17.44/5.27	WORST_CASE(Omega(n^1), O(n^1))
17.44/5.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.44/5.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.44/5.28	
17.44/5.28	
17.44/5.28	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
17.44/5.28	
17.44/5.28	(0) CpxTRS
17.44/5.28	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
17.44/5.28	(2) CpxWeightedTrs
17.44/5.28	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
17.44/5.28	    (4) CpxTypedWeightedTrs
17.44/5.28	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
17.44/5.28	    (6) CpxTypedWeightedCompleteTrs
17.44/5.28	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
17.44/5.28	    (8) CpxRNTS
17.44/5.28	    (9) CompleteCoflocoProof [FINISHED, 310 ms]
17.44/5.28	    (10) BOUNDS(1, n^1)
17.44/5.28	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
17.44/5.28	(12) TRS for Loop Detection
17.44/5.28	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
17.44/5.28	    (14) BEST
17.44/5.28	        (15) proven lower bound
17.44/5.28	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
17.44/5.28	            (17) BOUNDS(n^1, INF)
17.44/5.28	        (18) TRS for Loop Detection
17.44/5.28	
17.44/5.28	
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(0)
17.44/5.28	Obligation:
17.44/5.28	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
17.44/5.28	
17.44/5.28	
17.44/5.28	The TRS R consists of the following rules:
17.44/5.28	
17.44/5.28	   half(0) -> 0
17.44/5.28	   half(s(0)) -> 0
17.44/5.28	   half(s(s(x))) -> s(half(x))
17.44/5.28	   lastbit(0) -> 0
17.44/5.28	   lastbit(s(0)) -> s(0)
17.44/5.28	   lastbit(s(s(x))) -> lastbit(x)
17.44/5.28	   conv(0) -> cons(nil, 0)
17.44/5.28	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
17.44/5.28	
17.44/5.28	S is empty.
17.44/5.28	Rewrite Strategy: INNERMOST
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
17.44/5.28	Transformed relative TRS to weighted TRS
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(2)
17.44/5.28	Obligation:
17.44/5.28	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
17.44/5.28	
17.44/5.28	
17.44/5.28	The TRS R consists of the following rules:
17.44/5.28	
17.44/5.28	   half(0) -> 0 [1]
17.44/5.28	   half(s(0)) -> 0 [1]
17.44/5.28	   half(s(s(x))) -> s(half(x)) [1]
17.44/5.28	   lastbit(0) -> 0 [1]
17.44/5.28	   lastbit(s(0)) -> s(0) [1]
17.44/5.28	   lastbit(s(s(x))) -> lastbit(x) [1]
17.44/5.28	   conv(0) -> cons(nil, 0) [1]
17.44/5.28	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [1]
17.44/5.28	
17.44/5.28	Rewrite Strategy: INNERMOST
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
17.44/5.28	Infered types.
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(4)
17.44/5.28	Obligation:
17.44/5.28	Runtime Complexity Weighted TRS with Types.
17.44/5.28	The TRS R consists of the following rules:
17.44/5.28	
17.44/5.28	   half(0) -> 0 [1]
17.44/5.28	   half(s(0)) -> 0 [1]
17.44/5.28	   half(s(s(x))) -> s(half(x)) [1]
17.44/5.28	   lastbit(0) -> 0 [1]
17.44/5.28	   lastbit(s(0)) -> s(0) [1]
17.44/5.28	   lastbit(s(s(x))) -> lastbit(x) [1]
17.44/5.28	   conv(0) -> cons(nil, 0) [1]
17.44/5.28	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [1]
17.44/5.28	
17.44/5.28	The TRS has the following type information:
17.44/5.28	half :: 0:s -> 0:s
17.44/5.28	0 :: 0:s
17.44/5.28	s :: 0:s -> 0:s
17.44/5.28	lastbit :: 0:s -> 0:s
17.44/5.28	conv :: 0:s -> nil:cons
17.44/5.28	cons :: nil:cons -> 0:s -> nil:cons
17.44/5.28	nil :: nil:cons
17.44/5.28	
17.44/5.28	Rewrite Strategy: INNERMOST
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(5) CompletionProof (UPPER BOUND(ID))
17.44/5.28	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
17.44/5.28	none
17.44/5.28	
17.44/5.28	And the following fresh constants: none
17.44/5.28	
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(6)
17.44/5.28	Obligation:
17.44/5.28	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
17.44/5.28	
17.44/5.28	Runtime Complexity Weighted TRS with Types.
