325.17/291.51	WORST_CASE(Omega(n^1), O(n^3))
325.21/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
325.21/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
325.21/291.53	
325.21/291.53	
325.21/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
325.21/291.53	
325.21/291.53	(0) CpxTRS
325.21/291.53	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
325.21/291.53	(2) CpxWeightedTrs
325.21/291.53	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
325.21/291.53	    (4) CpxTypedWeightedTrs
325.21/291.53	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
325.21/291.53	    (6) CpxTypedWeightedCompleteTrs
325.21/291.53	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
325.21/291.53	    (8) CpxTypedWeightedCompleteTrs
325.21/291.53	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
325.21/291.53	    (10) CpxRNTS
325.21/291.53	    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
325.21/291.53	    (12) CpxRNTS
325.21/291.53	    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
325.21/291.53	    (14) CpxRNTS
325.21/291.53	    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
325.21/291.53	    (16) CpxRNTS
325.21/291.53	    (17) IntTrsBoundProof [UPPER BOUND(ID), 405 ms]
325.21/291.53	    (18) CpxRNTS
325.21/291.53	    (19) IntTrsBoundProof [UPPER BOUND(ID), 121 ms]
325.21/291.53	    (20) CpxRNTS
325.21/291.53	    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
325.21/291.53	    (22) CpxRNTS
325.21/291.53	    (23) IntTrsBoundProof [UPPER BOUND(ID), 949 ms]
325.21/291.53	    (24) CpxRNTS
325.21/291.53	    (25) IntTrsBoundProof [UPPER BOUND(ID), 334 ms]
325.21/291.53	    (26) CpxRNTS
325.21/291.53	    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
325.21/291.53	    (28) CpxRNTS
325.21/291.53	    (29) IntTrsBoundProof [UPPER BOUND(ID), 2432 ms]
325.21/291.53	    (30) CpxRNTS
325.21/291.53	    (31) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
325.21/291.53	    (32) CpxRNTS
325.21/291.53	    (33) FinalProof [FINISHED, 0 ms]
325.21/291.53	    (34) BOUNDS(1, n^3)
325.21/291.53	(35) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
325.21/291.53	(36) TRS for Loop Detection
325.21/291.53	    (37) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
325.21/291.53	    (38) BEST
325.21/291.53	        (39) proven lower bound
325.21/291.53	            (40) LowerBoundPropagationProof [FINISHED, 0 ms]
325.21/291.53	            (41) BOUNDS(n^1, INF)
325.21/291.53	        (42) TRS for Loop Detection
325.21/291.53	
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(0)
325.21/291.53	Obligation:
325.21/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
325.21/291.53	
325.21/291.53	
325.21/291.53	The TRS R consists of the following rules:
325.21/291.53	
325.21/291.53	   le(0, y) -> true
325.21/291.53	   le(s(x), 0) -> false
325.21/291.53	   le(s(x), s(y)) -> le(x, y)
325.21/291.53	   minus(0, y) -> 0
325.21/291.53	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
325.21/291.53	   if_minus(true, s(x), y) -> 0
325.21/291.53	   if_minus(false, s(x), y) -> s(minus(x, y))
325.21/291.53	   gcd(0, y) -> y
325.21/291.53	   gcd(s(x), 0) -> s(x)
325.21/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
325.21/291.53	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
325.21/291.53	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
325.21/291.53	
325.21/291.53	S is empty.
325.21/291.53	Rewrite Strategy: INNERMOST
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
325.21/291.53	Transformed relative TRS to weighted TRS
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(2)
325.21/291.53	Obligation:
325.21/291.53	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).
325.21/291.53	
325.21/291.53	
325.21/291.53	The TRS R consists of the following rules:
325.21/291.53	
325.21/291.53	   le(0, y) -> true [1]
325.21/291.53	   le(s(x), 0) -> false [1]
325.21/291.53	   le(s(x), s(y)) -> le(x, y) [1]
325.21/291.53	   minus(0, y) -> 0 [1]
325.21/291.53	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
325.21/291.53	   if_minus(true, s(x), y) -> 0 [1]
325.21/291.53	   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
325.21/291.53	   gcd(0, y) -> y [1]
325.21/291.53	   gcd(s(x), 0) -> s(x) [1]
325.21/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
325.21/291.53	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
325.21/291.53	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]
325.21/291.53	
325.21/291.53	Rewrite Strategy: INNERMOST
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
325.21/291.53	Infered types.
