320.67/291.57	WORST_CASE(Omega(n^1), O(n^2))
320.67/291.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
320.67/291.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
320.67/291.59	
320.67/291.59	
320.67/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.67/291.59	
320.67/291.59	(0) CpxTRS
320.67/291.59	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
320.67/291.59	(2) CpxWeightedTrs
320.67/291.59	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
320.67/291.59	    (4) CpxTypedWeightedTrs
320.67/291.59	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
320.67/291.59	    (6) CpxTypedWeightedCompleteTrs
320.67/291.59	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
320.67/291.59	    (8) CpxTypedWeightedCompleteTrs
320.67/291.59	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
320.67/291.59	    (10) CpxRNTS
320.67/291.59	    (11) InliningProof [UPPER BOUND(ID), 0 ms]
320.67/291.59	    (12) CpxRNTS
320.67/291.59	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
320.67/291.59	    (14) CpxRNTS
320.67/291.59	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
320.67/291.59	    (16) CpxRNTS
320.67/291.59	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
320.67/291.59	    (18) CpxRNTS
320.67/291.59	    (19) IntTrsBoundProof [UPPER BOUND(ID), 374 ms]
320.67/291.59	    (20) CpxRNTS
320.67/291.59	    (21) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
320.67/291.59	    (22) CpxRNTS
320.67/291.59	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
320.67/291.59	    (24) CpxRNTS
320.67/291.59	    (25) IntTrsBoundProof [UPPER BOUND(ID), 320 ms]
320.67/291.59	    (26) CpxRNTS
320.67/291.59	    (27) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
320.67/291.59	    (28) CpxRNTS
320.67/291.59	    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
320.67/291.59	    (30) CpxRNTS
320.67/291.59	    (31) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
320.67/291.59	    (32) CpxRNTS
320.67/291.59	    (33) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
320.67/291.59	    (34) CpxRNTS
320.67/291.59	    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
320.67/291.59	    (36) CpxRNTS
320.67/291.59	    (37) IntTrsBoundProof [UPPER BOUND(ID), 462 ms]
320.67/291.59	    (38) CpxRNTS
320.67/291.59	    (39) IntTrsBoundProof [UPPER BOUND(ID), 146 ms]
320.67/291.59	    (40) CpxRNTS
320.67/291.59	    (41) FinalProof [FINISHED, 0 ms]
320.67/291.59	    (42) BOUNDS(1, n^2)
320.67/291.59	(43) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
320.67/291.59	(44) TRS for Loop Detection
320.67/291.59	    (45) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
320.67/291.59	    (46) BEST
320.67/291.59	        (47) proven lower bound
320.67/291.59	            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
320.67/291.59	            (49) BOUNDS(n^1, INF)
320.67/291.59	        (50) TRS for Loop Detection
320.67/291.59	
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(0)
320.67/291.59	Obligation:
320.67/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.67/291.59	
320.67/291.59	
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0
320.67/291.59	   p(s(x)) -> x
320.67/291.59	   le(0, y) -> true
320.67/291.59	   le(s(x), 0) -> false
320.67/291.59	   le(s(x), s(y)) -> le(x, y)
320.67/291.59	   minus(x, 0) -> x
320.67/291.59	   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y)))))
320.67/291.59	   if(true, x, y) -> x
320.67/291.59	   if(false, x, y) -> y
320.67/291.59	
320.67/291.59	S is empty.
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
320.67/291.59	Transformed relative TRS to weighted TRS
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(2)
320.67/291.59	Obligation:
320.67/291.59	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
320.67/291.59	
320.67/291.59	
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0 [1]
320.67/291.59	   p(s(x)) -> x [1]
320.67/291.59	   le(0, y) -> true [1]
320.67/291.59	   le(s(x), 0) -> false [1]
320.67/291.59	   le(s(x), s(y)) -> le(x, y) [1]
320.67/291.59	   minus(x, 0) -> x [1]
320.67/291.59	   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y))))) [1]
320.67/291.59	   if(true, x, y) -> x [1]
320.67/291.59	   if(false, x, y) -> y [1]
320.67/291.59	
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
320.67/291.59	Infered types.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(4)
320.67/291.59	Obligation:
320.67/291.59	Runtime Complexity Weighted TRS with Types.
