22.95/8.94	WORST_CASE(Omega(n^1), O(n^1))
22.95/8.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.95/8.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.95/8.95	
22.95/8.95	
22.95/8.95	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.95/8.95	
22.95/8.95	(0) CpxTRS
22.95/8.95	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
22.95/8.95	(2) CpxTRS
22.95/8.95	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
22.95/8.95	    (4) CpxWeightedTrs
22.95/8.95	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
22.95/8.95	    (6) CpxTypedWeightedTrs
22.95/8.95	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
22.95/8.95	    (8) CpxTypedWeightedCompleteTrs
22.95/8.95	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
22.95/8.95	    (10) CpxTypedWeightedCompleteTrs
22.95/8.95	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
22.95/8.95	    (12) CpxRNTS
22.95/8.95	    (13) InliningProof [UPPER BOUND(ID), 2 ms]
22.95/8.95	    (14) CpxRNTS
22.95/8.95	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
22.95/8.95	    (16) CpxRNTS
22.95/8.95	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
22.95/8.95	    (18) CpxRNTS
22.95/8.95	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.95/8.95	    (20) CpxRNTS
22.95/8.95	    (21) IntTrsBoundProof [UPPER BOUND(ID), 263 ms]
22.95/8.95	    (22) CpxRNTS
22.95/8.95	    (23) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
22.95/8.95	    (24) CpxRNTS
22.95/8.95	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.95/8.95	    (26) CpxRNTS
22.95/8.95	    (27) IntTrsBoundProof [UPPER BOUND(ID), 341 ms]
22.95/8.95	    (28) CpxRNTS
22.95/8.95	    (29) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
22.95/8.95	    (30) CpxRNTS
22.95/8.95	    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.95/8.95	    (32) CpxRNTS
22.95/8.95	    (33) IntTrsBoundProof [UPPER BOUND(ID), 291 ms]
22.95/8.95	    (34) CpxRNTS
22.95/8.95	    (35) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
22.95/8.95	    (36) CpxRNTS
22.95/8.95	    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.95/8.95	    (38) CpxRNTS
22.95/8.95	    (39) IntTrsBoundProof [UPPER BOUND(ID), 467 ms]
22.95/8.95	    (40) CpxRNTS
22.95/8.95	    (41) IntTrsBoundProof [UPPER BOUND(ID), 249 ms]
22.95/8.95	    (42) CpxRNTS
22.95/8.95	    (43) FinalProof [FINISHED, 0 ms]
22.95/8.95	    (44) BOUNDS(1, n^1)
22.95/8.95	(45) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
22.95/8.95	(46) TRS for Loop Detection
22.95/8.95	    (47) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
22.95/8.95	    (48) BEST
22.95/8.95	        (49) proven lower bound
22.95/8.95	            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
22.95/8.95	            (51) BOUNDS(n^1, INF)
22.95/8.95	        (52) TRS for Loop Detection
22.95/8.95	
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(0)
22.95/8.95	Obligation:
22.95/8.95	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.95/8.95	
22.95/8.95	
22.95/8.95	The TRS R consists of the following rules:
22.95/8.95	
22.95/8.95	   minus(0, Y) -> 0
22.95/8.95	   minus(s(X), s(Y)) -> minus(X, Y)
22.95/8.95	   geq(X, 0) -> true
22.95/8.95	   geq(0, s(Y)) -> false
22.95/8.95	   geq(s(X), s(Y)) -> geq(X, Y)
22.95/8.95	   div(0, s(Y)) -> 0
22.95/8.95	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
22.95/8.95	   if(true, X, Y) -> X
22.95/8.95	   if(false, X, Y) -> Y
22.95/8.95	
22.95/8.95	S is empty.
22.95/8.95	Rewrite Strategy: FULL
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
22.95/8.95	Converted rc-obligation to irc-obligation.
