102.80/42.92	WORST_CASE(Omega(n^1), O(n^1))
102.80/42.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
102.80/42.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
102.80/42.93	
102.80/42.93	
102.80/42.93	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
102.80/42.93	
102.80/42.93	(0) CpxTRS
102.80/42.93	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
102.80/42.93	(2) CpxTRS
102.80/42.93	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
102.80/42.93	    (4) CpxTRS
102.80/42.93	    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 5 ms]
102.80/42.93	    (6) CdtProblem
102.80/42.93	    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
102.80/42.93	    (8) CdtProblem
102.80/42.93	    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
102.80/42.93	    (10) CdtProblem
102.80/42.93	    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms]
102.80/42.93	    (12) CdtProblem
102.80/42.93	    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
102.80/42.93	    (14) BOUNDS(1, 1)
102.80/42.93	(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
102.80/42.93	(16) TRS for Loop Detection
102.80/42.93	    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
102.80/42.93	    (18) BEST
102.80/42.93	        (19) proven lower bound
102.80/42.93	            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
102.80/42.93	            (21) BOUNDS(n^1, INF)
102.80/42.93	        (22) TRS for Loop Detection
102.80/42.93	
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(0)
102.80/42.93	Obligation:
102.80/42.93	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
102.80/42.93	
102.80/42.93	
102.80/42.93	The TRS R consists of the following rules:
102.80/42.93	
102.80/42.93	   :(:(x, y), z) -> :(x, :(y, z))
102.80/42.93	   :(+(x, y), z) -> +(:(x, z), :(y, z))
102.80/42.93	   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))
102.80/42.93	
102.80/42.93	S is empty.
102.80/42.93	Rewrite Strategy: FULL
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
102.80/42.93	The TRS does not nest defined symbols.
102.80/42.93	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
102.80/42.93	:(:(x, y), z) -> :(x, :(y, z))
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(2)
102.80/42.93	Obligation:
102.80/42.93	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
102.80/42.93	
102.80/42.93	
102.80/42.93	The TRS R consists of the following rules:
102.80/42.93	
102.80/42.93	   :(+(x, y), z) -> +(:(x, z), :(y, z))
102.80/42.93	   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))
102.80/42.93	
102.80/42.93	S is empty.
102.80/42.93	Rewrite Strategy: FULL
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
102.80/42.93	Converted rc-obligation to irc-obligation.
102.80/42.93	
102.80/42.93	As the TRS does not nest defined symbols, we have rc = irc.
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(4)
102.80/42.93	Obligation:
102.80/42.93	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
102.80/42.93	
102.80/42.93	
102.80/42.93	The TRS R consists of the following rules:
102.80/42.93	
102.80/42.93	   :(+(x, y), z) -> +(:(x, z), :(y, z))
102.80/42.93	   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))
102.80/42.93	
102.80/42.93	S is empty.
102.80/42.93	Rewrite Strategy: INNERMOST
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(5) CpxTrsToCdtProof (UPPER BOUND(ID))
102.80/42.93	Converted Cpx (relative) TRS to CDT
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(6)
102.80/42.93	Obligation:
102.80/42.93	Complexity Dependency Tuples Problem
102.80/42.93	
102.80/42.93	Rules:
102.80/42.93	   :(+(z0, z1), z2) -> +(:(z0, z2), :(z1, z2))
102.80/42.93	   :(z0, +(z1, f(z2))) -> :(g(z0, z2), +(z1, a))
102.80/42.93	Tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	   :'(z0, +(z1, f(z2))) -> c1(:'(g(z0, z2), +(z1, a)))
102.80/42.93	S tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	   :'(z0, +(z1, f(z2))) -> c1(:'(g(z0, z2), +(z1, a)))
102.80/42.93	K tuples:none
102.80/42.93	Defined Rule Symbols:   :_2
102.80/42.93	
102.80/42.93	Defined Pair Symbols:   :'_2
102.80/42.93	
102.80/42.93	Compound Symbols:   c_2, c1_1
102.80/42.93	
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
102.80/42.93	Removed 1 trailing nodes:
102.80/42.93	   :'(z0, +(z1, f(z2))) -> c1(:'(g(z0, z2), +(z1, a)))
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(8)
102.80/42.93	Obligation:
102.80/42.93	Complexity Dependency Tuples Problem
102.80/42.93	
102.80/42.93	Rules:
102.80/42.93	   :(+(z0, z1), z2) -> +(:(z0, z2), :(z1, z2))
102.80/42.93	   :(z0, +(z1, f(z2))) -> :(g(z0, z2), +(z1, a))
102.80/42.93	Tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	S tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	K tuples:none
102.80/42.93	Defined Rule Symbols:   :_2
