991.51/291.57	WORST_CASE(Omega(n^1), O(n^3))
991.57/291.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
991.57/291.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
991.57/291.60	
991.57/291.60	
991.57/291.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
991.57/291.60	
991.57/291.60	(0) CpxTRS
991.57/291.60	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	(2) CpxTRS
991.57/291.60	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	    (4) CpxWeightedTrs
991.57/291.60	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	    (6) CpxTypedWeightedTrs
991.57/291.60	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
991.57/291.60	    (8) CpxTypedWeightedCompleteTrs
991.57/291.60	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	    (10) CpxTypedWeightedCompleteTrs
991.57/291.60	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms]
991.57/291.60	    (12) CpxRNTS
991.57/291.60	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	    (14) CpxRNTS
991.57/291.60	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	    (16) CpxRNTS
991.57/291.60	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
991.57/291.60	    (18) CpxRNTS
991.57/291.60	    (19) IntTrsBoundProof [UPPER BOUND(ID), 507 ms]
991.57/291.60	    (20) CpxRNTS
991.57/291.60	    (21) IntTrsBoundProof [UPPER BOUND(ID), 179 ms]
991.57/291.60	    (22) CpxRNTS
991.57/291.60	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
991.57/291.60	    (24) CpxRNTS
991.57/291.60	    (25) IntTrsBoundProof [UPPER BOUND(ID), 841 ms]
991.57/291.60	    (26) CpxRNTS
991.57/291.60	    (27) IntTrsBoundProof [UPPER BOUND(ID), 342 ms]
991.57/291.60	    (28) CpxRNTS
991.57/291.60	    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
991.57/291.60	    (30) CpxRNTS
991.57/291.60	    (31) IntTrsBoundProof [UPPER BOUND(ID), 352 ms]
991.57/291.60	    (32) CpxRNTS
991.57/291.60	    (33) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
991.57/291.60	    (34) CpxRNTS
991.57/291.60	    (35) FinalProof [FINISHED, 0 ms]
991.57/291.60	    (36) BOUNDS(1, n^3)
991.57/291.60	(37) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	(38) CpxTRS
991.57/291.60	    (39) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
991.57/291.60	    (40) typed CpxTrs
991.57/291.60	    (41) OrderProof [LOWER BOUND(ID), 0 ms]
991.57/291.60	    (42) typed CpxTrs
991.57/291.60	    (43) RewriteLemmaProof [LOWER BOUND(ID), 240 ms]
991.57/291.60	    (44) BEST
991.57/291.60	        (45) proven lower bound
991.57/291.60	            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
991.57/291.60	            (47) BOUNDS(n^1, INF)
991.57/291.60	        (48) typed CpxTrs
991.57/291.60	
991.57/291.60	
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(0)
991.57/291.60	Obligation:
991.57/291.60	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
991.57/291.60	
991.57/291.60	
991.57/291.60	The TRS R consists of the following rules:
991.57/291.60	
991.57/291.60	   minus(x, x) -> 0
991.57/291.60	   minus(s(x), s(y)) -> minus(x, y)
991.57/291.60	   minus(0, x) -> 0
991.57/291.60	   minus(x, 0) -> x
991.57/291.60	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
991.57/291.60	   div(0, s(y)) -> 0
991.57/291.60	   f(x, 0, b) -> x
991.57/291.60	   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)
991.57/291.60	
991.57/291.60	S is empty.
991.57/291.60	Rewrite Strategy: FULL
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
991.57/291.60	Converted rc-obligation to irc-obligation.
991.57/291.60	
991.57/291.60	The duplicating contexts are:
991.57/291.60	div(s(x), s([]))
991.57/291.60	f(x, s(y), [])
991.57/291.60	
991.57/291.60	
991.57/291.60	The defined contexts are:
991.57/291.60	div([], x1)
991.57/291.60	f(x0, [], x2)
991.57/291.60	div([], s(x1))
991.57/291.60	minus([], x1)
991.57/291.60	minus(s([]), s(0))
991.57/291.60	
991.57/291.60	
991.57/291.60	[] just represents basic- or constructor-terms in the following defined contexts:
991.57/291.60	f(x0, [], x2)
991.57/291.60	div([], s(x1))
991.57/291.60	
991.57/291.60	
991.57/291.60	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(2)
991.57/291.60	Obligation:
991.57/291.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).
