23.31/6.84	WORST_CASE(Omega(n^1), O(n^1))
23.31/6.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.31/6.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.31/6.85	
23.31/6.85	
23.31/6.85	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.31/6.85	
23.31/6.85	(0) CpxTRS
23.31/6.85	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
23.31/6.85	(2) CpxTRS
23.31/6.85	    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
23.31/6.85	    (4) CdtProblem
23.31/6.85	    (5) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
23.31/6.85	    (6) CdtProblem
23.31/6.85	    (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
23.31/6.85	    (8) CdtProblem
23.31/6.85	    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
23.31/6.85	    (10) CdtProblem
23.31/6.85	    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms]
23.31/6.85	    (12) CdtProblem
23.31/6.85	    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
23.31/6.85	    (14) BOUNDS(1, 1)
23.31/6.85	(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
23.31/6.85	(16) TRS for Loop Detection
23.31/6.85	    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
23.31/6.85	    (18) BEST
23.31/6.85	        (19) proven lower bound
23.31/6.85	            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
23.31/6.85	            (21) BOUNDS(n^1, INF)
23.31/6.85	        (22) TRS for Loop Detection
23.31/6.85	
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(0)
23.31/6.85	Obligation:
23.31/6.85	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.31/6.85	
23.31/6.85	
23.31/6.85	The TRS R consists of the following rules:
23.31/6.85	
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(x))) -> prime1(s(s(x)), s(x))
23.31/6.85	   prime1(x, 0) -> false
23.31/6.85	   prime1(x, s(0)) -> true
23.31/6.85	   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
23.31/6.85	   divp(x, y) -> =(rem(x, y), 0)
23.31/6.85	
23.31/6.85	S is empty.
23.31/6.85	Rewrite Strategy: FULL
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
23.31/6.85	Converted rc-obligation to irc-obligation.
23.31/6.85	
23.31/6.85	As the TRS does not nest defined symbols, we have rc = irc.
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(2)
23.31/6.85	Obligation:
23.31/6.85	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
23.31/6.85	
23.31/6.85	
23.31/6.85	The TRS R consists of the following rules:
23.31/6.85	
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(x))) -> prime1(s(s(x)), s(x))
23.31/6.85	   prime1(x, 0) -> false
23.31/6.85	   prime1(x, s(0)) -> true
23.31/6.85	   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
23.31/6.85	   divp(x, y) -> =(rem(x, y), 0)
23.31/6.85	
23.31/6.85	S is empty.
23.31/6.85	Rewrite Strategy: INNERMOST
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(3) CpxTrsToCdtProof (UPPER BOUND(ID))
23.31/6.85	Converted Cpx (relative) TRS to CDT
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(4)
23.31/6.85	Obligation:
23.31/6.85	Complexity Dependency Tuples Problem
23.31/6.85	
23.31/6.85	Rules:
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
23.31/6.85	   prime1(z0, 0) -> false
23.31/6.85	   prime1(z0, s(0)) -> true
23.31/6.85	   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
23.31/6.85	   divp(z0, z1) -> =(rem(z0, z1), 0)
23.31/6.85	Tuples:
23.31/6.85	   PRIME(0) -> c
23.31/6.85	   PRIME(s(0)) -> c1
23.31/6.85	   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
23.31/6.85	   PRIME1(z0, 0) -> c3
23.31/6.85	   PRIME1(z0, s(0)) -> c4
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
23.31/6.85	   DIVP(z0, z1) -> c6
23.31/6.85	S tuples:
23.31/6.85	   PRIME(0) -> c
23.31/6.85	   PRIME(s(0)) -> c1
23.31/6.85	   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
23.31/6.85	   PRIME1(z0, 0) -> c3
23.31/6.85	   PRIME1(z0, s(0)) -> c4
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
23.31/6.85	   DIVP(z0, z1) -> c6
23.31/6.85	K tuples:none
23.31/6.85	Defined Rule Symbols:   prime_1, prime1_2, divp_2
23.31/6.85	
23.31/6.85	Defined Pair Symbols:   PRIME_1, PRIME1_2, DIVP_2
23.31/6.85	
23.31/6.85	Compound Symbols:   c, c1, c2_1, c3, c4, c5_2, c6
