14.22/4.51	WORST_CASE(Omega(n^1), O(n^1))
14.22/4.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
14.22/4.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
14.22/4.53	
14.22/4.53	
14.22/4.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
14.22/4.53	
14.22/4.53	(0) CpxTRS
14.22/4.53	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
14.22/4.53	(2) CpxWeightedTrs
14.22/4.53	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms]
14.22/4.53	    (4) CpxTypedWeightedTrs
14.22/4.53	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
14.22/4.53	    (6) CpxTypedWeightedCompleteTrs
14.22/4.53	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms]
14.22/4.53	    (8) CpxRNTS
14.22/4.53	    (9) CompleteCoflocoProof [FINISHED, 190 ms]
14.22/4.53	    (10) BOUNDS(1, n^1)
14.22/4.53	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
14.22/4.53	(12) TRS for Loop Detection
14.22/4.53	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
14.22/4.53	    (14) BEST
14.22/4.53	        (15) proven lower bound
14.22/4.53	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
14.22/4.53	            (17) BOUNDS(n^1, INF)
14.22/4.53	        (18) TRS for Loop Detection
14.22/4.53	
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(0)
14.22/4.53	Obligation:
14.22/4.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
14.22/4.53	
14.22/4.53	
14.22/4.53	The TRS R consists of the following rules:
14.22/4.53	
14.22/4.53	   half(0) -> 0
14.22/4.53	   half(s(0)) -> 0
14.22/4.53	   half(s(s(x))) -> s(half(x))
14.22/4.53	   bits(0) -> 0
14.22/4.53	   bits(s(x)) -> s(bits(half(s(x))))
14.22/4.53	
14.22/4.53	S is empty.
14.22/4.53	Rewrite Strategy: INNERMOST
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
14.22/4.53	Transformed relative TRS to weighted TRS
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(2)
14.22/4.53	Obligation:
14.22/4.53	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
14.22/4.53	
14.22/4.53	
14.22/4.53	The TRS R consists of the following rules:
14.22/4.53	
14.22/4.53	   half(0) -> 0 [1]
14.22/4.53	   half(s(0)) -> 0 [1]
14.22/4.53	   half(s(s(x))) -> s(half(x)) [1]
14.22/4.53	   bits(0) -> 0 [1]
14.22/4.53	   bits(s(x)) -> s(bits(half(s(x)))) [1]
14.22/4.53	
14.22/4.53	Rewrite Strategy: INNERMOST
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
14.22/4.53	Infered types.
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(4)
14.22/4.53	Obligation:
14.22/4.53	Runtime Complexity Weighted TRS with Types.
14.22/4.53	The TRS R consists of the following rules:
14.22/4.53	
14.22/4.53	   half(0) -> 0 [1]
14.22/4.53	   half(s(0)) -> 0 [1]
14.22/4.53	   half(s(s(x))) -> s(half(x)) [1]
14.22/4.53	   bits(0) -> 0 [1]
14.22/4.53	   bits(s(x)) -> s(bits(half(s(x)))) [1]
14.22/4.53	
14.22/4.53	The TRS has the following type information:
14.22/4.53	half :: 0:s -> 0:s
14.22/4.53	0 :: 0:s
14.22/4.53	s :: 0:s -> 0:s
14.22/4.53	bits :: 0:s -> 0:s
14.22/4.53	
14.22/4.53	Rewrite Strategy: INNERMOST
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(5) CompletionProof (UPPER BOUND(ID))
14.22/4.53	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
14.22/4.53	none
14.22/4.53	
14.22/4.53	And the following fresh constants: none
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(6)
14.22/4.53	Obligation:
14.22/4.53	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
14.22/4.53	
14.22/4.53	Runtime Complexity Weighted TRS with Types.