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0 [1]
320.67/291.59	   p(s(x)) -> x [1]
320.67/291.59	   le(0, y) -> true [1]
320.67/291.59	   le(s(x), 0) -> false [1]
320.67/291.59	   le(s(x), s(y)) -> le(x, y) [1]
320.67/291.59	   minus(x, 0) -> x [1]
320.67/291.59	   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y))))) [1]
320.67/291.59	   if(true, x, y) -> x [1]
320.67/291.59	   if(false, x, y) -> y [1]
320.67/291.59	
320.67/291.59	The TRS has the following type information:
320.67/291.59	p :: 0:s -> 0:s
320.67/291.59	0 :: 0:s
320.67/291.59	s :: 0:s -> 0:s
320.67/291.59	le :: 0:s -> 0:s -> true:false
320.67/291.59	true :: true:false
320.67/291.59	false :: true:false
320.67/291.59	minus :: 0:s -> 0:s -> 0:s
320.67/291.59	if :: true:false -> 0:s -> 0:s -> 0:s
320.67/291.59	
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(5) CompletionProof (UPPER BOUND(ID))
320.67/291.59	The transformation into a RNTS is sound, since:
320.67/291.59	
320.67/291.59	(a) The obligation is a constructor system where every type has a constant constructor,
320.67/291.59	
320.67/291.59	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
320.67/291.59	none
320.67/291.59	
320.67/291.59	(c) The following functions are completely defined:
320.67/291.59	
320.67/291.59	   le_2
320.67/291.59	   p_1
320.67/291.59	   minus_2
320.67/291.59	   if_3
320.67/291.59	
320.67/291.59	Due to the following rules being added:
320.67/291.59	none
320.67/291.59	
320.67/291.59	And the following fresh constants: none
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(6)
320.67/291.59	Obligation:
320.67/291.59	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
320.67/291.59	
320.67/291.59	Runtime Complexity Weighted TRS with Types.
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0 [1]
320.67/291.59	   p(s(x)) -> x [1]
320.67/291.59	   le(0, y) -> true [1]
320.67/291.59	   le(s(x), 0) -> false [1]
320.67/291.59	   le(s(x), s(y)) -> le(x, y) [1]
320.67/291.59	   minus(x, 0) -> x [1]
320.67/291.59	   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y))))) [1]
320.67/291.59	   if(true, x, y) -> x [1]
320.67/291.59	   if(false, x, y) -> y [1]
320.67/291.59	
320.67/291.59	The TRS has the following type information:
320.67/291.59	p :: 0:s -> 0:s
320.67/291.59	0 :: 0:s
320.67/291.59	s :: 0:s -> 0:s
320.67/291.59	le :: 0:s -> 0:s -> true:false
320.67/291.59	true :: true:false
320.67/291.59	false :: true:false
320.67/291.59	minus :: 0:s -> 0:s -> 0:s
320.67/291.59	if :: true:false -> 0:s -> 0:s -> 0:s
320.67/291.59	
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
320.67/291.59	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(8)
320.67/291.59	Obligation:
320.67/291.59	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
320.67/291.59	
320.67/291.59	Runtime Complexity Weighted TRS with Types.
