19.89/6.09	WORST_CASE(Omega(n^1), O(n^1))
20.12/6.11	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
20.12/6.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.12/6.11	
20.12/6.11	
20.12/6.11	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.12/6.11	
20.12/6.11	(0) CpxTRS
20.12/6.11	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
20.12/6.11	(2) CpxTRS
20.12/6.11	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
20.12/6.11	    (4) CpxWeightedTrs
20.12/6.11	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.12/6.11	    (6) CpxTypedWeightedTrs
20.12/6.11	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
20.12/6.11	    (8) CpxTypedWeightedCompleteTrs
20.12/6.11	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
20.12/6.11	    (10) CpxRNTS
20.12/6.11	    (11) CompleteCoflocoProof [FINISHED, 165 ms]
20.12/6.11	    (12) BOUNDS(1, n^1)
20.12/6.11	(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
20.12/6.11	(14) TRS for Loop Detection
20.12/6.11	    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
20.12/6.11	    (16) BEST
20.12/6.11	        (17) proven lower bound
20.12/6.11	            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
20.12/6.11	            (19) BOUNDS(n^1, INF)
20.12/6.11	        (20) TRS for Loop Detection
20.12/6.11	
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(0)
20.12/6.11	Obligation:
20.12/6.11	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.12/6.11	
20.12/6.11	
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0
20.12/6.11	   half(s(0)) -> 0
20.12/6.11	   half(s(s(x))) -> s(half(x))
20.12/6.11	   bits(0) -> 0
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x))))
20.12/6.11	
20.12/6.11	S is empty.
20.12/6.11	Rewrite Strategy: FULL
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.12/6.11	Converted rc-obligation to irc-obligation.
20.12/6.11	
20.12/6.11	As the TRS is a non-duplicating overlay system, we have rc = irc.
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(2)
20.12/6.11	Obligation:
20.12/6.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.12/6.11	
20.12/6.11	
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0
20.12/6.11	   half(s(0)) -> 0
20.12/6.11	   half(s(s(x))) -> s(half(x))
20.12/6.11	   bits(0) -> 0
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x))))
20.12/6.11	
20.12/6.11	S is empty.
20.12/6.11	Rewrite Strategy: INNERMOST
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
20.12/6.11	Transformed relative TRS to weighted TRS
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(4)
20.12/6.11	Obligation:
20.12/6.11	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.12/6.11	
20.12/6.11	
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0 [1]
20.12/6.11	   half(s(0)) -> 0 [1]
20.12/6.11	   half(s(s(x))) -> s(half(x)) [1]
20.12/6.11	   bits(0) -> 0 [1]
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x)))) [1]
20.12/6.11	
20.12/6.11	Rewrite Strategy: INNERMOST
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.12/6.11	Infered types.
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(6)
20.12/6.11	Obligation:
20.12/6.11	Runtime Complexity Weighted TRS with Types.
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0 [1]
20.12/6.11	   half(s(0)) -> 0 [1]
20.12/6.11	   half(s(s(x))) -> s(half(x)) [1]
20.12/6.11	   bits(0) -> 0 [1]
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x)))) [1]
20.12/6.11	
20.12/6.11	The TRS has the following type information:
20.12/6.11	half :: 0:s -> 0:s
20.12/6.11	0 :: 0:s
20.12/6.11	s :: 0:s -> 0:s
20.12/6.11	bits :: 0:s -> 0:s
20.12/6.11	
20.12/6.11	Rewrite Strategy: INNERMOST
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(7) CompletionProof (UPPER BOUND(ID))
20.12/6.11	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
20.12/6.11	none
20.12/6.11	
20.12/6.11	And the following fresh constants: none
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(8)
20.12/6.11	Obligation:
20.12/6.11	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
20.12/6.11	
20.12/6.11	Runtime Complexity Weighted TRS with Types.
