20.54/6.28	WORST_CASE(Omega(n^1), O(n^1))
20.54/6.29	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.54/6.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.54/6.29	
20.54/6.29	
20.54/6.29	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.54/6.29	
20.54/6.29	(0) CpxTRS
20.54/6.29	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
20.54/6.29	(2) CpxTRS
20.54/6.29	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	    (4) CpxTRS
20.54/6.29	    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	    (6) CpxWeightedTrs
20.54/6.29	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	    (8) CpxTypedWeightedTrs
20.54/6.29	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
20.54/6.29	    (10) CpxTypedWeightedCompleteTrs
20.54/6.29	    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	    (12) CpxTypedWeightedCompleteTrs
20.54/6.29	    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
20.54/6.29	    (14) CpxRNTS
20.54/6.29	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	    (16) CpxRNTS
20.54/6.29	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	    (18) CpxRNTS
20.54/6.29	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.54/6.29	    (20) CpxRNTS
20.54/6.29	    (21) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
20.54/6.29	    (22) CpxRNTS
20.54/6.29	    (23) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
20.54/6.29	    (24) CpxRNTS
20.54/6.29	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.54/6.29	    (26) CpxRNTS
20.54/6.29	    (27) IntTrsBoundProof [UPPER BOUND(ID), 271 ms]
20.54/6.29	    (28) CpxRNTS
20.54/6.29	    (29) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
20.54/6.29	    (30) CpxRNTS
20.54/6.29	    (31) FinalProof [FINISHED, 0 ms]
20.54/6.29	    (32) BOUNDS(1, n^1)
20.54/6.29	(33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	(34) CpxTRS
20.54/6.29	    (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.54/6.29	    (36) typed CpxTrs
20.54/6.29	    (37) OrderProof [LOWER BOUND(ID), 0 ms]
20.54/6.29	    (38) typed CpxTrs
20.54/6.29	    (39) RewriteLemmaProof [LOWER BOUND(ID), 404 ms]
20.54/6.29	    (40) BEST
20.54/6.29	        (41) proven lower bound
20.54/6.29	            (42) LowerBoundPropagationProof [FINISHED, 0 ms]
20.54/6.29	            (43) BOUNDS(n^1, INF)
20.54/6.29	        (44) typed CpxTrs
20.54/6.29	
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(0)
20.54/6.29	Obligation:
20.54/6.29	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.54/6.29	
20.54/6.29	
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z)
20.54/6.29	   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
20.54/6.29	   g(0, Y) -> 0
20.54/6.29	   g(X, s(Y)) -> g(X, Y)
20.54/6.29	
20.54/6.29	S is empty.
20.54/6.29	Rewrite Strategy: FULL
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
20.54/6.29	The TRS does not nest defined symbols.
20.54/6.29	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
20.54/6.29	f(X, Y, g(X, Y)) -> h(0, g(X, Y))
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(2)
20.54/6.29	Obligation:
20.54/6.29	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.54/6.29	
20.54/6.29	
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z)
20.54/6.29	   g(0, Y) -> 0
20.54/6.29	   g(X, s(Y)) -> g(X, Y)
20.54/6.29	
20.54/6.29	S is empty.
20.54/6.29	Rewrite Strategy: FULL
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Converted rc-obligation to irc-obligation.
20.54/6.29	
20.54/6.29	As the TRS does not nest defined symbols, we have rc = irc.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(4)
20.54/6.29	Obligation:
20.54/6.29	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.54/6.29	
20.54/6.29	
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z)
20.54/6.29	   g(0, Y) -> 0
20.54/6.29	   g(X, s(Y)) -> g(X, Y)
20.54/6.29	
20.54/6.29	S is empty.
20.54/6.29	Rewrite Strategy: INNERMOST
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Transformed relative TRS to weighted TRS
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(6)
20.54/6.29	Obligation:
20.54/6.29	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.54/6.29	
20.54/6.29	
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z) [1]
20.54/6.29	   g(0, Y) -> 0 [1]
20.54/6.29	   g(X, s(Y)) -> g(X, Y) [1]
20.54/6.29	
20.54/6.29	Rewrite Strategy: INNERMOST
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Infered types.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(8)
20.54/6.29	Obligation:
20.54/6.29	Runtime Complexity Weighted TRS with Types.