22.95/8.95	
22.95/8.95	The duplicating contexts are:
22.95/8.95	div(s([]), s(Y))
22.95/8.95	div(s(X), s([]))
22.95/8.95	
22.95/8.95	
22.95/8.95	The defined contexts are:
22.95/8.95	if([], s(x1), 0)
22.95/8.95	if(x0, s([]), 0)
22.95/8.95	div([], s(x1))
22.95/8.95	geq([], x1)
22.95/8.95	minus([], x1)
22.95/8.95	
22.95/8.95	
22.95/8.95	[] just represents basic- or constructor-terms in the following defined contexts:
22.95/8.95	if([], s(x1), 0)
22.95/8.95	div([], s(x1))
22.95/8.95	
22.95/8.95	
22.95/8.95	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(2)
22.95/8.95	Obligation:
22.95/8.95	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
22.95/8.95	
22.95/8.95	
22.95/8.95	The TRS R consists of the following rules:
22.95/8.95	
22.95/8.95	   minus(0, Y) -> 0
22.95/8.95	   minus(s(X), s(Y)) -> minus(X, Y)
22.95/8.95	   geq(X, 0) -> true
22.95/8.95	   geq(0, s(Y)) -> false
22.95/8.95	   geq(s(X), s(Y)) -> geq(X, Y)
22.95/8.95	   div(0, s(Y)) -> 0
22.95/8.95	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
22.95/8.95	   if(true, X, Y) -> X
22.95/8.95	   if(false, X, Y) -> Y
22.95/8.95	
22.95/8.95	S is empty.
22.95/8.95	Rewrite Strategy: INNERMOST
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
22.95/8.95	Transformed relative TRS to weighted TRS
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(4)
22.95/8.95	Obligation:
22.95/8.95	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
22.95/8.95	
22.95/8.95	
22.95/8.95	The TRS R consists of the following rules:
22.95/8.95	
22.95/8.95	   minus(0, Y) -> 0 [1]
22.95/8.95	   minus(s(X), s(Y)) -> minus(X, Y) [1]
22.95/8.95	   geq(X, 0) -> true [1]
22.95/8.95	   geq(0, s(Y)) -> false [1]
22.95/8.95	   geq(s(X), s(Y)) -> geq(X, Y) [1]
22.95/8.95	   div(0, s(Y)) -> 0 [1]
22.95/8.95	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
22.95/8.95	   if(true, X, Y) -> X [1]
22.95/8.95	   if(false, X, Y) -> Y [1]
22.95/8.95	
22.95/8.95	Rewrite Strategy: INNERMOST
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
22.95/8.95	Infered types.
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(6)
22.95/8.95	Obligation:
22.95/8.95	Runtime Complexity Weighted TRS with Types.
22.95/8.95	The TRS R consists of the following rules:
22.95/8.95	
22.95/8.95	   minus(0, Y) -> 0 [1]
22.95/8.95	   minus(s(X), s(Y)) -> minus(X, Y) [1]
22.95/8.95	   geq(X, 0) -> true [1]
22.95/8.95	   geq(0, s(Y)) -> false [1]
22.95/8.95	   geq(s(X), s(Y)) -> geq(X, Y) [1]
22.95/8.95	   div(0, s(Y)) -> 0 [1]
22.95/8.95	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
22.95/8.95	   if(true, X, Y) -> X [1]
22.95/8.95	   if(false, X, Y) -> Y [1]
22.95/8.95	
22.95/8.95	The TRS has the following type information:
22.95/8.95	minus :: 0:s -> 0:s -> 0:s
22.95/8.95	0 :: 0:s
22.95/8.95	s :: 0:s -> 0:s
22.95/8.95	geq :: 0:s -> 0:s -> true:false
22.95/8.95	true :: true:false
22.95/8.95	false :: true:false
22.95/8.95	div :: 0:s -> 0:s -> 0:s
22.95/8.95	if :: true:false -> 0:s -> 0:s -> 0:s
22.95/8.95	
22.95/8.95	Rewrite Strategy: INNERMOST
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(7) CompletionProof (UPPER BOUND(ID))
22.95/8.95	The transformation into a RNTS is sound, since:
22.95/8.95	
22.95/8.95	(a) The obligation is a constructor system where every type has a constant constructor,
22.95/8.95	
22.95/8.95	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
22.95/8.95	none
22.95/8.95	
22.95/8.95	(c) The following functions are completely defined:
22.95/8.95	
22.95/8.95	   geq_2
22.95/8.95	   div_2
22.95/8.95	   minus_2
22.95/8.95	   if_3
22.95/8.95	
22.95/8.95	Due to the following rules being added:
22.95/8.95	
22.95/8.95	   div(v0, v1) -> 0 [0]
22.95/8.95	   minus(v0, v1) -> 0 [0]
22.95/8.95	
22.95/8.95	And the following fresh constants: none
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(8)
22.95/8.95	Obligation:
22.95/8.95	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
22.95/8.95	
22.95/8.95	Runtime Complexity Weighted TRS with Types.