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(4)
325.21/291.53	Obligation:
325.21/291.53	Runtime Complexity Weighted TRS with Types.
325.21/291.53	The TRS R consists of the following rules:
325.21/291.53	
325.21/291.53	   le(0, y) -> true [1]
325.21/291.53	   le(s(x), 0) -> false [1]
325.21/291.53	   le(s(x), s(y)) -> le(x, y) [1]
325.21/291.53	   minus(0, y) -> 0 [1]
325.21/291.53	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
325.21/291.53	   if_minus(true, s(x), y) -> 0 [1]
325.21/291.53	   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
325.21/291.53	   gcd(0, y) -> y [1]
325.21/291.53	   gcd(s(x), 0) -> s(x) [1]
325.21/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
325.21/291.53	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
325.21/291.53	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]
325.21/291.53	
325.21/291.53	The TRS has the following type information:
325.21/291.53	le :: 0:s -> 0:s -> true:false
325.21/291.53	0 :: 0:s
325.21/291.53	true :: true:false
325.21/291.53	s :: 0:s -> 0:s
325.21/291.53	false :: true:false
325.21/291.53	minus :: 0:s -> 0:s -> 0:s
325.21/291.53	if_minus :: true:false -> 0:s -> 0:s -> 0:s
325.21/291.53	gcd :: 0:s -> 0:s -> 0:s
325.21/291.53	if_gcd :: true:false -> 0:s -> 0:s -> 0:s
325.21/291.53	
325.21/291.53	Rewrite Strategy: INNERMOST
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(5) CompletionProof (UPPER BOUND(ID))
325.21/291.53	The transformation into a RNTS is sound, since:
325.21/291.53	
325.21/291.53	(a) The obligation is a constructor system where every type has a constant constructor,
325.21/291.53	
325.21/291.53	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
325.21/291.53	
325.21/291.53	   gcd_2
325.21/291.53	   if_gcd_3
325.21/291.53	
325.21/291.53	(c) The following functions are completely defined:
325.21/291.53	
325.21/291.53	   le_2
325.21/291.53	   minus_2
325.21/291.53	   if_minus_3
325.21/291.53	
325.21/291.53	Due to the following rules being added:
325.21/291.53	
325.21/291.53	   if_minus(v0, v1, v2) -> 0 [0]
325.21/291.53	
325.21/291.53	And the following fresh constants: none
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(6)
325.21/291.53	Obligation:
325.21/291.53	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
325.21/291.53	
325.21/291.53	Runtime Complexity Weighted TRS with Types.
325.21/291.53	The TRS R consists of the following rules:
325.21/291.53	
325.21/291.53	   le(0, y) -> true [1]
325.21/291.53	   le(s(x), 0) -> false [1]
325.21/291.53	   le(s(x), s(y)) -> le(x, y) [1]
325.21/291.53	   minus(0, y) -> 0 [1]
325.21/291.53	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
325.21/291.53	   if_minus(true, s(x), y) -> 0 [1]
325.21/291.53	   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
325.21/291.53	   gcd(0, y) -> y [1]
325.21/291.53	   gcd(s(x), 0) -> s(x) [1]
325.21/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
325.21/291.53	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
325.21/291.53	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]
325.21/291.53	   if_minus(v0, v1, v2) -> 0 [0]
325.21/291.53	
325.21/291.53	The TRS has the following type information:
325.21/291.53	le :: 0:s -> 0:s -> true:false
325.21/291.53	0 :: 0:s
325.21/291.53	true :: true:false
325.21/291.53	s :: 0:s -> 0:s
325.21/291.53	false :: true:false
325.21/291.53	minus :: 0:s -> 0:s -> 0:s
325.21/291.53	if_minus :: true:false -> 0:s -> 0:s -> 0:s
325.21/291.53	gcd :: 0:s -> 0:s -> 0:s
325.21/291.53	if_gcd :: true:false -> 0:s -> 0:s -> 0:s
325.21/291.53	
325.21/291.53	Rewrite Strategy: INNERMOST
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
325.21/291.53	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(8)
325.21/291.53	Obligation:
325.21/291.53	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
325.21/291.53	
325.21/291.53	Runtime Complexity Weighted TRS with Types.