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z) [1]
20.54/6.29	   g(0, Y) -> 0 [1]
20.54/6.29	   g(X, s(Y)) -> g(X, Y) [1]
20.54/6.29	
20.54/6.29	The TRS has the following type information:
20.54/6.29	h :: s -> a -> f
20.54/6.29	f :: s -> s -> a -> f
20.54/6.29	s :: s -> s
20.54/6.29	g :: 0 -> s -> 0
20.54/6.29	0 :: 0
20.54/6.29	
20.54/6.29	Rewrite Strategy: INNERMOST
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(9) CompletionProof (UPPER BOUND(ID))
20.54/6.29	The transformation into a RNTS is sound, since:
20.54/6.29	
20.54/6.29	(a) The obligation is a constructor system where every type has a constant constructor,
20.54/6.29	
20.54/6.29	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
20.54/6.29	
20.54/6.29	   h_2
20.54/6.29	   g_2
20.54/6.29	
20.54/6.29	(c) The following functions are completely defined:
20.54/6.29	none
20.54/6.29	
20.54/6.29	Due to the following rules being added:
20.54/6.29	none
20.54/6.29	
20.54/6.29	And the following fresh constants:    const, const1, const2
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(10)
20.54/6.29	Obligation:
20.54/6.29	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.54/6.29	
20.54/6.29	Runtime Complexity Weighted TRS with Types.
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z) [1]
20.54/6.29	   g(0, Y) -> 0 [1]
20.54/6.29	   g(X, s(Y)) -> g(X, Y) [1]
20.54/6.29	
20.54/6.29	The TRS has the following type information:
20.54/6.29	h :: s -> a -> f
20.54/6.29	f :: s -> s -> a -> f
20.54/6.29	s :: s -> s
20.54/6.29	g :: 0 -> s -> 0
20.54/6.29	0 :: 0
20.54/6.29	const :: f
20.54/6.29	const1 :: s
20.54/6.29	const2 :: a
20.54/6.29	
20.54/6.29	Rewrite Strategy: INNERMOST
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(11) NarrowingProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(12)
20.54/6.29	Obligation:
20.54/6.29	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.54/6.29	
20.54/6.29	Runtime Complexity Weighted TRS with Types.
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z) [1]
20.54/6.29	   g(0, Y) -> 0 [1]
20.54/6.29	   g(X, s(Y)) -> g(X, Y) [1]
20.54/6.29	
20.54/6.29	The TRS has the following type information:
20.54/6.29	h :: s -> a -> f
20.54/6.29	f :: s -> s -> a -> f
20.54/6.29	s :: s -> s
20.54/6.29	g :: 0 -> s -> 0
20.54/6.29	0 :: 0
20.54/6.29	const :: f
20.54/6.29	const1 :: s
20.54/6.29	const2 :: a
20.54/6.29	
20.54/6.29	Rewrite Strategy: INNERMOST
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
20.54/6.29	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
20.54/6.29	The constant constructors are abstracted as follows:
20.54/6.29	
20.54/6.29	   0 => 0
20.54/6.29	   const => 0
20.54/6.29	   const1 => 0
20.54/6.29	   const2 => 0
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(14)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 }-> g(X, Y) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + X + (1 + X) + Z :|: Z >= 0, X >= 0, z' = Z, z = X
20.54/6.29	
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(16)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 }-> g(z, z' - 1) :|: z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Found the following analysis order by SCC decomposition:
20.54/6.29	
20.54/6.29	   { g }
20.54/6.29	   { h }
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(18)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 }-> g(z, z' - 1) :|: z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	Function symbols to be analyzed: {g}, {h}
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(19) ResultPropagationProof (UPPER BOUND(ID))
20.54/6.29	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(20)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 }-> g(z, z' - 1) :|: z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	Function symbols to be analyzed: {g}, {h}
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(21) IntTrsBoundProof (UPPER BOUND(ID))
20.54/6.29	
20.54/6.29	Computed SIZE bound using CoFloCo for: g
20.54/6.29	after applying outer abstraction to obtain an ITS,
20.54/6.29	resulting in: O(1) with polynomial bound: 0
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(22)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 }-> g(z, z' - 1) :|: z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	Function symbols to be analyzed: {g}, {h}
