380.31/291.56	WORST_CASE(Omega(n^1), O(n^2))
380.31/291.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
380.31/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
380.31/291.58	
380.31/291.58	
380.31/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
380.31/291.58	
380.31/291.58	(0) CpxTRS
380.31/291.58	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
380.31/291.58	(2) CpxWeightedTrs
380.31/291.58	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
380.31/291.58	    (4) CpxTypedWeightedTrs
380.31/291.58	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
380.31/291.58	    (6) CpxTypedWeightedCompleteTrs
380.31/291.58	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
380.31/291.58	    (8) CpxTypedWeightedCompleteTrs
380.31/291.58	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
380.31/291.58	    (10) CpxRNTS
380.31/291.58	    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
380.31/291.58	    (12) CpxRNTS
380.31/291.58	    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
380.31/291.58	    (14) CpxRNTS
380.31/291.58	    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
380.31/291.58	    (16) CpxRNTS
380.31/291.58	    (17) IntTrsBoundProof [UPPER BOUND(ID), 227 ms]
380.31/291.58	    (18) CpxRNTS
380.31/291.58	    (19) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
380.31/291.58	    (20) CpxRNTS
380.31/291.58	    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
380.31/291.58	    (22) CpxRNTS
380.31/291.58	    (23) IntTrsBoundProof [UPPER BOUND(ID), 385 ms]
380.31/291.58	    (24) CpxRNTS
380.31/291.58	    (25) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
380.31/291.58	    (26) CpxRNTS
380.31/291.58	    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
380.31/291.58	    (28) CpxRNTS
380.31/291.58	    (29) IntTrsBoundProof [UPPER BOUND(ID), 643 ms]
380.31/291.58	    (30) CpxRNTS
380.31/291.58	    (31) IntTrsBoundProof [UPPER BOUND(ID), 337 ms]
380.31/291.58	    (32) CpxRNTS
380.31/291.58	    (33) FinalProof [FINISHED, 0 ms]
380.31/291.58	    (34) BOUNDS(1, n^2)
380.31/291.58	(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
380.31/291.58	(36) CpxTRS
380.31/291.58	    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
380.31/291.58	    (38) typed CpxTrs
380.31/291.58	    (39) OrderProof [LOWER BOUND(ID), 0 ms]
380.31/291.58	    (40) typed CpxTrs
380.31/291.58	    (41) RewriteLemmaProof [LOWER BOUND(ID), 307 ms]
380.31/291.58	    (42) BEST
380.31/291.58	        (43) proven lower bound
380.31/291.58	            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
380.31/291.58	            (45) BOUNDS(n^1, INF)
380.31/291.58	        (46) typed CpxTrs
380.31/291.58	
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(0)
380.31/291.58	Obligation:
380.31/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
380.31/291.58	
380.31/291.58	
380.31/291.58	The TRS R consists of the following rules:
380.31/291.58	
380.31/291.58	   cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
380.31/291.58	   gr(0, x) -> false
380.31/291.58	   gr(s(x), 0) -> true
380.31/291.58	   gr(s(x), s(y)) -> gr(x, y)
380.31/291.58	   p(0) -> 0
380.31/291.58	   p(s(x)) -> x
380.31/291.58	
380.31/291.58	S is empty.
380.31/291.58	Rewrite Strategy: INNERMOST
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
380.31/291.58	Transformed relative TRS to weighted TRS
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(2)
380.31/291.58	Obligation:
380.31/291.58	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
380.31/291.58	
380.31/291.58	
380.31/291.58	The TRS R consists of the following rules:
380.31/291.58	
380.31/291.58	   cond(true, x, y) -> cond(gr(x, y), p(x), s(y)) [1]
380.31/291.58	   gr(0, x) -> false [1]
380.31/291.58	   gr(s(x), 0) -> true [1]
380.31/291.58	   gr(s(x), s(y)) -> gr(x, y) [1]
380.31/291.58	   p(0) -> 0 [1]
380.31/291.58	   p(s(x)) -> x [1]
380.31/291.58	
380.31/291.58	Rewrite Strategy: INNERMOST
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
380.31/291.58	Infered types.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(4)
380.31/291.58	Obligation:
380.31/291.58	Runtime Complexity Weighted TRS with Types.
