1076.78/291.51	WORST_CASE(Omega(n^1), O(n^2))
1085.25/293.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1085.25/293.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1085.25/293.73	
1085.25/293.73	
1085.25/293.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1085.25/293.73	
1085.25/293.73	(0) CpxTRS
1085.25/293.73	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1085.25/293.73	(2) CpxWeightedTrs
1085.25/293.73	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1085.25/293.73	    (4) CpxTypedWeightedTrs
1085.25/293.73	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
1085.25/293.73	    (6) CpxTypedWeightedCompleteTrs
1085.25/293.73	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
1085.25/293.73	    (8) CpxTypedWeightedCompleteTrs
1085.25/293.73	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1085.25/293.73	    (10) CpxRNTS
1085.25/293.73	    (11) InliningProof [UPPER BOUND(ID), 320 ms]
1085.25/293.73	    (12) CpxRNTS
1085.25/293.73	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
1085.25/293.73	    (14) CpxRNTS
1085.25/293.73	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
1085.25/293.73	    (16) CpxRNTS
1085.25/293.73	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1085.25/293.73	    (18) CpxRNTS
1085.25/293.73	    (19) IntTrsBoundProof [UPPER BOUND(ID), 210 ms]
1085.25/293.73	    (20) CpxRNTS
1085.25/293.73	    (21) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
1085.25/293.73	    (22) CpxRNTS
1085.25/293.73	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1085.25/293.73	    (24) CpxRNTS
1085.25/293.73	    (25) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
1085.25/293.73	    (26) CpxRNTS
1085.25/293.73	    (27) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
1085.25/293.73	    (28) CpxRNTS
1085.25/293.73	    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1085.25/293.73	    (30) CpxRNTS
1085.25/293.73	    (31) IntTrsBoundProof [UPPER BOUND(ID), 372 ms]
1085.25/293.73	    (32) CpxRNTS
1085.25/293.73	    (33) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
1085.25/293.73	    (34) CpxRNTS
1085.25/293.73	    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1085.25/293.73	    (36) CpxRNTS
1085.25/293.73	    (37) IntTrsBoundProof [UPPER BOUND(ID), 1971 ms]
1085.25/293.73	    (38) CpxRNTS
1085.25/293.73	    (39) IntTrsBoundProof [UPPER BOUND(ID), 1072 ms]
1085.25/293.73	    (40) CpxRNTS
1085.25/293.73	    (41) FinalProof [FINISHED, 0 ms]
1085.25/293.73	    (42) BOUNDS(1, n^2)
1085.25/293.73	(43) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1085.25/293.73	(44) TRS for Loop Detection
1085.25/293.73	    (45) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1085.25/293.73	    (46) BEST
1085.25/293.73	        (47) proven lower bound
1085.25/293.73	            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
1085.25/293.73	            (49) BOUNDS(n^1, INF)
1085.25/293.73	        (50) TRS for Loop Detection
1085.25/293.73	
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(0)
1085.25/293.73	Obligation:
1085.25/293.73	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1085.25/293.73	
1085.25/293.73	
1085.25/293.73	The TRS R consists of the following rules:
1085.25/293.73	
1085.25/293.73	   cond(true, x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z)
1085.25/293.73	   and(true, true) -> true
1085.25/293.73	   and(x, false) -> false
1085.25/293.73	   and(false, x) -> false
1085.25/293.73	   gr(0, 0) -> false
1085.25/293.73	   gr(0, x) -> false
1085.25/293.73	   gr(s(x), 0) -> true
1085.25/293.73	   gr(s(x), s(y)) -> gr(x, y)
1085.25/293.73	   p(0) -> 0
1085.25/293.73	   p(s(x)) -> x
1085.25/293.73	
1085.25/293.73	S is empty.