991.57/291.60	
991.57/291.60	
991.57/291.60	The TRS R consists of the following rules:
991.57/291.60	
991.57/291.60	   minus(x, x) -> 0
991.57/291.60	   minus(s(x), s(y)) -> minus(x, y)
991.57/291.60	   minus(0, x) -> 0
991.57/291.60	   minus(x, 0) -> x
991.57/291.60	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
991.57/291.60	   div(0, s(y)) -> 0
991.57/291.60	   f(x, 0, b) -> x
991.57/291.60	   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)
991.57/291.60	
991.57/291.60	S is empty.
991.57/291.60	Rewrite Strategy: INNERMOST
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
991.57/291.60	Transformed relative TRS to weighted TRS
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(4)
991.57/291.60	Obligation:
991.57/291.60	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).
991.57/291.60	
991.57/291.60	
991.57/291.60	The TRS R consists of the following rules:
991.57/291.60	
991.57/291.60	   minus(x, x) -> 0 [1]
991.57/291.60	   minus(s(x), s(y)) -> minus(x, y) [1]
991.57/291.60	   minus(0, x) -> 0 [1]
991.57/291.60	   minus(x, 0) -> x [1]
991.57/291.60	   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
991.57/291.60	   div(0, s(y)) -> 0 [1]
991.57/291.60	   f(x, 0, b) -> x [1]
991.57/291.60	   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]
991.57/291.60	
991.57/291.60	Rewrite Strategy: INNERMOST
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
991.57/291.60	Infered types.
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(6)
991.57/291.60	Obligation:
991.57/291.60	Runtime Complexity Weighted TRS with Types.
991.57/291.60	The TRS R consists of the following rules:
991.57/291.60	
991.57/291.60	   minus(x, x) -> 0 [1]
991.57/291.60	   minus(s(x), s(y)) -> minus(x, y) [1]
991.57/291.60	   minus(0, x) -> 0 [1]
991.57/291.60	   minus(x, 0) -> x [1]
991.57/291.60	   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
991.57/291.60	   div(0, s(y)) -> 0 [1]
991.57/291.60	   f(x, 0, b) -> x [1]
991.57/291.60	   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]
991.57/291.60	
991.57/291.60	The TRS has the following type information:
991.57/291.60	minus :: 0:s -> 0:s -> 0:s
991.57/291.60	0 :: 0:s
991.57/291.60	s :: 0:s -> 0:s
991.57/291.60	div :: 0:s -> 0:s -> 0:s
991.57/291.60	f :: 0:s -> 0:s -> 0:s -> 0:s
991.57/291.60	
991.57/291.60	Rewrite Strategy: INNERMOST
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(7) CompletionProof (UPPER BOUND(ID))
991.57/291.60	The transformation into a RNTS is sound, since:
991.57/291.60	
991.57/291.60	(a) The obligation is a constructor system where every type has a constant constructor,
991.57/291.60	
991.57/291.60	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
991.57/291.60	none
991.57/291.60	
991.57/291.60	(c) The following functions are completely defined:
991.57/291.60	
991.57/291.60	   f_3
991.57/291.60	   minus_2
991.57/291.60	   div_2
991.57/291.60	
991.57/291.60	Due to the following rules being added:
991.57/291.60	
991.57/291.60	   div(v0, v1) -> 0 [0]
991.57/291.60	
991.57/291.60	And the following fresh constants: none
991.57/291.60	
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(8)
991.57/291.60	Obligation:
991.57/291.60	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
991.57/291.60	
991.57/291.60	Runtime Complexity Weighted TRS with Types.