22.95/8.95	The TRS R consists of the following rules:
22.95/8.95	
22.95/8.95	   minus(0, Y) -> 0 [1]
22.95/8.95	   minus(s(X), s(Y)) -> minus(X, Y) [1]
22.95/8.95	   geq(X, 0) -> true [1]
22.95/8.95	   geq(0, s(Y)) -> false [1]
22.95/8.95	   geq(s(X), s(Y)) -> geq(X, Y) [1]
22.95/8.95	   div(0, s(Y)) -> 0 [1]
22.95/8.95	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
22.95/8.95	   if(true, X, Y) -> X [1]
22.95/8.95	   if(false, X, Y) -> Y [1]
22.95/8.95	   div(v0, v1) -> 0 [0]
22.95/8.95	   minus(v0, v1) -> 0 [0]
22.95/8.95	
22.95/8.95	The TRS has the following type information:
22.95/8.95	minus :: 0:s -> 0:s -> 0:s
22.95/8.95	0 :: 0:s
22.95/8.95	s :: 0:s -> 0:s
22.95/8.95	geq :: 0:s -> 0:s -> true:false
22.95/8.95	true :: true:false
22.95/8.95	false :: true:false
22.95/8.95	div :: 0:s -> 0:s -> 0:s
22.95/8.95	if :: true:false -> 0:s -> 0:s -> 0:s
22.95/8.95	
22.95/8.95	Rewrite Strategy: INNERMOST
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
22.95/8.95	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(10)
22.95/8.95	Obligation:
22.95/8.95	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
22.95/8.95	
22.95/8.95	Runtime Complexity Weighted TRS with Types.
22.95/8.95	The TRS R consists of the following rules:
22.95/8.95	
22.95/8.95	   minus(0, Y) -> 0 [1]
22.95/8.95	   minus(s(X), s(Y)) -> minus(X, Y) [1]
22.95/8.95	   geq(X, 0) -> true [1]
22.95/8.95	   geq(0, s(Y)) -> false [1]
22.95/8.95	   geq(s(X), s(Y)) -> geq(X, Y) [1]
22.95/8.95	   div(0, s(Y)) -> 0 [1]
22.95/8.95	   div(s(0), s(0)) -> if(true, s(div(0, s(0))), 0) [3]
22.95/8.95	   div(s(X), s(0)) -> if(true, s(div(0, s(0))), 0) [2]
22.95/8.95	   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [3]
22.95/8.95	   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [2]
22.95/8.95	   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(minus(X', Y''), s(s(Y'')))), 0) [3]
22.95/8.95	   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(0, s(s(Y'')))), 0) [2]
22.95/8.95	   if(true, X, Y) -> X [1]
22.95/8.95	   if(false, X, Y) -> Y [1]
22.95/8.95	   div(v0, v1) -> 0 [0]
22.95/8.95	   minus(v0, v1) -> 0 [0]
22.95/8.95	
22.95/8.95	The TRS has the following type information:
22.95/8.95	minus :: 0:s -> 0:s -> 0:s
22.95/8.95	0 :: 0:s
22.95/8.95	s :: 0:s -> 0:s
22.95/8.95	geq :: 0:s -> 0:s -> true:false
22.95/8.95	true :: true:false
22.95/8.95	false :: true:false
22.95/8.95	div :: 0:s -> 0:s -> 0:s
22.95/8.95	if :: true:false -> 0:s -> 0:s -> 0:s
22.95/8.95	
22.95/8.95	Rewrite Strategy: INNERMOST
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
22.95/8.95	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
22.95/8.95	The constant constructors are abstracted as follows:
22.95/8.95	
22.95/8.95	   0 => 0
22.95/8.95	   true => 1
22.95/8.95	   false => 0
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(12)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
22.95/8.95	   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
22.95/8.95	
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(13) InliningProof (UPPER BOUND(ID))
22.95/8.95	Inlined the following terminating rules on right-hand sides where appropriate:
22.95/8.95	
22.95/8.95	   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(14)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
22.95/8.95	   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
22.95/8.95	
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
22.95/8.95	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(16)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
22.95/8.95	Found the following analysis order by SCC decomposition:
22.95/8.95	
22.95/8.95	   { minus }
22.95/8.95	   { if }
22.95/8.95	   { geq }
22.95/8.95	   { div }
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(18)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	
22.95/8.95	Function symbols to be analyzed: {minus}, {if}, {geq}, {div}
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(19) ResultPropagationProof (UPPER BOUND(ID))
22.95/8.95	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(20)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	
22.95/8.95	Function symbols to be analyzed: {minus}, {if}, {geq}, {div}
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(21) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.95	