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0 [1]
320.67/291.59	   p(s(x)) -> x [1]
320.67/291.59	   le(0, y) -> true [1]
320.67/291.59	   le(s(x), 0) -> false [1]
320.67/291.59	   le(s(x), s(y)) -> le(x, y) [1]
320.67/291.59	   minus(x, 0) -> x [1]
320.67/291.59	   minus(0, s(y)) -> if(true, 0, p(minus(0, y))) [3]
320.67/291.59	   minus(s(x'), s(y)) -> if(le(x', y), 0, p(minus(s(x'), y))) [3]
320.67/291.59	   if(true, x, y) -> x [1]
320.67/291.59	   if(false, x, y) -> y [1]
320.67/291.59	
320.67/291.59	The TRS has the following type information:
320.67/291.59	p :: 0:s -> 0:s
320.67/291.59	0 :: 0:s
320.67/291.59	s :: 0:s -> 0:s
320.67/291.59	le :: 0:s -> 0:s -> true:false
320.67/291.59	true :: true:false
320.67/291.59	false :: true:false
320.67/291.59	minus :: 0:s -> 0:s -> 0:s
320.67/291.59	if :: true:false -> 0:s -> 0:s -> 0:s
320.67/291.59	
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
320.67/291.59	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
320.67/291.59	The constant constructors are abstracted as follows:
320.67/291.59	
320.67/291.59	   0 => 0
320.67/291.59	   true => 1
320.67/291.59	   false => 0
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(10)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(le(x', y), 0, p(minus(1 + x', y))) :|: z = 1 + x', z' = 1 + y, x' >= 0, y >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, y))) :|: z' = 1 + y, y >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(11) InliningProof (UPPER BOUND(ID))
320.67/291.59	Inlined the following terminating rules on right-hand sides where appropriate:
320.67/291.59	
320.67/291.59	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(12)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(le(x', y), 0, p(minus(1 + x', y))) :|: z = 1 + x', z' = 1 + y, x' >= 0, y >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, y))) :|: z' = 1 + y, y >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
320.67/291.59	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(14)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
320.67/291.59	Found the following analysis order by SCC decomposition:
320.67/291.59	
320.67/291.59	   { le }
320.67/291.59	   { if }
320.67/291.59	   { p }
320.67/291.59	   { minus }
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(16)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {le}, {if}, {p}, {minus}
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(17) ResultPropagationProof (UPPER BOUND(ID))
320.67/291.59	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(18)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {le}, {if}, {p}, {minus}
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(19) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed SIZE bound using CoFloCo for: le
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(1) with polynomial bound: 1
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(20)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {le}, {if}, {p}, {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: ?, size: O(1) [1]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(21) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed RUNTIME bound using KoAT for: le
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(n^1) with polynomial bound: 2 + z'
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(22)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {if}, {p}, {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(23) ResultPropagationProof (UPPER BOUND(ID))
320.67/291.59	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(24)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {if}, {p}, {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(25) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed SIZE bound using CoFloCo for: if
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(n^1) with polynomial bound: z' + z''
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(26)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {if}, {p}, {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: ?, size: O(n^1) [z' + z'']
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(27) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed RUNTIME bound using CoFloCo for: if
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(1) with polynomial bound: 1
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(28)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {p}, {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(29) ResultPropagationProof (UPPER BOUND(ID))
320.67/291.59	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(30)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {p}, {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(31) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed SIZE bound using KoAT for: p
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(n^1) with polynomial bound: z
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(32)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {p}, {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
320.67/291.59	  p: runtime: ?, size: O(n^1) [z]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(33) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed RUNTIME bound using CoFloCo for: p
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(1) with polynomial bound: 1
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(34)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
