20.46/7.53	WORST_CASE(Omega(n^1), O(n^1))
20.46/7.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.46/7.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.46/7.54	
20.46/7.54	
20.46/7.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.46/7.54	
20.46/7.54	(0) CpxTRS
20.46/7.54	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
20.46/7.54	(2) CpxTRS
20.46/7.54	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
20.46/7.54	    (4) CpxWeightedTrs
20.46/7.54	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.46/7.54	    (6) CpxTypedWeightedTrs
20.46/7.54	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
20.46/7.54	    (8) CpxTypedWeightedCompleteTrs
20.46/7.54	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.46/7.54	    (10) CpxTypedWeightedCompleteTrs
20.46/7.54	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
20.46/7.54	    (12) CpxRNTS
20.46/7.54	    (13) InliningProof [UPPER BOUND(ID), 125 ms]
20.46/7.54	    (14) CpxRNTS
20.46/7.54	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
20.46/7.54	    (16) CpxRNTS
20.46/7.54	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
20.46/7.54	    (18) CpxRNTS
20.46/7.54	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.46/7.54	    (20) CpxRNTS
20.46/7.54	    (21) IntTrsBoundProof [UPPER BOUND(ID), 124 ms]
20.46/7.54	    (22) CpxRNTS
20.46/7.54	    (23) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
20.46/7.54	    (24) CpxRNTS
20.46/7.54	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.46/7.54	    (26) CpxRNTS
20.46/7.54	    (27) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
20.46/7.54	    (28) CpxRNTS
20.46/7.54	    (29) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
20.46/7.54	    (30) CpxRNTS
20.46/7.54	    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.46/7.54	    (32) CpxRNTS
20.46/7.54	    (33) IntTrsBoundProof [UPPER BOUND(ID), 334 ms]
20.46/7.54	    (34) CpxRNTS
20.46/7.54	    (35) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
20.46/7.54	    (36) CpxRNTS
20.46/7.54	    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.46/7.54	    (38) CpxRNTS
20.46/7.54	    (39) IntTrsBoundProof [UPPER BOUND(ID), 422 ms]
20.46/7.54	    (40) CpxRNTS
20.46/7.54	    (41) IntTrsBoundProof [UPPER BOUND(ID), 51 ms]
20.46/7.54	    (42) CpxRNTS
20.46/7.54	    (43) FinalProof [FINISHED, 0 ms]
20.46/7.54	    (44) BOUNDS(1, n^1)
20.46/7.54	(45) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
20.46/7.54	(46) TRS for Loop Detection
20.46/7.54	    (47) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
20.46/7.54	    (48) BEST
20.46/7.54	        (49) proven lower bound
20.46/7.54	            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
20.46/7.54	            (51) BOUNDS(n^1, INF)
20.46/7.54	        (52) TRS for Loop Detection
20.46/7.54	
20.46/7.54	
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(0)
20.46/7.54	Obligation:
20.46/7.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.46/7.54	
20.46/7.54	
20.46/7.54	The TRS R consists of the following rules:
20.46/7.54	
20.46/7.54	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
20.46/7.54	   a__g(b) -> c
20.46/7.54	   a__b -> c
20.46/7.54	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
20.46/7.54	   mark(g(X)) -> a__g(mark(X))
20.46/7.54	   mark(b) -> a__b
20.46/7.54	   mark(c) -> c
20.46/7.54	   a__f(X1, X2, X3) -> f(X1, X2, X3)
20.46/7.54	   a__g(X) -> g(X)
20.46/7.54	   a__b -> b
20.46/7.54	
20.46/7.54	S is empty.
20.46/7.54	Rewrite Strategy: FULL
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.46/7.54	Converted rc-obligation to irc-obligation.