1085.25/293.73	Rewrite Strategy: INNERMOST
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1085.25/293.73	Transformed relative TRS to weighted TRS
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(2)
1085.25/293.73	Obligation:
1085.25/293.73	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1085.25/293.73	
1085.25/293.73	
1085.25/293.73	The TRS R consists of the following rules:
1085.25/293.73	
1085.25/293.73	   cond(true, x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) [1]
1085.25/293.73	   and(true, true) -> true [1]
1085.25/293.73	   and(x, false) -> false [1]
1085.25/293.73	   and(false, x) -> false [1]
1085.25/293.73	   gr(0, 0) -> false [1]
1085.25/293.73	   gr(0, x) -> false [1]
1085.25/293.73	   gr(s(x), 0) -> true [1]
1085.25/293.73	   gr(s(x), s(y)) -> gr(x, y) [1]
1085.25/293.73	   p(0) -> 0 [1]
1085.25/293.73	   p(s(x)) -> x [1]
1085.25/293.73	
1085.25/293.73	Rewrite Strategy: INNERMOST
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1085.25/293.73	Infered types.
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(4)
1085.25/293.73	Obligation:
1085.25/293.73	Runtime Complexity Weighted TRS with Types.
1085.25/293.73	The TRS R consists of the following rules:
1085.25/293.73	
1085.25/293.73	   cond(true, x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) [1]
1085.25/293.73	   and(true, true) -> true [1]
1085.25/293.73	   and(x, false) -> false [1]
1085.25/293.73	   and(false, x) -> false [1]
1085.25/293.73	   gr(0, 0) -> false [1]
1085.25/293.73	   gr(0, x) -> false [1]
1085.25/293.73	   gr(s(x), 0) -> true [1]
1085.25/293.73	   gr(s(x), s(y)) -> gr(x, y) [1]
1085.25/293.73	   p(0) -> 0 [1]
1085.25/293.73	   p(s(x)) -> x [1]
1085.25/293.73	
1085.25/293.73	The TRS has the following type information:
1085.25/293.73	cond :: true:false -> 0:s -> 0:s -> 0:s -> cond
1085.25/293.73	true :: true:false
1085.25/293.73	and :: true:false -> true:false -> true:false
1085.25/293.73	gr :: 0:s -> 0:s -> true:false
1085.25/293.73	p :: 0:s -> 0:s
1085.25/293.73	false :: true:false
1085.25/293.73	0 :: 0:s
1085.25/293.73	s :: 0:s -> 0:s
1085.25/293.73	
1085.25/293.73	Rewrite Strategy: INNERMOST
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(5) CompletionProof (UPPER BOUND(ID))
1085.25/293.73	The transformation into a RNTS is sound, since:
1085.25/293.73	
1085.25/293.73	(a) The obligation is a constructor system where every type has a constant constructor,
1085.25/293.73	
1085.25/293.73	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
1085.25/293.73	
1085.25/293.73	   cond_4
1085.25/293.73	
1085.25/293.73	(c) The following functions are completely defined:
1085.25/293.73	
1085.25/293.73	   and_2
1085.25/293.73	   gr_2
1085.25/293.73	   p_1
1085.25/293.73	
1085.25/293.73	Due to the following rules being added:
1085.25/293.73	none
1085.25/293.73	
1085.25/293.73	And the following fresh constants:    const
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(6)
1085.25/293.73	Obligation:
1085.25/293.73	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1085.25/293.73	
1085.25/293.73	Runtime Complexity Weighted TRS with Types.
