320.17/291.54	WORST_CASE(Omega(n^1), O(n^2))
320.17/291.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
320.17/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
320.17/291.55	
320.17/291.55	
320.17/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.17/291.55	
320.17/291.55	(0) CpxTRS
320.17/291.55	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
320.17/291.55	(2) CpxWeightedTrs
320.17/291.55	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
320.17/291.55	    (4) CpxTypedWeightedTrs
320.17/291.55	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
320.17/291.55	    (6) CpxTypedWeightedCompleteTrs
320.17/291.55	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
320.17/291.55	    (8) CpxTypedWeightedCompleteTrs
320.17/291.55	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
320.17/291.55	    (10) CpxRNTS
320.17/291.55	    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
320.17/291.55	    (12) CpxRNTS
320.17/291.55	    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
320.17/291.55	    (14) CpxRNTS
320.17/291.55	    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
320.17/291.55	    (16) CpxRNTS
320.17/291.55	    (17) IntTrsBoundProof [UPPER BOUND(ID), 333 ms]
320.17/291.55	    (18) CpxRNTS
320.17/291.55	    (19) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
320.17/291.55	    (20) CpxRNTS
320.17/291.55	    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
320.17/291.55	    (22) CpxRNTS
320.17/291.55	    (23) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
320.17/291.55	    (24) CpxRNTS
320.17/291.55	    (25) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
320.17/291.55	    (26) CpxRNTS
320.17/291.55	    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
320.17/291.55	    (28) CpxRNTS
320.17/291.55	    (29) IntTrsBoundProof [UPPER BOUND(ID), 1184 ms]
320.17/291.55	    (30) CpxRNTS
320.17/291.55	    (31) IntTrsBoundProof [UPPER BOUND(ID), 405 ms]
320.17/291.55	    (32) CpxRNTS
320.17/291.55	    (33) FinalProof [FINISHED, 0 ms]
320.17/291.55	    (34) BOUNDS(1, n^2)
320.17/291.55	(35) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
320.17/291.55	(36) TRS for Loop Detection
320.17/291.55	    (37) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
320.17/291.55	    (38) BEST
320.17/291.55	        (39) proven lower bound
320.17/291.55	            (40) LowerBoundPropagationProof [FINISHED, 0 ms]
320.17/291.55	            (41) BOUNDS(n^1, INF)
320.17/291.55	        (42) TRS for Loop Detection
320.17/291.55	
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(0)
320.17/291.55	Obligation:
320.17/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.17/291.55	
320.17/291.55	
320.17/291.55	The TRS R consists of the following rules:
320.17/291.55	
320.17/291.55	   p(0) -> 0
320.17/291.55	   p(s(x)) -> x
320.17/291.55	   le(0, y) -> true
320.17/291.55	   le(s(x), 0) -> false
320.17/291.55	   le(s(x), s(y)) -> le(x, y)
320.17/291.55	   minus(x, y) -> if(le(x, y), x, y)
320.17/291.55	   if(true, x, y) -> 0
320.17/291.55	   if(false, x, y) -> s(minus(p(x), y))
320.17/291.55	
320.17/291.55	S is empty.
320.17/291.55	Rewrite Strategy: INNERMOST
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
320.17/291.55	Transformed relative TRS to weighted TRS
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(2)
320.17/291.55	Obligation:
320.17/291.55	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
320.17/291.55	
320.17/291.55	
320.17/291.55	The TRS R consists of the following rules:
320.17/291.55	
320.17/291.55	   p(0) -> 0 [1]
320.17/291.55	   p(s(x)) -> x [1]
320.17/291.55	   le(0, y) -> true [1]
320.17/291.55	   le(s(x), 0) -> false [1]
320.17/291.55	   le(s(x), s(y)) -> le(x, y) [1]
320.17/291.55	   minus(x, y) -> if(le(x, y), x, y) [1]
320.17/291.55	   if(true, x, y) -> 0 [1]
320.17/291.55	   if(false, x, y) -> s(minus(p(x), y)) [1]
320.17/291.55	
320.17/291.55	Rewrite Strategy: INNERMOST
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
320.17/291.55	Infered types.
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(4)
320.17/291.55	Obligation:
320.17/291.55	Runtime Complexity Weighted TRS with Types.