991.57/291.60	The TRS R consists of the following rules:
991.57/291.60	
991.57/291.60	   minus(x, x) -> 0 [1]
991.57/291.60	   minus(s(x), s(y)) -> minus(x, y) [1]
991.57/291.60	   minus(0, x) -> 0 [1]
991.57/291.60	   minus(x, 0) -> x [1]
991.57/291.60	   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
991.57/291.60	   div(0, s(y)) -> 0 [1]
991.57/291.60	   f(x, 0, b) -> x [1]
991.57/291.60	   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]
991.57/291.60	   div(v0, v1) -> 0 [0]
991.57/291.60	
991.57/291.60	The TRS has the following type information:
991.57/291.60	minus :: 0:s -> 0:s -> 0:s
991.57/291.60	0 :: 0:s
991.57/291.60	s :: 0:s -> 0:s
991.57/291.60	div :: 0:s -> 0:s -> 0:s
991.57/291.60	f :: 0:s -> 0:s -> 0:s -> 0:s
991.57/291.60	
991.57/291.60	Rewrite Strategy: INNERMOST
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
991.57/291.60	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(10)
991.57/291.60	Obligation:
991.57/291.60	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
991.57/291.60	
991.57/291.60	Runtime Complexity Weighted TRS with Types.
991.57/291.60	The TRS R consists of the following rules:
991.57/291.60	
991.57/291.60	   minus(x, x) -> 0 [1]
991.57/291.60	   minus(s(x), s(y)) -> minus(x, y) [1]
991.57/291.60	   minus(0, x) -> 0 [1]
991.57/291.60	   minus(x, 0) -> x [1]
991.57/291.60	   div(s(x), s(x)) -> s(div(0, s(x))) [2]
991.57/291.60	   div(s(s(x')), s(s(y'))) -> s(div(minus(x', y'), s(s(y')))) [2]
991.57/291.60	   div(s(0), s(y)) -> s(div(0, s(y))) [2]
991.57/291.60	   div(s(x), s(0)) -> s(div(x, s(0))) [2]
991.57/291.60	   div(0, s(y)) -> 0 [1]
991.57/291.60	   f(x, 0, b) -> x [1]
991.57/291.60	   f(x, s(0), b) -> div(f(x, 0, b), b) [2]
991.57/291.60	   f(x, s(y), b) -> div(f(x, minus(y, 0), b), b) [2]
991.57/291.60	   div(v0, v1) -> 0 [0]
991.57/291.60	
991.57/291.60	The TRS has the following type information:
991.57/291.60	minus :: 0:s -> 0:s -> 0:s
991.57/291.60	0 :: 0:s
991.57/291.60	s :: 0:s -> 0:s
991.57/291.60	div :: 0:s -> 0:s -> 0:s
991.57/291.60	f :: 0:s -> 0:s -> 0:s -> 0:s
991.57/291.60	
991.57/291.60	Rewrite Strategy: INNERMOST
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
991.57/291.60	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
991.57/291.60	The constant constructors are abstracted as follows:
991.57/291.60	
991.57/291.60	   0 => 0
991.57/291.60	
991.57/291.60	----------------------------------------
991.57/291.60	
991.57/291.60	(12)
991.57/291.60	Obligation:
991.57/291.60	Complexity RNTS consisting of the following rules:
991.57/291.60	
991.57/291.60	   div(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
991.57/291.60	   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
991.57/291.60	   div(z, z') -{ 2 }-> 1 + div(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
991.57/291.60	   div(z, z') -{ 2 }-> 1 + div(minus(x', y'), 1 + (1 + y')) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
991.57/291.60	   div(z, z') -{ 2 }-> 1 + div(0, 1 + x) :|: z' = 1 + x, x >= 0, z = 1 + x
991.57/291.60	   div(z, z') -{ 2 }-> 1 + div(0, 1 + y) :|: z' = 1 + y, z = 1 + 0, y >= 0
991.57/291.60	   f(z, z', z'') -{ 1 }-> x :|: b >= 0, z'' = b, x >= 0, z = x, z' = 0
991.57/291.60	   f(z, z', z'') -{ 2 }-> div(f(x, minus(y, 0), b), b) :|: z' = 1 + y, b >= 0, z'' = b, x >= 0, y >= 0, z = x
991.57/291.60	   f(z, z', z'') -{ 2 }-> div(f(x, 0, b), b) :|: b >= 0, z'' = b, x >= 0, z' = 1 + 0, z = x
991.57/291.61	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = x
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
991.57/291.61	
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
991.57/291.61	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(14)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
991.57/291.61	Found the following analysis order by SCC decomposition:
991.57/291.61	
991.57/291.61	   { minus }
991.57/291.61	   { div }
991.57/291.61	   { f }
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(16)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {minus}, {div}, {f}
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(17) ResultPropagationProof (UPPER BOUND(ID))
991.57/291.61	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(18)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {minus}, {div}, {f}
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(19) IntTrsBoundProof (UPPER BOUND(ID))
991.57/291.61	
991.57/291.61	Computed SIZE bound using KoAT for: minus
991.57/291.61	after applying outer abstraction to obtain an ITS,
991.57/291.61	resulting in: O(n^1) with polynomial bound: z
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(20)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {minus}, {div}, {f}
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: ?, size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(21) IntTrsBoundProof (UPPER BOUND(ID))
991.57/291.61	
991.57/291.61	Computed RUNTIME bound using KoAT for: minus