17.44/5.28	The TRS R consists of the following rules:
17.44/5.28	
17.44/5.28	   half(0) -> 0 [1]
17.44/5.28	   half(s(0)) -> 0 [1]
17.44/5.28	   half(s(s(x))) -> s(half(x)) [1]
17.44/5.28	   lastbit(0) -> 0 [1]
17.44/5.28	   lastbit(s(0)) -> s(0) [1]
17.44/5.28	   lastbit(s(s(x))) -> lastbit(x) [1]
17.44/5.28	   conv(0) -> cons(nil, 0) [1]
17.44/5.28	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [1]
17.44/5.28	
17.44/5.28	The TRS has the following type information:
17.44/5.28	half :: 0:s -> 0:s
17.44/5.28	0 :: 0:s
17.44/5.28	s :: 0:s -> 0:s
17.44/5.28	lastbit :: 0:s -> 0:s
17.44/5.28	conv :: 0:s -> nil:cons
17.44/5.28	cons :: nil:cons -> 0:s -> nil:cons
17.44/5.28	nil :: nil:cons
17.44/5.28	
17.44/5.28	Rewrite Strategy: INNERMOST
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
17.44/5.28	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
17.44/5.28	The constant constructors are abstracted as follows:
17.44/5.28	
17.44/5.28	   0 => 0
17.44/5.28	   nil => 0
17.44/5.28	
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(8)
17.44/5.28	Obligation:
17.44/5.28	Complexity RNTS consisting of the following rules:
17.44/5.28	
17.44/5.28	   conv(z) -{ 1 }-> 1 + conv(half(1 + x)) + lastbit(1 + x) :|: x >= 0, z = 1 + x
17.44/5.28	   conv(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
17.44/5.28	   half(z) -{ 1 }-> 0 :|: z = 0
17.44/5.28	   half(z) -{ 1 }-> 0 :|: z = 1 + 0
17.44/5.28	   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
17.44/5.28	   lastbit(z) -{ 1 }-> lastbit(x) :|: x >= 0, z = 1 + (1 + x)
17.44/5.28	   lastbit(z) -{ 1 }-> 0 :|: z = 0
17.44/5.28	   lastbit(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
17.44/5.28	
17.44/5.28	Only complete derivations are relevant for the runtime complexity.
17.44/5.28	
17.44/5.28	----------------------------------------
17.44/5.28	
17.44/5.28	(9) CompleteCoflocoProof (FINISHED)
17.44/5.28	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
17.44/5.28	
17.44/5.28	eq(start(V),0,[half(V, Out)],[V >= 0]).
17.44/5.28	eq(start(V),0,[lastbit(V, Out)],[V >= 0]).
17.44/5.28	eq(start(V),0,[conv(V, Out)],[V >= 0]).
17.44/5.28	eq(half(V, Out),1,[],[Out = 0,V = 0]).
17.44/5.28	eq(half(V, Out),1,[],[Out = 0,V = 1]).
17.44/5.28	eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
17.44/5.28	eq(lastbit(V, Out),1,[],[Out = 0,V = 0]).
17.44/5.28	eq(lastbit(V, Out),1,[],[Out = 1,V = 1]).
17.44/5.29	eq(lastbit(V, Out),1,[lastbit(V2, Ret)],[Out = Ret,V2 >= 0,V = 2 + V2]).
17.44/5.29	eq(conv(V, Out),1,[],[Out = 1,V = 0]).
