32.04/9.47	WORST_CASE(Omega(n^1), O(n^1))
32.04/9.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.04/9.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.04/9.49	
32.04/9.49	
32.04/9.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.04/9.49	
32.04/9.49	(0) CpxTRS
32.04/9.49	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
32.04/9.49	(2) CpxWeightedTrs
32.04/9.49	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
32.04/9.49	    (4) CpxTypedWeightedTrs
32.04/9.49	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
32.04/9.49	    (6) CpxTypedWeightedCompleteTrs
32.04/9.49	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
32.04/9.49	    (8) CpxTypedWeightedCompleteTrs
32.04/9.49	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
32.04/9.49	    (10) CpxRNTS
32.04/9.49	    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
32.04/9.49	    (12) CpxRNTS
32.04/9.49	    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms]
32.04/9.49	    (14) CpxRNTS
32.04/9.49	    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
32.04/9.49	    (16) CpxRNTS
32.04/9.49	    (17) IntTrsBoundProof [UPPER BOUND(ID), 1165 ms]
32.04/9.49	    (18) CpxRNTS
32.04/9.49	    (19) IntTrsBoundProof [UPPER BOUND(ID), 609 ms]
32.04/9.49	    (20) CpxRNTS
32.04/9.49	    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
32.04/9.49	    (22) CpxRNTS
32.04/9.49	    (23) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
32.04/9.49	    (24) CpxRNTS
32.04/9.49	    (25) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
32.04/9.49	    (26) CpxRNTS
32.04/9.49	    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
32.04/9.49	    (28) CpxRNTS
32.04/9.49	    (29) IntTrsBoundProof [UPPER BOUND(ID), 280 ms]
32.04/9.49	    (30) CpxRNTS
32.04/9.49	    (31) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
32.04/9.49	    (32) CpxRNTS
32.04/9.49	    (33) FinalProof [FINISHED, 0 ms]
32.04/9.49	    (34) BOUNDS(1, n^1)
32.04/9.49	(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
32.04/9.49	(36) CpxTRS
32.04/9.49	    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
32.04/9.49	    (38) typed CpxTrs
32.04/9.49	    (39) OrderProof [LOWER BOUND(ID), 0 ms]
32.04/9.49	    (40) typed CpxTrs
32.04/9.49	    (41) RewriteLemmaProof [LOWER BOUND(ID), 309 ms]
32.04/9.49	    (42) BEST
32.04/9.49	        (43) proven lower bound
32.04/9.49	            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
32.04/9.49	            (45) BOUNDS(n^1, INF)
32.04/9.49	        (46) typed CpxTrs
32.04/9.49	
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(0)
32.04/9.49	Obligation:
32.04/9.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.04/9.49	
32.04/9.49	
32.04/9.49	The TRS R consists of the following rules:
32.04/9.49	
32.04/9.49	   cond1(true, x, y) -> cond2(gr(y, 0), x, y)
32.04/9.49	   cond2(true, x, y) -> cond2(gr(y, 0), x, p(y))
32.04/9.49	   cond2(false, x, y) -> cond1(gr(x, 0), p(x), y)
32.04/9.49	   gr(0, x) -> false
32.04/9.49	   gr(s(x), 0) -> true
32.04/9.49	   gr(s(x), s(y)) -> gr(x, y)
32.04/9.49	   p(0) -> 0
32.04/9.49	   p(s(x)) -> x
32.04/9.49	
32.04/9.49	S is empty.
32.04/9.49	Rewrite Strategy: INNERMOST
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
32.04/9.49	Transformed relative TRS to weighted TRS
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(2)
32.04/9.49	Obligation:
32.04/9.49	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
32.04/9.49	
32.04/9.49	
32.04/9.49	The TRS R consists of the following rules:
32.04/9.49	
32.04/9.49	   cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1]
32.04/9.49	   cond2(true, x, y) -> cond2(gr(y, 0), x, p(y)) [1]
32.04/9.49	   cond2(false, x, y) -> cond1(gr(x, 0), p(x), y) [1]
32.04/9.49	   gr(0, x) -> false [1]
32.04/9.49	   gr(s(x), 0) -> true [1]
32.04/9.49	   gr(s(x), s(y)) -> gr(x, y) [1]
32.04/9.49	   p(0) -> 0 [1]
32.04/9.49	   p(s(x)) -> x [1]
32.04/9.49	
32.04/9.49	Rewrite Strategy: INNERMOST
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
32.04/9.49	Infered types.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(4)
32.04/9.49	Obligation:
32.04/9.49	Runtime Complexity Weighted TRS with Types.