1085.25/293.73	The TRS R consists of the following rules:
1085.25/293.73	
1085.25/293.73	   cond(true, x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) [1]
1085.25/293.73	   and(true, true) -> true [1]
1085.25/293.73	   and(x, false) -> false [1]
1085.25/293.73	   and(false, x) -> false [1]
1085.25/293.73	   gr(0, 0) -> false [1]
1085.25/293.73	   gr(0, x) -> false [1]
1085.25/293.73	   gr(s(x), 0) -> true [1]
1085.25/293.73	   gr(s(x), s(y)) -> gr(x, y) [1]
1085.25/293.73	   p(0) -> 0 [1]
1085.25/293.73	   p(s(x)) -> x [1]
1085.25/293.73	
1085.25/293.73	The TRS has the following type information:
1085.25/293.73	cond :: true:false -> 0:s -> 0:s -> 0:s -> cond
1085.25/293.73	true :: true:false
1085.25/293.73	and :: true:false -> true:false -> true:false
1085.25/293.73	gr :: 0:s -> 0:s -> true:false
1085.25/293.73	p :: 0:s -> 0:s
1085.25/293.73	false :: true:false
1085.25/293.73	0 :: 0:s
1085.25/293.73	s :: 0:s -> 0:s
1085.25/293.73	const :: cond
1085.25/293.73	
1085.25/293.73	Rewrite Strategy: INNERMOST
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
1085.25/293.73	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(8)
1085.25/293.73	Obligation:
1085.25/293.73	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1085.25/293.73	
1085.25/293.73	Runtime Complexity Weighted TRS with Types.
1085.25/293.73	The TRS R consists of the following rules:
1085.25/293.73	
1085.25/293.73	   cond(true, 0, 0, 0) -> cond(and(false, false), 0, 0, 0) [5]
1085.25/293.73	   cond(true, 0, 0, 0) -> cond(and(false, false), 0, 0, 0) [5]
1085.25/293.73	   cond(true, 0, s(x1), 0) -> cond(and(false, true), 0, x1, 0) [5]
1085.25/293.73	   cond(true, 0, 0, 0) -> cond(and(false, false), 0, 0, 0) [5]
1085.25/293.73	   cond(true, 0, 0, z) -> cond(and(false, false), 0, 0, z) [5]
1085.25/293.73	   cond(true, 0, s(x2), 0) -> cond(and(false, true), 0, x2, 0) [5]
1085.25/293.73	   cond(true, 0, s(x3), s(y'')) -> cond(and(false, gr(x3, y'')), 0, x3, s(y'')) [5]
1085.25/293.73	   cond(true, s(x'), 0, 0) -> cond(and(true, false), x', 0, 0) [5]
1085.25/293.73	   cond(true, s(x'), 0, 0) -> cond(and(true, false), x', 0, 0) [5]
1085.25/293.73	   cond(true, s(x'), s(x4), 0) -> cond(and(true, true), x', x4, 0) [5]
1085.25/293.73	   cond(true, s(x''), 0, s(y')) -> cond(and(gr(x'', y'), false), x'', 0, s(y')) [5]
1085.25/293.73	   cond(true, s(x''), s(x5), s(y')) -> cond(and(gr(x'', y'), gr(x5, y')), x'', x5, s(y')) [5]
1085.25/293.73	   and(true, true) -> true [1]
1085.25/293.73	   and(x, false) -> false [1]
1085.25/293.73	   and(false, x) -> false [1]
1085.25/293.73	   gr(0, 0) -> false [1]
1085.25/293.73	   gr(0, x) -> false [1]
1085.25/293.73	   gr(s(x), 0) -> true [1]
1085.25/293.73	   gr(s(x), s(y)) -> gr(x, y) [1]
1085.25/293.73	   p(0) -> 0 [1]
1085.25/293.73	   p(s(x)) -> x [1]
1085.25/293.73	
1085.25/293.73	The TRS has the following type information:
1085.25/293.73	cond :: true:false -> 0:s -> 0:s -> 0:s -> cond
1085.25/293.73	true :: true:false
1085.25/293.73	and :: true:false -> true:false -> true:false
1085.25/293.73	gr :: 0:s -> 0:s -> true:false
1085.25/293.73	p :: 0:s -> 0:s
1085.25/293.73	false :: true:false
1085.25/293.73	0 :: 0:s
1085.25/293.73	s :: 0:s -> 0:s
1085.25/293.73	const :: cond
1085.25/293.73	
1085.25/293.73	Rewrite Strategy: INNERMOST
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1085.25/293.73	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1085.25/293.73	The constant constructors are abstracted as follows:
1085.25/293.73	
1085.25/293.73	   true => 1
1085.25/293.73	   false => 0
1085.25/293.73	   0 => 0
1085.25/293.73	   const => 0
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(10)
1085.25/293.73	Obligation:
1085.25/293.73	Complexity RNTS consisting of the following rules:
1085.25/293.73	