17.44/5.29	eq(conv(V, Out),1,[half(1 + V3, Ret010),conv(Ret010, Ret01),lastbit(1 + V3, Ret11)],[Out = 1 + Ret01 + Ret11,V3 >= 0,V = 1 + V3]).
17.44/5.29	input_output_vars(half(V,Out),[V],[Out]).
17.44/5.29	input_output_vars(lastbit(V,Out),[V],[Out]).
17.44/5.29	input_output_vars(conv(V,Out),[V],[Out]).
17.44/5.29	
17.44/5.29	
17.44/5.29	CoFloCo proof output:
17.44/5.29	Preprocessing Cost Relations
17.44/5.29	=====================================
17.44/5.29	
17.44/5.29	#### Computed strongly connected components 
17.44/5.29	0. recursive  : [half/2]
17.44/5.29	1. recursive  : [lastbit/2]
17.44/5.29	2. recursive [non_tail] : [conv/2]
17.44/5.29	3. non_recursive  : [start/1]
17.44/5.29	
17.44/5.29	#### Obtained direct recursion through partial evaluation 
17.44/5.29	0. SCC is partially evaluated into half/2
17.44/5.29	1. SCC is partially evaluated into lastbit/2
17.44/5.29	2. SCC is partially evaluated into conv/2
17.44/5.29	3. SCC is partially evaluated into start/1
17.44/5.29	
17.44/5.29	Control-Flow Refinement of Cost Relations
17.44/5.29	=====================================
17.44/5.29	
17.44/5.29	### Specialization of cost equations half/2 
17.44/5.29	* CE 6 is refined into CE [12] 
17.44/5.29	* CE 5 is refined into CE [13] 
17.44/5.29	* CE 4 is refined into CE [14] 
17.44/5.29	
17.44/5.29	
17.44/5.29	### Cost equations --> "Loop" of half/2 
17.44/5.29	* CEs [13] --> Loop 10 
17.44/5.29	* CEs [14] --> Loop 11 
17.44/5.29	* CEs [12] --> Loop 12 
17.44/5.29	
17.44/5.29	### Ranking functions of CR half(V,Out) 
17.44/5.29	* RF of phase [12]: [V-1]
17.44/5.29	
17.44/5.29	#### Partial ranking functions of CR half(V,Out) 
17.44/5.29	* Partial RF of phase [12]:
17.44/5.29	  - RF of loop [12:1]:
17.44/5.29	    V-1
17.44/5.29	
17.44/5.29	
17.44/5.29	### Specialization of cost equations lastbit/2 
17.44/5.29	* CE 9 is refined into CE [15] 
17.44/5.29	* CE 8 is refined into CE [16] 
17.44/5.29	* CE 7 is refined into CE [17] 
17.44/5.29	
17.44/5.29	
17.44/5.29	### Cost equations --> "Loop" of lastbit/2 
17.44/5.29	* CEs [16] --> Loop 13 
17.44/5.29	* CEs [17] --> Loop 14 
17.44/5.29	* CEs [15] --> Loop 15 
17.44/5.29	
17.44/5.29	### Ranking functions of CR lastbit(V,Out) 
17.44/5.29	* RF of phase [15]: [V-1]
17.44/5.29	
17.44/5.29	#### Partial ranking functions of CR lastbit(V,Out) 
17.44/5.29	* Partial RF of phase [15]:
17.44/5.29	  - RF of loop [15:1]:
17.44/5.29	    V-1
17.44/5.29	
17.44/5.29	
17.44/5.29	### Specialization of cost equations conv/2 
17.44/5.29	* CE 11 is refined into CE [18,19,20,21,22] 
17.44/5.29	* CE 10 is refined into CE [23] 
17.44/5.29	
17.44/5.29	