991.57/291.61	after applying outer abstraction to obtain an ITS,
991.57/291.61	resulting in: O(n^1) with polynomial bound: 3 + z'
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(22)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {div}, {f}
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(23) ResultPropagationProof (UPPER BOUND(ID))
991.57/291.61	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(24)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 3 + z' }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {div}, {f}
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(25) IntTrsBoundProof (UPPER BOUND(ID))
991.57/291.61	
991.57/291.61	Computed SIZE bound using KoAT for: div
991.57/291.61	after applying outer abstraction to obtain an ITS,
991.57/291.61	resulting in: O(n^1) with polynomial bound: z
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(26)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 3 + z' }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {div}, {f}
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
991.57/291.61	  div: runtime: ?, size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(27) IntTrsBoundProof (UPPER BOUND(ID))
991.57/291.61	
991.57/291.61	Computed RUNTIME bound using KoAT for: div
991.57/291.61	after applying outer abstraction to obtain an ITS,
991.57/291.61	resulting in: O(n^2) with polynomial bound: 1 + 9*z + z*z'
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(28)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 3 + z' }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {f}
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
991.57/291.61	  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(29) ResultPropagationProof (UPPER BOUND(ID))
991.57/291.61	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(30)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 4 + 9*s' + s'*z' + z' }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ -7 + 10*z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {f}
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
991.57/291.61	  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(31) IntTrsBoundProof (UPPER BOUND(ID))
991.57/291.61	
991.57/291.61	Computed SIZE bound using CoFloCo for: f
991.57/291.61	after applying outer abstraction to obtain an ITS,
991.57/291.61	resulting in: O(n^1) with polynomial bound: z
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(32)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 4 + 9*s' + s'*z' + z' }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ -7 + 10*z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: {f}
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
991.57/291.61	  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]
991.57/291.61	  f: runtime: ?, size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(33) IntTrsBoundProof (UPPER BOUND(ID))
991.57/291.61	
991.57/291.61	Computed RUNTIME bound using KoAT for: f
991.57/291.61	after applying outer abstraction to obtain an ITS,
991.57/291.61	resulting in: O(n^3) with polynomial bound: 1 + 18*z*z' + 2*z*z'*z'' + 9*z'
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(34)
991.57/291.61	Obligation:
991.57/291.61	Complexity RNTS consisting of the following rules:
991.57/291.61	
991.57/291.61	   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
991.57/291.61	   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
991.57/291.61	   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
991.57/291.61	   div(z, z') -{ 4 + 9*s' + s'*z' + z' }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
991.57/291.61	   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
991.57/291.61	   div(z, z') -{ -7 + 10*z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
991.57/291.61	   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
991.57/291.61	   f(z, z', z'') -{ 5 }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
991.57/291.61	   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
991.57/291.61	   minus(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
991.57/291.61	   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
991.57/291.61	   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
991.57/291.61	
991.57/291.61	Function symbols to be analyzed: 
991.57/291.61	Previous analysis results are:
991.57/291.61	  minus: runtime: O(n^1) [3 + z'], size: O(n^1) [z]
991.57/291.61	  div: runtime: O(n^2) [1 + 9*z + z*z'], size: O(n^1) [z]
991.57/291.61	  f: runtime: O(n^3) [1 + 18*z*z' + 2*z*z'*z'' + 9*z'], size: O(n^1) [z]
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(35) FinalProof (FINISHED)
991.57/291.61	Computed overall runtime complexity
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(36)
991.57/291.61	BOUNDS(1, n^3)
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(37) RenamingProof (BOTH BOUNDS(ID, ID))
991.57/291.61	Renamed function symbols to avoid clashes with predefined symbol.
