13.89/4.57	WORST_CASE(Omega(n^1), O(n^1))
13.89/4.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
13.89/4.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.89/4.58	
13.89/4.58	
13.89/4.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
13.89/4.58	
13.89/4.58	(0) CpxTRS
13.89/4.58	(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
13.89/4.58	(2) CdtProblem
13.89/4.58	    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
13.89/4.58	    (4) CdtProblem
13.89/4.58	    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 1 ms]
13.89/4.58	    (6) CdtProblem
13.89/4.58	    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms]
13.89/4.58	    (8) CdtProblem
13.89/4.58	    (9) CdtKnowledgeProof [FINISHED, 0 ms]
13.89/4.58	    (10) BOUNDS(1, 1)
13.89/4.58	(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
13.89/4.58	(12) CpxTRS
13.89/4.58	    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
13.89/4.58	    (14) typed CpxTrs
13.89/4.58	    (15) OrderProof [LOWER BOUND(ID), 0 ms]
13.89/4.58	    (16) typed CpxTrs
13.89/4.58	    (17) RewriteLemmaProof [LOWER BOUND(ID), 496 ms]
13.89/4.58	    (18) proven lower bound
13.89/4.58	    (19) LowerBoundPropagationProof [FINISHED, 0 ms]
13.89/4.58	    (20) BOUNDS(n^1, INF)
13.89/4.58	
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(0)
13.89/4.58	Obligation:
13.89/4.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
13.89/4.58	
13.89/4.58	
13.89/4.58	The TRS R consists of the following rules:
13.89/4.58	
13.89/4.58	   quot(0, s(y), s(z)) -> 0
13.89/4.58	   quot(s(x), s(y), z) -> quot(x, y, z)
13.89/4.58	   quot(x, 0, s(z)) -> s(quot(x, s(z), s(z)))
13.89/4.58	
13.89/4.58	S is empty.
13.89/4.58	Rewrite Strategy: INNERMOST
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(1) CpxTrsToCdtProof (UPPER BOUND(ID))
13.89/4.58	Converted Cpx (relative) TRS to CDT
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(2)
13.89/4.58	Obligation:
13.89/4.58	Complexity Dependency Tuples Problem
13.89/4.58	
13.89/4.58	Rules:
13.89/4.58	   quot(0, s(z0), s(z1)) -> 0
13.89/4.58	   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
13.89/4.58	   quot(z0, 0, s(z1)) -> s(quot(z0, s(z1), s(z1)))
13.89/4.58	Tuples:
13.89/4.58	   QUOT(0, s(z0), s(z1)) -> c
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	S tuples:
13.89/4.58	   QUOT(0, s(z0), s(z1)) -> c
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	K tuples:none
13.89/4.58	Defined Rule Symbols:   quot_3
13.89/4.58	
13.89/4.58	Defined Pair Symbols:   QUOT_3
13.89/4.58	
13.89/4.58	Compound Symbols:   c, c1_1, c2_1
13.89/4.58	
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
13.89/4.58	Removed 1 trailing nodes:
13.89/4.58	   QUOT(0, s(z0), s(z1)) -> c
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(4)
13.89/4.58	Obligation:
13.89/4.58	Complexity Dependency Tuples Problem
13.89/4.58	
13.89/4.58	Rules:
13.89/4.58	   quot(0, s(z0), s(z1)) -> 0
13.89/4.58	   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
13.89/4.58	   quot(z0, 0, s(z1)) -> s(quot(z0, s(z1), s(z1)))
13.89/4.58	Tuples:
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	S tuples:
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	K tuples:none
13.89/4.58	Defined Rule Symbols:   quot_3
13.89/4.58	
13.89/4.58	Defined Pair Symbols:   QUOT_3
13.89/4.58	
13.89/4.58	Compound Symbols:   c1_1, c2_1
13.89/4.58	
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
13.89/4.58	The following rules are not usable and were removed:
13.89/4.58	   quot(0, s(z0), s(z1)) -> 0
13.89/4.58	   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
13.89/4.58	   quot(z0, 0, s(z1)) -> s(quot(z0, s(z1), s(z1)))
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(6)
13.89/4.58	Obligation:
13.89/4.58	Complexity Dependency Tuples Problem
13.89/4.58	
13.89/4.58	Rules:none
13.89/4.58	Tuples:
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	S tuples:
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	K tuples:none
13.89/4.58	Defined Rule Symbols:none
13.89/4.58	
13.89/4.58	Defined Pair Symbols:   QUOT_3
13.89/4.58	
13.89/4.58	Compound Symbols:   c1_1, c2_1
13.89/4.58	
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
13.89/4.58	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	We considered the (Usable) Rules:none
13.89/4.58	And the Tuples:
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	The order we found is given by the following interpretation:
13.89/4.58	
13.89/4.58	Polynomial interpretation :
13.89/4.58	
13.89/4.58	   POL(0) = 0
13.89/4.58	   POL(QUOT(x_1, x_2, x_3)) = x_1
13.89/4.58	   POL(c1(x_1)) = x_1
13.89/4.58	   POL(c2(x_1)) = x_1
13.89/4.58	   POL(s(x_1)) = [1] + x_1
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(8)
13.89/4.58	Obligation:
13.89/4.58	Complexity Dependency Tuples Problem
13.89/4.58	
13.89/4.58	Rules:none
13.89/4.58	Tuples:
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	S tuples:
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	K tuples:
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	Defined Rule Symbols:none
13.89/4.58	
13.89/4.58	Defined Pair Symbols:   QUOT_3
13.89/4.58	
13.89/4.58	Compound Symbols:   c1_1, c2_1
13.89/4.58	
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(9) CdtKnowledgeProof (FINISHED)
13.89/4.58	The following tuples could be moved from S to K by knowledge propagation:
13.89/4.58	   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, s(z1), s(z1)))
13.89/4.58	   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
13.89/4.58	Now S is empty
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(10)
13.89/4.58	BOUNDS(1, 1)
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(11) RenamingProof (BOTH BOUNDS(ID, ID))
13.89/4.58	Renamed function symbols to avoid clashes with predefined symbol.