380.31/291.58	The TRS R consists of the following rules:
380.31/291.58	
380.31/291.58	   cond(true, x, y) -> cond(gr(x, y), p(x), s(y)) [1]
380.31/291.58	   gr(0, x) -> false [1]
380.31/291.58	   gr(s(x), 0) -> true [1]
380.31/291.58	   gr(s(x), s(y)) -> gr(x, y) [1]
380.31/291.58	   p(0) -> 0 [1]
380.31/291.58	   p(s(x)) -> x [1]
380.31/291.58	
380.31/291.58	The TRS has the following type information:
380.31/291.58	cond :: true:false -> s:0 -> s:0 -> cond
380.31/291.58	true :: true:false
380.31/291.58	gr :: s:0 -> s:0 -> true:false
380.31/291.58	p :: s:0 -> s:0
380.31/291.58	s :: s:0 -> s:0
380.31/291.58	0 :: s:0
380.31/291.58	false :: true:false
380.31/291.58	
380.31/291.58	Rewrite Strategy: INNERMOST
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(5) CompletionProof (UPPER BOUND(ID))
380.31/291.58	The transformation into a RNTS is sound, since:
380.31/291.58	
380.31/291.58	(a) The obligation is a constructor system where every type has a constant constructor,
380.31/291.58	
380.31/291.58	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
380.31/291.58	
380.31/291.58	   cond_3
380.31/291.58	
380.31/291.58	(c) The following functions are completely defined:
380.31/291.58	
380.31/291.58	   gr_2
380.31/291.58	   p_1
380.31/291.58	
380.31/291.58	Due to the following rules being added:
380.31/291.58	none
380.31/291.58	
380.31/291.58	And the following fresh constants:    const
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(6)
380.31/291.58	Obligation:
380.31/291.58	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
380.31/291.58	
380.31/291.58	Runtime Complexity Weighted TRS with Types.
380.31/291.58	The TRS R consists of the following rules:
380.31/291.58	
380.31/291.58	   cond(true, x, y) -> cond(gr(x, y), p(x), s(y)) [1]
380.31/291.58	   gr(0, x) -> false [1]
380.31/291.58	   gr(s(x), 0) -> true [1]
380.31/291.58	   gr(s(x), s(y)) -> gr(x, y) [1]
380.31/291.58	   p(0) -> 0 [1]
380.31/291.58	   p(s(x)) -> x [1]
380.31/291.58	
380.31/291.58	The TRS has the following type information:
380.31/291.58	cond :: true:false -> s:0 -> s:0 -> cond
380.31/291.58	true :: true:false
380.31/291.58	gr :: s:0 -> s:0 -> true:false
380.31/291.58	p :: s:0 -> s:0
380.31/291.58	s :: s:0 -> s:0
380.31/291.58	0 :: s:0
380.31/291.58	false :: true:false
380.31/291.58	const :: cond
380.31/291.58	
380.31/291.58	Rewrite Strategy: INNERMOST
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
380.31/291.58	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(8)
380.31/291.58	Obligation:
380.31/291.58	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
380.31/291.58	
380.31/291.58	Runtime Complexity Weighted TRS with Types.
380.31/291.58	The TRS R consists of the following rules:
380.31/291.58	
380.31/291.58	   cond(true, 0, y) -> cond(false, 0, s(y)) [3]
380.31/291.58	   cond(true, s(x'), 0) -> cond(true, x', s(0)) [3]
380.31/291.58	   cond(true, s(x''), s(y')) -> cond(gr(x'', y'), x'', s(s(y'))) [3]
380.31/291.58	   gr(0, x) -> false [1]
380.31/291.58	   gr(s(x), 0) -> true [1]
380.31/291.58	   gr(s(x), s(y)) -> gr(x, y) [1]
380.31/291.58	   p(0) -> 0 [1]
380.31/291.58	   p(s(x)) -> x [1]
380.31/291.58	
380.31/291.58	The TRS has the following type information:
380.31/291.58	cond :: true:false -> s:0 -> s:0 -> cond
380.31/291.58	true :: true:false
380.31/291.58	gr :: s:0 -> s:0 -> true:false
380.31/291.58	p :: s:0 -> s:0
380.31/291.58	s :: s:0 -> s:0
380.31/291.58	0 :: s:0
380.31/291.58	false :: true:false
380.31/291.58	const :: cond
380.31/291.58	
380.31/291.58	Rewrite Strategy: INNERMOST
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
380.31/291.58	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
380.31/291.58	The constant constructors are abstracted as follows:
380.31/291.58	
380.31/291.58	   true => 1
380.31/291.58	   0 => 0
380.31/291.58	   false => 0
380.31/291.58	   const => 0
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(10)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(x'', y'), x'', 1 + (1 + y')) :|: z' = 1 + x'', z = 1, y' >= 0, x'' >= 0, z'' = 1 + y'