17.44/5.29	### Cost equations --> "Loop" of conv/2 
17.44/5.29	* CEs [23] --> Loop 16 
17.44/5.29	* CEs [22] --> Loop 17 
17.44/5.29	* CEs [21] --> Loop 18 
17.44/5.29	* CEs [20] --> Loop 19 
17.44/5.29	* CEs [19] --> Loop 20 
17.44/5.29	* CEs [18] --> Loop 21 
17.44/5.29	
17.44/5.29	### Ranking functions of CR conv(V,Out) 
17.44/5.29	* RF of phase [17,18,19,20]: [V-1]
17.44/5.29	
17.44/5.29	#### Partial ranking functions of CR conv(V,Out) 
17.44/5.29	* Partial RF of phase [17,18,19,20]:
17.44/5.29	  - RF of loop [17:1,18:1]:
17.44/5.29	    V/2-1
17.44/5.29	  - RF of loop [19:1]:
17.44/5.29	    2*V-5
17.44/5.29	  - RF of loop [20:1]:
17.44/5.29	    V-1
17.44/5.29	
17.44/5.29	
17.44/5.29	### Specialization of cost equations start/1 
17.44/5.29	* CE 1 is refined into CE [24,25,26,27] 
17.44/5.29	* CE 2 is refined into CE [28,29,30,31] 
17.44/5.29	* CE 3 is refined into CE [32,33,34] 
17.44/5.29	
17.44/5.29	
17.44/5.29	### Cost equations --> "Loop" of start/1 
17.44/5.29	* CEs [26,27,30,31,34] --> Loop 22 
17.44/5.29	* CEs [25,29,33] --> Loop 23 
17.44/5.29	* CEs [24,28,32] --> Loop 24 
17.44/5.29	
17.44/5.29	### Ranking functions of CR start(V) 
17.44/5.29	
17.44/5.29	#### Partial ranking functions of CR start(V) 
17.44/5.29	
17.44/5.29	
17.44/5.29	Computing Bounds
17.44/5.29	=====================================
17.44/5.29	
17.44/5.29	#### Cost of chains of half(V,Out):
17.44/5.29	* Chain [[12],11]: 1*it(12)+1
17.44/5.29	  Such that:it(12) =< 2*Out
17.44/5.29	
17.44/5.29	  with precondition: [V=2*Out,V>=2] 
17.44/5.29	
17.44/5.29	* Chain [[12],10]: 1*it(12)+1
17.44/5.29	  Such that:it(12) =< 2*Out
17.44/5.29	
17.44/5.29	  with precondition: [V=2*Out+1,V>=3] 
17.44/5.29	
17.44/5.29	* Chain [11]: 1
17.44/5.29	  with precondition: [V=0,Out=0] 
17.44/5.29	
17.44/5.29	* Chain [10]: 1
17.44/5.29	  with precondition: [V=1,Out=0] 
17.44/5.29	
17.44/5.29	
17.44/5.29	#### Cost of chains of lastbit(V,Out):
17.44/5.29	* Chain [[15],14]: 1*it(15)+1
17.44/5.29	  Such that:it(15) =< V
17.44/5.29	
17.44/5.29	  with precondition: [Out=0,V>=2] 
17.44/5.29	
17.44/5.29	* Chain [[15],13]: 1*it(15)+1
17.44/5.29	  Such that:it(15) =< V
17.44/5.29	
17.44/5.29	  with precondition: [Out=1,V>=3] 
17.44/5.29	
17.44/5.29	* Chain [14]: 1
17.44/5.29	  with precondition: [V=0,Out=0] 
17.44/5.29	
17.44/5.29	* Chain [13]: 1
17.44/5.29	  with precondition: [V=1,Out=1] 
17.44/5.29	
17.44/5.29	
17.44/5.29	#### Cost of chains of conv(V,Out):
17.44/5.29	* Chain [[17,18,19,20],21,16]: 3*it(17)+3*it(18)+3*it(19)+3*it(20)+2*s(17)+2*s(18)+4*s(21)+4
17.44/5.29	  Such that:aux(5) =< 2*V+4
17.44/5.29	aux(13) =< V