1085.25/293.73	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(x'', y'), gr(x5, y')), x'', x5, 1 + y') :|: x5 >= 0, y' >= 0, z2 = 1 + y', z' = 1, z'' = 1 + x'', z1 = 1 + x5, x'' >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(x'', y'), 0), x'', 0, 1 + y') :|: z1 = 0, y' >= 0, z2 = 1 + y', z' = 1, z'' = 1 + x'', x'' >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(1, 1), x', x4, 0) :|: x4 >= 0, z'' = 1 + x', z2 = 0, x' >= 0, z1 = 1 + x4, z' = 1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(1, 0), x', 0, 0) :|: z1 = 0, z'' = 1 + x', z2 = 0, x' >= 0, z' = 1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(x3, y'')), 0, x3, 1 + y'') :|: z'' = 0, z2 = 1 + y'', y'' >= 0, z' = 1, z1 = 1 + x3, x3 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, 1), 0, x1, 0) :|: z'' = 0, x1 >= 0, z2 = 0, z' = 1, z1 = 1 + x1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, 1), 0, x2, 0) :|: z'' = 0, z2 = 0, z' = 1, x2 >= 0, z1 = 1 + x2
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, 0), 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, z' = 1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, 0), 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1
1085.25/293.73	   gr(z', z'') -{ 1 }-> gr(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
1085.25/293.73	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
1085.25/293.73	   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
1085.25/293.73	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.73	
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(11) InliningProof (UPPER BOUND(ID))
1085.25/293.73	Inlined the following terminating rules on right-hand sides where appropriate:
1085.25/293.73	
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
1085.25/293.73	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(12)
1085.25/293.73	Obligation:
1085.25/293.73	Complexity RNTS consisting of the following rules:
1085.25/293.73	
1085.25/293.73	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(x'', y'), gr(x5, y')), x'', x5, 1 + y') :|: x5 >= 0, y' >= 0, z2 = 1 + y', z' = 1, z'' = 1 + x'', z1 = 1 + x5, x'' >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(x'', y'), 0), x'', 0, 1 + y') :|: z1 = 0, y' >= 0, z2 = 1 + y', z' = 1, z'' = 1 + x'', x'' >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(x3, y'')), 0, x3, 1 + y'') :|: z'' = 0, z2 = 1 + y'', y'' >= 0, z' = 1, z1 = 1 + x3, x3 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, x', x4, 0) :|: x4 >= 0, z'' = 1 + x', z2 = 0, x' >= 0, z1 = 1 + x4, z' = 1, 1 = 1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, x', 0, 0) :|: z1 = 0, z'' = 1 + x', z2 = 0, x' >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, x1, 0) :|: z'' = 0, x1 >= 0, z2 = 0, z' = 1, z1 = 1 + x1, x >= 0, 1 = x, 0 = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, x2, 0) :|: z'' = 0, z2 = 0, z' = 1, x2 >= 0, z1 = 1 + x2, x >= 0, 1 = x, 0 = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> gr(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
1085.25/293.73	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
1085.25/293.73	   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
1085.25/293.73	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.73	
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
1085.25/293.73	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(14)
1085.25/293.73	Obligation:
1085.25/293.73	Complexity RNTS consisting of the following rules:
1085.25/293.73	
1085.25/293.73	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.73	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.73	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.25/293.73	
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
1085.25/293.73	Found the following analysis order by SCC decomposition:
1085.25/293.73	
1085.25/293.73	   { and }
1085.25/293.73	   { p }
1085.25/293.73	   { gr }
1085.25/293.73	   { cond }
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(16)
1085.25/293.73	Obligation:
1085.25/293.73	Complexity RNTS consisting of the following rules:
1085.25/293.73	
1085.25/293.73	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.73	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.73	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.73	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.25/293.73	
1085.25/293.73	Function symbols to be analyzed: {and}, {p}, {gr}, {cond}
1085.25/293.73	
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(17) ResultPropagationProof (UPPER BOUND(ID))
1085.25/293.73	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1085.25/293.73	----------------------------------------