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(38)
991.57/291.61	Obligation:
991.57/291.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
991.57/291.61	
991.57/291.61	
991.57/291.61	The TRS R consists of the following rules:
991.57/291.61	
991.57/291.61	   minus(x, x) -> 0'
991.57/291.61	   minus(s(x), s(y)) -> minus(x, y)
991.57/291.61	   minus(0', x) -> 0'
991.57/291.61	   minus(x, 0') -> x
991.57/291.61	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
991.57/291.61	   div(0', s(y)) -> 0'
991.57/291.61	   f(x, 0', b) -> x
991.57/291.61	   f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)
991.57/291.61	
991.57/291.61	S is empty.
991.57/291.61	Rewrite Strategy: FULL
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(39) TypeInferenceProof (BOTH BOUNDS(ID, ID))
991.57/291.61	Infered types.
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(40)
991.57/291.61	Obligation:
991.57/291.61	TRS:
991.57/291.61	Rules:
991.57/291.61	minus(x, x) -> 0'
991.57/291.61	minus(s(x), s(y)) -> minus(x, y)
991.57/291.61	minus(0', x) -> 0'
991.57/291.61	minus(x, 0') -> x
991.57/291.61	div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
991.57/291.61	div(0', s(y)) -> 0'
991.57/291.61	f(x, 0', b) -> x
991.57/291.61	f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)
991.57/291.61	
991.57/291.61	Types:
991.57/291.61	minus :: 0':s -> 0':s -> 0':s
991.57/291.61	0' :: 0':s
991.57/291.61	s :: 0':s -> 0':s
991.57/291.61	div :: 0':s -> 0':s -> 0':s
991.57/291.61	f :: 0':s -> 0':s -> 0':s -> 0':s
991.57/291.61	hole_0':s1_0 :: 0':s
991.57/291.61	gen_0':s2_0 :: Nat -> 0':s
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(41) OrderProof (LOWER BOUND(ID))
991.57/291.61	Heuristically decided to analyse the following defined symbols:
991.57/291.61	minus, div, f
991.57/291.61	
991.57/291.61	They will be analysed ascendingly in the following order:
991.57/291.61	minus < div
991.57/291.61	minus < f
991.57/291.61	div < f
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(42)
991.57/291.61	Obligation:
991.57/291.61	TRS:
991.57/291.61	Rules:
991.57/291.61	minus(x, x) -> 0'
991.57/291.61	minus(s(x), s(y)) -> minus(x, y)
991.57/291.61	minus(0', x) -> 0'
991.57/291.61	minus(x, 0') -> x
991.57/291.61	div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
991.57/291.61	div(0', s(y)) -> 0'
991.57/291.61	f(x, 0', b) -> x
991.57/291.61	f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)
991.57/291.61	
991.57/291.61	Types:
991.57/291.61	minus :: 0':s -> 0':s -> 0':s
991.57/291.61	0' :: 0':s
991.57/291.61	s :: 0':s -> 0':s
991.57/291.61	div :: 0':s -> 0':s -> 0':s
991.57/291.61	f :: 0':s -> 0':s -> 0':s -> 0':s
991.57/291.61	hole_0':s1_0 :: 0':s
991.57/291.61	gen_0':s2_0 :: Nat -> 0':s
991.57/291.61	
991.57/291.61	
991.57/291.61	Generator Equations:
991.57/291.61	gen_0':s2_0(0) <=> 0'
991.57/291.61	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
991.57/291.61	
991.57/291.61	
991.57/291.61	The following defined symbols remain to be analysed:
991.57/291.61	minus, div, f
991.57/291.61	
991.57/291.61	They will be analysed ascendingly in the following order:
991.57/291.61	minus < div
991.57/291.61	minus < f
991.57/291.61	div < f
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(43) RewriteLemmaProof (LOWER BOUND(ID))
991.57/291.61	Proved the following rewrite lemma:
991.57/291.61	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