32.04/9.49	The TRS R consists of the following rules:
32.04/9.49	
32.04/9.49	   cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1]
32.04/9.49	   cond2(true, x, y) -> cond2(gr(y, 0), x, p(y)) [1]
32.04/9.49	   cond2(false, x, y) -> cond1(gr(x, 0), p(x), y) [1]
32.04/9.49	   gr(0, x) -> false [1]
32.04/9.49	   gr(s(x), 0) -> true [1]
32.04/9.49	   gr(s(x), s(y)) -> gr(x, y) [1]
32.04/9.49	   p(0) -> 0 [1]
32.04/9.49	   p(s(x)) -> x [1]
32.04/9.49	
32.04/9.49	The TRS has the following type information:
32.04/9.49	cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2
32.04/9.49	true :: true:false
32.04/9.49	cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2
32.04/9.49	gr :: 0:s -> 0:s -> true:false
32.04/9.49	0 :: 0:s
32.04/9.49	p :: 0:s -> 0:s
32.04/9.49	false :: true:false
32.04/9.49	s :: 0:s -> 0:s
32.04/9.49	
32.04/9.49	Rewrite Strategy: INNERMOST
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(5) CompletionProof (UPPER BOUND(ID))
32.04/9.49	The transformation into a RNTS is sound, since:
32.04/9.49	
32.04/9.49	(a) The obligation is a constructor system where every type has a constant constructor,
32.04/9.49	
32.04/9.49	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
32.04/9.49	
32.04/9.49	   cond1_3
32.04/9.49	   cond2_3
32.04/9.49	
32.04/9.49	(c) The following functions are completely defined:
32.04/9.49	
32.04/9.49	   gr_2
32.04/9.49	   p_1
32.04/9.49	
32.04/9.49	Due to the following rules being added:
32.04/9.49	none
32.04/9.49	
32.04/9.49	And the following fresh constants:    const
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(6)
32.04/9.49	Obligation:
32.04/9.49	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
32.04/9.49	
32.04/9.49	Runtime Complexity Weighted TRS with Types.
32.04/9.49	The TRS R consists of the following rules:
32.04/9.49	
32.04/9.49	   cond1(true, x, y) -> cond2(gr(y, 0), x, y) [1]
32.04/9.49	   cond2(true, x, y) -> cond2(gr(y, 0), x, p(y)) [1]
32.04/9.49	   cond2(false, x, y) -> cond1(gr(x, 0), p(x), y) [1]
32.04/9.49	   gr(0, x) -> false [1]
32.04/9.49	   gr(s(x), 0) -> true [1]
32.04/9.49	   gr(s(x), s(y)) -> gr(x, y) [1]
32.04/9.49	   p(0) -> 0 [1]
32.04/9.49	   p(s(x)) -> x [1]
32.04/9.49	
32.04/9.49	The TRS has the following type information:
32.04/9.49	cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2
32.04/9.49	true :: true:false
32.04/9.49	cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2
32.04/9.49	gr :: 0:s -> 0:s -> true:false
32.04/9.49	0 :: 0:s
32.04/9.49	p :: 0:s -> 0:s
32.04/9.49	false :: true:false
32.04/9.49	s :: 0:s -> 0:s
32.04/9.49	const :: cond1:cond2
32.04/9.49	
32.04/9.49	Rewrite Strategy: INNERMOST
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
32.04/9.49	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(8)
32.04/9.49	Obligation:
32.04/9.49	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
32.04/9.49	
32.04/9.49	Runtime Complexity Weighted TRS with Types.
