40.39/11.45	WORST_CASE(Omega(n^1), O(n^1))
48.39/13.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
48.39/13.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
48.39/13.49	
48.39/13.49	
48.39/13.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
48.39/13.49	
48.39/13.49	(0) CpxTRS
48.39/13.49	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
48.39/13.49	(2) CpxWeightedTrs
48.39/13.49	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
48.39/13.49	    (4) CpxTypedWeightedTrs
48.39/13.49	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
48.39/13.49	    (6) CpxTypedWeightedCompleteTrs
48.39/13.49	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
48.39/13.49	    (8) CpxTypedWeightedCompleteTrs
48.39/13.49	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
48.39/13.49	    (10) CpxRNTS
48.39/13.49	    (11) InliningProof [UPPER BOUND(ID), 297 ms]
48.39/13.49	    (12) CpxRNTS
48.39/13.49	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
48.39/13.49	    (14) CpxRNTS
48.39/13.49	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
48.39/13.49	    (16) CpxRNTS
48.39/13.49	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
48.39/13.49	    (18) CpxRNTS
48.39/13.49	    (19) IntTrsBoundProof [UPPER BOUND(ID), 243 ms]
48.39/13.49	    (20) CpxRNTS
48.39/13.49	    (21) IntTrsBoundProof [UPPER BOUND(ID), 79 ms]
48.39/13.49	    (22) CpxRNTS
48.39/13.49	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
48.39/13.49	    (24) CpxRNTS
48.39/13.49	    (25) IntTrsBoundProof [UPPER BOUND(ID), 647 ms]
48.39/13.49	    (26) CpxRNTS
48.39/13.49	    (27) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
48.39/13.49	    (28) CpxRNTS
48.39/13.49	    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
48.39/13.49	    (30) CpxRNTS
48.39/13.49	    (31) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
48.39/13.49	    (32) CpxRNTS
48.39/13.49	    (33) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
48.39/13.49	    (34) CpxRNTS
48.39/13.49	    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
48.39/13.49	    (36) CpxRNTS
48.39/13.49	    (37) IntTrsBoundProof [UPPER BOUND(ID), 386 ms]
48.39/13.49	    (38) CpxRNTS
48.39/13.49	    (39) IntTrsBoundProof [UPPER BOUND(ID), 166 ms]
48.39/13.49	    (40) CpxRNTS
48.39/13.49	    (41) FinalProof [FINISHED, 0 ms]
48.39/13.49	    (42) BOUNDS(1, n^1)
48.39/13.49	(43) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
48.39/13.49	(44) TRS for Loop Detection
48.39/13.49	    (45) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
48.39/13.49	    (46) BEST
48.39/13.49	        (47) proven lower bound
48.39/13.49	            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
48.39/13.49	            (49) BOUNDS(n^1, INF)
48.39/13.49	        (50) TRS for Loop Detection
48.39/13.49	
48.39/13.49	
48.39/13.49	----------------------------------------
48.39/13.49	
48.39/13.49	(0)
48.39/13.49	Obligation:
48.39/13.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
48.39/13.49	
48.39/13.49	
48.39/13.49	The TRS R consists of the following rules:
48.39/13.49	
48.39/13.49	   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y))
48.39/13.49	   and(true, true) -> true
48.39/13.49	   and(x, false) -> false
48.39/13.49	   and(false, x) -> false
48.39/13.49	   gr(0, 0) -> false
48.39/13.49	   gr(0, x) -> false
48.39/13.49	   gr(s(x), 0) -> true
48.39/13.49	   gr(s(x), s(y)) -> gr(x, y)
48.39/13.49	   p(0) -> 0
48.39/13.49	   p(s(x)) -> x
48.39/13.49	
48.39/13.49	S is empty.
48.39/13.49	Rewrite Strategy: INNERMOST
48.39/13.49	----------------------------------------
48.39/13.49	
48.39/13.49	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
48.39/13.49	Transformed relative TRS to weighted TRS
48.39/13.49	----------------------------------------
48.39/13.49	
48.39/13.49	(2)
48.39/13.49	Obligation:
48.39/13.49	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
48.39/13.49	
48.39/13.49	
48.39/13.49	The TRS R consists of the following rules:
48.39/13.49	
48.39/13.49	   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y)) [1]
48.39/13.49	   and(true, true) -> true [1]
48.39/13.49	   and(x, false) -> false [1]
48.39/13.49	   and(false, x) -> false [1]
48.39/13.49	   gr(0, 0) -> false [1]
48.39/13.49	   gr(0, x) -> false [1]
48.39/13.49	   gr(s(x), 0) -> true [1]
48.39/13.49	   gr(s(x), s(y)) -> gr(x, y) [1]
48.39/13.49	   p(0) -> 0 [1]
48.39/13.49	   p(s(x)) -> x [1]
48.39/13.49	
48.39/13.49	Rewrite Strategy: INNERMOST
48.39/13.49	----------------------------------------
48.39/13.49	
48.39/13.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
48.39/13.49	Infered types.
