20.36/7.30	WORST_CASE(Omega(n^1), O(n^1))
20.36/7.31	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
20.36/7.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.36/7.31	
20.36/7.31	
20.36/7.31	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/7.31	
20.36/7.31	(0) CpxTRS
20.36/7.31	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
20.36/7.31	(2) CpxTRS
20.36/7.31	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/7.31	    (4) CpxTRS
20.36/7.31	    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/7.31	    (6) CpxWeightedTrs
20.36/7.31	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/7.31	    (8) CpxTypedWeightedTrs
20.36/7.31	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
20.36/7.31	    (10) CpxTypedWeightedCompleteTrs
20.36/7.31	    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/7.31	    (12) CpxTypedWeightedCompleteTrs
20.36/7.31	    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
20.36/7.31	    (14) CpxRNTS
20.36/7.31	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/7.31	    (16) CpxRNTS
20.36/7.31	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/7.31	    (18) CpxRNTS
20.36/7.31	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.36/7.31	    (20) CpxRNTS
20.36/7.31	    (21) IntTrsBoundProof [UPPER BOUND(ID), 1093 ms]
20.36/7.31	    (22) CpxRNTS
20.36/7.31	    (23) IntTrsBoundProof [UPPER BOUND(ID), 202 ms]
20.36/7.31	    (24) CpxRNTS
20.36/7.31	    (25) FinalProof [FINISHED, 0 ms]
20.36/7.31	    (26) BOUNDS(1, n^1)
20.36/7.31	(27) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
20.36/7.31	(28) TRS for Loop Detection
20.36/7.31	    (29) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
20.36/7.31	    (30) BEST
20.36/7.31	        (31) proven lower bound
20.36/7.31	            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
20.36/7.31	            (33) BOUNDS(n^1, INF)
20.36/7.31	        (34) TRS for Loop Detection
20.36/7.31	
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(0)
20.36/7.31	Obligation:
20.36/7.31	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/7.31	
20.36/7.31	
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y
20.36/7.31	   f(x, 0) -> x
20.36/7.31	   f(i(x), y) -> i(x)
20.36/7.31	   f(f(x, y), z) -> f(x, f(y, z))
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z))
20.36/7.31	   f(1, g(x, y)) -> x
20.36/7.31	   f(2, g(x, y)) -> y
20.36/7.31	
20.36/7.31	S is empty.
20.36/7.31	Rewrite Strategy: FULL
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
20.36/7.31	The TRS does not nest defined symbols.
20.36/7.31	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
20.36/7.31	f(f(x, y), z) -> f(x, f(y, z))
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(2)
20.36/7.31	Obligation:
20.36/7.31	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.36/7.31	
20.36/7.31	
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y
20.36/7.31	   f(x, 0) -> x
20.36/7.31	   f(i(x), y) -> i(x)
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z))
20.36/7.31	   f(1, g(x, y)) -> x
20.36/7.31	   f(2, g(x, y)) -> y
20.36/7.31	
20.36/7.31	S is empty.
20.36/7.31	Rewrite Strategy: FULL
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.36/7.31	Converted rc-obligation to irc-obligation.
20.36/7.31	
20.36/7.31	As the TRS does not nest defined symbols, we have rc = irc.
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(4)
20.36/7.31	Obligation:
20.36/7.31	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.36/7.31	
20.36/7.31	
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y
20.36/7.31	   f(x, 0) -> x
20.36/7.31	   f(i(x), y) -> i(x)
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z))
20.36/7.31	   f(1, g(x, y)) -> x
20.36/7.31	   f(2, g(x, y)) -> y
20.36/7.31	
20.36/7.31	S is empty.
20.36/7.31	Rewrite Strategy: INNERMOST
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
20.36/7.31	Transformed relative TRS to weighted TRS
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(6)
20.36/7.31	Obligation:
20.36/7.31	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.36/7.31	
20.36/7.31	
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y [1]
20.36/7.31	   f(x, 0) -> x [1]
20.36/7.31	   f(i(x), y) -> i(x) [1]
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
20.36/7.31	   f(1, g(x, y)) -> x [1]
20.36/7.31	   f(2, g(x, y)) -> y [1]
20.36/7.31	
20.36/7.31	Rewrite Strategy: INNERMOST
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.36/7.31	Infered types.
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(8)
20.36/7.31	Obligation:
20.36/7.31	Runtime Complexity Weighted TRS with Types.