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, x', 1 + 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + y) :|: z'' = y, z = 1, y >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(11) SimplificationProof (BOTH BOUNDS(ID, ID))
380.31/291.58	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(12)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
380.31/291.58	Found the following analysis order by SCC decomposition:
380.31/291.58	
380.31/291.58	   { p }
380.31/291.58	   { gr }
380.31/291.58	   { cond }
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(14)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {p}, {gr}, {cond}
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(15) ResultPropagationProof (UPPER BOUND(ID))
380.31/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(16)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {p}, {gr}, {cond}
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(17) IntTrsBoundProof (UPPER BOUND(ID))
380.31/291.58	
380.31/291.58	Computed SIZE bound using KoAT for: p
380.31/291.58	after applying outer abstraction to obtain an ITS,
380.31/291.58	resulting in: O(n^1) with polynomial bound: z
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(18)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {p}, {gr}, {cond}
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: ?, size: O(n^1) [z]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(19) IntTrsBoundProof (UPPER BOUND(ID))
380.31/291.58	
380.31/291.58	Computed RUNTIME bound using CoFloCo for: p
380.31/291.58	after applying outer abstraction to obtain an ITS,
380.31/291.58	resulting in: O(1) with polynomial bound: 1
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(20)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {gr}, {cond}
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: O(1) [1], size: O(n^1) [z]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(21) ResultPropagationProof (UPPER BOUND(ID))
380.31/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(22)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {gr}, {cond}
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: O(1) [1], size: O(n^1) [z]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(23) IntTrsBoundProof (UPPER BOUND(ID))
380.31/291.58	
380.31/291.58	Computed SIZE bound using CoFloCo for: gr
380.31/291.58	after applying outer abstraction to obtain an ITS,
380.31/291.58	resulting in: O(1) with polynomial bound: 1
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(24)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {gr}, {cond}
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: O(1) [1], size: O(n^1) [z]
380.31/291.58	  gr: runtime: ?, size: O(1) [1]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(25) IntTrsBoundProof (UPPER BOUND(ID))
380.31/291.58	
380.31/291.58	Computed RUNTIME bound using KoAT for: gr
380.31/291.58	after applying outer abstraction to obtain an ITS,
380.31/291.58	resulting in: O(n^1) with polynomial bound: 2 + z'
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(26)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {cond}
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: O(1) [1], size: O(n^1) [z]
380.31/291.58	  gr: runtime: O(n^1) [2 + z'], size: O(1) [1]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(27) ResultPropagationProof (UPPER BOUND(ID))
380.31/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(28)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 4 + z'' }-> cond(s, z' - 1, 1 + (1 + (z'' - 1))) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {cond}
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: O(1) [1], size: O(n^1) [z]
380.31/291.58	  gr: runtime: O(n^1) [2 + z'], size: O(1) [1]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(29) IntTrsBoundProof (UPPER BOUND(ID))
380.31/291.58	
380.31/291.58	Computed SIZE bound using CoFloCo for: cond
380.31/291.58	after applying outer abstraction to obtain an ITS,
380.31/291.58	resulting in: O(1) with polynomial bound: 0
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(30)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 4 + z'' }-> cond(s, z' - 1, 1 + (1 + (z'' - 1))) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: {cond}