32.04/9.49	The TRS R consists of the following rules:
32.04/9.49	
32.04/9.49	   cond1(true, x, 0) -> cond2(false, x, 0) [2]
32.04/9.49	   cond1(true, x, s(x')) -> cond2(true, x, s(x')) [2]
32.04/9.49	   cond2(true, x, 0) -> cond2(false, x, 0) [3]
32.04/9.49	   cond2(true, x, s(x'')) -> cond2(true, x, x'') [3]
32.04/9.49	   cond2(false, 0, y) -> cond1(false, 0, y) [3]
32.04/9.49	   cond2(false, s(x1), y) -> cond1(true, x1, y) [3]
32.04/9.49	   gr(0, x) -> false [1]
32.04/9.49	   gr(s(x), 0) -> true [1]
32.04/9.49	   gr(s(x), s(y)) -> gr(x, y) [1]
32.04/9.49	   p(0) -> 0 [1]
32.04/9.49	   p(s(x)) -> x [1]
32.04/9.49	
32.04/9.49	The TRS has the following type information:
32.04/9.49	cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2
32.04/9.49	true :: true:false
32.04/9.49	cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2
32.04/9.49	gr :: 0:s -> 0:s -> true:false
32.04/9.49	0 :: 0:s
32.04/9.49	p :: 0:s -> 0:s
32.04/9.49	false :: true:false
32.04/9.49	s :: 0:s -> 0:s
32.04/9.49	const :: cond1:cond2
32.04/9.49	
32.04/9.49	Rewrite Strategy: INNERMOST
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
32.04/9.49	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
32.04/9.49	The constant constructors are abstracted as follows:
32.04/9.49	
32.04/9.49	   true => 1
32.04/9.49	   0 => 0
32.04/9.49	   false => 0
32.04/9.49	   const => 0
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(10)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(1, x, 1 + x') :|: z' = x, z = 1, z'' = 1 + x', x >= 0, x' >= 0
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(1, x, x'') :|: z' = x, z = 1, x >= 0, z'' = 1 + x'', x'' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(1, x1, y) :|: x1 >= 0, z'' = y, y >= 0, z' = 1 + x1, z = 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(0, 0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(11) SimplificationProof (BOTH BOUNDS(ID, ID))
32.04/9.49	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(12)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
32.04/9.49	Found the following analysis order by SCC decomposition:
32.04/9.49	
32.04/9.49	   { cond2, cond1 }
32.04/9.49	   { p }
32.04/9.49	   { gr }
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(14)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {cond2,cond1}, {p}, {gr}
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(15) ResultPropagationProof (UPPER BOUND(ID))
32.04/9.49	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(16)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {cond2,cond1}, {p}, {gr}
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(17) IntTrsBoundProof (UPPER BOUND(ID))
32.04/9.49	
32.04/9.49	Computed SIZE bound using CoFloCo for: cond2
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(1) with polynomial bound: 0
32.04/9.49	
32.04/9.49	Computed SIZE bound using CoFloCo for: cond1
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(1) with polynomial bound: 0
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(18)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {cond2,cond1}, {p}, {gr}
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: ?, size: O(1) [0]
32.04/9.49	  cond1: runtime: ?, size: O(1) [0]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(19) IntTrsBoundProof (UPPER BOUND(ID))
32.04/9.49	
32.04/9.49	Computed RUNTIME bound using CoFloCo for: cond2
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(n^1) with polynomial bound: 21 + 5*z' + 3*z''
32.04/9.49	
32.04/9.49	Computed RUNTIME bound using CoFloCo for: cond1
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(n^1) with polynomial bound: 23 + 5*z' + 3*z''
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(20)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond1(z, z', z'') -{ 2 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   cond2(z, z', z'') -{ 3 }-> cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {p}, {gr}
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: O(n^1) [21 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  cond1: runtime: O(n^1) [23 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(21) ResultPropagationProof (UPPER BOUND(ID))
32.04/9.49	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(22)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' + 3*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 24 + 5*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 26 + 3*z'' }-> s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {p}, {gr}
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: O(n^1) [21 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  cond1: runtime: O(n^1) [23 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(23) IntTrsBoundProof (UPPER BOUND(ID))
32.04/9.49	
32.04/9.49	Computed SIZE bound using KoAT for: p
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(n^1) with polynomial bound: z
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(24)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' + 3*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 24 + 5*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 26 + 3*z'' }-> s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {p}, {gr}