17.44/5.29	aux(14) =< 2*V
17.44/5.29	aux(15) =< 3*V
17.44/5.29	aux(16) =< 4*V
17.44/5.29	aux(17) =< V/2
17.44/5.29	it(19) =< aux(14)
17.44/5.29	it(17) =< aux(13)
17.44/5.29	it(18) =< aux(13)
17.44/5.29	it(19) =< aux(13)
17.44/5.29	it(20) =< aux(13)
17.44/5.29	it(20) =< aux(5)
17.44/5.29	s(22) =< aux(5)
17.44/5.29	it(20) =< aux(14)
17.44/5.29	s(22) =< aux(14)
17.44/5.29	it(18) =< aux(15)
17.44/5.29	it(19) =< aux(15)
17.44/5.29	s(18) =< aux(15)
17.44/5.29	it(20) =< aux(15)
17.44/5.29	it(18) =< aux(16)
17.44/5.29	it(19) =< aux(16)
17.44/5.29	s(17) =< aux(16)
17.44/5.29	it(20) =< aux(16)
17.44/5.29	it(17) =< aux(17)
17.44/5.29	it(18) =< aux(17)
17.44/5.29	s(21) =< s(22)
17.44/5.29	
17.44/5.29	  with precondition: [Out>=4,V+8>=2*Out,V+2>=Out] 
17.44/5.29	
17.44/5.29	* Chain [21,16]: 4
17.44/5.29	  with precondition: [V=1,Out=3] 
17.44/5.29	
17.44/5.29	* Chain [16]: 1
17.44/5.29	  with precondition: [V=0,Out=1] 
17.44/5.29	
17.44/5.29	
17.44/5.29	#### Cost of chains of start(V):
17.44/5.29	* Chain [24]: 1
17.44/5.29	  with precondition: [V=0] 
17.44/5.29	
17.44/5.29	* Chain [23]: 4
17.44/5.29	  with precondition: [V=1] 
17.44/5.29	
17.44/5.29	* Chain [22]: 4*s(25)+3*s(35)+3*s(36)+3*s(37)+3*s(38)+2*s(40)+2*s(41)+4*s(42)+4
17.44/5.29	  Such that:s(31) =< 2*V
17.44/5.29	s(29) =< 2*V+4
17.44/5.29	s(32) =< 3*V
17.44/5.29	s(33) =< 4*V
17.44/5.29	s(34) =< V/2
17.44/5.29	aux(18) =< V
17.44/5.29	s(25) =< aux(18)
17.44/5.29	s(35) =< s(31)
17.44/5.29	s(36) =< aux(18)
17.44/5.29	s(37) =< aux(18)
17.44/5.29	s(35) =< aux(18)
17.44/5.29	s(38) =< aux(18)
17.44/5.29	s(38) =< s(29)
17.44/5.29	s(39) =< s(29)
17.44/5.29	s(38) =< s(31)
17.44/5.29	s(39) =< s(31)
17.44/5.29	s(37) =< s(32)
17.44/5.29	s(35) =< s(32)
17.44/5.29	s(40) =< s(32)
17.44/5.29	s(38) =< s(32)
17.44/5.29	s(37) =< s(33)
17.44/5.29	s(35) =< s(33)
17.44/5.29	s(41) =< s(33)
17.44/5.29	s(38) =< s(33)
17.44/5.29	s(36) =< s(34)
17.44/5.29	s(37) =< s(34)
17.44/5.29	s(42) =< s(39)
17.44/5.29	
17.44/5.29	  with precondition: [V>=2] 
17.44/5.29	
17.44/5.29	
17.44/5.29	Closed-form bounds of start(V): 
17.44/5.29	-------------------------------------
17.44/5.29	* Chain [24] with precondition: [V=0] 
17.44/5.29	    - Upper bound: 1 
17.44/5.29	    - Complexity: constant 
17.44/5.29	* Chain [23] with precondition: [V=1] 
17.44/5.29	    - Upper bound: 4 
17.44/5.29	    - Complexity: constant 
17.44/5.29	* Chain [22] with precondition: [V>=2] 
17.44/5.29	    - Upper bound: 41*V+20 
17.44/5.29	    - Complexity: n 
17.44/5.29	
17.44/5.29	### Maximum cost of start(V): 41*V+20 
17.44/5.29	Asymptotic class: n 
17.44/5.29	* Total analysis performed in 228 ms.
