21.46/7.73	WORST_CASE(Omega(n^1), O(n^1))
21.47/7.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
21.47/7.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.47/7.74	
21.47/7.74	
21.47/7.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.47/7.74	
21.47/7.74	(0) CpxTRS
21.47/7.74	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	(2) CpxTRS
21.47/7.74	    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
21.47/7.74	    (4) CdtProblem
21.47/7.74	    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	    (6) CdtProblem
21.47/7.74	    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	    (8) CdtProblem
21.47/7.74	    (9) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	    (10) CdtProblem
21.47/7.74	    (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	    (12) CdtProblem
21.47/7.74	    (13) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
21.47/7.74	    (14) CdtProblem
21.47/7.74	    (15) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
21.47/7.74	    (16) CdtProblem
21.47/7.74	    (17) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	    (18) CdtProblem
21.47/7.74	    (19) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
21.47/7.74	    (20) CdtProblem
21.47/7.74	    (21) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	    (22) CdtProblem
21.47/7.74	    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
21.47/7.74	    (24) CdtProblem
21.47/7.74	    (25) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
21.47/7.74	    (26) BOUNDS(1, 1)
21.47/7.74	(27) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
21.47/7.74	(28) TRS for Loop Detection
21.47/7.74	    (29) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
21.47/7.74	    (30) BEST
21.47/7.74	        (31) proven lower bound
21.47/7.74	            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
21.47/7.74	            (33) BOUNDS(n^1, INF)
21.47/7.74	        (34) TRS for Loop Detection
21.47/7.74	
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(0)
21.47/7.74	Obligation:
21.47/7.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.47/7.74	
21.47/7.74	
21.47/7.74	The TRS R consists of the following rules:
21.47/7.74	
21.47/7.74	   +(X, 0) -> X
21.47/7.74	   +(X, s(Y)) -> s(+(X, Y))
21.47/7.74	   double(X) -> +(X, X)
21.47/7.74	   f(0, s(0), X) -> f(X, double(X), X)
21.47/7.74	   g(X, Y) -> X
21.47/7.74	   g(X, Y) -> Y
21.47/7.74	
21.47/7.74	S is empty.
21.47/7.74	Rewrite Strategy: FULL
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
21.47/7.74	Converted rc-obligation to irc-obligation.
21.47/7.74	
21.47/7.74	The duplicating contexts are:
21.47/7.74	double([])
21.47/7.74	f(0, s(0), [])
21.47/7.74	
21.47/7.74	
21.47/7.74	The defined contexts are:
21.47/7.74	f(x0, [], x2)
21.47/7.74	
21.47/7.74	
21.47/7.74	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(2)
21.47/7.74	Obligation:
21.47/7.74	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
21.47/7.74	
21.47/7.74	
21.47/7.74	The TRS R consists of the following rules:
21.47/7.74	
21.47/7.74	   +(X, 0) -> X
21.47/7.74	   +(X, s(Y)) -> s(+(X, Y))
21.47/7.74	   double(X) -> +(X, X)
21.47/7.74	   f(0, s(0), X) -> f(X, double(X), X)
21.47/7.74	   g(X, Y) -> X
21.47/7.74	   g(X, Y) -> Y
21.47/7.74	
21.47/7.74	S is empty.
21.47/7.74	Rewrite Strategy: INNERMOST
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(3) CpxTrsToCdtProof (UPPER BOUND(ID))
21.47/7.74	Converted Cpx (relative) TRS to CDT
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(4)
21.47/7.74	Obligation:
21.47/7.74	Complexity Dependency Tuples Problem
21.47/7.74	
21.47/7.74	Rules:
21.47/7.74	   +(z0, 0) -> z0
21.47/7.74	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.74	   double(z0) -> +(z0, z0)
21.47/7.74	   f(0, s(0), z0) -> f(z0, double(z0), z0)
21.47/7.74	   g(z0, z1) -> z0
21.47/7.74	   g(z0, z1) -> z1
21.47/7.74	Tuples:
21.47/7.74	   +'(z0, 0) -> c
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
21.47/7.74	   G(z0, z1) -> c4
21.47/7.74	   G(z0, z1) -> c5
21.47/7.74	S tuples:
21.47/7.74	   +'(z0, 0) -> c
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
21.47/7.74	   G(z0, z1) -> c4
21.47/7.74	   G(z0, z1) -> c5
21.47/7.74	K tuples:none
21.47/7.74	Defined Rule Symbols:   +_2, double_1, f_3, g_2
21.47/7.74	
21.47/7.74	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3, G_2
21.47/7.74	
21.47/7.74	Compound Symbols:   c, c1_1, c2_1, c3_2, c4, c5
21.47/7.74	
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
21.47/7.74	Removed 3 trailing nodes:
21.47/7.74	   G(z0, z1) -> c4
21.47/7.74	   G(z0, z1) -> c5
21.47/7.74	   +'(z0, 0) -> c
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(6)
21.47/7.74	Obligation:
