352.37/291.88	WORST_CASE(Omega(n^1), O(n^2))
352.37/291.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
352.37/291.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
352.37/291.89	
352.37/291.89	
352.37/291.89	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
352.37/291.89	
352.37/291.89	(0) CpxTRS
352.37/291.89	(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
352.37/291.89	(2) CdtProblem
352.37/291.89	    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
352.37/291.89	    (4) CdtProblem
352.37/291.89	    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
352.37/291.89	    (6) CdtProblem
352.37/291.89	    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 25 ms]
352.37/291.89	    (8) CdtProblem
352.37/291.89	    (9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
352.37/291.89	    (10) BOUNDS(1, 1)
352.37/291.89	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
352.37/291.89	(12) TRS for Loop Detection
352.37/291.89	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
352.37/291.89	    (14) BEST
352.37/291.89	        (15) proven lower bound
352.37/291.89	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
352.37/291.89	            (17) BOUNDS(n^1, INF)
352.37/291.89	        (18) TRS for Loop Detection
352.37/291.89	
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(0)
352.37/291.89	Obligation:
352.37/291.89	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
352.37/291.89	
352.37/291.89	
352.37/291.89	The TRS R consists of the following rules:
352.37/291.89	
352.37/291.89	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
352.37/291.89	   *(+(x, y), z) -> +(*(x, z), *(y, z))
352.37/291.89	   *(x, 1) -> x
352.37/291.89	   *(1, y) -> y
352.37/291.89	
352.37/291.89	S is empty.
352.37/291.89	Rewrite Strategy: INNERMOST
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(1) CpxTrsToCdtProof (UPPER BOUND(ID))
352.37/291.89	Converted Cpx (relative) TRS to CDT
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(2)
352.37/291.89	Obligation:
352.37/291.89	Complexity Dependency Tuples Problem
352.37/291.89	
352.37/291.89	Rules:
352.37/291.89	   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
352.37/291.89	   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
352.37/291.89	   *(z0, 1) -> z0
352.37/291.89	   *(1, z0) -> z0
352.37/291.89	Tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	   *'(z0, 1) -> c2
352.37/291.89	   *'(1, z0) -> c3
352.37/291.89	S tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	   *'(z0, 1) -> c2
352.37/291.89	   *'(1, z0) -> c3
352.37/291.89	K tuples:none
352.37/291.89	Defined Rule Symbols:   *_2
352.37/291.89	
352.37/291.89	Defined Pair Symbols:   *'_2
352.37/291.89	
352.37/291.89	Compound Symbols:   c_2, c1_2, c2, c3
352.37/291.89	
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
352.37/291.89	Removed 2 trailing nodes:
352.37/291.89	   *'(1, z0) -> c3
352.37/291.89	   *'(z0, 1) -> c2
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(4)
352.37/291.89	Obligation:
352.37/291.89	Complexity Dependency Tuples Problem
352.37/291.89	
352.37/291.89	Rules:
352.37/291.89	   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
352.37/291.89	   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
352.37/291.89	   *(z0, 1) -> z0
352.37/291.89	   *(1, z0) -> z0
352.37/291.89	Tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	S tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	K tuples:none
352.37/291.89	Defined Rule Symbols:   *_2
352.37/291.89	
352.37/291.89	Defined Pair Symbols:   *'_2
352.37/291.89	
352.37/291.89	Compound Symbols:   c_2, c1_2
352.37/291.89	
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
352.37/291.89	The following rules are not usable and were removed:
352.37/291.89	   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
352.37/291.89	   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
352.37/291.89	   *(z0, 1) -> z0
352.37/291.89	   *(1, z0) -> z0
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(6)
352.37/291.89	Obligation:
352.37/291.89	Complexity Dependency Tuples Problem
352.37/291.89	
352.37/291.89	Rules:none
352.37/291.89	Tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	S tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	K tuples:none
352.37/291.89	Defined Rule Symbols:none
352.37/291.89	
352.37/291.89	Defined Pair Symbols:   *'_2