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y [1]
20.36/7.31	   f(x, 0) -> x [1]
20.36/7.31	   f(i(x), y) -> i(x) [1]
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
20.36/7.31	   f(1, g(x, y)) -> x [1]
20.36/7.31	   f(2, g(x, y)) -> y [1]
20.36/7.31	
20.36/7.31	The TRS has the following type information:
20.36/7.31	f :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
20.36/7.31	0 :: 0:i:g:1:2
20.36/7.31	i :: a -> 0:i:g:1:2
20.36/7.31	g :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
20.36/7.31	1 :: 0:i:g:1:2
20.36/7.31	2 :: 0:i:g:1:2
20.36/7.31	
20.36/7.31	Rewrite Strategy: INNERMOST
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(9) CompletionProof (UPPER BOUND(ID))
20.36/7.31	The transformation into a RNTS is sound, since:
20.36/7.31	
20.36/7.31	(a) The obligation is a constructor system where every type has a constant constructor,
20.36/7.31	
20.36/7.31	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
20.36/7.31	
20.36/7.31	   f_2
20.36/7.31	
20.36/7.31	(c) The following functions are completely defined:
20.36/7.31	none
20.36/7.31	
20.36/7.31	Due to the following rules being added:
20.36/7.31	none
20.36/7.31	
20.36/7.31	And the following fresh constants:    const
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(10)
20.36/7.31	Obligation:
20.36/7.31	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.36/7.31	
20.36/7.31	Runtime Complexity Weighted TRS with Types.
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y [1]
20.36/7.31	   f(x, 0) -> x [1]
20.36/7.31	   f(i(x), y) -> i(x) [1]
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
20.36/7.31	   f(1, g(x, y)) -> x [1]
20.36/7.31	   f(2, g(x, y)) -> y [1]
20.36/7.31	
20.36/7.31	The TRS has the following type information:
20.36/7.31	f :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
20.36/7.31	0 :: 0:i:g:1:2
20.36/7.31	i :: a -> 0:i:g:1:2
20.36/7.31	g :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
20.36/7.31	1 :: 0:i:g:1:2
20.36/7.31	2 :: 0:i:g:1:2
20.36/7.31	const :: a
20.36/7.31	
20.36/7.31	Rewrite Strategy: INNERMOST
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(11) NarrowingProof (BOTH BOUNDS(ID, ID))
20.36/7.31	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(12)
20.36/7.31	Obligation:
20.36/7.31	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.36/7.31	
20.36/7.31	Runtime Complexity Weighted TRS with Types.
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y [1]
20.36/7.31	   f(x, 0) -> x [1]
20.36/7.31	   f(i(x), y) -> i(x) [1]
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
20.36/7.31	   f(1, g(x, y)) -> x [1]
20.36/7.31	   f(2, g(x, y)) -> y [1]
20.36/7.31	
20.36/7.31	The TRS has the following type information:
20.36/7.31	f :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
20.36/7.31	0 :: 0:i:g:1:2
20.36/7.31	i :: a -> 0:i:g:1:2
20.36/7.31	g :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
20.36/7.31	1 :: 0:i:g:1:2
20.36/7.31	2 :: 0:i:g:1:2
20.36/7.31	const :: a
20.36/7.31	
20.36/7.31	Rewrite Strategy: INNERMOST
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
20.36/7.31	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
20.36/7.31	The constant constructors are abstracted as follows:
20.36/7.31	
20.36/7.31	   0 => 0
20.36/7.31	   1 => 1
20.36/7.31	   2 => 2
20.36/7.31	   const => 0
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(14)
20.36/7.31	Obligation:
20.36/7.31	Complexity RNTS consisting of the following rules:
20.36/7.31	
20.36/7.31	   f(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
20.36/7.31	   f(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
20.36/7.31	   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + f(x, z) + f(y, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
20.36/7.31	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(16)
20.36/7.31	Obligation:
20.36/7.31	Complexity RNTS consisting of the following rules:
20.36/7.31	
20.36/7.31	   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
20.36/7.31	   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
20.36/7.31	Found the following analysis order by SCC decomposition:
20.36/7.31	
20.36/7.31	   { f }
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(18)
20.36/7.31	Obligation:
20.36/7.31	Complexity RNTS consisting of the following rules:
20.36/7.31	
20.36/7.31	   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
20.36/7.31	   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	
20.36/7.31	Function symbols to be analyzed: {f}
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(19) ResultPropagationProof (UPPER BOUND(ID))
20.36/7.31	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(20)
20.36/7.31	Obligation:
20.36/7.31	Complexity RNTS consisting of the following rules:
20.36/7.31	
20.36/7.31	   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