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: O(n^1) [21 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  cond1: runtime: O(n^1) [23 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  p: runtime: ?, size: O(n^1) [z]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(25) IntTrsBoundProof (UPPER BOUND(ID))
32.04/9.49	
32.04/9.49	Computed RUNTIME bound using CoFloCo for: p
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(1) with polynomial bound: 1
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(26)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' + 3*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 24 + 5*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 26 + 3*z'' }-> s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {gr}
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: O(n^1) [21 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  cond1: runtime: O(n^1) [23 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  p: runtime: O(1) [1], size: O(n^1) [z]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(27) ResultPropagationProof (UPPER BOUND(ID))
32.04/9.49	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(28)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' + 3*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 24 + 5*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 26 + 3*z'' }-> s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {gr}
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: O(n^1) [21 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  cond1: runtime: O(n^1) [23 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  p: runtime: O(1) [1], size: O(n^1) [z]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(29) IntTrsBoundProof (UPPER BOUND(ID))
32.04/9.49	
32.04/9.49	Computed SIZE bound using CoFloCo for: gr
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(1) with polynomial bound: 1
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(30)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' + 3*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 24 + 5*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 26 + 3*z'' }-> s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: {gr}
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: O(n^1) [21 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  cond1: runtime: O(n^1) [23 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  p: runtime: O(1) [1], size: O(n^1) [z]
32.04/9.49	  gr: runtime: ?, size: O(1) [1]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(31) IntTrsBoundProof (UPPER BOUND(ID))
32.04/9.49	
32.04/9.49	Computed RUNTIME bound using KoAT for: gr
32.04/9.49	after applying outer abstraction to obtain an ITS,
32.04/9.49	resulting in: O(n^1) with polynomial bound: 2 + z'
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(32)
32.04/9.49	Obligation:
32.04/9.49	Complexity RNTS consisting of the following rules:
32.04/9.49	
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond1(z, z', z'') -{ 23 + 5*z' + 3*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 24 + 5*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
32.04/9.49	   cond2(z, z', z'') -{ 26 + 3*z'' }-> s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
32.04/9.49	   cond2(z, z', z'') -{ 21 + 5*z' + 3*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
32.04/9.49	   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
32.04/9.49	   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
32.04/9.49	   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
32.04/9.49	   p(z) -{ 1 }-> 0 :|: z = 0
32.04/9.49	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
32.04/9.49	
32.04/9.49	Function symbols to be analyzed: 
32.04/9.49	Previous analysis results are:
32.04/9.49	  cond2: runtime: O(n^1) [21 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  cond1: runtime: O(n^1) [23 + 5*z' + 3*z''], size: O(1) [0]
32.04/9.49	  p: runtime: O(1) [1], size: O(n^1) [z]
32.04/9.49	  gr: runtime: O(n^1) [2 + z'], size: O(1) [1]
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(33) FinalProof (FINISHED)
32.04/9.49	Computed overall runtime complexity
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(34)
32.04/9.49	BOUNDS(1, n^1)
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(35) RenamingProof (BOTH BOUNDS(ID, ID))
32.04/9.49	Renamed function symbols to avoid clashes with predefined symbol.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(36)
32.04/9.49	Obligation:
32.04/9.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
32.04/9.49	
32.04/9.49	
32.04/9.49	The TRS R consists of the following rules:
32.04/9.49	
32.04/9.49	   cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
32.04/9.49	   cond2(true, x, y) -> cond2(gr(y, 0'), x, p(y))
32.04/9.49	   cond2(false, x, y) -> cond1(gr(x, 0'), p(x), y)
32.04/9.49	   gr(0', x) -> false
32.04/9.49	   gr(s(x), 0') -> true
32.04/9.49	   gr(s(x), s(y)) -> gr(x, y)
32.04/9.49	   p(0') -> 0'
32.04/9.49	   p(s(x)) -> x
32.04/9.49	
32.04/9.49	S is empty.
