6.37/2.47	WORST_CASE(Omega(n^1), O(n^1))
6.37/2.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.37/2.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.37/2.48	
6.37/2.48	
6.37/2.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.37/2.48	
6.37/2.48	(0) CpxTRS
6.37/2.48	(1) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
6.37/2.48	(2) CpxRelTRS
6.37/2.48	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
6.37/2.48	    (4) CpxRelTRS
6.37/2.48	    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 2 ms]
6.37/2.48	    (6) CdtProblem
6.37/2.48	    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
6.37/2.48	    (8) CdtProblem
6.37/2.48	    (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
6.37/2.48	    (10) CdtProblem
6.37/2.48	    (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
6.37/2.48	    (12) CdtProblem
6.37/2.48	    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 61 ms]
6.37/2.48	    (14) CdtProblem
6.37/2.48	    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
6.37/2.48	    (16) BOUNDS(1, 1)
6.37/2.48	(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
6.37/2.48	(18) TRS for Loop Detection
6.37/2.48	    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
6.37/2.48	    (20) BEST
6.37/2.48	        (21) proven lower bound
6.37/2.48	            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
6.37/2.48	            (23) BOUNDS(n^1, INF)
6.37/2.48	        (24) TRS for Loop Detection
6.37/2.48	
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(0)
6.37/2.48	Obligation:
6.37/2.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.37/2.48	
6.37/2.48	
6.37/2.48	The TRS R consists of the following rules:
6.37/2.48	
6.37/2.48	   f(cons(nil, y)) -> y
6.37/2.48	   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
6.37/2.48	   copy(0, y, z) -> f(z)
6.37/2.48	   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
6.37/2.48	
6.37/2.48	S is empty.
6.37/2.48	Rewrite Strategy: FULL
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(1) NonCtorToCtorProof (UPPER BOUND(ID))
6.37/2.48	transformed non-ctor to ctor-system
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(2)
6.37/2.48	Obligation:
6.37/2.48	The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).
6.37/2.48	
6.37/2.48	
6.37/2.48	The TRS R consists of the following rules:
6.37/2.48	
6.37/2.48	   f(cons(nil, y)) -> y
6.37/2.48	   copy(0, y, z) -> f(z)
6.37/2.48	   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
6.37/2.48	   f(cons(c_f(cons(nil, y)), z)) -> copy(n, y, z)
6.37/2.48	
6.37/2.48	The (relative) TRS S consists of the following rules:
6.37/2.48	
6.37/2.48	   f(x0) -> c_f(x0)
6.37/2.48	
6.37/2.48	Rewrite Strategy: FULL
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
6.37/2.48	Converted rc-obligation to irc-obligation.
6.37/2.48	
6.37/2.48	The duplicating contexts are:
6.37/2.48	copy(s(x), [], z)
6.37/2.48	
6.37/2.48	
6.37/2.48	The defined contexts are:
6.37/2.48	copy(x0, x1, cons([], x3))
6.37/2.48	f([])
6.37/2.48	copy(x0, x1, cons(x2, []))
6.37/2.48	copy(n, [], x1)
6.37/2.48	copy(n, x0, [])
6.37/2.48	
6.37/2.48	
6.37/2.48	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(4)
6.37/2.48	Obligation:
6.37/2.48	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).