20.46/7.54	
20.46/7.54	The duplicating contexts are:
20.46/7.54	a__f(X, g(X), [])
20.46/7.54	
20.46/7.54	
20.46/7.54	The defined contexts are:
20.46/7.54	a__g([])
20.46/7.54	
20.46/7.54	
20.46/7.54	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(2)
20.46/7.54	Obligation:
20.46/7.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.46/7.54	
20.46/7.54	
20.46/7.54	The TRS R consists of the following rules:
20.46/7.54	
20.46/7.54	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
20.46/7.54	   a__g(b) -> c
20.46/7.54	   a__b -> c
20.46/7.54	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
20.46/7.54	   mark(g(X)) -> a__g(mark(X))
20.46/7.54	   mark(b) -> a__b
20.46/7.54	   mark(c) -> c
20.46/7.54	   a__f(X1, X2, X3) -> f(X1, X2, X3)
20.46/7.54	   a__g(X) -> g(X)
20.46/7.54	   a__b -> b
20.46/7.54	
20.46/7.54	S is empty.
20.46/7.54	Rewrite Strategy: INNERMOST
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
20.46/7.54	Transformed relative TRS to weighted TRS
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(4)
20.46/7.54	Obligation:
20.46/7.54	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.46/7.54	
20.46/7.54	
20.46/7.54	The TRS R consists of the following rules:
20.46/7.54	
20.46/7.54	   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
20.46/7.54	   a__g(b) -> c [1]
20.46/7.54	   a__b -> c [1]
20.46/7.54	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
20.46/7.54	   mark(g(X)) -> a__g(mark(X)) [1]
20.46/7.54	   mark(b) -> a__b [1]
20.46/7.54	   mark(c) -> c [1]
20.46/7.54	   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
20.46/7.54	   a__g(X) -> g(X) [1]
20.46/7.54	   a__b -> b [1]
20.46/7.54	
20.46/7.54	Rewrite Strategy: INNERMOST
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.46/7.54	Infered types.
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(6)
20.46/7.54	Obligation:
20.46/7.54	Runtime Complexity Weighted TRS with Types.
20.46/7.54	The TRS R consists of the following rules:
20.46/7.54	
20.46/7.54	   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
20.46/7.54	   a__g(b) -> c [1]
20.46/7.54	   a__b -> c [1]
20.46/7.54	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
20.46/7.54	   mark(g(X)) -> a__g(mark(X)) [1]
20.46/7.54	   mark(b) -> a__b [1]
20.46/7.54	   mark(c) -> c [1]
20.46/7.54	   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
20.46/7.54	   a__g(X) -> g(X) [1]
20.46/7.54	   a__b -> b [1]
20.46/7.54	
20.46/7.54	The TRS has the following type information:
20.46/7.54	a__f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
20.46/7.54	g :: g:b:c:f -> g:b:c:f
20.46/7.54	a__g :: g:b:c:f -> g:b:c:f
20.46/7.54	b :: g:b:c:f
20.46/7.54	c :: g:b:c:f
20.46/7.54	a__b :: g:b:c:f
20.46/7.54	mark :: g:b:c:f -> g:b:c:f
20.46/7.54	f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
20.46/7.54	
20.46/7.54	Rewrite Strategy: INNERMOST
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(7) CompletionProof (UPPER BOUND(ID))
20.46/7.54	The transformation into a RNTS is sound, since:
20.46/7.54	
20.46/7.54	(a) The obligation is a constructor system where every type has a constant constructor,
20.46/7.54	
20.46/7.54	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
20.46/7.54	none
20.46/7.54	
20.46/7.54	(c) The following functions are completely defined:
20.46/7.54	
20.46/7.54	   mark_1
20.46/7.54	   a__b
20.46/7.54	   a__f_3
20.46/7.54	   a__g_1
20.46/7.54	
20.46/7.54	Due to the following rules being added:
20.46/7.54	none
20.46/7.54	
20.46/7.54	And the following fresh constants: none
20.46/7.54	
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(8)
20.46/7.54	Obligation:
20.46/7.54	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.46/7.54	
20.46/7.54	Runtime Complexity Weighted TRS with Types.