48.39/13.49	----------------------------------------
48.39/13.49	
48.39/13.49	(4)
48.39/13.49	Obligation:
48.39/13.49	Runtime Complexity Weighted TRS with Types.
48.39/13.49	The TRS R consists of the following rules:
48.39/13.49	
48.39/13.49	   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y)) [1]
48.39/13.49	   and(true, true) -> true [1]
48.39/13.49	   and(x, false) -> false [1]
48.39/13.49	   and(false, x) -> false [1]
48.39/13.49	   gr(0, 0) -> false [1]
48.39/13.49	   gr(0, x) -> false [1]
48.39/13.49	   gr(s(x), 0) -> true [1]
48.39/13.49	   gr(s(x), s(y)) -> gr(x, y) [1]
48.39/13.49	   p(0) -> 0 [1]
48.39/13.49	   p(s(x)) -> x [1]
48.39/13.49	
48.39/13.49	The TRS has the following type information:
48.39/13.49	cond :: true:false -> 0:s -> 0:s -> cond
48.39/13.49	true :: true:false
48.39/13.49	and :: true:false -> true:false -> true:false
48.39/13.49	gr :: 0:s -> 0:s -> true:false
48.39/13.49	0 :: 0:s
48.39/13.49	p :: 0:s -> 0:s
48.39/13.49	false :: true:false
48.39/13.49	s :: 0:s -> 0:s
48.39/13.50	
48.39/13.50	Rewrite Strategy: INNERMOST
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(5) CompletionProof (UPPER BOUND(ID))
48.39/13.50	The transformation into a RNTS is sound, since:
48.39/13.50	
48.39/13.50	(a) The obligation is a constructor system where every type has a constant constructor,
48.39/13.50	
48.39/13.50	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
48.39/13.50	
48.39/13.50	   cond_3
48.39/13.50	
48.39/13.50	(c) The following functions are completely defined:
48.39/13.50	
48.39/13.50	   and_2
48.39/13.50	   gr_2
48.39/13.50	   p_1
48.39/13.50	
48.39/13.50	Due to the following rules being added:
48.39/13.50	none
48.39/13.50	
48.39/13.50	And the following fresh constants:    const
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(6)
48.39/13.50	Obligation:
48.39/13.50	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
48.39/13.50	
48.39/13.50	Runtime Complexity Weighted TRS with Types.
48.39/13.50	The TRS R consists of the following rules:
48.39/13.50	
48.39/13.50	   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y)) [1]
48.39/13.50	   and(true, true) -> true [1]
48.39/13.50	   and(x, false) -> false [1]
48.39/13.50	   and(false, x) -> false [1]
48.39/13.50	   gr(0, 0) -> false [1]
48.39/13.50	   gr(0, x) -> false [1]
48.39/13.50	   gr(s(x), 0) -> true [1]
48.39/13.50	   gr(s(x), s(y)) -> gr(x, y) [1]
48.39/13.50	   p(0) -> 0 [1]
48.39/13.50	   p(s(x)) -> x [1]
48.39/13.50	
48.39/13.50	The TRS has the following type information:
48.39/13.50	cond :: true:false -> 0:s -> 0:s -> cond
48.39/13.50	true :: true:false
48.39/13.50	and :: true:false -> true:false -> true:false
48.39/13.50	gr :: 0:s -> 0:s -> true:false
48.39/13.50	0 :: 0:s
48.39/13.50	p :: 0:s -> 0:s
48.39/13.50	false :: true:false
48.39/13.50	s :: 0:s -> 0:s
48.39/13.50	const :: cond
48.39/13.50	
48.39/13.50	Rewrite Strategy: INNERMOST
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
48.39/13.50	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(8)
48.39/13.50	Obligation:
48.39/13.50	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
48.39/13.50	
48.39/13.50	Runtime Complexity Weighted TRS with Types.