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(12)
13.89/4.58	Obligation:
13.89/4.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
13.89/4.58	
13.89/4.58	
13.89/4.58	The TRS R consists of the following rules:
13.89/4.58	
13.89/4.58	   quot(0', s(y), s(z)) -> 0'
13.89/4.58	   quot(s(x), s(y), z) -> quot(x, y, z)
13.89/4.58	   quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
13.89/4.58	
13.89/4.58	S is empty.
13.89/4.58	Rewrite Strategy: INNERMOST
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
13.89/4.58	Infered types.
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(14)
13.89/4.58	Obligation:
13.89/4.58	Innermost TRS:
13.89/4.58	Rules:
13.89/4.58	quot(0', s(y), s(z)) -> 0'
13.89/4.58	quot(s(x), s(y), z) -> quot(x, y, z)
13.89/4.58	quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
13.89/4.58	
13.89/4.58	Types:
13.89/4.58	quot :: 0':s -> 0':s -> 0':s -> 0':s
13.89/4.58	0' :: 0':s
13.89/4.58	s :: 0':s -> 0':s
13.89/4.58	hole_0':s1_0 :: 0':s
13.89/4.58	gen_0':s2_0 :: Nat -> 0':s
13.89/4.58	
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(15) OrderProof (LOWER BOUND(ID))
13.89/4.58	Heuristically decided to analyse the following defined symbols:
13.89/4.58	quot
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(16)
13.89/4.58	Obligation:
13.89/4.58	Innermost TRS:
13.89/4.58	Rules:
13.89/4.58	quot(0', s(y), s(z)) -> 0'
13.89/4.58	quot(s(x), s(y), z) -> quot(x, y, z)
13.89/4.58	quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
13.89/4.58	
13.89/4.58	Types:
13.89/4.58	quot :: 0':s -> 0':s -> 0':s -> 0':s
13.89/4.58	0' :: 0':s
13.89/4.58	s :: 0':s -> 0':s
13.89/4.58	hole_0':s1_0 :: 0':s
13.89/4.58	gen_0':s2_0 :: Nat -> 0':s
13.89/4.58	
13.89/4.58	
13.89/4.58	Generator Equations:
13.89/4.58	gen_0':s2_0(0) <=> 0'
13.89/4.58	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
13.89/4.58	
13.89/4.58	
13.89/4.58	The following defined symbols remain to be analysed:
13.89/4.58	quot
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(17) RewriteLemmaProof (LOWER BOUND(ID))
13.89/4.58	Proved the following rewrite lemma:
13.89/4.58	quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(1)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
13.89/4.58	
13.89/4.58	Induction Base:
13.89/4.58	quot(gen_0':s2_0(0), gen_0':s2_0(+(1, 0)), gen_0':s2_0(1)) ->_R^Omega(1)
13.89/4.58	0'
13.89/4.58	
13.89/4.58	Induction Step:
13.89/4.58	quot(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(1)) ->_R^Omega(1)
13.89/4.58	quot(gen_0':s2_0(n4_0), gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(1)) ->_IH
13.89/4.58	gen_0':s2_0(0)
13.89/4.58	
13.89/4.58	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(18)
13.89/4.58	Obligation:
13.89/4.58	Proved the lower bound n^1 for the following obligation:
13.89/4.58	
13.89/4.58	Innermost TRS:
13.89/4.58	Rules:
13.89/4.58	quot(0', s(y), s(z)) -> 0'
13.89/4.58	quot(s(x), s(y), z) -> quot(x, y, z)
13.89/4.58	quot(x, 0', s(z)) -> s(quot(x, s(z), s(z)))
13.89/4.58	
13.89/4.58	Types:
13.89/4.58	quot :: 0':s -> 0':s -> 0':s -> 0':s
13.89/4.58	0' :: 0':s
13.89/4.58	s :: 0':s -> 0':s
13.89/4.58	hole_0':s1_0 :: 0':s
13.89/4.58	gen_0':s2_0 :: Nat -> 0':s
13.89/4.58	
13.89/4.58	
13.89/4.58	Generator Equations:
13.89/4.58	gen_0':s2_0(0) <=> 0'
13.89/4.58	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
13.89/4.58	
13.89/4.58	
13.89/4.58	The following defined symbols remain to be analysed:
13.89/4.58	quot
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(19) LowerBoundPropagationProof (FINISHED)
13.89/4.58	Propagated lower bound.
13.89/4.58	----------------------------------------
13.89/4.58	
13.89/4.58	(20)
13.89/4.58	BOUNDS(n^1, INF)
14.09/4.63	EOF