20.46/7.54	The TRS R consists of the following rules:
20.46/7.54	
20.46/7.54	   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
20.46/7.54	   a__g(b) -> c [1]
20.46/7.54	   a__b -> c [1]
20.46/7.54	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
20.46/7.54	   mark(g(X)) -> a__g(mark(X)) [1]
20.46/7.54	   mark(b) -> a__b [1]
20.46/7.54	   mark(c) -> c [1]
20.46/7.54	   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
20.46/7.54	   a__g(X) -> g(X) [1]
20.46/7.54	   a__b -> b [1]
20.46/7.54	
20.46/7.54	The TRS has the following type information:
20.46/7.54	a__f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
20.46/7.54	g :: g:b:c:f -> g:b:c:f
20.46/7.54	a__g :: g:b:c:f -> g:b:c:f
20.46/7.54	b :: g:b:c:f
20.46/7.54	c :: g:b:c:f
20.46/7.54	a__b :: g:b:c:f
20.46/7.54	mark :: g:b:c:f -> g:b:c:f
20.46/7.54	f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
20.46/7.54	
20.46/7.54	Rewrite Strategy: INNERMOST
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
20.46/7.54	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
20.46/7.54	----------------------------------------
20.46/7.54	
20.46/7.54	(10)
20.46/7.54	Obligation:
20.46/7.54	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.46/7.54	
20.46/7.54	Runtime Complexity Weighted TRS with Types.
20.46/7.54	The TRS R consists of the following rules:
20.46/7.54	
20.46/7.54	   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
20.46/7.54	   a__g(b) -> c [1]
20.46/7.54	   a__b -> c [1]
20.46/7.54	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
20.46/7.54	   mark(g(f(X1', X2', X3'))) -> a__g(a__f(X1', X2', X3')) [2]
20.46/7.54	   mark(g(g(X'))) -> a__g(a__g(mark(X'))) [2]
20.46/7.54	   mark(g(b)) -> a__g(a__b) [2]
20.46/7.54	   mark(g(c)) -> a__g(c) [2]
20.46/7.54	   mark(b) -> a__b [1]
20.46/7.54	   mark(c) -> c [1]
20.46/7.54	   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
20.46/7.54	   a__g(X) -> g(X) [1]
20.46/7.54	   a__b -> b [1]
20.46/7.54	
20.46/7.54	The TRS has the following type information:
20.46/7.55	a__f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
20.46/7.55	g :: g:b:c:f -> g:b:c:f
20.46/7.55	a__g :: g:b:c:f -> g:b:c:f
20.46/7.55	b :: g:b:c:f
20.46/7.55	c :: g:b:c:f
20.46/7.55	a__b :: g:b:c:f
20.46/7.55	mark :: g:b:c:f -> g:b:c:f
20.46/7.55	f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
20.46/7.55	
20.46/7.55	Rewrite Strategy: INNERMOST
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
20.46/7.55	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
20.46/7.55	The constant constructors are abstracted as follows:
20.46/7.55	
20.46/7.55	   b => 0
20.46/7.55	   c => 1
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(12)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(Y, Y, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__b) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(1) :|: z = 1 + 1
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> a__b :|: z = 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(13) InliningProof (UPPER BOUND(ID))
20.46/7.55	Inlined the following terminating rules on right-hand sides where appropriate:
20.46/7.55	
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(14)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(Y, Y, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
20.46/7.55	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(16)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
20.46/7.55	Found the following analysis order by SCC decomposition:
20.46/7.55	
20.46/7.55	   { a__b }
20.46/7.55	   { a__g }
20.46/7.55	   { a__f }
20.46/7.55	   { mark }
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(18)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__b}, {a__g}, {a__f}, {mark}
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(19) ResultPropagationProof (UPPER BOUND(ID))
20.46/7.55	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(20)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__b}, {a__g}, {a__f}, {mark}
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(21) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed SIZE bound using CoFloCo for: a__b
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(1) with polynomial bound: 1
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(22)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__b}, {a__g}, {a__f}, {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: ?, size: O(1) [1]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(23) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed RUNTIME bound using CoFloCo for: a__b
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(1) with polynomial bound: 1
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(24)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__g}, {a__f}, {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(25) ResultPropagationProof (UPPER BOUND(ID))
20.46/7.55	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(26)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__g}, {a__f}, {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(27) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed SIZE bound using CoFloCo for: a__g