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0 [1]
20.12/6.11	   half(s(0)) -> 0 [1]
20.12/6.11	   half(s(s(x))) -> s(half(x)) [1]
20.12/6.11	   bits(0) -> 0 [1]
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x)))) [1]
20.12/6.11	
20.12/6.11	The TRS has the following type information:
20.12/6.11	half :: 0:s -> 0:s
20.12/6.11	0 :: 0:s
20.12/6.11	s :: 0:s -> 0:s
20.12/6.11	bits :: 0:s -> 0:s
20.12/6.11	
20.12/6.11	Rewrite Strategy: INNERMOST
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
20.12/6.11	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
20.12/6.11	The constant constructors are abstracted as follows:
20.12/6.11	
20.12/6.11	   0 => 0
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(10)
20.12/6.11	Obligation:
20.12/6.11	Complexity RNTS consisting of the following rules:
20.12/6.11	
20.12/6.11	   bits(z) -{ 1 }-> 0 :|: z = 0
20.12/6.11	   bits(z) -{ 1 }-> 1 + bits(half(1 + x)) :|: x >= 0, z = 1 + x
20.12/6.11	   half(z) -{ 1 }-> 0 :|: z = 0
20.12/6.11	   half(z) -{ 1 }-> 0 :|: z = 1 + 0
20.12/6.11	   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
20.12/6.11	
20.12/6.11	Only complete derivations are relevant for the runtime complexity.
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(11) CompleteCoflocoProof (FINISHED)
20.12/6.11	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
20.12/6.11	
20.12/6.11	eq(start(V),0,[half(V, Out)],[V >= 0]).
20.12/6.11	eq(start(V),0,[bits(V, Out)],[V >= 0]).
20.12/6.11	eq(half(V, Out),1,[],[Out = 0,V = 0]).
20.12/6.11	eq(half(V, Out),1,[],[Out = 0,V = 1]).
20.12/6.11	eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
20.12/6.11	eq(bits(V, Out),1,[],[Out = 0,V = 0]).
20.12/6.11	eq(bits(V, Out),1,[half(1 + V2, Ret10),bits(Ret10, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + V2]).
20.12/6.11	input_output_vars(half(V,Out),[V],[Out]).
20.12/6.11	input_output_vars(bits(V,Out),[V],[Out]).
20.12/6.11	
20.12/6.11	
20.12/6.11	CoFloCo proof output:
20.12/6.11	Preprocessing Cost Relations
20.12/6.11	=====================================
20.12/6.11	
20.12/6.11	#### Computed strongly connected components 
20.12/6.11	0. recursive  : [half/2]
20.12/6.11	1. recursive  : [bits/2]
20.12/6.11	2. non_recursive  : [start/1]
20.12/6.11	
20.12/6.11	#### Obtained direct recursion through partial evaluation 
20.12/6.11	0. SCC is partially evaluated into half/2
20.12/6.11	1. SCC is partially evaluated into bits/2
20.12/6.11	2. SCC is partially evaluated into start/1
20.12/6.11	
20.12/6.11	Control-Flow Refinement of Cost Relations
20.12/6.11	=====================================
20.12/6.11	
20.12/6.11	### Specialization of cost equations half/2 
20.12/6.11	* CE 5 is refined into CE [8] 
20.12/6.11	* CE 4 is refined into CE [9] 
20.12/6.11	* CE 3 is refined into CE [10] 
20.12/6.11	
20.12/6.11	
20.12/6.11	### Cost equations --> "Loop" of half/2 
20.12/6.11	* CEs [9] --> Loop 7 
20.12/6.11	* CEs [10] --> Loop 8 
20.12/6.11	* CEs [8] --> Loop 9 
20.12/6.11	
20.12/6.11	### Ranking functions of CR half(V,Out) 
20.12/6.11	* RF of phase [9]: [V-1]
20.12/6.11	
20.12/6.11	#### Partial ranking functions of CR half(V,Out) 
20.12/6.11	* Partial RF of phase [9]:
20.12/6.11	  - RF of loop [9:1]:
20.12/6.11	    V-1
20.12/6.11	
20.12/6.11	
20.12/6.11	### Specialization of cost equations bits/2 
20.12/6.11	* CE 7 is refined into CE [11,12,13] 
20.12/6.11	* CE 6 is refined into CE [14] 
20.12/6.11	
20.12/6.11	
20.12/6.11	### Cost equations --> "Loop" of bits/2 
20.12/6.11	* CEs [14] --> Loop 10 
20.12/6.11	* CEs [13] --> Loop 11 
20.12/6.11	* CEs [12] --> Loop 12 
20.12/6.11	* CEs [11] --> Loop 13 
20.12/6.11	
20.12/6.11	### Ranking functions of CR bits(V,Out) 
20.12/6.11	* RF of phase [11,12]: [V-1]
20.12/6.11	
20.12/6.11	#### Partial ranking functions of CR bits(V,Out) 
20.12/6.11	* Partial RF of phase [11,12]:
20.12/6.11	  - RF of loop [11:1]:
20.12/6.11	    V/2-1
20.12/6.11	  - RF of loop [12:1]:
20.12/6.11	    V-1
20.12/6.11	
20.12/6.11	
20.12/6.11	### Specialization of cost equations start/1 
20.12/6.11	* CE 1 is refined into CE [15,16,17,18] 
20.12/6.11	* CE 2 is refined into CE [19,20,21] 
20.12/6.11	
20.12/6.11	
20.12/6.11	### Cost equations --> "Loop" of start/1 
20.12/6.11	* CEs [17,18,21] --> Loop 14 
20.12/6.11	* CEs [16,20] --> Loop 15 
20.12/6.11	* CEs [15,19] --> Loop 16 
20.12/6.11	
20.12/6.11	### Ranking functions of CR start(V) 
20.12/6.11	
20.12/6.11	#### Partial ranking functions of CR start(V) 
20.12/6.11	
20.12/6.11	
20.12/6.11	Computing Bounds
20.12/6.11	=====================================
20.12/6.11	
20.12/6.11	#### Cost of chains of half(V,Out):
20.12/6.11	* Chain [[9],8]: 1*it(9)+1
20.12/6.11	  Such that:it(9) =< 2*Out
20.12/6.11	
20.12/6.11	  with precondition: [V=2*Out,V>=2] 
20.12/6.11	
20.12/6.11	* Chain [[9],7]: 1*it(9)+1
20.12/6.11	  Such that:it(9) =< 2*Out
20.12/6.11	
20.12/6.11	  with precondition: [V=2*Out+1,V>=3] 
20.12/6.11	
20.12/6.11	* Chain [8]: 1
20.12/6.11	  with precondition: [V=0,Out=0] 
20.12/6.11	
20.12/6.11	* Chain [7]: 1
20.12/6.11	  with precondition: [V=1,Out=0] 
20.12/6.11	
20.12/6.11	
20.12/6.11	#### Cost of chains of bits(V,Out):
20.12/6.11	* Chain [[11,12],13,10]: 2*it(11)+2*it(12)+2*s(5)+3
20.12/6.11	  Such that:it(11) =< V/2
20.12/6.11	aux(5) =< V
20.12/6.11	aux(6) =< 2*V
20.12/6.11	it(11) =< aux(5)
20.12/6.11	it(12) =< aux(5)
20.12/6.11	it(12) =< aux(6)
20.12/6.11	s(5) =< aux(6)
20.12/6.11	
20.12/6.11	  with precondition: [Out>=2,V+2>=2*Out] 
20.12/6.11	
20.12/6.11	* Chain [13,10]: 3
20.12/6.11	  with precondition: [V=1,Out=1] 
20.12/6.11	
20.12/6.11	* Chain [10]: 1
20.12/6.11	  with precondition: [V=0,Out=0] 
20.12/6.11	
20.12/6.11	
20.12/6.11	#### Cost of chains of start(V):
20.12/6.11	* Chain [16]: 1
20.12/6.11	  with precondition: [V=0] 
20.12/6.11	
20.12/6.11	* Chain [15]: 3
20.12/6.11	  with precondition: [V=1] 
20.12/6.11	
20.12/6.11	* Chain [14]: 2*s(7)+2*s(9)+2*s(12)+2*s(13)+3
20.12/6.11	  Such that:s(11) =< 2*V
20.12/6.11	s(9) =< V/2
20.12/6.11	aux(7) =< V
20.12/6.11	s(7) =< aux(7)
20.12/6.11	s(9) =< aux(7)
20.12/6.11	s(12) =< aux(7)
20.12/6.11	s(12) =< s(11)
20.12/6.11	s(13) =< s(11)
20.12/6.11	
20.12/6.11	  with precondition: [V>=2] 
20.12/6.11	
20.12/6.11	
20.12/6.11	Closed-form bounds of start(V): 
20.12/6.11	-------------------------------------
20.12/6.11	* Chain [16] with precondition: [V=0] 