325.21/291.53	The TRS R consists of the following rules:
325.21/291.53	
325.21/291.53	   le(0, y) -> true [1]
325.21/291.53	   le(s(x), 0) -> false [1]
325.21/291.53	   le(s(x), s(y)) -> le(x, y) [1]
325.21/291.53	   minus(0, y) -> 0 [1]
325.21/291.53	   minus(s(x), 0) -> if_minus(false, s(x), 0) [2]
325.21/291.53	   minus(s(x), s(y')) -> if_minus(le(x, y'), s(x), s(y')) [2]
325.21/291.53	   if_minus(true, s(x), y) -> 0 [1]
325.21/291.53	   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
325.21/291.53	   gcd(0, y) -> y [1]
325.21/291.53	   gcd(s(x), 0) -> s(x) [1]
325.21/291.53	   gcd(s(x), s(0)) -> if_gcd(true, s(x), s(0)) [2]
325.21/291.53	   gcd(s(0), s(s(x'))) -> if_gcd(false, s(0), s(s(x'))) [2]
325.21/291.53	   gcd(s(s(y'')), s(s(x''))) -> if_gcd(le(x'', y''), s(s(y'')), s(s(x''))) [2]
325.21/291.53	   if_gcd(true, s(0), s(y)) -> gcd(0, s(y)) [2]
325.21/291.53	   if_gcd(true, s(s(x1)), s(y)) -> gcd(if_minus(le(s(x1), y), s(x1), y), s(y)) [2]
325.21/291.53	   if_gcd(false, s(x), s(0)) -> gcd(0, s(x)) [2]
325.21/291.53	   if_gcd(false, s(x), s(s(x2))) -> gcd(if_minus(le(s(x2), x), s(x2), x), s(x)) [2]
325.21/291.53	   if_minus(v0, v1, v2) -> 0 [0]
325.21/291.53	
325.21/291.53	The TRS has the following type information:
325.21/291.53	le :: 0:s -> 0:s -> true:false
325.21/291.53	0 :: 0:s
325.21/291.53	true :: true:false
325.21/291.53	s :: 0:s -> 0:s
325.21/291.53	false :: true:false
325.21/291.53	minus :: 0:s -> 0:s -> 0:s
325.21/291.53	if_minus :: true:false -> 0:s -> 0:s -> 0:s
325.21/291.53	gcd :: 0:s -> 0:s -> 0:s
325.21/291.53	if_gcd :: true:false -> 0:s -> 0:s -> 0:s
325.21/291.53	
325.21/291.53	Rewrite Strategy: INNERMOST
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
325.21/291.53	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
325.21/291.53	The constant constructors are abstracted as follows:
325.21/291.53	
325.21/291.53	   0 => 0
325.21/291.53	   true => 1
325.21/291.53	   false => 0
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(10)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(le(x'', y''), 1 + (1 + y''), 1 + (1 + x'')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y''), y'' >= 0, x'' >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + x')) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + x1, y), 1 + x1, y), 1 + y) :|: x1 >= 0, z' = 1 + (1 + x1), z = 1, y >= 0, z'' = 1 + y
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + x2, x), 1 + x2, x), 1 + x) :|: z' = 1 + x, x >= 0, z'' = 1 + (1 + x2), z = 0, x2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + x) :|: z' = 1 + x, x >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + y) :|: z = 1, y >= 0, z'' = 1 + y, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(le(x, y'), 1 + x, 1 + y') :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + x, 0) :|: x >= 0, z = 1 + x, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
325.21/291.53	
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(11) SimplificationProof (BOTH BOUNDS(ID, ID))
325.21/291.53	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(12)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
325.21/291.53	Found the following analysis order by SCC decomposition:
325.21/291.53	
325.21/291.53	   { le }
325.21/291.53	   { minus, if_minus }
325.21/291.53	   { gcd, if_gcd }
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(14)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(15) ResultPropagationProof (UPPER BOUND(ID))
325.21/291.53	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(16)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(17) IntTrsBoundProof (UPPER BOUND(ID))
325.21/291.53	
325.21/291.53	Computed SIZE bound using CoFloCo for: le
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(1) with polynomial bound: 1
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(18)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: ?, size: O(1) [1]
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(19) IntTrsBoundProof (UPPER BOUND(ID))
325.21/291.53	
325.21/291.53	Computed RUNTIME bound using KoAT for: le
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^1) with polynomial bound: 2 + z'
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(20)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(21) ResultPropagationProof (UPPER BOUND(ID))
325.21/291.53	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(22)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(23) IntTrsBoundProof (UPPER BOUND(ID))
325.21/291.53	
325.21/291.53	Computed SIZE bound using KoAT for: minus
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^1) with polynomial bound: z
325.21/291.53	
325.21/291.53	Computed SIZE bound using CoFloCo for: if_minus
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^1) with polynomial bound: z'
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(24)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
325.21/291.53	  minus: runtime: ?, size: O(n^1) [z]
325.21/291.53	  if_minus: runtime: ?, size: O(n^1) [z']