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: O(1) [1], size: O(n^1) [z]
380.31/291.58	  gr: runtime: O(n^1) [2 + z'], size: O(1) [1]
380.31/291.58	  cond: runtime: ?, size: O(1) [0]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(31) IntTrsBoundProof (UPPER BOUND(ID))
380.31/291.58	
380.31/291.58	Computed RUNTIME bound using CoFloCo for: cond
380.31/291.58	after applying outer abstraction to obtain an ITS,
380.31/291.58	resulting in: O(n^2) with polynomial bound: 9 + 4*z' + z'*z'' + z'^2
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(32)
380.31/291.58	Obligation:
380.31/291.58	Complexity RNTS consisting of the following rules:
380.31/291.58	
380.31/291.58	   cond(z, z', z'') -{ 4 + z'' }-> cond(s, z' - 1, 1 + (1 + (z'' - 1))) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
380.31/291.58	   cond(z, z', z'') -{ 3 }-> cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
380.31/291.58	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
380.31/291.58	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
380.31/291.58	   p(z) -{ 1 }-> 0 :|: z = 0
380.31/291.58	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
380.31/291.58	
380.31/291.58	Function symbols to be analyzed: 
380.31/291.58	Previous analysis results are:
380.31/291.58	  p: runtime: O(1) [1], size: O(n^1) [z]
380.31/291.58	  gr: runtime: O(n^1) [2 + z'], size: O(1) [1]
380.31/291.58	  cond: runtime: O(n^2) [9 + 4*z' + z'*z'' + z'^2], size: O(1) [0]
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(33) FinalProof (FINISHED)
380.31/291.58	Computed overall runtime complexity
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(34)
380.31/291.58	BOUNDS(1, n^2)
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(35) RenamingProof (BOTH BOUNDS(ID, ID))
380.31/291.58	Renamed function symbols to avoid clashes with predefined symbol.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(36)
380.31/291.58	Obligation:
380.31/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
380.31/291.58	
380.31/291.58	
380.31/291.58	The TRS R consists of the following rules:
380.31/291.58	
380.31/291.58	   cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
380.31/291.58	   gr(0', x) -> false
380.31/291.58	   gr(s(x), 0') -> true
380.31/291.58	   gr(s(x), s(y)) -> gr(x, y)
380.31/291.58	   p(0') -> 0'
380.31/291.58	   p(s(x)) -> x
380.31/291.58	
380.31/291.58	S is empty.
380.31/291.58	Rewrite Strategy: INNERMOST
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(37) TypeInferenceProof (BOTH BOUNDS(ID, ID))
380.31/291.58	Infered types.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(38)
380.31/291.58	Obligation:
380.31/291.58	Innermost TRS:
380.31/291.58	Rules:
380.31/291.58	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
380.31/291.58	gr(0', x) -> false
380.31/291.58	gr(s(x), 0') -> true
380.31/291.58	gr(s(x), s(y)) -> gr(x, y)
380.31/291.58	p(0') -> 0'
380.31/291.58	p(s(x)) -> x
380.31/291.58	
380.31/291.58	Types:
380.31/291.58	cond :: true:false -> s:0' -> s:0' -> cond
380.31/291.58	true :: true:false
380.31/291.58	gr :: s:0' -> s:0' -> true:false
380.31/291.58	p :: s:0' -> s:0'
380.31/291.58	s :: s:0' -> s:0'
380.31/291.58	0' :: s:0'
380.31/291.58	false :: true:false
380.31/291.58	hole_cond1_0 :: cond
380.31/291.58	hole_true:false2_0 :: true:false
380.31/291.58	hole_s:0'3_0 :: s:0'
380.31/291.58	gen_s:0'4_0 :: Nat -> s:0'
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(39) OrderProof (LOWER BOUND(ID))
380.31/291.58	Heuristically decided to analyse the following defined symbols:
380.31/291.58	cond, gr
380.31/291.58	
380.31/291.58	They will be analysed ascendingly in the following order:
380.31/291.58	gr < cond
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(40)
380.31/291.58	Obligation:
380.31/291.58	Innermost TRS:
380.31/291.58	Rules:
380.31/291.58	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
380.31/291.58	gr(0', x) -> false
380.31/291.58	gr(s(x), 0') -> true
380.31/291.58	gr(s(x), s(y)) -> gr(x, y)
380.31/291.58	p(0') -> 0'
380.31/291.58	p(s(x)) -> x
380.31/291.58	
380.31/291.58	Types:
380.31/291.58	cond :: true:false -> s:0' -> s:0' -> cond
380.31/291.58	true :: true:false
380.31/291.58	gr :: s:0' -> s:0' -> true:false