17.44/5.29	
17.44/5.29	
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(10)
17.44/5.29	BOUNDS(1, n^1)
17.44/5.29	
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
17.44/5.29	Transformed a relative TRS into a decreasing-loop problem.
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(12)
17.44/5.29	Obligation:
17.44/5.29	Analyzing the following TRS for decreasing loops:
17.44/5.29	
17.44/5.29	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
17.44/5.29	
17.44/5.29	
17.44/5.29	The TRS R consists of the following rules:
17.44/5.29	
17.44/5.29	   half(0) -> 0
17.44/5.29	   half(s(0)) -> 0
17.44/5.29	   half(s(s(x))) -> s(half(x))
17.44/5.29	   lastbit(0) -> 0
17.44/5.29	   lastbit(s(0)) -> s(0)
17.44/5.29	   lastbit(s(s(x))) -> lastbit(x)
17.44/5.29	   conv(0) -> cons(nil, 0)
17.44/5.29	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
17.44/5.29	
17.44/5.29	S is empty.
17.44/5.29	Rewrite Strategy: INNERMOST
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(13) DecreasingLoopProof (LOWER BOUND(ID))
17.44/5.29	The following loop(s) give(s) rise to the lower bound Omega(n^1):
17.44/5.29	
17.44/5.29	The rewrite sequence
17.44/5.29	
17.44/5.29	lastbit(s(s(x))) ->^+ lastbit(x)
17.44/5.29	
17.44/5.29	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
17.44/5.29	
17.44/5.29	The pumping substitution is [x / s(s(x))].
17.44/5.29	
17.44/5.29	The result substitution is [ ].
17.44/5.29	
17.44/5.29	
17.44/5.29	
17.44/5.29	
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(14)
17.44/5.29	Complex Obligation (BEST)
17.44/5.29	
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(15)
17.44/5.29	Obligation:
17.44/5.29	Proved the lower bound n^1 for the following obligation:
17.44/5.29	
17.44/5.29	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
17.44/5.29	
17.44/5.29	
17.44/5.29	The TRS R consists of the following rules:
17.44/5.29	
17.44/5.29	   half(0) -> 0
17.44/5.29	   half(s(0)) -> 0
17.44/5.29	   half(s(s(x))) -> s(half(x))
17.44/5.29	   lastbit(0) -> 0
17.44/5.29	   lastbit(s(0)) -> s(0)
17.44/5.29	   lastbit(s(s(x))) -> lastbit(x)
17.44/5.29	   conv(0) -> cons(nil, 0)
17.44/5.29	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
17.44/5.29	
17.44/5.29	S is empty.
17.44/5.29	Rewrite Strategy: INNERMOST
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(16) LowerBoundPropagationProof (FINISHED)
17.44/5.29	Propagated lower bound.
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(17)
17.44/5.29	BOUNDS(n^1, INF)
17.44/5.29	
17.44/5.29	----------------------------------------
17.44/5.29	
17.44/5.29	(18)
17.44/5.29	Obligation:
17.44/5.29	Analyzing the following TRS for decreasing loops:
17.44/5.29	
17.44/5.29	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
17.44/5.29	
17.44/5.29	
17.44/5.29	The TRS R consists of the following rules:
17.44/5.29	
17.44/5.29	   half(0) -> 0
17.44/5.29	   half(s(0)) -> 0
17.44/5.29	   half(s(s(x))) -> s(half(x))
17.44/5.29	   lastbit(0) -> 0
17.44/5.29	   lastbit(s(0)) -> s(0)
17.44/5.29	   lastbit(s(s(x))) -> lastbit(x)
17.44/5.29	   conv(0) -> cons(nil, 0)
17.44/5.29	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
17.44/5.29	
17.44/5.29	S is empty.
17.44/5.29	Rewrite Strategy: INNERMOST
17.67/5.33	EOF