14.22/4.53	The TRS R consists of the following rules:
14.22/4.53	
14.22/4.53	   half(0) -> 0 [1]
14.22/4.53	   half(s(0)) -> 0 [1]
14.22/4.53	   half(s(s(x))) -> s(half(x)) [1]
14.22/4.53	   bits(0) -> 0 [1]
14.22/4.53	   bits(s(x)) -> s(bits(half(s(x)))) [1]
14.22/4.53	
14.22/4.53	The TRS has the following type information:
14.22/4.53	half :: 0:s -> 0:s
14.22/4.53	0 :: 0:s
14.22/4.53	s :: 0:s -> 0:s
14.22/4.53	bits :: 0:s -> 0:s
14.22/4.53	
14.22/4.53	Rewrite Strategy: INNERMOST
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
14.22/4.53	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
14.22/4.53	The constant constructors are abstracted as follows:
14.22/4.53	
14.22/4.53	   0 => 0
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(8)
14.22/4.53	Obligation:
14.22/4.53	Complexity RNTS consisting of the following rules:
14.22/4.53	
14.22/4.53	   bits(z) -{ 1 }-> 0 :|: z = 0
14.22/4.53	   bits(z) -{ 1 }-> 1 + bits(half(1 + x)) :|: x >= 0, z = 1 + x
14.22/4.53	   half(z) -{ 1 }-> 0 :|: z = 0
14.22/4.53	   half(z) -{ 1 }-> 0 :|: z = 1 + 0
14.22/4.53	   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
14.22/4.53	
14.22/4.53	Only complete derivations are relevant for the runtime complexity.
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(9) CompleteCoflocoProof (FINISHED)
14.22/4.53	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
14.22/4.53	
14.22/4.53	eq(start(V),0,[half(V, Out)],[V >= 0]).
14.22/4.53	eq(start(V),0,[bits(V, Out)],[V >= 0]).
14.22/4.53	eq(half(V, Out),1,[],[Out = 0,V = 0]).
14.22/4.53	eq(half(V, Out),1,[],[Out = 0,V = 1]).
14.22/4.53	eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
14.22/4.53	eq(bits(V, Out),1,[],[Out = 0,V = 0]).
14.22/4.53	eq(bits(V, Out),1,[half(1 + V2, Ret10),bits(Ret10, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + V2]).
14.22/4.53	input_output_vars(half(V,Out),[V],[Out]).
14.22/4.53	input_output_vars(bits(V,Out),[V],[Out]).
14.22/4.53	
14.22/4.53	
14.22/4.53	CoFloCo proof output:
14.22/4.53	Preprocessing Cost Relations
14.22/4.53	=====================================
14.22/4.53	
14.22/4.53	#### Computed strongly connected components 
14.22/4.53	0. recursive  : [half/2]
14.22/4.53	1. recursive  : [bits/2]
14.22/4.53	2. non_recursive  : [start/1]
14.22/4.53	
14.22/4.53	#### Obtained direct recursion through partial evaluation 
14.22/4.53	0. SCC is partially evaluated into half/2
14.22/4.53	1. SCC is partially evaluated into bits/2
14.22/4.53	2. SCC is partially evaluated into start/1
14.22/4.53	
14.22/4.53	Control-Flow Refinement of Cost Relations
14.22/4.53	=====================================
14.22/4.53	
14.22/4.53	### Specialization of cost equations half/2 
14.22/4.53	* CE 5 is refined into CE [8] 
14.22/4.53	* CE 4 is refined into CE [9] 
14.22/4.53	* CE 3 is refined into CE [10] 
14.22/4.53	
14.22/4.53	
14.22/4.53	### Cost equations --> "Loop" of half/2 
14.22/4.53	* CEs [9] --> Loop 7 
14.22/4.53	* CEs [10] --> Loop 8 
14.22/4.53	* CEs [8] --> Loop 9 
14.22/4.53	
14.22/4.53	### Ranking functions of CR half(V,Out) 
14.22/4.53	* RF of phase [9]: [V-1]
14.22/4.53	
14.22/4.53	#### Partial ranking functions of CR half(V,Out) 
14.22/4.53	* Partial RF of phase [9]:
14.22/4.53	  - RF of loop [9:1]:
14.22/4.53	    V-1
14.22/4.53	
14.22/4.53	
14.22/4.53	### Specialization of cost equations bits/2 
14.22/4.53	* CE 7 is refined into CE [11,12,13] 
14.22/4.53	* CE 6 is refined into CE [14] 