6.37/2.48	
6.37/2.48	
6.37/2.48	The TRS R consists of the following rules:
6.37/2.48	
6.37/2.48	   f(cons(nil, y)) -> y
6.37/2.48	   copy(0, y, z) -> f(z)
6.37/2.48	   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
6.37/2.48	   f(cons(c_f(cons(nil, y)), z)) -> copy(n, y, z)
6.37/2.48	
6.37/2.48	The (relative) TRS S consists of the following rules:
6.37/2.48	
6.37/2.48	   f(x0) -> c_f(x0)
6.37/2.48	
6.37/2.48	Rewrite Strategy: INNERMOST
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(5) CpxTrsToCdtProof (UPPER BOUND(ID))
6.37/2.48	Converted Cpx (relative) TRS to CDT
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(6)
6.37/2.48	Obligation:
6.37/2.48	Complexity Dependency Tuples Problem
6.37/2.48	
6.37/2.48	Rules:
6.37/2.48	   f(z0) -> c_f(z0)
6.37/2.48	   f(cons(nil, z0)) -> z0
6.37/2.48	   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
6.37/2.48	   copy(0, z0, z1) -> f(z1)
6.37/2.48	   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))
6.37/2.48	Tuples:
6.37/2.48	   F(z0) -> c
6.37/2.48	   F(cons(nil, z0)) -> c1
6.37/2.48	   F(cons(c_f(cons(nil, z0)), z1)) -> c2(COPY(n, z0, z1))
6.37/2.48	   COPY(0, z0, z1) -> c3(F(z1))
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
6.37/2.48	S tuples:
6.37/2.48	   F(cons(nil, z0)) -> c1
6.37/2.48	   F(cons(c_f(cons(nil, z0)), z1)) -> c2(COPY(n, z0, z1))
6.37/2.48	   COPY(0, z0, z1) -> c3(F(z1))
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
6.37/2.48	K tuples:none
6.37/2.48	Defined Rule Symbols:   f_1, copy_3
6.37/2.48	
6.37/2.48	Defined Pair Symbols:   F_1, COPY_3
6.37/2.48	
6.37/2.48	Compound Symbols:   c, c1, c2_1, c3_1, c4_2
6.37/2.48	
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
6.37/2.48	Removed 4 trailing nodes:
6.37/2.48	   F(z0) -> c
6.37/2.48	   F(cons(c_f(cons(nil, z0)), z1)) -> c2(COPY(n, z0, z1))
6.37/2.48	   F(cons(nil, z0)) -> c1
6.37/2.48	   COPY(0, z0, z1) -> c3(F(z1))
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(8)
6.37/2.48	Obligation:
6.37/2.48	Complexity Dependency Tuples Problem
6.37/2.48	
6.37/2.48	Rules:
6.37/2.48	   f(z0) -> c_f(z0)
6.37/2.48	   f(cons(nil, z0)) -> z0
6.37/2.48	   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
6.37/2.48	   copy(0, z0, z1) -> f(z1)
6.37/2.48	   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))
6.37/2.48	Tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
6.37/2.48	S tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
6.37/2.48	K tuples:none
6.37/2.48	Defined Rule Symbols:   f_1, copy_3
6.37/2.48	
6.37/2.48	Defined Pair Symbols:   COPY_3
6.37/2.48	
6.37/2.48	Compound Symbols:   c4_2
6.37/2.48	
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
6.37/2.48	Removed 1 trailing tuple parts
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(10)
6.37/2.48	Obligation:
6.37/2.48	Complexity Dependency Tuples Problem
6.37/2.48	
6.37/2.48	Rules:
6.37/2.48	   f(z0) -> c_f(z0)
6.37/2.48	   f(cons(nil, z0)) -> z0
6.37/2.48	   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
6.37/2.48	   copy(0, z0, z1) -> f(z1)
6.37/2.48	   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))
6.37/2.48	Tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	S tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	K tuples:none
6.37/2.48	Defined Rule Symbols:   f_1, copy_3
6.37/2.48	
6.37/2.48	Defined Pair Symbols:   COPY_3
6.37/2.48	
6.37/2.48	Compound Symbols:   c4_1
6.37/2.48	
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
6.37/2.48	The following rules are not usable and were removed:
6.37/2.48	   copy(0, z0, z1) -> f(z1)
6.37/2.48	   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(12)
6.37/2.48	Obligation:
6.37/2.48	Complexity Dependency Tuples Problem
6.37/2.48	
6.37/2.48	Rules:
6.37/2.48	   f(z0) -> c_f(z0)
6.37/2.48	   f(cons(nil, z0)) -> z0
6.37/2.48	   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
6.37/2.48	Tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	S tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	K tuples:none
6.37/2.48	Defined Rule Symbols:   f_1
6.37/2.48	
6.37/2.48	Defined Pair Symbols:   COPY_3
6.37/2.48	
6.37/2.48	Compound Symbols:   c4_1
6.37/2.48	
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