23.31/6.85	
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(5) CdtLeafRemovalProof (ComplexityIfPolyImplication)
23.31/6.85	Removed 1 leading nodes:
23.31/6.85	   PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)), s(z0)))
23.31/6.85	Removed 5 trailing nodes:
23.31/6.85	   DIVP(z0, z1) -> c6
23.31/6.85	   PRIME(s(0)) -> c1
23.31/6.85	   PRIME1(z0, s(0)) -> c4
23.31/6.85	   PRIME(0) -> c
23.31/6.85	   PRIME1(z0, 0) -> c3
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(6)
23.31/6.85	Obligation:
23.31/6.85	Complexity Dependency Tuples Problem
23.31/6.85	
23.31/6.85	Rules:
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
23.31/6.85	   prime1(z0, 0) -> false
23.31/6.85	   prime1(z0, s(0)) -> true
23.31/6.85	   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
23.31/6.85	   divp(z0, z1) -> =(rem(z0, z1), 0)
23.31/6.85	Tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
23.31/6.85	S tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(DIVP(s(s(z1)), z0), PRIME1(z0, s(z1)))
23.31/6.85	K tuples:none
23.31/6.85	Defined Rule Symbols:   prime_1, prime1_2, divp_2
23.31/6.85	
23.31/6.85	Defined Pair Symbols:   PRIME1_2
23.31/6.85	
23.31/6.85	Compound Symbols:   c5_2
23.31/6.85	
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
23.31/6.85	Removed 1 trailing tuple parts
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(8)
23.31/6.85	Obligation:
23.31/6.85	Complexity Dependency Tuples Problem
23.31/6.85	
23.31/6.85	Rules:
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
23.31/6.85	   prime1(z0, 0) -> false
23.31/6.85	   prime1(z0, s(0)) -> true
23.31/6.85	   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
23.31/6.85	   divp(z0, z1) -> =(rem(z0, z1), 0)
23.31/6.85	Tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	S tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	K tuples:none
23.31/6.85	Defined Rule Symbols:   prime_1, prime1_2, divp_2
23.31/6.85	
23.31/6.85	Defined Pair Symbols:   PRIME1_2
23.31/6.85	
23.31/6.85	Compound Symbols:   c5_1
23.31/6.85	
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
23.31/6.85	The following rules are not usable and were removed:
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(z0))) -> prime1(s(s(z0)), s(z0))
23.31/6.85	   prime1(z0, 0) -> false
23.31/6.85	   prime1(z0, s(0)) -> true
23.31/6.85	   prime1(z0, s(s(z1))) -> and(not(divp(s(s(z1)), z0)), prime1(z0, s(z1)))
23.31/6.85	   divp(z0, z1) -> =(rem(z0, z1), 0)
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(10)
23.31/6.85	Obligation:
23.31/6.85	Complexity Dependency Tuples Problem
23.31/6.85	
23.31/6.85	Rules:none
23.31/6.85	Tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	S tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	K tuples:none
23.31/6.85	Defined Rule Symbols:none
23.31/6.85	
23.31/6.85	Defined Pair Symbols:   PRIME1_2
23.31/6.85	
23.31/6.85	Compound Symbols:   c5_1
23.31/6.85	
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
23.31/6.85	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	We considered the (Usable) Rules:none
23.31/6.85	And the Tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	The order we found is given by the following interpretation:
23.31/6.85	
23.31/6.85	Polynomial interpretation :
23.31/6.85	
23.31/6.85	   POL(PRIME1(x_1, x_2)) = x_2
23.31/6.85	   POL(c5(x_1)) = x_1
23.31/6.85	   POL(s(x_1)) = [1] + x_1
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(12)
23.31/6.85	Obligation:
23.31/6.85	Complexity Dependency Tuples Problem
23.31/6.85	
23.31/6.85	Rules:none
23.31/6.85	Tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	S tuples:none
23.31/6.85	K tuples:
23.31/6.85	   PRIME1(z0, s(s(z1))) -> c5(PRIME1(z0, s(z1)))
23.31/6.85	Defined Rule Symbols:none
23.31/6.85	
23.31/6.85	Defined Pair Symbols:   PRIME1_2
23.31/6.85	
23.31/6.85	Compound Symbols:   c5_1
23.31/6.85	
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
23.31/6.85	The set S is empty
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(14)
23.31/6.85	BOUNDS(1, 1)
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
23.31/6.85	Transformed a relative TRS into a decreasing-loop problem.