320.67/291.59	  p: runtime: O(1) [1], size: O(n^1) [z]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(35) ResultPropagationProof (UPPER BOUND(ID))
320.67/291.59	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(36)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
320.67/291.59	  p: runtime: O(1) [1], size: O(n^1) [z]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(37) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed SIZE bound using CoFloCo for: minus
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(n^1) with polynomial bound: z
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(38)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: {minus}
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
320.67/291.59	  p: runtime: O(1) [1], size: O(n^1) [z]
320.67/291.59	  minus: runtime: ?, size: O(n^1) [z]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(39) IntTrsBoundProof (UPPER BOUND(ID))
320.67/291.59	
320.67/291.59	Computed RUNTIME bound using CoFloCo for: minus
320.67/291.59	after applying outer abstraction to obtain an ITS,
320.67/291.59	resulting in: O(n^2) with polynomial bound: 1 + 6*z' + z'^2
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(40)
320.67/291.59	Obligation:
320.67/291.59	Complexity RNTS consisting of the following rules:
320.67/291.59	
320.67/291.59	   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
320.67/291.59	   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
320.67/291.59	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
320.67/291.59	   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
320.67/291.59	   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
320.67/291.59	   p(z) -{ 1 }-> 0 :|: z = 0
320.67/291.59	   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
320.67/291.59	
320.67/291.59	Function symbols to be analyzed: 
320.67/291.59	Previous analysis results are:
320.67/291.59	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
320.67/291.59	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
320.67/291.59	  p: runtime: O(1) [1], size: O(n^1) [z]
320.67/291.59	  minus: runtime: O(n^2) [1 + 6*z' + z'^2], size: O(n^1) [z]
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(41) FinalProof (FINISHED)
320.67/291.59	Computed overall runtime complexity
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(42)
320.67/291.59	BOUNDS(1, n^2)
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(43) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
320.67/291.59	Transformed a relative TRS into a decreasing-loop problem.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(44)
320.67/291.59	Obligation:
320.67/291.59	Analyzing the following TRS for decreasing loops:
320.67/291.59	
320.67/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.67/291.59	
320.67/291.59	
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0
320.67/291.59	   p(s(x)) -> x
320.67/291.59	   le(0, y) -> true
320.67/291.59	   le(s(x), 0) -> false
320.67/291.59	   le(s(x), s(y)) -> le(x, y)
320.67/291.59	   minus(x, 0) -> x
320.67/291.59	   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y)))))
320.67/291.59	   if(true, x, y) -> x
320.67/291.59	   if(false, x, y) -> y
320.67/291.59	
320.67/291.59	S is empty.
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(45) DecreasingLoopProof (LOWER BOUND(ID))
320.67/291.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
320.67/291.59	
320.67/291.59	The rewrite sequence
320.67/291.59	
320.67/291.59	le(s(x), s(y)) ->^+ le(x, y)
320.67/291.59	
320.67/291.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
320.67/291.59	
320.67/291.59	The pumping substitution is [x / s(x), y / s(y)].
320.67/291.59	
320.67/291.59	The result substitution is [ ].
320.67/291.59	
320.67/291.59	
320.67/291.59	
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(46)
320.67/291.59	Complex Obligation (BEST)
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(47)
320.67/291.59	Obligation:
320.67/291.59	Proved the lower bound n^1 for the following obligation:
320.67/291.59	
320.67/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.67/291.59	
320.67/291.59	
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0
320.67/291.59	   p(s(x)) -> x
320.67/291.59	   le(0, y) -> true
320.67/291.59	   le(s(x), 0) -> false
320.67/291.59	   le(s(x), s(y)) -> le(x, y)
320.67/291.59	   minus(x, 0) -> x
320.67/291.59	   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y)))))
320.67/291.59	   if(true, x, y) -> x
320.67/291.59	   if(false, x, y) -> y
320.67/291.59	
320.67/291.59	S is empty.
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(48) LowerBoundPropagationProof (FINISHED)
320.67/291.59	Propagated lower bound.
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(49)
320.67/291.59	BOUNDS(n^1, INF)
320.67/291.59	
320.67/291.59	----------------------------------------
320.67/291.59	
320.67/291.59	(50)
320.67/291.59	Obligation:
320.67/291.59	Analyzing the following TRS for decreasing loops:
320.67/291.59	
320.67/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
320.67/291.59	
320.67/291.59	
320.67/291.59	The TRS R consists of the following rules:
320.67/291.59	
320.67/291.59	   p(0) -> 0
320.67/291.59	   p(s(x)) -> x
320.67/291.59	   le(0, y) -> true
320.67/291.59	   le(s(x), 0) -> false
320.67/291.59	   le(s(x), s(y)) -> le(x, y)
320.67/291.59	   minus(x, 0) -> x
320.67/291.59	   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y)))))
320.67/291.59	   if(true, x, y) -> x
320.67/291.59	   if(false, x, y) -> y
320.67/291.59	
320.67/291.59	S is empty.
320.67/291.59	Rewrite Strategy: INNERMOST
320.67/291.63	EOF