22.95/8.95	Computed SIZE bound using CoFloCo for: minus
22.95/8.95	after applying outer abstraction to obtain an ITS,
22.95/8.95	resulting in: O(1) with polynomial bound: 0
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(22)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	
22.95/8.95	Function symbols to be analyzed: {minus}, {if}, {geq}, {div}
22.95/8.95	Previous analysis results are:
22.95/8.95	  minus: runtime: ?, size: O(1) [0]
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(23) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.95	
22.95/8.95	Computed RUNTIME bound using KoAT for: minus
22.95/8.95	after applying outer abstraction to obtain an ITS,
22.95/8.95	resulting in: O(n^1) with polynomial bound: 1 + z'
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(24)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	
22.95/8.95	Function symbols to be analyzed: {if}, {geq}, {div}
22.95/8.95	Previous analysis results are:
22.95/8.95	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(25) ResultPropagationProof (UPPER BOUND(ID))
22.95/8.95	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(26)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	
22.95/8.95	Function symbols to be analyzed: {if}, {geq}, {div}
22.95/8.95	Previous analysis results are:
22.95/8.95	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(27) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.95	
22.95/8.95	Computed SIZE bound using CoFloCo for: if
22.95/8.95	after applying outer abstraction to obtain an ITS,
22.95/8.95	resulting in: O(n^1) with polynomial bound: z' + z''
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(28)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.95	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.95	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.95	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.95	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.95	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.95	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.95	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.95	
22.95/8.95	Function symbols to be analyzed: {if}, {geq}, {div}
22.95/8.95	Previous analysis results are:
22.95/8.95	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.95	  if: runtime: ?, size: O(n^1) [z' + z'']
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(29) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.95	
22.95/8.95	Computed RUNTIME bound using CoFloCo for: if
22.95/8.95	after applying outer abstraction to obtain an ITS,
22.95/8.95	resulting in: O(1) with polynomial bound: 1
22.95/8.95	
22.95/8.95	----------------------------------------
22.95/8.95	
22.95/8.95	(30)
22.95/8.95	Obligation:
22.95/8.95	Complexity RNTS consisting of the following rules:
22.95/8.95	
22.95/8.95	   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.96	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	
22.95/8.96	Function symbols to be analyzed: {geq}, {div}
22.95/8.96	Previous analysis results are:
22.95/8.96	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.96	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(31) ResultPropagationProof (UPPER BOUND(ID))
22.95/8.96	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(32)
22.95/8.96	Obligation:
22.95/8.96	Complexity RNTS consisting of the following rules:
22.95/8.96	
22.95/8.96	   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.96	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	
22.95/8.96	Function symbols to be analyzed: {geq}, {div}
22.95/8.96	Previous analysis results are:
22.95/8.96	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.96	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(33) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.96	
22.95/8.96	Computed SIZE bound using CoFloCo for: geq
22.95/8.96	after applying outer abstraction to obtain an ITS,
22.95/8.96	resulting in: O(1) with polynomial bound: 1
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(34)
22.95/8.96	Obligation:
22.95/8.96	Complexity RNTS consisting of the following rules:
22.95/8.96	
22.95/8.96	   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.96	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	
22.95/8.96	Function symbols to be analyzed: {geq}, {div}