32.04/9.49	Rewrite Strategy: INNERMOST
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(37) TypeInferenceProof (BOTH BOUNDS(ID, ID))
32.04/9.49	Infered types.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(38)
32.04/9.49	Obligation:
32.04/9.49	Innermost TRS:
32.04/9.49	Rules:
32.04/9.49	cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
32.04/9.49	cond2(true, x, y) -> cond2(gr(y, 0'), x, p(y))
32.04/9.49	cond2(false, x, y) -> cond1(gr(x, 0'), p(x), y)
32.04/9.49	gr(0', x) -> false
32.04/9.49	gr(s(x), 0') -> true
32.04/9.49	gr(s(x), s(y)) -> gr(x, y)
32.04/9.49	p(0') -> 0'
32.04/9.49	p(s(x)) -> x
32.04/9.49	
32.04/9.49	Types:
32.04/9.49	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	true :: true:false
32.04/9.49	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	gr :: 0':s -> 0':s -> true:false
32.04/9.49	0' :: 0':s
32.04/9.49	p :: 0':s -> 0':s
32.04/9.49	false :: true:false
32.04/9.49	s :: 0':s -> 0':s
32.04/9.49	hole_cond1:cond21_0 :: cond1:cond2
32.04/9.49	hole_true:false2_0 :: true:false
32.04/9.49	hole_0':s3_0 :: 0':s
32.04/9.49	gen_0':s4_0 :: Nat -> 0':s
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(39) OrderProof (LOWER BOUND(ID))
32.04/9.49	Heuristically decided to analyse the following defined symbols:
32.04/9.49	cond1, cond2, gr
32.04/9.49	
32.04/9.49	They will be analysed ascendingly in the following order:
32.04/9.49	cond1 = cond2
32.04/9.49	gr < cond1
32.04/9.49	gr < cond2
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(40)
32.04/9.49	Obligation:
32.04/9.49	Innermost TRS:
32.04/9.49	Rules:
32.04/9.49	cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
32.04/9.49	cond2(true, x, y) -> cond2(gr(y, 0'), x, p(y))
32.04/9.49	cond2(false, x, y) -> cond1(gr(x, 0'), p(x), y)
32.04/9.49	gr(0', x) -> false
32.04/9.49	gr(s(x), 0') -> true
32.04/9.49	gr(s(x), s(y)) -> gr(x, y)
32.04/9.49	p(0') -> 0'
32.04/9.49	p(s(x)) -> x
32.04/9.49	
32.04/9.49	Types:
32.04/9.49	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	true :: true:false
32.04/9.49	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	gr :: 0':s -> 0':s -> true:false
32.04/9.49	0' :: 0':s
32.04/9.49	p :: 0':s -> 0':s
32.04/9.49	false :: true:false
32.04/9.49	s :: 0':s -> 0':s
32.04/9.49	hole_cond1:cond21_0 :: cond1:cond2
32.04/9.49	hole_true:false2_0 :: true:false
32.04/9.49	hole_0':s3_0 :: 0':s
32.04/9.49	gen_0':s4_0 :: Nat -> 0':s
32.04/9.49	
32.04/9.49	
32.04/9.49	Generator Equations:
32.04/9.49	gen_0':s4_0(0) <=> 0'
32.04/9.49	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
32.04/9.49	
32.04/9.49	
32.04/9.49	The following defined symbols remain to be analysed:
32.04/9.49	gr, cond1, cond2
32.04/9.49	
32.04/9.49	They will be analysed ascendingly in the following order:
32.04/9.49	cond1 = cond2
32.04/9.49	gr < cond1
32.04/9.49	gr < cond2
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(41) RewriteLemmaProof (LOWER BOUND(ID))
32.04/9.49	Proved the following rewrite lemma:
32.04/9.49	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
32.04/9.49	
32.04/9.49	Induction Base:
32.04/9.49	gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
32.04/9.49	false
32.04/9.49	
32.04/9.49	Induction Step:
32.04/9.49	gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
32.04/9.49	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