48.39/13.50	The TRS R consists of the following rules:
48.39/13.50	
48.39/13.50	   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
48.39/13.50	   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
48.39/13.50	   cond(true, 0, s(x'')) -> cond(and(false, true), 0, x'') [5]
48.39/13.50	   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
48.39/13.50	   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
48.39/13.50	   cond(true, 0, s(x1)) -> cond(and(false, true), 0, x1) [5]
48.39/13.50	   cond(true, s(x'), 0) -> cond(and(true, false), x', 0) [5]
48.39/13.50	   cond(true, s(x'), 0) -> cond(and(true, false), x', 0) [5]
48.39/13.50	   cond(true, s(x'), s(x2)) -> cond(and(true, true), x', x2) [5]
48.39/13.50	   and(true, true) -> true [1]
48.39/13.50	   and(x, false) -> false [1]
48.39/13.50	   and(false, x) -> false [1]
48.39/13.50	   gr(0, 0) -> false [1]
48.39/13.50	   gr(0, x) -> false [1]
48.39/13.50	   gr(s(x), 0) -> true [1]
48.39/13.50	   gr(s(x), s(y)) -> gr(x, y) [1]
48.39/13.50	   p(0) -> 0 [1]
48.39/13.50	   p(s(x)) -> x [1]
48.39/13.50	
48.39/13.50	The TRS has the following type information:
48.39/13.50	cond :: true:false -> 0:s -> 0:s -> cond
48.39/13.50	true :: true:false
48.39/13.50	and :: true:false -> true:false -> true:false
48.39/13.50	gr :: 0:s -> 0:s -> true:false
48.39/13.50	0 :: 0:s
48.39/13.50	p :: 0:s -> 0:s
48.39/13.50	false :: true:false
48.39/13.50	s :: 0:s -> 0:s
48.39/13.50	const :: cond
48.39/13.50	
48.39/13.50	Rewrite Strategy: INNERMOST
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
48.39/13.50	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
48.39/13.50	The constant constructors are abstracted as follows:
48.39/13.50	
48.39/13.50	   true => 1
48.39/13.50	   0 => 0
48.39/13.50	   false => 0
48.39/13.50	   const => 0
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(10)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 5 }-> cond(and(1, 1), x', x2) :|: z' = 1 + x', z = 1, x' >= 0, z'' = 1 + x2, x2 >= 0
48.39/13.50	   cond(z, z', z'') -{ 5 }-> cond(and(1, 0), x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
48.39/13.50	   cond(z, z', z'') -{ 5 }-> cond(and(0, 1), 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0
48.39/13.50	   cond(z, z', z'') -{ 5 }-> cond(and(0, 1), 0, x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0
48.39/13.50	   cond(z, z', z'') -{ 5 }-> cond(and(0, 0), 0, 0) :|: z'' = 0, z = 1, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(11) InliningProof (UPPER BOUND(ID))
48.39/13.50	Inlined the following terminating rules on right-hand sides where appropriate:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(12)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, x', x2) :|: z' = 1 + x', z = 1, x' >= 0, z'' = 1 + x2, x2 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
48.39/13.50	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(14)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
48.39/13.50	Found the following analysis order by SCC decomposition:
48.39/13.50	
48.39/13.50	   { and }
48.39/13.50	   { cond }
48.39/13.50	   { p }
48.39/13.50	   { gr }
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(16)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {and}, {cond}, {p}, {gr}
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(17) ResultPropagationProof (UPPER BOUND(ID))
48.39/13.50	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(18)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {and}, {cond}, {p}, {gr}
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(19) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed SIZE bound using CoFloCo for: and
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(1) with polynomial bound: 1
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(20)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {and}, {cond}, {p}, {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: ?, size: O(1) [1]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(21) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed RUNTIME bound using CoFloCo for: and
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(1) with polynomial bound: 1
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(22)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {cond}, {p}, {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(23) ResultPropagationProof (UPPER BOUND(ID))
48.39/13.50	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(24)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {cond}, {p}, {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(25) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed SIZE bound using CoFloCo for: cond
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(1) with polynomial bound: 0
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(26)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {cond}, {p}, {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: ?, size: O(1) [0]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(27) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed RUNTIME bound using KoAT for: cond
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(n^1) with polynomial bound: 24*z + 24*z'
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(28)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {p}, {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(29) ResultPropagationProof (UPPER BOUND(ID))
48.39/13.50	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(30)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {p}, {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(31) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed SIZE bound using KoAT for: p
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(n^1) with polynomial bound: z
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(32)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {p}, {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