14.22/4.53	
14.22/4.53	
14.22/4.53	### Cost equations --> "Loop" of bits/2 
14.22/4.53	* CEs [14] --> Loop 10 
14.22/4.53	* CEs [13] --> Loop 11 
14.22/4.53	* CEs [12] --> Loop 12 
14.22/4.53	* CEs [11] --> Loop 13 
14.22/4.53	
14.22/4.53	### Ranking functions of CR bits(V,Out) 
14.22/4.53	* RF of phase [11,12]: [V-1]
14.22/4.53	
14.22/4.53	#### Partial ranking functions of CR bits(V,Out) 
14.22/4.53	* Partial RF of phase [11,12]:
14.22/4.53	  - RF of loop [11:1]:
14.22/4.53	    V/2-1
14.22/4.53	  - RF of loop [12:1]:
14.22/4.53	    V-1
14.22/4.53	
14.22/4.53	
14.22/4.53	### Specialization of cost equations start/1 
14.22/4.53	* CE 1 is refined into CE [15,16,17,18] 
14.22/4.53	* CE 2 is refined into CE [19,20,21] 
14.22/4.53	
14.22/4.53	
14.22/4.53	### Cost equations --> "Loop" of start/1 
14.22/4.53	* CEs [17,18,21] --> Loop 14 
14.22/4.53	* CEs [16,20] --> Loop 15 
14.22/4.53	* CEs [15,19] --> Loop 16 
14.22/4.53	
14.22/4.53	### Ranking functions of CR start(V) 
14.22/4.53	
14.22/4.53	#### Partial ranking functions of CR start(V) 
14.22/4.53	
14.22/4.53	
14.22/4.53	Computing Bounds
14.22/4.53	=====================================
14.22/4.53	
14.22/4.53	#### Cost of chains of half(V,Out):
14.22/4.53	* Chain [[9],8]: 1*it(9)+1
14.22/4.53	  Such that:it(9) =< 2*Out
14.22/4.53	
14.22/4.53	  with precondition: [V=2*Out,V>=2] 
14.22/4.53	
14.22/4.53	* Chain [[9],7]: 1*it(9)+1
14.22/4.53	  Such that:it(9) =< 2*Out
14.22/4.53	
14.22/4.53	  with precondition: [V=2*Out+1,V>=3] 
14.22/4.53	
14.22/4.53	* Chain [8]: 1
14.22/4.53	  with precondition: [V=0,Out=0] 
14.22/4.53	
14.22/4.53	* Chain [7]: 1
14.22/4.53	  with precondition: [V=1,Out=0] 
14.22/4.53	
14.22/4.53	
14.22/4.53	#### Cost of chains of bits(V,Out):
14.22/4.53	* Chain [[11,12],13,10]: 2*it(11)+2*it(12)+2*s(5)+3
14.22/4.53	  Such that:it(11) =< V/2
14.22/4.53	aux(5) =< V
14.22/4.53	aux(6) =< 2*V
14.22/4.53	it(11) =< aux(5)
14.22/4.53	it(12) =< aux(5)
14.22/4.53	it(12) =< aux(6)
14.22/4.53	s(5) =< aux(6)
14.22/4.53	
14.22/4.53	  with precondition: [Out>=2,V+2>=2*Out] 
14.22/4.53	
14.22/4.53	* Chain [13,10]: 3
14.22/4.53	  with precondition: [V=1,Out=1] 
14.22/4.53	
14.22/4.53	* Chain [10]: 1
14.22/4.53	  with precondition: [V=0,Out=0] 
14.22/4.53	
14.22/4.53	
14.22/4.53	#### Cost of chains of start(V):
14.22/4.53	* Chain [16]: 1
14.22/4.53	  with precondition: [V=0] 
14.22/4.53	
14.22/4.53	* Chain [15]: 3
14.22/4.53	  with precondition: [V=1] 
14.22/4.53	
14.22/4.53	* Chain [14]: 2*s(7)+2*s(9)+2*s(12)+2*s(13)+3
14.22/4.53	  Such that:s(11) =< 2*V
14.22/4.53	s(9) =< V/2
14.22/4.53	aux(7) =< V
14.22/4.53	s(7) =< aux(7)
14.22/4.53	s(9) =< aux(7)
14.22/4.53	s(12) =< aux(7)
14.22/4.53	s(12) =< s(11)
14.22/4.53	s(13) =< s(11)
14.22/4.53	
14.22/4.53	  with precondition: [V>=2] 
14.22/4.53	
14.22/4.53	
14.22/4.53	Closed-form bounds of start(V): 
14.22/4.53	-------------------------------------
14.22/4.53	* Chain [16] with precondition: [V=0] 
14.22/4.53	    - Upper bound: 1 
14.22/4.53	    - Complexity: constant 
14.22/4.53	* Chain [15] with precondition: [V=1] 
14.22/4.53	    - Upper bound: 3 
14.22/4.53	    - Complexity: constant 
14.22/4.53	* Chain [14] with precondition: [V>=2] 
14.22/4.53	    - Upper bound: 9*V+3 
14.22/4.53	    - Complexity: n 
14.22/4.53	
14.22/4.53	### Maximum cost of start(V): 9*V+3 
14.22/4.53	Asymptotic class: n 
14.22/4.53	* Total analysis performed in 109 ms.