22.95/8.96	Previous analysis results are:
22.95/8.96	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.96	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
22.95/8.96	  geq: runtime: ?, size: O(1) [1]
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(35) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.96	
22.95/8.96	Computed RUNTIME bound using KoAT for: geq
22.95/8.96	after applying outer abstraction to obtain an ITS,
22.95/8.96	resulting in: O(n^1) with polynomial bound: 2 + z'
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(36)
22.95/8.96	Obligation:
22.95/8.96	Complexity RNTS consisting of the following rules:
22.95/8.96	
22.95/8.96	   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.96	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	
22.95/8.96	Function symbols to be analyzed: {div}
22.95/8.96	Previous analysis results are:
22.95/8.96	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.96	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
22.95/8.96	  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(37) ResultPropagationProof (UPPER BOUND(ID))
22.95/8.96	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(38)
22.95/8.96	Obligation:
22.95/8.96	Complexity RNTS consisting of the following rules:
22.95/8.96	
22.95/8.96	   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.96	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	
22.95/8.96	Function symbols to be analyzed: {div}
22.95/8.96	Previous analysis results are:
22.95/8.96	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.96	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
22.95/8.96	  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(39) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.96	
22.95/8.96	Computed SIZE bound using CoFloCo for: div
22.95/8.96	after applying outer abstraction to obtain an ITS,
22.95/8.96	resulting in: O(1) with polynomial bound: 1
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(40)
22.95/8.96	Obligation:
22.95/8.96	Complexity RNTS consisting of the following rules:
22.95/8.96	
22.95/8.96	   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.96	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	
22.95/8.96	Function symbols to be analyzed: {div}
22.95/8.96	Previous analysis results are:
22.95/8.96	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.96	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
22.95/8.96	  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
22.95/8.96	  div: runtime: ?, size: O(1) [1]
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(41) IntTrsBoundProof (UPPER BOUND(ID))
22.95/8.96	
22.95/8.96	Computed RUNTIME bound using CoFloCo for: div
22.95/8.96	after applying outer abstraction to obtain an ITS,
22.95/8.96	resulting in: O(n^1) with polynomial bound: 5 + 2*z'
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(42)
22.95/8.96	Obligation:
22.95/8.96	Complexity RNTS consisting of the following rules:
22.95/8.96	
22.95/8.96	   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
22.95/8.96	   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
22.95/8.96	   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
22.95/8.96	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
22.95/8.96	   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
22.95/8.96	   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
22.95/8.96	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
22.95/8.96	   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
22.95/8.96	
22.95/8.96	Function symbols to be analyzed: 
22.95/8.96	Previous analysis results are:
22.95/8.96	  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
22.95/8.96	  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
22.95/8.96	  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
22.95/8.96	  div: runtime: O(n^1) [5 + 2*z'], size: O(1) [1]
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(43) FinalProof (FINISHED)
22.95/8.96	Computed overall runtime complexity
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(44)
22.95/8.96	BOUNDS(1, n^1)
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(45) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
22.95/8.96	Transformed a relative TRS into a decreasing-loop problem.