1085.25/293.73	
1085.25/293.73	(18)
1085.25/293.73	Obligation:
1085.25/293.73	Complexity RNTS consisting of the following rules:
1085.25/293.73	
1085.25/293.73	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.73	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.25/293.73	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.76	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.25/293.76	
1085.25/293.76	Function symbols to be analyzed: {and}, {p}, {gr}, {cond}
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(19) IntTrsBoundProof (UPPER BOUND(ID))
1085.25/293.76	
1085.25/293.76	Computed SIZE bound using CoFloCo for: and
1085.25/293.76	after applying outer abstraction to obtain an ITS,
1085.25/293.76	resulting in: O(1) with polynomial bound: 1
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(20)
1085.25/293.76	Obligation:
1085.25/293.76	Complexity RNTS consisting of the following rules:
1085.25/293.76	
1085.25/293.76	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.76	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.25/293.76	
1085.25/293.76	Function symbols to be analyzed: {and}, {p}, {gr}, {cond}
1085.25/293.76	Previous analysis results are:
1085.25/293.76	  and: runtime: ?, size: O(1) [1]
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(21) IntTrsBoundProof (UPPER BOUND(ID))
1085.25/293.76	
1085.25/293.76	Computed RUNTIME bound using CoFloCo for: and
1085.25/293.76	after applying outer abstraction to obtain an ITS,
1085.25/293.76	resulting in: O(1) with polynomial bound: 1
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(22)
1085.25/293.76	Obligation:
1085.25/293.76	Complexity RNTS consisting of the following rules:
1085.25/293.76	
1085.25/293.76	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.76	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.25/293.76	
1085.25/293.76	Function symbols to be analyzed: {p}, {gr}, {cond}
1085.25/293.76	Previous analysis results are:
1085.25/293.76	  and: runtime: O(1) [1], size: O(1) [1]
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(23) ResultPropagationProof (UPPER BOUND(ID))
1085.25/293.76	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(24)
1085.25/293.76	Obligation:
1085.25/293.76	Complexity RNTS consisting of the following rules:
1085.25/293.76	
1085.25/293.76	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.76	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.25/293.76	
1085.25/293.76	Function symbols to be analyzed: {p}, {gr}, {cond}
1085.25/293.76	Previous analysis results are:
1085.25/293.76	  and: runtime: O(1) [1], size: O(1) [1]
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(25) IntTrsBoundProof (UPPER BOUND(ID))
1085.25/293.76	
1085.25/293.76	Computed SIZE bound using KoAT for: p
1085.25/293.76	after applying outer abstraction to obtain an ITS,
1085.25/293.76	resulting in: O(n^1) with polynomial bound: z'
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(26)
1085.25/293.76	Obligation:
1085.25/293.76	Complexity RNTS consisting of the following rules:
1085.25/293.76	
1085.25/293.76	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.76	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.25/293.76	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.25/293.76	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.76	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.25/293.76	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.25/293.76	
1085.25/293.76	Function symbols to be analyzed: {p}, {gr}, {cond}
1085.25/293.76	Previous analysis results are:
1085.25/293.76	  and: runtime: O(1) [1], size: O(1) [1]
1085.25/293.76	  p: runtime: ?, size: O(n^1) [z']
1085.25/293.76	
1085.25/293.76	----------------------------------------
1085.25/293.76	
1085.25/293.76	(27) IntTrsBoundProof (UPPER BOUND(ID))
1085.25/293.76	
1085.25/293.76	Computed RUNTIME bound using CoFloCo for: p
1085.25/293.76	after applying outer abstraction to obtain an ITS,
1085.25/293.76	resulting in: O(1) with polynomial bound: 1
1085.25/293.80	
1085.25/293.80	----------------------------------------
1085.25/293.80	
1085.25/293.80	(28)
1085.25/293.80	Obligation:
1085.25/293.80	Complexity RNTS consisting of the following rules:
1085.25/293.80	
1085.25/293.80	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.25/293.80	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.25/293.80	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.25/293.80	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.80	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.25/293.80	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.25/293.80	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.25/293.80	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.25/293.80	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.57/293.82	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.57/293.82	
1085.57/293.82	Function symbols to be analyzed: {gr}, {cond}
1085.57/293.82	Previous analysis results are:
1085.57/293.82	  and: runtime: O(1) [1], size: O(1) [1]
1085.57/293.82	  p: runtime: O(1) [1], size: O(n^1) [z']
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(29) ResultPropagationProof (UPPER BOUND(ID))
1085.57/293.82	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(30)
1085.57/293.82	Obligation:
1085.57/293.82	Complexity RNTS consisting of the following rules:
1085.57/293.82	
1085.57/293.82	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.57/293.82	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.57/293.82	
1085.57/293.82	Function symbols to be analyzed: {gr}, {cond}
1085.57/293.82	Previous analysis results are:
1085.57/293.82	  and: runtime: O(1) [1], size: O(1) [1]
1085.57/293.82	  p: runtime: O(1) [1], size: O(n^1) [z']
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(31) IntTrsBoundProof (UPPER BOUND(ID))
1085.57/293.82	
1085.57/293.82	Computed SIZE bound using CoFloCo for: gr
1085.57/293.82	after applying outer abstraction to obtain an ITS,
1085.57/293.82	resulting in: O(1) with polynomial bound: 1
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(32)
1085.57/293.82	Obligation:
1085.57/293.82	Complexity RNTS consisting of the following rules:
1085.57/293.82	
1085.57/293.82	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.57/293.82	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.57/293.82	
1085.57/293.82	Function symbols to be analyzed: {gr}, {cond}
1085.57/293.82	Previous analysis results are:
1085.57/293.82	  and: runtime: O(1) [1], size: O(1) [1]
1085.57/293.82	  p: runtime: O(1) [1], size: O(n^1) [z']
1085.57/293.82	  gr: runtime: ?, size: O(1) [1]
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(33) IntTrsBoundProof (UPPER BOUND(ID))
1085.57/293.82	
1085.57/293.82	Computed RUNTIME bound using KoAT for: gr
1085.57/293.82	after applying outer abstraction to obtain an ITS,
1085.57/293.82	resulting in: O(n^1) with polynomial bound: 3 + z''
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(34)
1085.57/293.82	Obligation:
1085.57/293.82	Complexity RNTS consisting of the following rules:
1085.57/293.82	
1085.57/293.82	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 5 }-> cond(and(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.57/293.82	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.57/293.82	
1085.57/293.82	Function symbols to be analyzed: {cond}
1085.57/293.82	Previous analysis results are:
1085.57/293.82	  and: runtime: O(1) [1], size: O(1) [1]
1085.57/293.82	  p: runtime: O(1) [1], size: O(n^1) [z']
1085.57/293.82	  gr: runtime: O(n^1) [3 + z''], size: O(1) [1]
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(35) ResultPropagationProof (UPPER BOUND(ID))
1085.57/293.82	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(36)
1085.57/293.82	Obligation:
1085.57/293.82	Complexity RNTS consisting of the following rules:
1085.57/293.82	
1085.57/293.82	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 8 + z2 }-> cond(s'', 0, z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 8 + z2 }-> cond(s2, z'' - 1, 0, 1 + (z2 - 1)) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 10 + 2*z2 }-> cond(s5, z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.57/293.82	   gr(z', z'') -{ 3 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.57/293.82	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.57/293.82	
1085.57/293.82	Function symbols to be analyzed: {cond}