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(25) IntTrsBoundProof (UPPER BOUND(ID))
325.21/291.53	
325.21/291.53	Computed RUNTIME bound using CoFloCo for: minus
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^2) with polynomial bound: 8 + 4*z + z*z' + 2*z'
325.21/291.53	
325.21/291.53	Computed RUNTIME bound using KoAT for: if_minus
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^2) with polynomial bound: 6 + 4*z' + z'*z'' + z''
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(26)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 2 }-> if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {gcd,if_gcd}
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
325.21/291.53	  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
325.21/291.53	  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(27) ResultPropagationProof (UPPER BOUND(ID))
325.21/291.53	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(28)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 + (z'' - 2), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {gcd,if_gcd}
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
325.21/291.53	  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
325.21/291.53	  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(29) IntTrsBoundProof (UPPER BOUND(ID))
325.21/291.53	
325.21/291.53	Computed SIZE bound using KoAT for: gcd
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^1) with polynomial bound: z + z'
325.21/291.53	
325.21/291.53	Computed SIZE bound using KoAT for: if_gcd
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^1) with polynomial bound: z' + z''
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(30)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 + (z'' - 2), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: {gcd,if_gcd}
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
325.21/291.53	  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
325.21/291.53	  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
325.21/291.53	  gcd: runtime: ?, size: O(n^1) [z + z']
325.21/291.53	  if_gcd: runtime: ?, size: O(n^1) [z' + z'']
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(31) IntTrsBoundProof (UPPER BOUND(ID))
325.21/291.53	
325.21/291.53	Computed RUNTIME bound using KoAT for: gcd
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^3) with polynomial bound: 2 + 58*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 58*z' + 26*z'^2 + 4*z'^3
325.21/291.53	
325.21/291.53	Computed RUNTIME bound using KoAT for: if_gcd
325.21/291.53	after applying outer abstraction to obtain an ITS,
325.21/291.53	resulting in: O(n^3) with polynomial bound: 8 + 98*z' + 58*z'*z'' + 24*z'*z''^2 + 54*z'^2 + 24*z'^2*z'' + 12*z'^3 + 98*z'' + 54*z''^2 + 12*z''^3
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(32)
325.21/291.53	Obligation:
325.21/291.53	Complexity RNTS consisting of the following rules:
325.21/291.53	
325.21/291.53	   gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
325.21/291.53	   gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
325.21/291.53	   gcd(z, z') -{ 2 }-> if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
325.21/291.53	   gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
325.21/291.53	   if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 2), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 + (z'' - 2), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
325.21/291.53	   if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
325.21/291.53	   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
325.21/291.53	   if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z'' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
325.21/291.53	   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0
325.21/291.53	   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
325.21/291.53	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
325.21/291.53	
325.21/291.53	Function symbols to be analyzed: 
325.21/291.53	Previous analysis results are:
325.21/291.53	  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
325.21/291.53	  minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z]
325.21/291.53	  if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
325.21/291.53	  gcd: runtime: O(n^3) [2 + 58*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 58*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z']
325.21/291.53	  if_gcd: runtime: O(n^3) [8 + 98*z' + 58*z'*z'' + 24*z'*z''^2 + 54*z'^2 + 24*z'^2*z'' + 12*z'^3 + 98*z'' + 54*z''^2 + 12*z''^3], size: O(n^1) [z' + z'']
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(33) FinalProof (FINISHED)
325.21/291.53	Computed overall runtime complexity
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(34)
325.21/291.53	BOUNDS(1, n^3)
325.21/291.53	
325.21/291.53	----------------------------------------
325.21/291.53	
325.21/291.53	(35) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
325.21/291.54	Transformed a relative TRS into a decreasing-loop problem.