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(46)
22.95/8.96	Obligation:
22.95/8.96	Analyzing the following TRS for decreasing loops:
22.95/8.96	
22.95/8.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.95/8.96	
22.95/8.96	
22.95/8.96	The TRS R consists of the following rules:
22.95/8.96	
22.95/8.96	   minus(0, Y) -> 0
22.95/8.96	   minus(s(X), s(Y)) -> minus(X, Y)
22.95/8.96	   geq(X, 0) -> true
22.95/8.96	   geq(0, s(Y)) -> false
22.95/8.96	   geq(s(X), s(Y)) -> geq(X, Y)
22.95/8.96	   div(0, s(Y)) -> 0
22.95/8.96	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
22.95/8.96	   if(true, X, Y) -> X
22.95/8.96	   if(false, X, Y) -> Y
22.95/8.96	
22.95/8.96	S is empty.
22.95/8.96	Rewrite Strategy: FULL
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(47) DecreasingLoopProof (LOWER BOUND(ID))
22.95/8.96	The following loop(s) give(s) rise to the lower bound Omega(n^1):
22.95/8.96	
22.95/8.96	The rewrite sequence
22.95/8.96	
22.95/8.96	minus(s(X), s(Y)) ->^+ minus(X, Y)
22.95/8.96	
22.95/8.96	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
22.95/8.96	
22.95/8.96	The pumping substitution is [X / s(X), Y / s(Y)].
22.95/8.96	
22.95/8.96	The result substitution is [ ].
22.95/8.96	
22.95/8.96	
22.95/8.96	
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(48)
22.95/8.96	Complex Obligation (BEST)
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(49)
22.95/8.96	Obligation:
22.95/8.96	Proved the lower bound n^1 for the following obligation:
22.95/8.96	
22.95/8.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.95/8.96	
22.95/8.96	
22.95/8.96	The TRS R consists of the following rules:
22.95/8.96	
22.95/8.96	   minus(0, Y) -> 0
22.95/8.96	   minus(s(X), s(Y)) -> minus(X, Y)
22.95/8.96	   geq(X, 0) -> true
22.95/8.96	   geq(0, s(Y)) -> false
22.95/8.96	   geq(s(X), s(Y)) -> geq(X, Y)
22.95/8.96	   div(0, s(Y)) -> 0
22.95/8.96	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
22.95/8.96	   if(true, X, Y) -> X
22.95/8.96	   if(false, X, Y) -> Y
22.95/8.96	
22.95/8.96	S is empty.
22.95/8.96	Rewrite Strategy: FULL
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(50) LowerBoundPropagationProof (FINISHED)
22.95/8.96	Propagated lower bound.
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(51)
22.95/8.96	BOUNDS(n^1, INF)
22.95/8.96	
22.95/8.96	----------------------------------------
22.95/8.96	
22.95/8.96	(52)
22.95/8.96	Obligation:
22.95/8.96	Analyzing the following TRS for decreasing loops:
22.95/8.96	
22.95/8.96	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.95/8.96	
22.95/8.96	
22.95/8.96	The TRS R consists of the following rules:
22.95/8.96	
22.95/8.96	   minus(0, Y) -> 0
22.95/8.96	   minus(s(X), s(Y)) -> minus(X, Y)
22.95/8.96	   geq(X, 0) -> true
22.95/8.96	   geq(0, s(Y)) -> false
22.95/8.96	   geq(s(X), s(Y)) -> geq(X, Y)
22.95/8.96	   div(0, s(Y)) -> 0
22.95/8.96	   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
22.95/8.96	   if(true, X, Y) -> X
22.95/8.96	   if(false, X, Y) -> Y
22.95/8.96	
22.95/8.96	S is empty.
22.95/8.96	Rewrite Strategy: FULL
22.95/8.98	EOF