991.57/291.61	
991.57/291.61	Induction Base:
991.57/291.61	minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
991.57/291.61	0'
991.57/291.61	
991.57/291.61	Induction Step:
991.57/291.61	minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
991.57/291.61	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
991.57/291.61	gen_0':s2_0(0)
991.57/291.61	
991.57/291.61	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(44)
991.57/291.61	Complex Obligation (BEST)
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(45)
991.57/291.61	Obligation:
991.57/291.61	Proved the lower bound n^1 for the following obligation:
991.57/291.61	
991.57/291.61	TRS:
991.57/291.61	Rules:
991.57/291.61	minus(x, x) -> 0'
991.57/291.61	minus(s(x), s(y)) -> minus(x, y)
991.57/291.61	minus(0', x) -> 0'
991.57/291.61	minus(x, 0') -> x
991.57/291.61	div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
991.57/291.61	div(0', s(y)) -> 0'
991.57/291.61	f(x, 0', b) -> x
991.57/291.61	f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)
991.57/291.61	
991.57/291.61	Types:
991.57/291.61	minus :: 0':s -> 0':s -> 0':s
991.57/291.61	0' :: 0':s
991.57/291.61	s :: 0':s -> 0':s
991.57/291.61	div :: 0':s -> 0':s -> 0':s
991.57/291.61	f :: 0':s -> 0':s -> 0':s -> 0':s
991.57/291.61	hole_0':s1_0 :: 0':s
991.57/291.61	gen_0':s2_0 :: Nat -> 0':s
991.57/291.61	
991.57/291.61	
991.57/291.61	Generator Equations:
991.57/291.61	gen_0':s2_0(0) <=> 0'
991.57/291.61	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
991.57/291.61	
991.57/291.61	
991.57/291.61	The following defined symbols remain to be analysed:
991.57/291.61	minus, div, f
991.57/291.61	
991.57/291.61	They will be analysed ascendingly in the following order:
991.57/291.61	minus < div
991.57/291.61	minus < f
991.57/291.61	div < f
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(46) LowerBoundPropagationProof (FINISHED)
991.57/291.61	Propagated lower bound.
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(47)
991.57/291.61	BOUNDS(n^1, INF)
991.57/291.61	
991.57/291.61	----------------------------------------
991.57/291.61	
991.57/291.61	(48)
991.57/291.61	Obligation:
991.57/291.61	TRS:
991.57/291.61	Rules:
991.57/291.61	minus(x, x) -> 0'
991.57/291.61	minus(s(x), s(y)) -> minus(x, y)
991.57/291.61	minus(0', x) -> 0'
991.57/291.61	minus(x, 0') -> x
991.57/291.61	div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
991.57/291.61	div(0', s(y)) -> 0'
991.57/291.61	f(x, 0', b) -> x
991.57/291.61	f(x, s(y), b) -> div(f(x, minus(s(y), s(0')), b), b)
991.57/291.61	
991.57/291.61	Types:
991.57/291.61	minus :: 0':s -> 0':s -> 0':s
991.57/291.61	0' :: 0':s
991.57/291.61	s :: 0':s -> 0':s
991.57/291.61	div :: 0':s -> 0':s -> 0':s
991.57/291.61	f :: 0':s -> 0':s -> 0':s -> 0':s
991.57/291.61	hole_0':s1_0 :: 0':s
991.57/291.61	gen_0':s2_0 :: Nat -> 0':s
991.57/291.61	
991.57/291.61	
991.57/291.61	Lemmas:
991.57/291.61	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
991.57/291.61	
991.57/291.61	
991.57/291.61	Generator Equations:
991.57/291.61	gen_0':s2_0(0) <=> 0'
991.57/291.61	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
991.57/291.61	
991.57/291.61	
991.57/291.61	The following defined symbols remain to be analysed:
991.57/291.61	div, f
991.57/291.61	
991.57/291.61	They will be analysed ascendingly in the following order:
991.57/291.61	div < f
991.81/291.70	EOF