6.37/2.48	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	We considered the (Usable) Rules:none
6.37/2.48	And the Tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	The order we found is given by the following interpretation:
6.37/2.48	
6.37/2.48	Polynomial interpretation :
6.37/2.48	
6.37/2.48	   POL(COPY(x_1, x_2, x_3)) = x_1
6.37/2.48	   POL(c4(x_1)) = x_1
6.37/2.48	   POL(c_f(x_1)) = [1] + x_1
6.37/2.48	   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
6.37/2.48	   POL(copy(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
6.37/2.48	   POL(f(x_1)) = [1] + x_1
6.37/2.48	   POL(n) = 0
6.37/2.48	   POL(nil) = 0
6.37/2.48	   POL(s(x_1)) = [1] + x_1
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(14)
6.37/2.48	Obligation:
6.37/2.48	Complexity Dependency Tuples Problem
6.37/2.48	
6.37/2.48	Rules:
6.37/2.48	   f(z0) -> c_f(z0)
6.37/2.48	   f(cons(nil, z0)) -> z0
6.37/2.48	   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
6.37/2.48	Tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	S tuples:none
6.37/2.48	K tuples:
6.37/2.48	   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
6.37/2.48	Defined Rule Symbols:   f_1
6.37/2.48	
6.37/2.48	Defined Pair Symbols:   COPY_3
6.37/2.48	
6.37/2.48	Compound Symbols:   c4_1
6.37/2.48	
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
6.37/2.48	The set S is empty
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(16)
6.37/2.48	BOUNDS(1, 1)
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(17) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
6.37/2.48	Transformed a relative TRS into a decreasing-loop problem.
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(18)
6.37/2.48	Obligation:
6.37/2.48	Analyzing the following TRS for decreasing loops:
6.37/2.48	
6.37/2.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.37/2.48	
6.37/2.48	
6.37/2.48	The TRS R consists of the following rules:
6.37/2.48	
6.37/2.48	   f(cons(nil, y)) -> y
6.37/2.48	   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
6.37/2.48	   copy(0, y, z) -> f(z)
6.37/2.48	   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
6.37/2.48	
6.37/2.48	S is empty.
6.37/2.48	Rewrite Strategy: FULL
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(19) DecreasingLoopProof (LOWER BOUND(ID))
6.37/2.48	The following loop(s) give(s) rise to the lower bound Omega(n^1):
6.37/2.48	
6.37/2.48	The rewrite sequence
6.37/2.48	
6.37/2.48	copy(s(x), y, z) ->^+ copy(x, y, cons(f(y), z))
6.37/2.48	
6.37/2.48	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
6.37/2.48	
6.37/2.48	The pumping substitution is [x / s(x)].
6.37/2.48	
6.37/2.48	The result substitution is [z / cons(f(y), z)].
6.37/2.48	
6.37/2.48	
6.37/2.48	
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(20)
6.37/2.48	Complex Obligation (BEST)
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(21)
6.37/2.48	Obligation:
6.37/2.48	Proved the lower bound n^1 for the following obligation:
6.37/2.48	
6.37/2.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.37/2.48	
6.37/2.48	
6.37/2.48	The TRS R consists of the following rules:
6.37/2.48	
6.37/2.48	   f(cons(nil, y)) -> y
6.37/2.48	   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
6.37/2.48	   copy(0, y, z) -> f(z)
6.37/2.48	   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
6.37/2.48	
6.37/2.48	S is empty.
6.37/2.48	Rewrite Strategy: FULL
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(22) LowerBoundPropagationProof (FINISHED)
6.37/2.48	Propagated lower bound.
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(23)
6.37/2.48	BOUNDS(n^1, INF)
6.37/2.48	
6.37/2.48	----------------------------------------
6.37/2.48	
6.37/2.48	(24)
6.37/2.48	Obligation:
6.37/2.48	Analyzing the following TRS for decreasing loops:
6.37/2.48	
6.37/2.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
6.37/2.48	
6.37/2.48	
6.37/2.48	The TRS R consists of the following rules:
6.37/2.48	
6.37/2.48	   f(cons(nil, y)) -> y
6.37/2.48	   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
6.37/2.48	   copy(0, y, z) -> f(z)
6.37/2.48	   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
6.37/2.48	
6.37/2.48	S is empty.
6.37/2.48	Rewrite Strategy: FULL
6.65/2.51	EOF