20.54/6.29	Previous analysis results are:
20.54/6.29	  g: runtime: ?, size: O(1) [0]
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(23) IntTrsBoundProof (UPPER BOUND(ID))
20.54/6.29	
20.54/6.29	Computed RUNTIME bound using CoFloCo for: g
20.54/6.29	after applying outer abstraction to obtain an ITS,
20.54/6.29	resulting in: O(n^1) with polynomial bound: 1 + z'
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(24)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 }-> g(z, z' - 1) :|: z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	Function symbols to be analyzed: {h}
20.54/6.29	Previous analysis results are:
20.54/6.29	  g: runtime: O(n^1) [1 + z'], size: O(1) [0]
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(25) ResultPropagationProof (UPPER BOUND(ID))
20.54/6.29	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(26)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	Function symbols to be analyzed: {h}
20.54/6.29	Previous analysis results are:
20.54/6.29	  g: runtime: O(n^1) [1 + z'], size: O(1) [0]
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(27) IntTrsBoundProof (UPPER BOUND(ID))
20.54/6.29	
20.54/6.29	Computed SIZE bound using CoFloCo for: h
20.54/6.29	after applying outer abstraction to obtain an ITS,
20.54/6.29	resulting in: O(n^1) with polynomial bound: 2 + 2*z + z'
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(28)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	Function symbols to be analyzed: {h}
20.54/6.29	Previous analysis results are:
20.54/6.29	  g: runtime: O(n^1) [1 + z'], size: O(1) [0]
20.54/6.29	  h: runtime: ?, size: O(n^1) [2 + 2*z + z']
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(29) IntTrsBoundProof (UPPER BOUND(ID))
20.54/6.29	
20.54/6.29	Computed RUNTIME bound using CoFloCo for: h
20.54/6.29	after applying outer abstraction to obtain an ITS,
20.54/6.29	resulting in: O(1) with polynomial bound: 1
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(30)
20.54/6.29	Obligation:
20.54/6.29	Complexity RNTS consisting of the following rules:
20.54/6.29	
20.54/6.29	   g(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
20.54/6.29	   g(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
20.54/6.29	   h(z, z') -{ 1 }-> 1 + z + (1 + z) + z' :|: z' >= 0, z >= 0
20.54/6.29	
20.54/6.29	Function symbols to be analyzed: 
20.54/6.29	Previous analysis results are:
20.54/6.29	  g: runtime: O(n^1) [1 + z'], size: O(1) [0]
20.54/6.29	  h: runtime: O(1) [1], size: O(n^1) [2 + 2*z + z']
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(31) FinalProof (FINISHED)
20.54/6.29	Computed overall runtime complexity
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(32)
20.54/6.29	BOUNDS(1, n^1)
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(33) RenamingProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Renamed function symbols to avoid clashes with predefined symbol.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(34)
20.54/6.29	Obligation:
20.54/6.29	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
20.54/6.29	
20.54/6.29	
20.54/6.29	The TRS R consists of the following rules:
20.54/6.29	
20.54/6.29	   h(X, Z) -> f(X, s(X), Z)
20.54/6.29	   f(X, Y, g(X, Y)) -> h(0', g(X, Y))
20.54/6.29	   g(0', Y) -> 0'
20.54/6.29	   g(X, s(Y)) -> g(X, Y)
20.54/6.29	
20.54/6.29	S is empty.
20.54/6.29	Rewrite Strategy: FULL
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(35) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.54/6.29	Infered types.
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(36)
20.54/6.29	Obligation:
20.54/6.29	TRS:
20.54/6.29	Rules:
20.54/6.29	h(X, Z) -> f(X, s(X), Z)
20.54/6.29	f(X, Y, g(X, Y)) -> h(0', g(X, Y))
20.54/6.29	g(0', Y) -> 0'
20.54/6.29	g(X, s(Y)) -> g(X, Y)
20.54/6.29	
20.54/6.29	Types:
20.54/6.29	h :: s:0' -> s:0' -> h:f
20.54/6.29	f :: s:0' -> s:0' -> s:0' -> h:f
20.54/6.29	s :: s:0' -> s:0'
20.54/6.29	g :: s:0' -> s:0' -> s:0'
20.54/6.29	0' :: s:0'
20.54/6.29	hole_h:f1_0 :: h:f
20.54/6.29	hole_s:0'2_0 :: s:0'
20.54/6.29	gen_s:0'3_0 :: Nat -> s:0'
20.54/6.29	
20.54/6.29	----------------------------------------