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(n^1) with polynomial bound: 1 + z
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(28)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__g}, {a__f}, {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: ?, size: O(n^1) [1 + z]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(29) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed RUNTIME bound using CoFloCo for: a__g
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(1) with polynomial bound: 1
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(30)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__f}, {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(31) ResultPropagationProof (UPPER BOUND(ID))
20.46/7.55	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(32)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__f}, {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(33) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed SIZE bound using CoFloCo for: a__f
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(n^1) with polynomial bound: 1 + z + z' + 3*z''
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(34)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {a__f}, {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
20.46/7.55	  a__f: runtime: ?, size: O(n^1) [1 + z + z' + 3*z'']
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(35) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed RUNTIME bound using CoFloCo for: a__f
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(1) with polynomial bound: 2
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(36)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
20.46/7.55	  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(37) ResultPropagationProof (UPPER BOUND(ID))
20.46/7.55	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(38)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 3 }-> s'' :|: s'' >= 0, s'' <= z'' + z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 5 }-> s3 :|: s2 >= 0, s2 <= X1' + X2' + 3 * X3' + 1, s3 >= 0, s3 <= s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
20.46/7.55	  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(39) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed SIZE bound using KoAT for: mark
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(n^1) with polynomial bound: 1 + 3*z
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(40)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 3 }-> s'' :|: s'' >= 0, s'' <= z'' + z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 5 }-> s3 :|: s2 >= 0, s2 <= X1' + X2' + 3 * X3' + 1, s3 >= 0, s3 <= s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: {mark}
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
20.46/7.55	  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']
20.46/7.55	  mark: runtime: ?, size: O(n^1) [1 + 3*z]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(41) IntTrsBoundProof (UPPER BOUND(ID))
20.46/7.55	
20.46/7.55	Computed RUNTIME bound using KoAT for: mark
20.46/7.55	after applying outer abstraction to obtain an ITS,
20.46/7.55	resulting in: O(n^1) with polynomial bound: 22 + 4*z
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(42)
20.46/7.55	Obligation:
20.46/7.55	Complexity RNTS consisting of the following rules:
20.46/7.55	
20.46/7.55	   a__b -{ 1 }-> 1 :|: 
20.46/7.55	   a__b -{ 1 }-> 0 :|: 
20.46/7.55	   a__f(z, z', z'') -{ 3 }-> s'' :|: s'' >= 0, s'' <= z'' + z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
20.46/7.55	   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 :|: z = 0
20.46/7.55	   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
20.46/7.55	   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
20.46/7.55	   mark(z) -{ 3 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
20.46/7.55	   mark(z) -{ 5 }-> s3 :|: s2 >= 0, s2 <= X1' + X2' + 3 * X3' + 1, s3 >= 0, s3 <= s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
20.46/7.55	   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
20.46/7.55	   mark(z) -{ 1 }-> 1 :|: z = 1
20.46/7.55	   mark(z) -{ 2 }-> 1 :|: z = 0
20.46/7.55	   mark(z) -{ 2 }-> 0 :|: z = 0
20.46/7.55	   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
20.46/7.55	
20.46/7.55	Function symbols to be analyzed: 
20.46/7.55	Previous analysis results are:
20.46/7.55	  a__b: runtime: O(1) [1], size: O(1) [1]
20.46/7.55	  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
20.46/7.55	  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']
20.46/7.55	  mark: runtime: O(n^1) [22 + 4*z], size: O(n^1) [1 + 3*z]
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(43) FinalProof (FINISHED)
20.46/7.55	Computed overall runtime complexity
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(44)
20.46/7.55	BOUNDS(1, n^1)
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(45) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
20.46/7.55	Transformed a relative TRS into a decreasing-loop problem.