20.36/7.31	   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	
20.36/7.31	Function symbols to be analyzed: {f}
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(21) IntTrsBoundProof (UPPER BOUND(ID))
20.36/7.31	
20.36/7.31	Computed SIZE bound using CoFloCo for: f
20.36/7.31	after applying outer abstraction to obtain an ITS,
20.36/7.31	resulting in: O(n^2) with polynomial bound: 2 + 8*z' + 4*z'*z'' + 2*z''
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(22)
20.36/7.31	Obligation:
20.36/7.31	Complexity RNTS consisting of the following rules:
20.36/7.31	
20.36/7.31	   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
20.36/7.31	   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	
20.36/7.31	Function symbols to be analyzed: {f}
20.36/7.31	Previous analysis results are:
20.36/7.31	  f: runtime: ?, size: O(n^2) [2 + 8*z' + 4*z'*z'' + 2*z'']
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(23) IntTrsBoundProof (UPPER BOUND(ID))
20.36/7.31	
20.36/7.31	Computed RUNTIME bound using CoFloCo for: f
20.36/7.31	after applying outer abstraction to obtain an ITS,
20.36/7.31	resulting in: O(n^1) with polynomial bound: 1 + 2*z'
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(24)
20.36/7.31	Obligation:
20.36/7.31	Complexity RNTS consisting of the following rules:
20.36/7.31	
20.36/7.31	   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
20.36/7.31	   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
20.36/7.31	   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
20.36/7.31	
20.36/7.31	Function symbols to be analyzed: 
20.36/7.31	Previous analysis results are:
20.36/7.31	  f: runtime: O(n^1) [1 + 2*z'], size: O(n^2) [2 + 8*z' + 4*z'*z'' + 2*z'']
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(25) FinalProof (FINISHED)
20.36/7.31	Computed overall runtime complexity
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(26)
20.36/7.31	BOUNDS(1, n^1)
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(27) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
20.36/7.31	Transformed a relative TRS into a decreasing-loop problem.
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(28)
20.36/7.31	Obligation:
20.36/7.31	Analyzing the following TRS for decreasing loops:
20.36/7.31	
20.36/7.31	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/7.31	
20.36/7.31	
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y
20.36/7.31	   f(x, 0) -> x
20.36/7.31	   f(i(x), y) -> i(x)
20.36/7.31	   f(f(x, y), z) -> f(x, f(y, z))
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z))
20.36/7.31	   f(1, g(x, y)) -> x
20.36/7.31	   f(2, g(x, y)) -> y
20.36/7.31	
20.36/7.31	S is empty.
20.36/7.31	Rewrite Strategy: FULL
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(29) DecreasingLoopProof (LOWER BOUND(ID))
20.36/7.31	The following loop(s) give(s) rise to the lower bound Omega(n^1):
20.36/7.31	
20.36/7.31	The rewrite sequence
20.36/7.31	
20.36/7.31	f(g(x, y), z) ->^+ g(f(x, z), f(y, z))
20.36/7.31	
20.36/7.31	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
20.36/7.31	
20.36/7.31	The pumping substitution is [x / g(x, y)].
20.36/7.31	
20.36/7.31	The result substitution is [ ].
20.36/7.31	
20.36/7.31	
20.36/7.31	
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(30)
20.36/7.31	Complex Obligation (BEST)
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(31)
20.36/7.31	Obligation:
20.36/7.31	Proved the lower bound n^1 for the following obligation:
20.36/7.31	
20.36/7.31	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/7.31	
20.36/7.31	
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y
20.36/7.31	   f(x, 0) -> x
20.36/7.31	   f(i(x), y) -> i(x)
20.36/7.31	   f(f(x, y), z) -> f(x, f(y, z))
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z))
20.36/7.31	   f(1, g(x, y)) -> x
20.36/7.31	   f(2, g(x, y)) -> y
20.36/7.31	
20.36/7.31	S is empty.
20.36/7.31	Rewrite Strategy: FULL
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(32) LowerBoundPropagationProof (FINISHED)
20.36/7.31	Propagated lower bound.
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(33)
20.36/7.31	BOUNDS(n^1, INF)
20.36/7.31	
20.36/7.31	----------------------------------------
20.36/7.31	
20.36/7.31	(34)
20.36/7.31	Obligation:
20.36/7.31	Analyzing the following TRS for decreasing loops:
20.36/7.31	
20.36/7.31	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/7.31	
20.36/7.31	
20.36/7.31	The TRS R consists of the following rules:
20.36/7.31	
20.36/7.31	   f(0, y) -> y
20.36/7.31	   f(x, 0) -> x
20.36/7.31	   f(i(x), y) -> i(x)
20.36/7.31	   f(f(x, y), z) -> f(x, f(y, z))
20.36/7.31	   f(g(x, y), z) -> g(f(x, z), f(y, z))
20.36/7.31	   f(1, g(x, y)) -> x
20.36/7.31	   f(2, g(x, y)) -> y
20.36/7.31	
20.36/7.31	S is empty.
20.36/7.31	Rewrite Strategy: FULL
20.52/8.20	EOF