102.80/42.93	
102.80/42.93	Defined Pair Symbols:   :'_2
102.80/42.93	
102.80/42.93	Compound Symbols:   c_2
102.80/42.93	
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
102.80/42.93	The following rules are not usable and were removed:
102.80/42.93	   :(+(z0, z1), z2) -> +(:(z0, z2), :(z1, z2))
102.80/42.93	   :(z0, +(z1, f(z2))) -> :(g(z0, z2), +(z1, a))
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(10)
102.80/42.93	Obligation:
102.80/42.93	Complexity Dependency Tuples Problem
102.80/42.93	
102.80/42.93	Rules:none
102.80/42.93	Tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	S tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	K tuples:none
102.80/42.93	Defined Rule Symbols:none
102.80/42.93	
102.80/42.93	Defined Pair Symbols:   :'_2
102.80/42.93	
102.80/42.93	Compound Symbols:   c_2
102.80/42.93	
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
102.80/42.93	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	We considered the (Usable) Rules:none
102.80/42.93	And the Tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	The order we found is given by the following interpretation:
102.80/42.93	
102.80/42.93	Polynomial interpretation :
102.80/42.93	
102.80/42.93	   POL(+(x_1, x_2)) = [1] + x_1 + x_2
102.80/42.93	   POL(:'(x_1, x_2)) = x_1
102.80/42.93	   POL(c(x_1, x_2)) = x_1 + x_2
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(12)
102.80/42.93	Obligation:
102.80/42.93	Complexity Dependency Tuples Problem
102.80/42.93	
102.80/42.93	Rules:none
102.80/42.93	Tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	S tuples:none
102.80/42.93	K tuples:
102.80/42.93	   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
102.80/42.93	Defined Rule Symbols:none
102.80/42.93	
102.80/42.93	Defined Pair Symbols:   :'_2
102.80/42.93	
102.80/42.93	Compound Symbols:   c_2
102.80/42.93	
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
102.80/42.93	The set S is empty
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(14)
102.80/42.93	BOUNDS(1, 1)
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
102.80/42.93	Transformed a relative TRS into a decreasing-loop problem.
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(16)
102.80/42.93	Obligation:
102.80/42.93	Analyzing the following TRS for decreasing loops:
102.80/42.93	
102.80/42.93	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
102.80/42.93	
102.80/42.93	
102.80/42.93	The TRS R consists of the following rules:
102.80/42.93	
102.80/42.93	   :(:(x, y), z) -> :(x, :(y, z))
102.80/42.93	   :(+(x, y), z) -> +(:(x, z), :(y, z))
102.80/42.93	   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))
102.80/42.93	
102.80/42.93	S is empty.
102.80/42.93	Rewrite Strategy: FULL
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(17) DecreasingLoopProof (LOWER BOUND(ID))
102.80/42.93	The following loop(s) give(s) rise to the lower bound Omega(n^1):
102.80/42.93	
102.80/42.93	The rewrite sequence
102.80/42.93	
102.80/42.93	:(+(x, y), z) ->^+ +(:(x, z), :(y, z))
102.80/42.93	
102.80/42.93	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
102.80/42.93	
102.80/42.93	The pumping substitution is [x / +(x, y)].
102.80/42.93	
102.80/42.93	The result substitution is [ ].
102.80/42.93	
102.80/42.93	
102.80/42.93	
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(18)
102.80/42.93	Complex Obligation (BEST)
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(19)
102.80/42.93	Obligation:
102.80/42.93	Proved the lower bound n^1 for the following obligation:
102.80/42.93	
102.80/42.93	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
102.80/42.93	
102.80/42.93	
102.80/42.93	The TRS R consists of the following rules:
102.80/42.93	
102.80/42.93	   :(:(x, y), z) -> :(x, :(y, z))
102.80/42.93	   :(+(x, y), z) -> +(:(x, z), :(y, z))
102.80/42.93	   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))
102.80/42.93	
102.80/42.93	S is empty.
102.80/42.93	Rewrite Strategy: FULL
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(20) LowerBoundPropagationProof (FINISHED)
102.80/42.93	Propagated lower bound.
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(21)
102.80/42.93	BOUNDS(n^1, INF)
102.80/42.93	
102.80/42.93	----------------------------------------
102.80/42.93	
102.80/42.93	(22)
102.80/42.93	Obligation:
102.80/42.93	Analyzing the following TRS for decreasing loops:
102.80/42.93	
102.80/42.93	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
102.80/42.93	
102.80/42.93	
102.80/42.93	The TRS R consists of the following rules:
102.80/42.93	
102.80/42.93	   :(:(x, y), z) -> :(x, :(y, z))
102.80/42.93	   :(+(x, y), z) -> +(:(x, z), :(y, z))
102.80/42.93	   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))
102.80/42.93	
102.80/42.93	S is empty.
102.80/42.93	Rewrite Strategy: FULL
102.97/42.96	EOF