20.12/6.11	    - Upper bound: 1 
20.12/6.11	    - Complexity: constant 
20.12/6.11	* Chain [15] with precondition: [V=1] 
20.12/6.11	    - Upper bound: 3 
20.12/6.11	    - Complexity: constant 
20.12/6.11	* Chain [14] with precondition: [V>=2] 
20.12/6.11	    - Upper bound: 9*V+3 
20.12/6.11	    - Complexity: n 
20.12/6.11	
20.12/6.11	### Maximum cost of start(V): 9*V+3 
20.12/6.11	Asymptotic class: n 
20.12/6.11	* Total analysis performed in 105 ms.
20.12/6.11	
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(12)
20.12/6.11	BOUNDS(1, n^1)
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
20.12/6.11	Transformed a relative TRS into a decreasing-loop problem.
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(14)
20.12/6.11	Obligation:
20.12/6.11	Analyzing the following TRS for decreasing loops:
20.12/6.11	
20.12/6.11	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.12/6.11	
20.12/6.11	
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0
20.12/6.11	   half(s(0)) -> 0
20.12/6.11	   half(s(s(x))) -> s(half(x))
20.12/6.11	   bits(0) -> 0
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x))))
20.12/6.11	
20.12/6.11	S is empty.
20.12/6.11	Rewrite Strategy: FULL
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(15) DecreasingLoopProof (LOWER BOUND(ID))
20.12/6.11	The following loop(s) give(s) rise to the lower bound Omega(n^1):
20.12/6.11	
20.12/6.11	The rewrite sequence
20.12/6.11	
20.12/6.11	half(s(s(x))) ->^+ s(half(x))
20.12/6.11	
20.12/6.11	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
20.12/6.11	
20.12/6.11	The pumping substitution is [x / s(s(x))].
20.12/6.11	
20.12/6.11	The result substitution is [ ].
20.12/6.11	
20.12/6.11	
20.12/6.11	
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(16)
20.12/6.11	Complex Obligation (BEST)
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(17)
20.12/6.11	Obligation:
20.12/6.11	Proved the lower bound n^1 for the following obligation:
20.12/6.11	
20.12/6.11	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.12/6.11	
20.12/6.11	
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0
20.12/6.11	   half(s(0)) -> 0
20.12/6.11	   half(s(s(x))) -> s(half(x))
20.12/6.11	   bits(0) -> 0
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x))))
20.12/6.11	
20.12/6.11	S is empty.
20.12/6.11	Rewrite Strategy: FULL
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(18) LowerBoundPropagationProof (FINISHED)
20.12/6.11	Propagated lower bound.
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(19)
20.12/6.11	BOUNDS(n^1, INF)
20.12/6.11	
20.12/6.11	----------------------------------------
20.12/6.11	
20.12/6.11	(20)
20.12/6.11	Obligation:
20.12/6.11	Analyzing the following TRS for decreasing loops:
20.12/6.11	
20.12/6.11	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.12/6.11	
20.12/6.11	
20.12/6.11	The TRS R consists of the following rules:
20.12/6.11	
20.12/6.11	   half(0) -> 0
20.12/6.11	   half(s(0)) -> 0
20.12/6.11	   half(s(s(x))) -> s(half(x))
20.12/6.11	   bits(0) -> 0
20.12/6.11	   bits(s(x)) -> s(bits(half(s(x))))
20.12/6.11	
20.12/6.11	S is empty.
20.12/6.11	Rewrite Strategy: FULL
20.12/6.16	EOF