320.17/291.55	The TRS R consists of the following rules:
320.17/291.55	
320.17/291.55	   p(0) -> 0 [1]
320.17/291.55	   p(s(x)) -> x [1]
320.17/291.55	   le(0, y) -> true [1]
320.17/291.55	   le(s(x), 0) -> false [1]
320.17/291.55	   le(s(x), s(y)) -> le(x, y) [1]
320.17/291.55	   minus(x, y) -> if(le(x, y), x, y) [1]
320.17/291.55	   if(true, x, y) -> 0 [1]
320.17/291.55	   if(false, x, y) -> s(minus(p(x), y)) [1]
320.17/291.55	
320.17/291.55	The TRS has the following type information:
320.17/291.55	p :: 0:s -> 0:s
320.17/291.55	0 :: 0:s
320.17/291.55	s :: 0:s -> 0:s
320.17/291.55	le :: 0:s -> 0:s -> true:false
320.17/291.55	true :: true:false
320.17/291.55	false :: true:false
320.17/291.55	minus :: 0:s -> 0:s -> 0:s
320.17/291.55	if :: true:false -> 0:s -> 0:s -> 0:s
320.17/291.55	
320.17/291.55	Rewrite Strategy: INNERMOST
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(5) CompletionProof (UPPER BOUND(ID))
320.17/291.55	The transformation into a RNTS is sound, since:
320.17/291.55	
320.17/291.55	(a) The obligation is a constructor system where every type has a constant constructor,
320.17/291.55	
320.17/291.55	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
320.17/291.55	
320.17/291.55	   minus_2
320.17/291.55	   if_3
320.17/291.55	
320.17/291.55	(c) The following functions are completely defined:
320.17/291.55	
320.17/291.55	   p_1
320.17/291.55	   le_2
320.17/291.55	
320.17/291.55	Due to the following rules being added:
320.17/291.55	none
320.17/291.55	
320.17/291.55	And the following fresh constants: none
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(6)
320.17/291.55	Obligation:
320.17/291.55	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
320.17/291.55	
320.17/291.55	Runtime Complexity Weighted TRS with Types.
320.17/291.55	The TRS R consists of the following rules:
320.17/291.55	
320.17/291.55	   p(0) -> 0 [1]
320.17/291.55	   p(s(x)) -> x [1]
320.17/291.55	   le(0, y) -> true [1]
320.17/291.55	   le(s(x), 0) -> false [1]
320.17/291.55	   le(s(x), s(y)) -> le(x, y) [1]
320.17/291.55	   minus(x, y) -> if(le(x, y), x, y) [1]
320.17/291.55	   if(true, x, y) -> 0 [1]
320.17/291.55	   if(false, x, y) -> s(minus(p(x), y)) [1]
320.17/291.55	
320.17/291.55	The TRS has the following type information:
320.17/291.55	p :: 0:s -> 0:s
320.17/291.55	0 :: 0:s
320.17/291.55	s :: 0:s -> 0:s
320.17/291.55	le :: 0:s -> 0:s -> true:false
320.17/291.55	true :: true:false
320.17/291.55	false :: true:false
320.17/291.55	minus :: 0:s -> 0:s -> 0:s
320.17/291.55	if :: true:false -> 0:s -> 0:s -> 0:s
320.17/291.55	
320.17/291.55	Rewrite Strategy: INNERMOST
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
320.17/291.55	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(8)
320.17/291.55	Obligation:
320.17/291.55	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
320.17/291.55	
320.17/291.55	Runtime Complexity Weighted TRS with Types.