352.37/291.89	
352.37/291.89	Compound Symbols:   c_2, c1_2
352.37/291.89	
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
352.37/291.89	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	We considered the (Usable) Rules:none
352.37/291.89	And the Tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	The order we found is given by the following interpretation:
352.37/291.89	
352.37/291.89	Polynomial interpretation :
352.37/291.89	
352.37/291.89	   POL(*'(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_1*x_2
352.37/291.89	   POL(+(x_1, x_2)) = [2] + x_1 + x_2
352.37/291.89	   POL(c(x_1, x_2)) = x_1 + x_2
352.37/291.89	   POL(c1(x_1, x_2)) = x_1 + x_2
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(8)
352.37/291.89	Obligation:
352.37/291.89	Complexity Dependency Tuples Problem
352.37/291.89	
352.37/291.89	Rules:none
352.37/291.89	Tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	S tuples:none
352.37/291.89	K tuples:
352.37/291.89	   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
352.37/291.89	   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
352.37/291.89	Defined Rule Symbols:none
352.37/291.89	
352.37/291.89	Defined Pair Symbols:   *'_2
352.37/291.89	
352.37/291.89	Compound Symbols:   c_2, c1_2
352.37/291.89	
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(9) SIsEmptyProof (BOTH BOUNDS(ID, ID))
352.37/291.89	The set S is empty
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(10)
352.37/291.89	BOUNDS(1, 1)
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
352.37/291.89	Transformed a relative TRS into a decreasing-loop problem.
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(12)
352.37/291.89	Obligation:
352.37/291.89	Analyzing the following TRS for decreasing loops:
352.37/291.89	
352.37/291.89	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
352.37/291.89	
352.37/291.89	
352.37/291.89	The TRS R consists of the following rules:
352.37/291.89	
352.37/291.89	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
352.37/291.89	   *(+(x, y), z) -> +(*(x, z), *(y, z))
352.37/291.89	   *(x, 1) -> x
352.37/291.89	   *(1, y) -> y
352.37/291.89	
352.37/291.89	S is empty.
352.37/291.89	Rewrite Strategy: INNERMOST
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(13) DecreasingLoopProof (LOWER BOUND(ID))
352.37/291.89	The following loop(s) give(s) rise to the lower bound Omega(n^1):
352.37/291.89	
352.37/291.89	The rewrite sequence
352.37/291.89	
352.37/291.89	*(+(x, y), z) ->^+ +(*(x, z), *(y, z))
352.37/291.89	
352.37/291.89	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
352.37/291.89	
352.37/291.89	The pumping substitution is [x / +(x, y)].
352.37/291.89	
352.37/291.89	The result substitution is [ ].
352.37/291.89	
352.37/291.89	
352.37/291.89	
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(14)
352.37/291.89	Complex Obligation (BEST)
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(15)
352.37/291.89	Obligation:
352.37/291.89	Proved the lower bound n^1 for the following obligation:
352.37/291.89	
352.37/291.89	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
352.37/291.89	
352.37/291.89	
352.37/291.89	The TRS R consists of the following rules:
352.37/291.89	
352.37/291.89	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
352.37/291.89	   *(+(x, y), z) -> +(*(x, z), *(y, z))
352.37/291.89	   *(x, 1) -> x
352.37/291.89	   *(1, y) -> y
352.37/291.89	
352.37/291.89	S is empty.
352.37/291.89	Rewrite Strategy: INNERMOST
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(16) LowerBoundPropagationProof (FINISHED)
352.37/291.89	Propagated lower bound.
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(17)
352.37/291.89	BOUNDS(n^1, INF)
352.37/291.89	
352.37/291.89	----------------------------------------
352.37/291.89	
352.37/291.89	(18)
352.37/291.89	Obligation:
352.37/291.89	Analyzing the following TRS for decreasing loops:
352.37/291.89	
352.37/291.89	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
352.37/291.89	
352.37/291.89	
352.37/291.89	The TRS R consists of the following rules:
352.37/291.89	
352.37/291.89	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
352.37/291.89	   *(+(x, y), z) -> +(*(x, z), *(y, z))
352.37/291.89	   *(x, 1) -> x
352.37/291.89	   *(1, y) -> y
352.37/291.89	
352.37/291.89	S is empty.
352.37/291.89	Rewrite Strategy: INNERMOST
352.53/291.94	EOF