1085.57/293.82	Previous analysis results are:
1085.57/293.82	  and: runtime: O(1) [1], size: O(1) [1]
1085.57/293.82	  p: runtime: O(1) [1], size: O(n^1) [z']
1085.57/293.82	  gr: runtime: O(n^1) [3 + z''], size: O(1) [1]
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(37) IntTrsBoundProof (UPPER BOUND(ID))
1085.57/293.82	
1085.57/293.82	Computed SIZE bound using CoFloCo for: cond
1085.57/293.82	after applying outer abstraction to obtain an ITS,
1085.57/293.82	resulting in: O(1) with polynomial bound: 0
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(38)
1085.57/293.82	Obligation:
1085.57/293.82	Complexity RNTS consisting of the following rules:
1085.57/293.82	
1085.57/293.82	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 8 + z2 }-> cond(s'', 0, z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 8 + z2 }-> cond(s2, z'' - 1, 0, 1 + (z2 - 1)) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 10 + 2*z2 }-> cond(s5, z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.57/293.82	   gr(z', z'') -{ 3 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.57/293.82	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.57/293.82	
1085.57/293.82	Function symbols to be analyzed: {cond}
1085.57/293.82	Previous analysis results are:
1085.57/293.82	  and: runtime: O(1) [1], size: O(1) [1]
1085.57/293.82	  p: runtime: O(1) [1], size: O(n^1) [z']
1085.57/293.82	  gr: runtime: O(n^1) [3 + z''], size: O(1) [1]
1085.57/293.82	  cond: runtime: ?, size: O(1) [0]
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(39) IntTrsBoundProof (UPPER BOUND(ID))
1085.57/293.82	
1085.57/293.82	Computed RUNTIME bound using CoFloCo for: cond
1085.57/293.82	after applying outer abstraction to obtain an ITS,
1085.57/293.82	resulting in: O(n^2) with polynomial bound: 6 + 18*z'' + 3*z''*z2 + 18*z1 + 3*z1*z2
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(40)
1085.57/293.82	Obligation:
1085.57/293.82	Complexity RNTS consisting of the following rules:
1085.57/293.82	
1085.57/293.82	   and(z', z'') -{ 1 }-> 1 :|: z' = 1, z'' = 1
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
1085.57/293.82	   and(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 8 + z2 }-> cond(s'', 0, z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 8 + z2 }-> cond(s2, z'' - 1, 0, 1 + (z2 - 1)) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z1 = 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 10 + 2*z2 }-> cond(s5, z'' - 1, z1 - 1, 1 + (z2 - 1)) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, s5 >= 0, s5 <= 1, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1, z'' - 1 >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(1, z'' - 1, z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' - 1 >= 0, z' = 1, 1 = 1
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, 0, 0) :|: z'' = 0, z1 = 0, z2 = 0, z' = 1, 0 = 0, 0 = x, x >= 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, 0, z1 - 1, 0) :|: z'' = 0, z1 - 1 >= 0, z2 = 0, z' = 1, x >= 0, 1 = x, 0 = 0
1085.57/293.82	   cond(z', z'', z1, z2) -{ 6 }-> cond(0, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z'' - 1 >= 0, z' = 1, 0 = 0, 1 = x, x >= 0
1085.57/293.82	   gr(z', z'') -{ 3 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
1085.57/293.82	   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
1085.57/293.82	   p(z') -{ 1 }-> 0 :|: z' = 0
1085.57/293.82	   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
1085.57/293.82	
1085.57/293.82	Function symbols to be analyzed: 
1085.57/293.82	Previous analysis results are:
1085.57/293.82	  and: runtime: O(1) [1], size: O(1) [1]
1085.57/293.82	  p: runtime: O(1) [1], size: O(n^1) [z']
1085.57/293.82	  gr: runtime: O(n^1) [3 + z''], size: O(1) [1]
1085.57/293.82	  cond: runtime: O(n^2) [6 + 18*z'' + 3*z''*z2 + 18*z1 + 3*z1*z2], size: O(1) [0]
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(41) FinalProof (FINISHED)
1085.57/293.82	Computed overall runtime complexity
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(42)
1085.57/293.82	BOUNDS(1, n^2)
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(43) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1085.57/293.82	Transformed a relative TRS into a decreasing-loop problem.
