: 0':s:+'
19.54/5.73	gen_0':s:+'2_0 :: Nat -> 0':s:+'
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(19) OrderProof (LOWER BOUND(ID))
19.54/5.73	Heuristically decided to analyse the following defined symbols:
19.54/5.73	sum, sum1
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(20)
19.54/5.73	Obligation:
19.54/5.73	Innermost TRS:
19.54/5.73	Rules:
19.54/5.73	sum(0') -> 0'
19.54/5.73	sum(s(x)) -> +'(sum(x))
19.54/5.73	sum1(0') -> 0'
19.54/5.73	sum1(s(x)) -> s(+'(sum1(x)))
19.54/5.73	
19.54/5.73	Types:
19.54/5.73	sum :: 0':s:+' -> 0':s:+'
19.54/5.73	0' :: 0':s:+'
19.54/5.73	s :: 0':s:+' -> 0':s:+'
19.54/5.73	+' :: 0':s:+' -> 0':s:+'
19.54/5.73	sum1 :: 0':s:+' -> 0':s:+'
19.54/5.73	hole_0':s:+'1_0 :: 0':s:+'
19.54/5.73	gen_0':s:+'2_0 :: Nat -> 0':s:+'
19.54/5.73	
19.54/5.73	
19.54/5.73	Generator Equations:
19.54/5.73	gen_0':s:+'2_0(0) <=> 0'
19.54/5.73	gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))
19.54/5.73	
19.54/5.73	
19.54/5.73	The following defined symbols remain to be analysed:
19.54/5.73	sum, sum1
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(21) RewriteLemmaProof (LOWER BOUND(ID))
19.54/5.73	Proved the following rewrite lemma:
19.54/5.73	sum(gen_0':s:+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)
19.54/5.73	
19.54/5.73	Induction Base:
19.54/5.73	sum(gen_0':s:+'2_0(+(1, 0)))
19.54/5.73	
19.54/5.73	Induction Step:
19.54/5.73	sum(gen_0':s:+'2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
19.54/5.73	+'(sum(gen_0':s:+'2_0(+(1, n4_0)))) ->_IH
19.54/5.73	+'(*3_0)
19.54/5.73	
19.54/5.73	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(22)
19.54/5.73	Complex Obligation (BEST)
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(23)
19.54/5.73	Obligation:
19.54/5.73	Proved the lower bound n^1 for the following obligation:
19.54/5.73	
19.54/5.73	Innermost TRS:
19.54/5.73	Rules:
19.54/5.73	sum(0') -> 0'
19.54/5.73	sum(s(x)) -> +'(sum(x))
19.54/5.73	sum1(0') -> 0'
19.54/5.73	sum1(s(x)) -> s(+'(sum1(x)))
19.54/5.73	
19.54/5.73	Types:
19.54/5.73	sum :: 0':s:+' -> 0':s:+'
19.54/5.73	0' :: 0':s:+'
19.54/5.73	s :: 0':s:+' -> 0':s:+'
19.54/5.73	+' :: 0':s:+' -> 0':s:+'
19.54/5.73	sum1 :: 0':s:+' -> 0':s:+'
19.54/5.73	hole_0':s:+'1_0 :: 0':s:+'
19.54/5.73	gen_0':s:+'2_0 :: Nat -> 0':s:+'
19.54/5.73	
19.54/5.73	
19.54/5.73	Generator Equations:
19.54/5.73	gen_0':s:+'2_0(0) <=> 0'
19.54/5.73	gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))
19.54/5.73	
19.54/5.73	
19.54/5.73	The following defined symbols remain to be analysed:
19.54/5.73	sum, sum1
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(24) LowerBoundPropagationProof (FINISHED)
19.54/5.73	Propagated lower bound.
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(25)
19.54/5.73	BOUNDS(n^1, INF)
19.54/5.73	
19.54/5.73	----------------------------------------
19.54/5.73	
19.54/5.73	(26)
19.54/5.73	Obligation:
19.54/5.73	Innermost TRS:
19.54/5.73	Rules:
19.54/5.73	sum(0') -> 0'
19.54/5.73	sum(s(x)) -> +'(sum(x))
19.54/5.73	sum1(0') -> 0'
19.54/5.73	sum1(s(x)) -> s(+'(sum1(x)))
19.54/5.73	
19.54/5.73	Types:
19.54/5.73	sum :: 0':s:+' -> 0':s:+'
19.54/5.73	0' :: 0':s:+'
19.54/5.73	s :: 0':s:+' -> 0':s:+'
19.54/5.73	+' :: 0':s:+' -> 0':s:+'
19.54/5.73	sum1 :: 0':s:+' -> 0':s:+'
19.54/5.73	hole_0':s:+'1_0 :: 0':s:+'
19.54/5.73	gen_0':s:+'2_0 :: Nat -> 0':s:+'
19.54/5.73	
19.54/5.73	
19.54/5.73	Lemmas:
19.54/5.73	sum(gen_0':s:+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)
19.54/5.73	
19.54/5.73	
19.54/5.73	Generator Equations:
19.54/5.73	gen_0':s:+'2_0(0) <=> 0'
19.54/5.73	gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))
19.54/5.73	
19.54/5.73	
19.54/5.73	The following defined symbols remain to be analysed:
19.54/5.73	sum1
19.54/5.80	EOF