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(46)
20.46/7.55	Obligation:
20.46/7.55	Analyzing the following TRS for decreasing loops:
20.46/7.55	
20.46/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.46/7.55	
20.46/7.55	
20.46/7.55	The TRS R consists of the following rules:
20.46/7.55	
20.46/7.55	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
20.46/7.55	   a__g(b) -> c
20.46/7.55	   a__b -> c
20.46/7.55	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
20.46/7.55	   mark(g(X)) -> a__g(mark(X))
20.46/7.55	   mark(b) -> a__b
20.46/7.55	   mark(c) -> c
20.46/7.55	   a__f(X1, X2, X3) -> f(X1, X2, X3)
20.46/7.55	   a__g(X) -> g(X)
20.46/7.55	   a__b -> b
20.46/7.55	
20.46/7.55	S is empty.
20.46/7.55	Rewrite Strategy: FULL
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(47) DecreasingLoopProof (LOWER BOUND(ID))
20.46/7.55	The following loop(s) give(s) rise to the lower bound Omega(n^1):
20.46/7.55	
20.46/7.55	The rewrite sequence
20.46/7.55	
20.46/7.55	mark(g(X)) ->^+ a__g(mark(X))
20.46/7.55	
20.46/7.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
20.46/7.55	
20.46/7.55	The pumping substitution is [X / g(X)].
20.46/7.55	
20.46/7.55	The result substitution is [ ].
20.46/7.55	
20.46/7.55	
20.46/7.55	
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(48)
20.46/7.55	Complex Obligation (BEST)
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(49)
20.46/7.55	Obligation:
20.46/7.55	Proved the lower bound n^1 for the following obligation:
20.46/7.55	
20.46/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.46/7.55	
20.46/7.55	
20.46/7.55	The TRS R consists of the following rules:
20.46/7.55	
20.46/7.55	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
20.46/7.55	   a__g(b) -> c
20.46/7.55	   a__b -> c
20.46/7.55	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
20.46/7.55	   mark(g(X)) -> a__g(mark(X))
20.46/7.55	   mark(b) -> a__b
20.46/7.55	   mark(c) -> c
20.46/7.55	   a__f(X1, X2, X3) -> f(X1, X2, X3)
20.46/7.55	   a__g(X) -> g(X)
20.46/7.55	   a__b -> b
20.46/7.55	
20.46/7.55	S is empty.
20.46/7.55	Rewrite Strategy: FULL
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(50) LowerBoundPropagationProof (FINISHED)
20.46/7.55	Propagated lower bound.
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(51)
20.46/7.55	BOUNDS(n^1, INF)
20.46/7.55	
20.46/7.55	----------------------------------------
20.46/7.55	
20.46/7.55	(52)
20.46/7.55	Obligation:
20.46/7.55	Analyzing the following TRS for decreasing loops:
20.46/7.55	
20.46/7.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.46/7.55	
20.46/7.55	
20.46/7.55	The TRS R consists of the following rules:
20.46/7.55	
20.46/7.55	   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
20.46/7.55	   a__g(b) -> c
20.46/7.55	   a__b -> c
20.46/7.55	   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
20.46/7.55	   mark(g(X)) -> a__g(mark(X))
20.46/7.55	   mark(b) -> a__b
20.46/7.55	   mark(c) -> c
20.46/7.55	   a__f(X1, X2, X3) -> f(X1, X2, X3)
20.46/7.55	   a__g(X) -> g(X)
20.46/7.55	   a__b -> b
20.46/7.55	
20.46/7.55	S is empty.
20.46/7.55	Rewrite Strategy: FULL
20.67/7.58	EOF