14.22/4.53	
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(10)
14.22/4.53	BOUNDS(1, n^1)
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
14.22/4.53	Transformed a relative TRS into a decreasing-loop problem.
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(12)
14.22/4.53	Obligation:
14.22/4.53	Analyzing the following TRS for decreasing loops:
14.22/4.53	
14.22/4.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
14.22/4.53	
14.22/4.53	
14.22/4.53	The TRS R consists of the following rules:
14.22/4.53	
14.22/4.53	   half(0) -> 0
14.22/4.53	   half(s(0)) -> 0
14.22/4.53	   half(s(s(x))) -> s(half(x))
14.22/4.53	   bits(0) -> 0
14.22/4.53	   bits(s(x)) -> s(bits(half(s(x))))
14.22/4.53	
14.22/4.53	S is empty.
14.22/4.53	Rewrite Strategy: INNERMOST
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(13) DecreasingLoopProof (LOWER BOUND(ID))
14.22/4.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
14.22/4.53	
14.22/4.53	The rewrite sequence
14.22/4.53	
14.22/4.53	half(s(s(x))) ->^+ s(half(x))
14.22/4.53	
14.22/4.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
14.22/4.53	
14.22/4.53	The pumping substitution is [x / s(s(x))].
14.22/4.53	
14.22/4.53	The result substitution is [ ].
14.22/4.53	
14.22/4.53	
14.22/4.53	
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(14)
14.22/4.53	Complex Obligation (BEST)
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(15)
14.22/4.53	Obligation:
14.22/4.53	Proved the lower bound n^1 for the following obligation:
14.22/4.53	
14.22/4.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
14.22/4.53	
14.22/4.53	
14.22/4.53	The TRS R consists of the following rules:
14.22/4.53	
14.22/4.53	   half(0) -> 0
14.22/4.53	   half(s(0)) -> 0
14.22/4.53	   half(s(s(x))) -> s(half(x))
14.22/4.53	   bits(0) -> 0
14.22/4.53	   bits(s(x)) -> s(bits(half(s(x))))
14.22/4.53	
14.22/4.53	S is empty.
14.22/4.53	Rewrite Strategy: INNERMOST
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(16) LowerBoundPropagationProof (FINISHED)
14.22/4.53	Propagated lower bound.
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(17)
14.22/4.53	BOUNDS(n^1, INF)
14.22/4.53	
14.22/4.53	----------------------------------------
14.22/4.53	
14.22/4.53	(18)
14.22/4.53	Obligation:
14.22/4.53	Analyzing the following TRS for decreasing loops:
14.22/4.53	
14.22/4.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
14.22/4.53	
14.22/4.53	
14.22/4.53	The TRS R consists of the following rules:
14.22/4.53	
14.22/4.53	   half(0) -> 0
14.22/4.53	   half(s(0)) -> 0
14.22/4.53	   half(s(s(x))) -> s(half(x))
14.22/4.53	   bits(0) -> 0
14.22/4.53	   bits(s(x)) -> s(bits(half(s(x))))
14.22/4.53	
14.22/4.53	S is empty.
14.22/4.53	Rewrite Strategy: INNERMOST
14.22/4.57	EOF