20.54/6.29	
20.54/6.29	(37) OrderProof (LOWER BOUND(ID))
20.54/6.29	Heuristically decided to analyse the following defined symbols:
20.54/6.30	h, g
20.54/6.30	
20.54/6.30	They will be analysed ascendingly in the following order:
20.54/6.30	g < h
20.54/6.30	
20.54/6.30	----------------------------------------
20.54/6.30	
20.54/6.30	(38)
20.54/6.30	Obligation:
20.54/6.30	TRS:
20.54/6.30	Rules:
20.54/6.30	h(X, Z) -> f(X, s(X), Z)
20.54/6.30	f(X, Y, g(X, Y)) -> h(0', g(X, Y))
20.54/6.30	g(0', Y) -> 0'
20.54/6.30	g(X, s(Y)) -> g(X, Y)
20.54/6.30	
20.54/6.30	Types:
20.54/6.30	h :: s:0' -> s:0' -> h:f
20.54/6.30	f :: s:0' -> s:0' -> s:0' -> h:f
20.54/6.30	s :: s:0' -> s:0'
20.54/6.30	g :: s:0' -> s:0' -> s:0'
20.54/6.30	0' :: s:0'
20.54/6.30	hole_h:f1_0 :: h:f
20.54/6.30	hole_s:0'2_0 :: s:0'
20.54/6.30	gen_s:0'3_0 :: Nat -> s:0'
20.54/6.30	
20.54/6.30	
20.54/6.30	Generator Equations:
20.54/6.30	gen_s:0'3_0(0) <=> 0'
20.54/6.30	gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))
20.54/6.30	
20.54/6.30	
20.54/6.30	The following defined symbols remain to be analysed:
20.54/6.30	g, h
20.54/6.30	
20.54/6.30	They will be analysed ascendingly in the following order:
20.54/6.30	g < h
20.54/6.30	
20.54/6.30	----------------------------------------
20.54/6.30	
20.54/6.30	(39) RewriteLemmaProof (LOWER BOUND(ID))
20.54/6.30	Proved the following rewrite lemma:
20.54/6.30	g(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
20.54/6.30	
20.54/6.30	Induction Base:
20.54/6.30	g(gen_s:0'3_0(a), gen_s:0'3_0(+(1, 0)))
20.54/6.30	
20.54/6.30	Induction Step:
20.54/6.30	g(gen_s:0'3_0(a), gen_s:0'3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
20.54/6.30	g(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) ->_IH
20.54/6.30	*4_0
20.54/6.30	
20.54/6.30	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
20.54/6.30	----------------------------------------
20.54/6.30	
20.54/6.30	(40)
20.54/6.30	Complex Obligation (BEST)
20.54/6.30	
20.54/6.30	----------------------------------------
20.54/6.30	
20.54/6.30	(41)
20.54/6.30	Obligation:
20.54/6.30	Proved the lower bound n^1 for the following obligation:
20.54/6.30	
20.54/6.30	TRS:
20.54/6.30	Rules:
20.54/6.30	h(X, Z) -> f(X, s(X), Z)
20.54/6.30	f(X, Y, g(X, Y)) -> h(0', g(X, Y))
20.54/6.30	g(0', Y) -> 0'
20.54/6.30	g(X, s(Y)) -> g(X, Y)
20.54/6.30	
20.54/6.30	Types:
20.54/6.30	h :: s:0' -> s:0' -> h:f
20.54/6.30	f :: s:0' -> s:0' -> s:0' -> h:f
20.54/6.30	s :: s:0' -> s:0'
20.54/6.30	g :: s:0' -> s:0' -> s:0'
20.54/6.30	0' :: s:0'
20.54/6.30	hole_h:f1_0 :: h:f
20.54/6.30	hole_s:0'2_0 :: s:0'
20.54/6.30	gen_s:0'3_0 :: Nat -> s:0'
20.54/6.30	
20.54/6.30	
20.54/6.30	Generator Equations:
20.54/6.30	gen_s:0'3_0(0) <=> 0'
20.54/6.30	gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))
20.54/6.30	
20.54/6.30	
20.54/6.30	The following defined symbols remain to be analysed:
20.54/6.30	g, h
20.54/6.30	
20.54/6.30	They will be analysed ascendingly in the following order:
20.54/6.30	g < h
20.54/6.30	
20.54/6.30	----------------------------------------
20.54/6.30	
20.54/6.30	(42) LowerBoundPropagationProof (FINISHED)
20.54/6.30	Propagated lower bound.
20.54/6.30	----------------------------------------
20.54/6.30	
20.54/6.30	(43)
20.54/6.30	BOUNDS(n^1, INF)
20.54/6.30	
20.54/6.30	----------------------------------------
20.54/6.30	
20.54/6.30	(44)
20.54/6.30	Obligation:
20.54/6.30	TRS:
20.54/6.30	Rules:
20.54/6.30	h(X, Z) -> f(X, s(X), Z)
20.54/6.30	f(X, Y, g(X, Y)) -> h(0', g(X, Y))
20.54/6.30	g(0', Y) -> 0'
20.54/6.30	g(X, s(Y)) -> g(X, Y)
20.54/6.30	
20.54/6.30	Types:
20.54/6.30	h :: s:0' -> s:0' -> h:f
20.54/6.30	f :: s:0' -> s:0' -> s:0' -> h:f
20.54/6.30	s :: s:0' -> s:0'
20.54/6.30	g :: s:0' -> s:0' -> s:0'
20.54/6.30	0' :: s:0'
20.54/6.30	hole_h:f1_0 :: h:f
20.54/6.30	hole_s:0'2_0 :: s:0'
20.54/6.30	gen_s:0'3_0 :: Nat -> s:0'
20.54/6.30	
20.54/6.30	
20.54/6.30	Lemmas:
20.54/6.30	g(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
20.54/6.30	
20.54/6.30	
20.54/6.30	Generator Equations:
20.54/6.30	gen_s:0'3_0(0) <=> 0'
20.54/6.30	gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))
20.54/6.30	
20.54/6.30	
20.54/6.30	The following defined symbols remain to be analysed:
20.54/6.30	h
20.59/8.70	EOF