325.21/291.54	----------------------------------------
325.21/291.54	
325.21/291.54	(36)
325.21/291.54	Obligation:
325.21/291.54	Analyzing the following TRS for decreasing loops:
325.21/291.54	
325.21/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
325.21/291.54	
325.21/291.54	
325.21/291.54	The TRS R consists of the following rules:
325.21/291.54	
325.21/291.54	   le(0, y) -> true
325.21/291.54	   le(s(x), 0) -> false
325.21/291.54	   le(s(x), s(y)) -> le(x, y)
325.21/291.54	   minus(0, y) -> 0
325.21/291.54	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
325.21/291.54	   if_minus(true, s(x), y) -> 0
325.21/291.54	   if_minus(false, s(x), y) -> s(minus(x, y))
325.21/291.54	   gcd(0, y) -> y
325.21/291.54	   gcd(s(x), 0) -> s(x)
325.21/291.54	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
325.21/291.54	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
325.21/291.54	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
325.21/291.54	
325.21/291.54	S is empty.
325.21/291.54	Rewrite Strategy: INNERMOST
325.21/291.54	----------------------------------------
325.21/291.54	
325.21/291.54	(37) DecreasingLoopProof (LOWER BOUND(ID))
325.21/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
325.21/291.54	
325.21/291.54	The rewrite sequence
325.21/291.54	
325.21/291.54	le(s(x), s(y)) ->^+ le(x, y)
325.21/291.54	
325.21/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
325.21/291.54	
325.21/291.54	The pumping substitution is [x / s(x), y / s(y)].
325.21/291.54	
325.21/291.54	The result substitution is [ ].
325.21/291.54	
325.21/291.54	
325.21/291.54	
325.21/291.54	
325.21/291.54	----------------------------------------
325.21/291.54	
325.21/291.54	(38)
325.21/291.54	Complex Obligation (BEST)
325.21/291.54	
325.21/291.54	----------------------------------------
325.21/291.54	
325.21/291.54	(39)
325.21/291.54	Obligation:
325.21/291.54	Proved the lower bound n^1 for the following obligation:
325.21/291.54	
325.21/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
325.21/291.54	
325.21/291.54	
325.21/291.54	The TRS R consists of the following rules:
325.21/291.54	
325.21/291.54	   le(0, y) -> true
325.21/291.54	   le(s(x), 0) -> false
325.21/291.54	   le(s(x), s(y)) -> le(x, y)
325.21/291.54	   minus(0, y) -> 0
325.21/291.54	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
325.21/291.54	   if_minus(true, s(x), y) -> 0
325.21/291.54	   if_minus(false, s(x), y) -> s(minus(x, y))
325.21/291.54	   gcd(0, y) -> y
325.21/291.54	   gcd(s(x), 0) -> s(x)
325.21/291.54	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
325.21/291.54	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
325.21/291.54	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
325.21/291.54	
325.21/291.54	S is empty.
325.21/291.54	Rewrite Strategy: INNERMOST
325.21/291.54	----------------------------------------
325.21/291.54	
325.21/291.54	(40) LowerBoundPropagationProof (FINISHED)
325.21/291.54	Propagated lower bound.
325.21/291.54	----------------------------------------
325.21/291.54	
325.21/291.54	(41)
325.21/291.54	BOUNDS(n^1, INF)
325.21/291.54	
325.21/291.54	----------------------------------------
325.21/291.54	
325.21/291.54	(42)
325.21/291.54	Obligation:
325.21/291.54	Analyzing the following TRS for decreasing loops:
325.21/291.54	
325.21/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
325.21/291.54	
325.21/291.54	
325.21/291.54	The TRS R consists of the following rules:
325.21/291.54	
325.21/291.54	   le(0, y) -> true
325.21/291.54	   le(s(x), 0) -> false
325.21/291.54	   le(s(x), s(y)) -> le(x, y)
325.21/291.54	   minus(0, y) -> 0
325.21/291.54	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
325.21/291.54	   if_minus(true, s(x), y) -> 0
325.21/291.54	   if_minus(false, s(x), y) -> s(minus(x, y))
325.21/291.54	   gcd(0, y) -> y
325.21/291.54	   gcd(s(x), 0) -> s(x)
325.21/291.54	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
325.21/291.54	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
325.21/291.54	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
325.21/291.54	
325.21/291.54	S is empty.
325.21/291.54	Rewrite Strategy: INNERMOST
325.27/291.57	EOF