320.17/291.55	The TRS R consists of the following rules:
320.17/291.55	
320.17/291.55	   p(0) -> 0 [1]
320.17/291.55	   p(s(x)) -> x [1]
320.17/291.55	   le(0, y) -> true [1]
320.17/291.55	   le(s(x), 0) -> false [1]
320.17/291.55	   le(s(x), s(y)) -> le(x, y) [1]
320.17/291.55	   minus(0, y) -> if(true, 0, y) [2]
320.17/291.55	   minus(s(x'), 0) -> if(false, s(x'), 0) [2]
320.17/291.55	   minus(s(x''), s(y')) -> if(le(x'', y'), s(x''), s(y')) [2]
320.17/291.55	   if(true, x, y) -> 0 [1]
320.17/291.55	   if(false, 0, y) -> s(minus(0, y)) [2]
320.17/291.55	   if(false, s(x1), y) -> s(minus(x1, y)) [2]
320.17/291.55	
320.17/291.55	The TRS has the following type information:
320.17/291.55	p :: 0:s -> 0:s
320.17/291.55	0 :: 0:s
320.17/291.55	s :: 0:s -> 0:s
320.17/291.55	le :: 0:s -> 0:s -> true:false
320.17/291.55	true :: true:false
320.17/291.55	false :: true:false
320.17/291.55	minus :: 0:s -> 0:s -> 0:s
320.17/291.55	if :: true:false -> 0:s -> 0:s -> 0:s
320.17/291.55	
320.17/291.55	Rewrite Strategy: INNERMOST
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
320.17/291.55	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
320.17/291.55	The constant constructors are abstracted as follows:
320.17/291.55	
320.17/291.55	   0 => 0
320.17/291.55	   true => 1
320.17/291.55	   false => 0
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(10)
320.17/291.55	Obligation:
320.17/291.55	Complexity RNTS consisting of the following rules:
320.17/291.55	
320.17/291.55	   if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(x1, y) :|: x1 >= 0, z'' = y, y >= 0, z' = 1 + x1, z = 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
320.17/291.55	   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
320.17/291.55	   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
320.17/291.55	   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(le(x'', y'), 1 + x'', 1 + y') :|: z = 1 + x'', y' >= 0, z' = 1 + y', x'' >= 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(1, 0, y) :|: y >= 0, z = 0, z' = y
320.17/291.55	   minus(z, z') -{ 2 }-> if(0, 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0
320.17/291.55	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
320.17/291.55	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.55	
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(11) SimplificationProof (BOTH BOUNDS(ID, ID))
320.17/291.55	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(12)
320.17/291.55	Obligation:
320.17/291.55	Complexity RNTS consisting of the following rules:
320.17/291.55	
320.17/291.55	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.17/291.55	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.55	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.55	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.55	
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
320.17/291.55	Found the following analysis order by SCC decomposition:
320.17/291.55	
320.17/291.55	   { le }
320.17/291.55	   { p }
320.17/291.55	   { if, minus }
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(14)
320.17/291.55	Obligation:
320.17/291.55	Complexity RNTS consisting of the following rules:
320.17/291.55	
320.17/291.55	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.17/291.55	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.55	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.55	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.55	
320.17/291.55	Function symbols to be analyzed: {le}, {p}, {if,minus}
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(15) ResultPropagationProof (UPPER BOUND(ID))
320.17/291.55	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(16)
320.17/291.55	Obligation:
320.17/291.55	Complexity RNTS consisting of the following rules:
320.17/291.55	
320.17/291.55	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.17/291.55	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.55	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.55	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.55	
320.17/291.55	Function symbols to be analyzed: {le}, {p}, {if,minus}
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(17) IntTrsBoundProof (UPPER BOUND(ID))
320.17/291.55	
320.17/291.55	Computed SIZE bound using CoFloCo for: le
320.17/291.55	after applying outer abstraction to obtain an ITS,
320.17/291.55	resulting in: O(1) with polynomial bound: 1
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(18)
320.17/291.55	Obligation:
320.17/291.55	Complexity RNTS consisting of the following rules:
320.17/291.55	
320.17/291.55	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.17/291.55	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.55	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.55	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.55	
320.17/291.55	Function symbols to be analyzed: {le}, {p}, {if,minus}
320.17/291.55	Previous analysis results are:
320.17/291.55	  le: runtime: ?, size: O(1) [1]
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(19) IntTrsBoundProof (UPPER BOUND(ID))
320.17/291.55	
320.17/291.55	Computed RUNTIME bound using KoAT for: le
320.17/291.55	after applying outer abstraction to obtain an ITS,
320.17/291.55	resulting in: O(n^1) with polynomial bound: 2 + z'
320.17/291.55	
320.17/291.55	----------------------------------------
320.17/291.55	
320.17/291.55	(20)
320.17/291.55	Obligation:
320.17/291.55	Complexity RNTS consisting of the following rules:
320.17/291.55	
320.17/291.55	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.55	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.17/291.55	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.55	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.55	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.55	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.55	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.55	
320.17/291.55	Function symbols to be analyzed: {p}, {if,minus}
320.17/291.55	Previous analysis results are:
320.17/291.55	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(21) ResultPropagationProof (UPPER BOUND(ID))
320.17/291.56	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(22)
320.17/291.56	Obligation:
320.17/291.56	Complexity RNTS consisting of the following rules:
320.17/291.56	