48.39/13.50	  p: runtime: ?, size: O(n^1) [z]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(33) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed RUNTIME bound using CoFloCo for: p
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(1) with polynomial bound: 1
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(34)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
48.39/13.50	  p: runtime: O(1) [1], size: O(n^1) [z]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(35) ResultPropagationProof (UPPER BOUND(ID))
48.39/13.50	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(36)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
48.39/13.50	  p: runtime: O(1) [1], size: O(n^1) [z]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(37) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed SIZE bound using CoFloCo for: gr
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(1) with polynomial bound: 1
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(38)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: {gr}
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
48.39/13.50	  p: runtime: O(1) [1], size: O(n^1) [z]
48.39/13.50	  gr: runtime: ?, size: O(1) [1]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(39) IntTrsBoundProof (UPPER BOUND(ID))
48.39/13.50	
48.39/13.50	Computed RUNTIME bound using KoAT for: gr
48.39/13.50	after applying outer abstraction to obtain an ITS,
48.39/13.50	resulting in: O(n^1) with polynomial bound: 3 + z'
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(40)
48.39/13.50	Obligation:
48.39/13.50	Complexity RNTS consisting of the following rules:
48.39/13.50	
48.39/13.50	   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
48.39/13.50	   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
48.39/13.50	   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
48.39/13.50	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
48.39/13.50	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
48.39/13.50	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
48.39/13.50	   p(z) -{ 1 }-> 0 :|: z = 0
48.39/13.50	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
48.39/13.50	
48.39/13.50	Function symbols to be analyzed: 
48.39/13.50	Previous analysis results are:
48.39/13.50	  and: runtime: O(1) [1], size: O(1) [1]
48.39/13.50	  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
48.39/13.50	  p: runtime: O(1) [1], size: O(n^1) [z]
48.39/13.50	  gr: runtime: O(n^1) [3 + z'], size: O(1) [1]
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(41) FinalProof (FINISHED)
48.39/13.50	Computed overall runtime complexity
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(42)
48.39/13.50	BOUNDS(1, n^1)
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(43) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
48.39/13.50	Transformed a relative TRS into a decreasing-loop problem.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(44)
48.39/13.50	Obligation:
48.39/13.50	Analyzing the following TRS for decreasing loops:
48.39/13.50	
48.39/13.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
48.39/13.50	
48.39/13.50	
48.39/13.50	The TRS R consists of the following rules:
48.39/13.50	
48.39/13.50	   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y))
48.39/13.50	   and(true, true) -> true
48.39/13.50	   and(x, false) -> false
48.39/13.50	   and(false, x) -> false
48.39/13.50	   gr(0, 0) -> false
48.39/13.50	   gr(0, x) -> false
48.39/13.50	   gr(s(x), 0) -> true
48.39/13.50	   gr(s(x), s(y)) -> gr(x, y)
48.39/13.50	   p(0) -> 0
48.39/13.50	   p(s(x)) -> x
48.39/13.50	
48.39/13.50	S is empty.
48.39/13.50	Rewrite Strategy: INNERMOST
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(45) DecreasingLoopProof (LOWER BOUND(ID))
48.39/13.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
48.39/13.50	
48.39/13.50	The rewrite sequence
48.39/13.50	
48.39/13.50	gr(s(x), s(y)) ->^+ gr(x, y)
48.39/13.50	
48.39/13.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
48.39/13.50	
48.39/13.50	The pumping substitution is [x / s(x), y / s(y)].
48.39/13.50	
48.39/13.50	The result substitution is [ ].
48.39/13.50	
48.39/13.50	
48.39/13.50	
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(46)
48.39/13.50	Complex Obligation (BEST)
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(47)
48.39/13.50	Obligation:
48.39/13.50	Proved the lower bound n^1 for the following obligation:
48.39/13.50	
48.39/13.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
48.39/13.50	
48.39/13.50	
48.39/13.50	The TRS R consists of the following rules:
48.39/13.50	
48.39/13.50	   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y))
48.39/13.50	   and(true, true) -> true
48.39/13.50	   and(x, false) -> false
48.39/13.50	   and(false, x) -> false
48.39/13.50	   gr(0, 0) -> false
48.39/13.50	   gr(0, x) -> false
48.39/13.50	   gr(s(x), 0) -> true
48.39/13.50	   gr(s(x), s(y)) -> gr(x, y)
48.39/13.50	   p(0) -> 0
48.39/13.50	   p(s(x)) -> x
48.39/13.50	
48.39/13.50	S is empty.
48.39/13.50	Rewrite Strategy: INNERMOST
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(48) LowerBoundPropagationProof (FINISHED)
48.39/13.50	Propagated lower bound.
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(49)
48.39/13.50	BOUNDS(n^1, INF)
48.39/13.50	
48.39/13.50	----------------------------------------
48.39/13.50	
48.39/13.50	(50)
48.39/13.50	Obligation:
48.39/13.50	Analyzing the following TRS for decreasing loops:
48.39/13.50	
48.39/13.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
48.39/13.50	
48.39/13.50	
48.39/13.50	The TRS R consists of the following rules:
48.39/13.50	
48.39/13.50	   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y))
48.39/13.50	   and(true, true) -> true
48.39/13.50	   and(x, false) -> false
48.39/13.50	   and(false, x) -> false
48.39/13.50	   gr(0, 0) -> false
48.39/13.50	   gr(0, x) -> false
48.39/13.50	   gr(s(x), 0) -> true
48.39/13.50	   gr(s(x), s(y)) -> gr(x, y)
48.39/13.50	   p(0) -> 0
48.39/13.50	   p(s(x)) -> x
48.39/13.50	
48.39/13.50	S is empty.
48.39/13.50	Rewrite Strategy: INNERMOST
48.70/13.57	EOF