21.47/7.74	Complexity Dependency Tuples Problem
21.47/7.74	
21.47/7.74	Rules:
21.47/7.74	   +(z0, 0) -> z0
21.47/7.74	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.74	   double(z0) -> +(z0, z0)
21.47/7.74	   f(0, s(0), z0) -> f(z0, double(z0), z0)
21.47/7.74	   g(z0, z1) -> z0
21.47/7.74	   g(z0, z1) -> z1
21.47/7.74	Tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
21.47/7.74	S tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
21.47/7.74	K tuples:none
21.47/7.74	Defined Rule Symbols:   +_2, double_1, f_3, g_2
21.47/7.74	
21.47/7.74	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3
21.47/7.74	
21.47/7.74	Compound Symbols:   c1_1, c2_1, c3_2
21.47/7.74	
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
21.47/7.74	The following rules are not usable and were removed:
21.47/7.74	   f(0, s(0), z0) -> f(z0, double(z0), z0)
21.47/7.74	   g(z0, z1) -> z0
21.47/7.74	   g(z0, z1) -> z1
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(8)
21.47/7.74	Obligation:
21.47/7.74	Complexity Dependency Tuples Problem
21.47/7.74	
21.47/7.74	Rules:
21.47/7.74	   double(z0) -> +(z0, z0)
21.47/7.74	   +(z0, 0) -> z0
21.47/7.74	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.74	Tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
21.47/7.74	S tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0))
21.47/7.74	K tuples:none
21.47/7.74	Defined Rule Symbols:   double_1, +_2
21.47/7.74	
21.47/7.74	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3
21.47/7.74	
21.47/7.74	Compound Symbols:   c1_1, c2_1, c3_2
21.47/7.74	
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
21.47/7.74	Use narrowing to replace F(0, s(0), z0) -> c3(F(z0, double(z0), z0), DOUBLE(z0)) by 
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, +(z0, z0), z0), DOUBLE(z0))
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(10)
21.47/7.74	Obligation:
21.47/7.74	Complexity Dependency Tuples Problem
21.47/7.74	
21.47/7.74	Rules:
21.47/7.74	   double(z0) -> +(z0, z0)
21.47/7.74	   +(z0, 0) -> z0
21.47/7.74	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.74	Tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, +(z0, z0), z0), DOUBLE(z0))
21.47/7.74	S tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, +(z0, z0), z0), DOUBLE(z0))
21.47/7.74	K tuples:none
21.47/7.74	Defined Rule Symbols:   double_1, +_2
21.47/7.74	
21.47/7.74	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3
21.47/7.74	
21.47/7.74	Compound Symbols:   c1_1, c2_1, c3_2
21.47/7.74	
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
21.47/7.74	The following rules are not usable and were removed:
21.47/7.74	   double(z0) -> +(z0, z0)
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(12)
21.47/7.74	Obligation:
21.47/7.74	Complexity Dependency Tuples Problem
21.47/7.74	
21.47/7.74	Rules:
21.47/7.74	   +(z0, 0) -> z0
21.47/7.74	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.74	Tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, +(z0, z0), z0), DOUBLE(z0))
21.47/7.74	S tuples:
21.47/7.74	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.74	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.74	   F(0, s(0), z0) -> c3(F(z0, +(z0, z0), z0), DOUBLE(z0))
21.47/7.74	K tuples:none
21.47/7.74	Defined Rule Symbols:   +_2
21.47/7.74	
21.47/7.74	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3
21.47/7.74	
21.47/7.74	Compound Symbols:   c1_1, c2_1, c3_2
21.47/7.74	
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
21.47/7.74	Use narrowing to replace F(0, s(0), z0) -> c3(F(z0, +(z0, z0), z0), DOUBLE(z0)) by 
21.47/7.74	   F(0, s(0), 0) -> c3(F(0, 0, 0), DOUBLE(0))
21.47/7.74	   F(0, s(0), s(z1)) -> c3(F(s(z1), s(+(s(z1), z1)), s(z1)), DOUBLE(s(z1)))
21.47/7.74	   F(0, s(0), x0) -> c3(DOUBLE(x0))
21.47/7.74	
21.47/7.74	----------------------------------------
21.47/7.74	
21.47/7.74	(14)
21.47/7.74	Obligation:
21.47/7.75	Complexity Dependency Tuples Problem
21.47/7.75	
21.47/7.75	Rules:
21.47/7.75	   +(z0, 0) -> z0
21.47/7.75	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.75	Tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.75	   F(0, s(0), 0) -> c3(F(0, 0, 0), DOUBLE(0))
21.47/7.75	   F(0, s(0), s(z1)) -> c3(F(s(z1), s(+(s(z1), z1)), s(z1)), DOUBLE(s(z1)))
21.47/7.75	   F(0, s(0), x0) -> c3(DOUBLE(x0))
21.47/7.75	S tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.75	   F(0, s(0), 0) -> c3(F(0, 0, 0), DOUBLE(0))
21.47/7.75	   F(0, s(0), s(z1)) -> c3(F(s(z1), s(+(s(z1), z1)), s(z1)), DOUBLE(s(z1)))
21.47/7.75	   F(0, s(0), x0) -> c3(DOUBLE(x0))
21.47/7.75	K tuples:none
21.47/7.75	Defined Rule Symbols:   +_2
21.47/7.75	
21.47/7.75	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3
21.47/7.75	