320.17/291.56	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.17/291.56	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.56	   minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.56	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.56	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.56	
320.17/291.56	Function symbols to be analyzed: {p}, {if,minus}
320.17/291.56	Previous analysis results are:
320.17/291.56	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(23) IntTrsBoundProof (UPPER BOUND(ID))
320.17/291.56	
320.17/291.56	Computed SIZE bound using KoAT for: p
320.17/291.56	after applying outer abstraction to obtain an ITS,
320.17/291.56	resulting in: O(n^1) with polynomial bound: z
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(24)
320.17/291.56	Obligation:
320.17/291.56	Complexity RNTS consisting of the following rules:
320.17/291.56	
320.17/291.56	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.17/291.56	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.56	   minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.56	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.56	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.56	
320.17/291.56	Function symbols to be analyzed: {p}, {if,minus}
320.17/291.56	Previous analysis results are:
320.17/291.56	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.17/291.56	  p: runtime: ?, size: O(n^1) [z]
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(25) IntTrsBoundProof (UPPER BOUND(ID))
320.17/291.56	
320.17/291.56	Computed RUNTIME bound using CoFloCo for: p
320.17/291.56	after applying outer abstraction to obtain an ITS,
320.17/291.56	resulting in: O(1) with polynomial bound: 1
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(26)
320.17/291.56	Obligation:
320.17/291.56	Complexity RNTS consisting of the following rules:
320.17/291.56	
320.17/291.56	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.17/291.56	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.56	   minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.56	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.56	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.56	
320.17/291.56	Function symbols to be analyzed: {if,minus}
320.17/291.56	Previous analysis results are:
320.17/291.56	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.17/291.56	  p: runtime: O(1) [1], size: O(n^1) [z]
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(27) ResultPropagationProof (UPPER BOUND(ID))
320.17/291.56	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(28)
320.17/291.56	Obligation:
320.17/291.56	Complexity RNTS consisting of the following rules:
320.17/291.56	
320.17/291.56	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.17/291.56	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.56	   minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.56	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.56	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.56	
320.17/291.56	Function symbols to be analyzed: {if,minus}
320.17/291.56	Previous analysis results are:
320.17/291.56	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.17/291.56	  p: runtime: O(1) [1], size: O(n^1) [z]
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(29) IntTrsBoundProof (UPPER BOUND(ID))
320.17/291.56	
320.17/291.56	Computed SIZE bound using CoFloCo for: if
320.17/291.56	after applying outer abstraction to obtain an ITS,
320.17/291.56	resulting in: O(n^1) with polynomial bound: 1 + z'
320.17/291.56	
320.17/291.56	Computed SIZE bound using CoFloCo for: minus
320.17/291.56	after applying outer abstraction to obtain an ITS,
320.17/291.56	resulting in: O(n^1) with polynomial bound: 1 + z
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(30)
320.17/291.56	Obligation:
320.17/291.56	Complexity RNTS consisting of the following rules:
320.17/291.56	
320.17/291.56	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.17/291.56	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.56	   minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.56	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.56	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.56	
320.17/291.56	Function symbols to be analyzed: {if,minus}
320.17/291.56	Previous analysis results are:
320.17/291.56	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.17/291.56	  p: runtime: O(1) [1], size: O(n^1) [z]
320.17/291.56	  if: runtime: ?, size: O(n^1) [1 + z']
320.17/291.56	  minus: runtime: ?, size: O(n^1) [1 + z]
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(31) IntTrsBoundProof (UPPER BOUND(ID))
320.17/291.56	
320.17/291.56	Computed RUNTIME bound using CoFloCo for: if
320.17/291.56	after applying outer abstraction to obtain an ITS,
320.17/291.56	resulting in: O(n^2) with polynomial bound: 10 + 5*z' + z'*z'' + z''
320.17/291.56	
320.17/291.56	Computed RUNTIME bound using KoAT for: minus
320.17/291.56	after applying outer abstraction to obtain an ITS,
320.17/291.56	resulting in: O(n^2) with polynomial bound: 37 + 10*z + z*z' + 3*z'
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(32)
320.17/291.56	Obligation:
320.17/291.56	Complexity RNTS consisting of the following rules:
320.17/291.56	
320.17/291.56	   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
320.17/291.56	   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.17/291.56	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.17/291.56	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.17/291.56	   minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
320.17/291.56	   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
320.17/291.56	   p(z) -{ 1 }-> 0 :|: z = 0
320.17/291.56	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.17/291.56	
320.17/291.56	Function symbols to be analyzed: 
320.17/291.56	Previous analysis results are:
320.17/291.56	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.17/291.56	  p: runtime: O(1) [1], size: O(n^1) [z]
320.17/291.56	  if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z']
320.17/291.56	  minus: runtime: O(n^2) [37 + 10*z + z*z' + 3*z'], size: O(n^1) [1 + z]
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(33) FinalProof (FINISHED)
320.17/291.56	Computed overall runtime complexity
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(34)
320.17/291.56	BOUNDS(1, n^2)
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(35) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
320.17/291.56	Transformed a relative TRS into a decreasing-loop problem.