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(16)
23.31/6.85	Obligation:
23.31/6.85	Analyzing the following TRS for decreasing loops:
23.31/6.85	
23.31/6.85	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.31/6.85	
23.31/6.85	
23.31/6.85	The TRS R consists of the following rules:
23.31/6.85	
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(x))) -> prime1(s(s(x)), s(x))
23.31/6.85	   prime1(x, 0) -> false
23.31/6.85	   prime1(x, s(0)) -> true
23.31/6.85	   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
23.31/6.85	   divp(x, y) -> =(rem(x, y), 0)
23.31/6.85	
23.31/6.85	S is empty.
23.31/6.85	Rewrite Strategy: FULL
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(17) DecreasingLoopProof (LOWER BOUND(ID))
23.31/6.85	The following loop(s) give(s) rise to the lower bound Omega(n^1):
23.31/6.85	
23.31/6.85	The rewrite sequence
23.31/6.85	
23.31/6.85	prime1(x, s(s(y))) ->^+ and(not(divp(s(s(y)), x)), prime1(x, s(y)))
23.31/6.85	
23.31/6.85	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
23.31/6.85	
23.31/6.85	The pumping substitution is [y / s(y)].
23.31/6.85	
23.31/6.85	The result substitution is [ ].
23.31/6.85	
23.31/6.85	
23.31/6.85	
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(18)
23.31/6.85	Complex Obligation (BEST)
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(19)
23.31/6.85	Obligation:
23.31/6.85	Proved the lower bound n^1 for the following obligation:
23.31/6.85	
23.31/6.85	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.31/6.85	
23.31/6.85	
23.31/6.85	The TRS R consists of the following rules:
23.31/6.85	
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(x))) -> prime1(s(s(x)), s(x))
23.31/6.85	   prime1(x, 0) -> false
23.31/6.85	   prime1(x, s(0)) -> true
23.31/6.85	   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
23.31/6.85	   divp(x, y) -> =(rem(x, y), 0)
23.31/6.85	
23.31/6.85	S is empty.
23.31/6.85	Rewrite Strategy: FULL
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(20) LowerBoundPropagationProof (FINISHED)
23.31/6.85	Propagated lower bound.
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(21)
23.31/6.85	BOUNDS(n^1, INF)
23.31/6.85	
23.31/6.85	----------------------------------------
23.31/6.85	
23.31/6.85	(22)
23.31/6.85	Obligation:
23.31/6.85	Analyzing the following TRS for decreasing loops:
23.31/6.85	
23.31/6.85	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.31/6.85	
23.31/6.85	
23.31/6.85	The TRS R consists of the following rules:
23.31/6.85	
23.31/6.85	   prime(0) -> false
23.31/6.85	   prime(s(0)) -> false
23.31/6.85	   prime(s(s(x))) -> prime1(s(s(x)), s(x))
23.31/6.85	   prime1(x, 0) -> false
23.31/6.85	   prime1(x, s(0)) -> true
23.31/6.85	   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
23.31/6.85	   divp(x, y) -> =(rem(x, y), 0)
23.31/6.85	
23.31/6.85	S is empty.
23.31/6.85	Rewrite Strategy: FULL
23.53/6.93	EOF