32.04/9.49	false
32.04/9.49	
32.04/9.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(42)
32.04/9.49	Complex Obligation (BEST)
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(43)
32.04/9.49	Obligation:
32.04/9.49	Proved the lower bound n^1 for the following obligation:
32.04/9.49	
32.04/9.49	Innermost TRS:
32.04/9.49	Rules:
32.04/9.49	cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
32.04/9.49	cond2(true, x, y) -> cond2(gr(y, 0'), x, p(y))
32.04/9.49	cond2(false, x, y) -> cond1(gr(x, 0'), p(x), y)
32.04/9.49	gr(0', x) -> false
32.04/9.49	gr(s(x), 0') -> true
32.04/9.49	gr(s(x), s(y)) -> gr(x, y)
32.04/9.49	p(0') -> 0'
32.04/9.49	p(s(x)) -> x
32.04/9.49	
32.04/9.49	Types:
32.04/9.49	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	true :: true:false
32.04/9.49	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	gr :: 0':s -> 0':s -> true:false
32.04/9.49	0' :: 0':s
32.04/9.49	p :: 0':s -> 0':s
32.04/9.49	false :: true:false
32.04/9.49	s :: 0':s -> 0':s
32.04/9.49	hole_cond1:cond21_0 :: cond1:cond2
32.04/9.49	hole_true:false2_0 :: true:false
32.04/9.49	hole_0':s3_0 :: 0':s
32.04/9.49	gen_0':s4_0 :: Nat -> 0':s
32.04/9.49	
32.04/9.49	
32.04/9.49	Generator Equations:
32.04/9.49	gen_0':s4_0(0) <=> 0'
32.04/9.49	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
32.04/9.49	
32.04/9.49	
32.04/9.49	The following defined symbols remain to be analysed:
32.04/9.49	gr, cond1, cond2
32.04/9.49	
32.04/9.49	They will be analysed ascendingly in the following order:
32.04/9.49	cond1 = cond2
32.04/9.49	gr < cond1
32.04/9.49	gr < cond2
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(44) LowerBoundPropagationProof (FINISHED)
32.04/9.49	Propagated lower bound.
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(45)
32.04/9.49	BOUNDS(n^1, INF)
32.04/9.49	
32.04/9.49	----------------------------------------
32.04/9.49	
32.04/9.49	(46)
32.04/9.49	Obligation:
32.04/9.49	Innermost TRS:
32.04/9.49	Rules:
32.04/9.49	cond1(true, x, y) -> cond2(gr(y, 0'), x, y)
32.04/9.49	cond2(true, x, y) -> cond2(gr(y, 0'), x, p(y))
32.04/9.49	cond2(false, x, y) -> cond1(gr(x, 0'), p(x), y)
32.04/9.49	gr(0', x) -> false
32.04/9.49	gr(s(x), 0') -> true
32.04/9.49	gr(s(x), s(y)) -> gr(x, y)
32.04/9.49	p(0') -> 0'
32.04/9.49	p(s(x)) -> x
32.04/9.49	
32.04/9.49	Types:
32.04/9.49	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	true :: true:false
32.04/9.49	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
32.04/9.49	gr :: 0':s -> 0':s -> true:false
32.04/9.49	0' :: 0':s
32.04/9.49	p :: 0':s -> 0':s
32.04/9.49	false :: true:false
32.04/9.49	s :: 0':s -> 0':s
32.04/9.49	hole_cond1:cond21_0 :: cond1:cond2
32.04/9.49	hole_true:false2_0 :: true:false
32.04/9.49	hole_0':s3_0 :: 0':s
32.04/9.49	gen_0':s4_0 :: Nat -> 0':s
32.04/9.49	
32.04/9.49	
32.04/9.49	Lemmas:
32.04/9.49	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
32.04/9.49	
32.04/9.49	
32.04/9.49	Generator Equations:
32.04/9.49	gen_0':s4_0(0) <=> 0'
32.04/9.49	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
32.04/9.49	
32.04/9.49	
32.04/9.49	The following defined symbols remain to be analysed:
32.04/9.49	cond2, cond1
32.04/9.49	
32.04/9.49	They will be analysed ascendingly in the following order:
32.04/9.49	cond1 = cond2
32.13/12.11	EOF