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(44)
1085.57/293.82	Obligation:
1085.57/293.82	Analyzing the following TRS for decreasing loops:
1085.57/293.82	
1085.57/293.82	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1085.57/293.82	
1085.57/293.82	
1085.57/293.82	The TRS R consists of the following rules:
1085.57/293.82	
1085.57/293.82	   cond(true, x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z)
1085.57/293.82	   and(true, true) -> true
1085.57/293.82	   and(x, false) -> false
1085.57/293.82	   and(false, x) -> false
1085.57/293.82	   gr(0, 0) -> false
1085.57/293.82	   gr(0, x) -> false
1085.57/293.82	   gr(s(x), 0) -> true
1085.57/293.82	   gr(s(x), s(y)) -> gr(x, y)
1085.57/293.82	   p(0) -> 0
1085.57/293.82	   p(s(x)) -> x
1085.57/293.82	
1085.57/293.82	S is empty.
1085.57/293.82	Rewrite Strategy: INNERMOST
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(45) DecreasingLoopProof (LOWER BOUND(ID))
1085.57/293.82	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1085.57/293.82	
1085.57/293.82	The rewrite sequence
1085.57/293.82	
1085.57/293.82	gr(s(x), s(y)) ->^+ gr(x, y)
1085.57/293.82	
1085.57/293.82	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1085.57/293.82	
1085.57/293.82	The pumping substitution is [x / s(x), y / s(y)].
1085.57/293.82	
1085.57/293.82	The result substitution is [ ].
1085.57/293.82	
1085.57/293.82	
1085.57/293.82	
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(46)
1085.57/293.82	Complex Obligation (BEST)
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(47)
1085.57/293.82	Obligation:
1085.57/293.82	Proved the lower bound n^1 for the following obligation:
1085.57/293.82	
1085.57/293.82	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1085.57/293.82	
1085.57/293.82	
1085.57/293.82	The TRS R consists of the following rules:
1085.57/293.82	
1085.57/293.82	   cond(true, x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z)
1085.57/293.82	   and(true, true) -> true
1085.57/293.82	   and(x, false) -> false
1085.57/293.82	   and(false, x) -> false
1085.57/293.82	   gr(0, 0) -> false
1085.57/293.82	   gr(0, x) -> false
1085.57/293.82	   gr(s(x), 0) -> true
1085.57/293.82	   gr(s(x), s(y)) -> gr(x, y)
1085.57/293.82	   p(0) -> 0
1085.57/293.82	   p(s(x)) -> x
1085.57/293.82	
1085.57/293.82	S is empty.
1085.57/293.82	Rewrite Strategy: INNERMOST
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(48) LowerBoundPropagationProof (FINISHED)
1085.57/293.82	Propagated lower bound.
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(49)
1085.57/293.82	BOUNDS(n^1, INF)
1085.57/293.82	
1085.57/293.82	----------------------------------------
1085.57/293.82	
1085.57/293.82	(50)
1085.57/293.82	Obligation:
1085.57/293.82	Analyzing the following TRS for decreasing loops:
1085.57/293.82	
1085.57/293.82	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1085.57/293.82	
1085.57/293.82	
1085.57/293.82	The TRS R consists of the following rules:
1085.57/293.82	
1085.57/293.82	   cond(true, x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z)
1085.57/293.82	   and(true, true) -> true
1085.57/293.82	   and(x, false) -> false
1085.57/293.82	   and(false, x) -> false
1085.57/293.82	   gr(0, 0) -> false
1085.57/293.82	   gr(0, x) -> false
1085.57/293.82	   gr(s(x), 0) -> true
1085.57/293.82	   gr(s(x), s(y)) -> gr(x, y)
1085.57/293.82	   p(0) -> 0
1085.57/293.82	   p(s(x)) -> x
1085.57/293.82	
1085.57/293.82	S is empty.
1085.57/293.82	Rewrite Strategy: INNERMOST
1085.69/293.92	EOF