21.47/7.75	Compound Symbols:   c1_1, c2_1, c3_2, c3_1
21.47/7.75	
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(15) CdtLeafRemovalProof (ComplexityIfPolyImplication)
21.47/7.75	Removed 2 leading nodes:
21.47/7.75	   F(0, s(0), 0) -> c3(F(0, 0, 0), DOUBLE(0))
21.47/7.75	   F(0, s(0), x0) -> c3(DOUBLE(x0))
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(16)
21.47/7.75	Obligation:
21.47/7.75	Complexity Dependency Tuples Problem
21.47/7.75	
21.47/7.75	Rules:
21.47/7.75	   +(z0, 0) -> z0
21.47/7.75	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.75	Tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.75	   F(0, s(0), s(z1)) -> c3(F(s(z1), s(+(s(z1), z1)), s(z1)), DOUBLE(s(z1)))
21.47/7.75	S tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.75	   F(0, s(0), s(z1)) -> c3(F(s(z1), s(+(s(z1), z1)), s(z1)), DOUBLE(s(z1)))
21.47/7.75	K tuples:none
21.47/7.75	Defined Rule Symbols:   +_2
21.47/7.75	
21.47/7.75	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3
21.47/7.75	
21.47/7.75	Compound Symbols:   c1_1, c2_1, c3_2
21.47/7.75	
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(17) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
21.47/7.75	Removed 1 trailing tuple parts
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(18)
21.47/7.75	Obligation:
21.47/7.75	Complexity Dependency Tuples Problem
21.47/7.75	
21.47/7.75	Rules:
21.47/7.75	   +(z0, 0) -> z0
21.47/7.75	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.75	Tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.75	   F(0, s(0), s(z1)) -> c3(DOUBLE(s(z1)))
21.47/7.75	S tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.75	   F(0, s(0), s(z1)) -> c3(DOUBLE(s(z1)))
21.47/7.75	K tuples:none
21.47/7.75	Defined Rule Symbols:   +_2
21.47/7.75	
21.47/7.75	Defined Pair Symbols:   +'_2, DOUBLE_1, F_3
21.47/7.75	
21.47/7.75	Compound Symbols:   c1_1, c2_1, c3_1
21.47/7.75	
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(19) CdtLeafRemovalProof (ComplexityIfPolyImplication)
21.47/7.75	Removed 2 leading nodes:
21.47/7.75	   F(0, s(0), s(z1)) -> c3(DOUBLE(s(z1)))
21.47/7.75	   DOUBLE(z0) -> c2(+'(z0, z0))
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(20)
21.47/7.75	Obligation:
21.47/7.75	Complexity Dependency Tuples Problem
21.47/7.75	
21.47/7.75	Rules:
21.47/7.75	   +(z0, 0) -> z0
21.47/7.75	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.75	Tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	S tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	K tuples:none
21.47/7.75	Defined Rule Symbols:   +_2
21.47/7.75	
21.47/7.75	Defined Pair Symbols:   +'_2
21.47/7.75	
21.47/7.75	Compound Symbols:   c1_1
21.47/7.75	
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(21) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
21.47/7.75	The following rules are not usable and were removed:
21.47/7.75	   +(z0, 0) -> z0
21.47/7.75	   +(z0, s(z1)) -> s(+(z0, z1))
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(22)
21.47/7.75	Obligation:
21.47/7.75	Complexity Dependency Tuples Problem
21.47/7.75	
21.47/7.75	Rules:none
21.47/7.75	Tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	S tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	K tuples:none
21.47/7.75	Defined Rule Symbols:none
21.47/7.75	
21.47/7.75	Defined Pair Symbols:   +'_2
21.47/7.75	
21.47/7.75	Compound Symbols:   c1_1
21.47/7.75	
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
21.47/7.75	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	We considered the (Usable) Rules:none
21.47/7.75	And the Tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	The order we found is given by the following interpretation:
21.47/7.75	
21.47/7.75	Polynomial interpretation :
21.47/7.75	
21.47/7.75	   POL(+'(x_1, x_2)) = x_2
21.47/7.75	   POL(c1(x_1)) = x_1
21.47/7.75	   POL(s(x_1)) = [1] + x_1
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(24)
21.47/7.75	Obligation:
21.47/7.75	Complexity Dependency Tuples Problem
21.47/7.75	
21.47/7.75	Rules:none
21.47/7.75	Tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	S tuples:none
21.47/7.75	K tuples:
21.47/7.75	   +'(z0, s(z1)) -> c1(+'(z0, z1))
21.47/7.75	Defined Rule Symbols:none
21.47/7.75	
21.47/7.75	Defined Pair Symbols:   +'_2
21.47/7.75	
21.47/7.75	Compound Symbols:   c1_1
21.47/7.75	
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(25) SIsEmptyProof (BOTH BOUNDS(ID, ID))
21.47/7.75	The set S is empty
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(26)
21.47/7.75	BOUNDS(1, 1)
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(27) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
21.47/7.75	Transformed a relative TRS into a decreasing-loop problem.