380.31/291.58	p :: s:0' -> s:0'
380.31/291.58	s :: s:0' -> s:0'
380.31/291.58	0' :: s:0'
380.31/291.58	false :: true:false
380.31/291.58	hole_cond1_0 :: cond
380.31/291.58	hole_true:false2_0 :: true:false
380.31/291.58	hole_s:0'3_0 :: s:0'
380.31/291.58	gen_s:0'4_0 :: Nat -> s:0'
380.31/291.58	
380.31/291.58	
380.31/291.58	Generator Equations:
380.31/291.58	gen_s:0'4_0(0) <=> 0'
380.31/291.58	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
380.31/291.58	
380.31/291.58	
380.31/291.58	The following defined symbols remain to be analysed:
380.31/291.58	gr, cond
380.31/291.58	
380.31/291.58	They will be analysed ascendingly in the following order:
380.31/291.58	gr < cond
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(41) RewriteLemmaProof (LOWER BOUND(ID))
380.31/291.58	Proved the following rewrite lemma:
380.31/291.58	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
380.31/291.58	
380.31/291.58	Induction Base:
380.31/291.58	gr(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
380.31/291.58	false
380.31/291.58	
380.31/291.58	Induction Step:
380.31/291.58	gr(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
380.31/291.58	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
380.31/291.58	false
380.31/291.58	
380.31/291.58	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(42)
380.31/291.58	Complex Obligation (BEST)
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(43)
380.31/291.58	Obligation:
380.31/291.58	Proved the lower bound n^1 for the following obligation:
380.31/291.58	
380.31/291.58	Innermost TRS:
380.31/291.58	Rules:
380.31/291.58	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
380.31/291.58	gr(0', x) -> false
380.31/291.58	gr(s(x), 0') -> true
380.31/291.58	gr(s(x), s(y)) -> gr(x, y)
380.31/291.58	p(0') -> 0'
380.31/291.58	p(s(x)) -> x
380.31/291.58	
380.31/291.58	Types:
380.31/291.58	cond :: true:false -> s:0' -> s:0' -> cond
380.31/291.58	true :: true:false
380.31/291.58	gr :: s:0' -> s:0' -> true:false
380.31/291.58	p :: s:0' -> s:0'
380.31/291.58	s :: s:0' -> s:0'
380.31/291.58	0' :: s:0'
380.31/291.58	false :: true:false
380.31/291.58	hole_cond1_0 :: cond
380.31/291.58	hole_true:false2_0 :: true:false
380.31/291.58	hole_s:0'3_0 :: s:0'
380.31/291.58	gen_s:0'4_0 :: Nat -> s:0'
380.31/291.58	
380.31/291.58	
380.31/291.58	Generator Equations:
380.31/291.58	gen_s:0'4_0(0) <=> 0'
380.31/291.58	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
380.31/291.58	
380.31/291.58	
380.31/291.58	The following defined symbols remain to be analysed:
380.31/291.58	gr, cond
380.31/291.58	
380.31/291.58	They will be analysed ascendingly in the following order:
380.31/291.58	gr < cond
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(44) LowerBoundPropagationProof (FINISHED)
380.31/291.58	Propagated lower bound.
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(45)
380.31/291.58	BOUNDS(n^1, INF)
380.31/291.58	
380.31/291.58	----------------------------------------
380.31/291.58	
380.31/291.58	(46)
380.31/291.58	Obligation:
380.31/291.58	Innermost TRS:
380.31/291.58	Rules:
380.31/291.58	cond(true, x, y) -> cond(gr(x, y), p(x), s(y))
380.31/291.58	gr(0', x) -> false
380.31/291.58	gr(s(x), 0') -> true
380.31/291.58	gr(s(x), s(y)) -> gr(x, y)
380.31/291.58	p(0') -> 0'
380.31/291.58	p(s(x)) -> x
380.31/291.58	
380.31/291.58	Types:
380.31/291.58	cond :: true:false -> s:0' -> s:0' -> cond
380.31/291.58	true :: true:false
380.31/291.58	gr :: s:0' -> s:0' -> true:false
380.31/291.58	p :: s:0' -> s:0'
380.31/291.58	s :: s:0' -> s:0'
380.31/291.58	0' :: s:0'
380.31/291.58	false :: true:false
380.31/291.58	hole_cond1_0 :: cond
380.31/291.58	hole_true:false2_0 :: true:false
380.31/291.58	hole_s:0'3_0 :: s:0'
380.31/291.58	gen_s:0'4_0 :: Nat -> s:0'
380.31/291.58	
380.31/291.58	
380.31/291.58	Lemmas:
380.31/291.58	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
380.31/291.58	
380.31/291.58	
380.31/291.58	Generator Equations:
380.31/291.58	gen_s:0'4_0(0) <=> 0'
380.31/291.58	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
380.31/291.58	
380.31/291.58	
380.31/291.58	The following defined symbols remain to be analysed:
380.31/291.58	cond
380.37/291.62	EOF