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(36)
320.17/291.56	Obligation:
320.17/291.56	Analyzing the following TRS for decreasing loops:
320.17/291.56	
320.17/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.17/291.56	
320.17/291.56	
320.17/291.56	The TRS R consists of the following rules:
320.17/291.56	
320.17/291.56	   p(0) -> 0
320.17/291.56	   p(s(x)) -> x
320.17/291.56	   le(0, y) -> true
320.17/291.56	   le(s(x), 0) -> false
320.17/291.56	   le(s(x), s(y)) -> le(x, y)
320.17/291.56	   minus(x, y) -> if(le(x, y), x, y)
320.17/291.56	   if(true, x, y) -> 0
320.17/291.56	   if(false, x, y) -> s(minus(p(x), y))
320.17/291.56	
320.17/291.56	S is empty.
320.17/291.56	Rewrite Strategy: INNERMOST
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(37) DecreasingLoopProof (LOWER BOUND(ID))
320.17/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
320.17/291.56	
320.17/291.56	The rewrite sequence
320.17/291.56	
320.17/291.56	le(s(x), s(y)) ->^+ le(x, y)
320.17/291.56	
320.17/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
320.17/291.56	
320.17/291.56	The pumping substitution is [x / s(x), y / s(y)].
320.17/291.56	
320.17/291.56	The result substitution is [ ].
320.17/291.56	
320.17/291.56	
320.17/291.56	
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(38)
320.17/291.56	Complex Obligation (BEST)
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(39)
320.17/291.56	Obligation:
320.17/291.56	Proved the lower bound n^1 for the following obligation:
320.17/291.56	
320.17/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.17/291.56	
320.17/291.56	
320.17/291.56	The TRS R consists of the following rules:
320.17/291.56	
320.17/291.56	   p(0) -> 0
320.17/291.56	   p(s(x)) -> x
320.17/291.56	   le(0, y) -> true
320.17/291.56	   le(s(x), 0) -> false
320.17/291.56	   le(s(x), s(y)) -> le(x, y)
320.17/291.56	   minus(x, y) -> if(le(x, y), x, y)
320.17/291.56	   if(true, x, y) -> 0
320.17/291.56	   if(false, x, y) -> s(minus(p(x), y))
320.17/291.56	
320.17/291.56	S is empty.
320.17/291.56	Rewrite Strategy: INNERMOST
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(40) LowerBoundPropagationProof (FINISHED)
320.17/291.56	Propagated lower bound.
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(41)
320.17/291.56	BOUNDS(n^1, INF)
320.17/291.56	
320.17/291.56	----------------------------------------
320.17/291.56	
320.17/291.56	(42)
320.17/291.56	Obligation:
320.17/291.56	Analyzing the following TRS for decreasing loops:
320.17/291.56	
320.17/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.17/291.56	
320.17/291.56	
320.17/291.56	The TRS R consists of the following rules:
320.17/291.56	
320.17/291.56	   p(0) -> 0
320.17/291.56	   p(s(x)) -> x
320.17/291.56	   le(0, y) -> true
320.17/291.56	   le(s(x), 0) -> false
320.17/291.56	   le(s(x), s(y)) -> le(x, y)
320.17/291.56	   minus(x, y) -> if(le(x, y), x, y)
320.17/291.56	   if(true, x, y) -> 0
320.17/291.56	   if(false, x, y) -> s(minus(p(x), y))
320.17/291.56	
320.17/291.56	S is empty.
320.17/291.56	Rewrite Strategy: INNERMOST
320.31/291.59	EOF