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(28)
21.47/7.75	Obligation:
21.47/7.75	Analyzing the following TRS for decreasing loops:
21.47/7.75	
21.47/7.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.47/7.75	
21.47/7.75	
21.47/7.75	The TRS R consists of the following rules:
21.47/7.75	
21.47/7.75	   +(X, 0) -> X
21.47/7.75	   +(X, s(Y)) -> s(+(X, Y))
21.47/7.75	   double(X) -> +(X, X)
21.47/7.75	   f(0, s(0), X) -> f(X, double(X), X)
21.47/7.75	   g(X, Y) -> X
21.47/7.75	   g(X, Y) -> Y
21.47/7.75	
21.47/7.75	S is empty.
21.47/7.75	Rewrite Strategy: FULL
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(29) DecreasingLoopProof (LOWER BOUND(ID))
21.47/7.75	The following loop(s) give(s) rise to the lower bound Omega(n^1):
21.47/7.75	
21.47/7.75	The rewrite sequence
21.47/7.75	
21.47/7.75	+(X, s(Y)) ->^+ s(+(X, Y))
21.47/7.75	
21.47/7.75	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
21.47/7.75	
21.47/7.75	The pumping substitution is [Y / s(Y)].
21.47/7.75	
21.47/7.75	The result substitution is [ ].
21.47/7.75	
21.47/7.75	
21.47/7.75	
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(30)
21.47/7.75	Complex Obligation (BEST)
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(31)
21.47/7.75	Obligation:
21.47/7.75	Proved the lower bound n^1 for the following obligation:
21.47/7.75	
21.47/7.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.47/7.75	
21.47/7.75	
21.47/7.75	The TRS R consists of the following rules:
21.47/7.75	
21.47/7.75	   +(X, 0) -> X
21.47/7.75	   +(X, s(Y)) -> s(+(X, Y))
21.47/7.75	   double(X) -> +(X, X)
21.47/7.75	   f(0, s(0), X) -> f(X, double(X), X)
21.47/7.75	   g(X, Y) -> X
21.47/7.75	   g(X, Y) -> Y
21.47/7.75	
21.47/7.75	S is empty.
21.47/7.75	Rewrite Strategy: FULL
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(32) LowerBoundPropagationProof (FINISHED)
21.47/7.75	Propagated lower bound.
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(33)
21.47/7.75	BOUNDS(n^1, INF)
21.47/7.75	
21.47/7.75	----------------------------------------
21.47/7.75	
21.47/7.75	(34)
21.47/7.75	Obligation:
21.47/7.75	Analyzing the following TRS for decreasing loops:
21.47/7.75	
21.47/7.75	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
21.47/7.75	
21.47/7.75	
21.47/7.75	The TRS R consists of the following rules:
21.47/7.75	
21.47/7.75	   +(X, 0) -> X
21.47/7.75	   +(X, s(Y)) -> s(+(X, Y))
21.47/7.75	   double(X) -> +(X, X)
21.47/7.75	   f(0, s(0), X) -> f(X, double(X), X)
21.47/7.75	   g(X, Y) -> X
21.47/7.75	   g(X, Y) -> Y
21.47/7.75	
21.47/7.75	S is empty.
21.47/7.75	Rewrite Strategy: FULL
21.47/7.80	EOF