1078.27/291.49	WORST_CASE(Omega(n^1), O(n^2))
1078.27/291.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1078.27/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1078.27/291.51	
1078.27/291.51	(0) CpxTRS
1078.27/291.51	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1078.27/291.51	(2) CpxWeightedTrs
1078.27/291.51	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1078.27/291.51	    (4) CpxTypedWeightedTrs
1078.27/291.51	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
1078.27/291.51	    (6) CpxTypedWeightedCompleteTrs
1078.27/291.51	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms]
1078.27/291.51	    (8) CpxRNTS
1078.27/291.51	    (9) CompleteCoflocoProof [FINISHED, 493 ms]
1078.27/291.51	    (10) BOUNDS(1, n^2)
1078.27/291.51	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1078.27/291.51	(12) TRS for Loop Detection
1078.27/291.51	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1078.27/291.51	    (14) BEST
1078.27/291.51	        (15) proven lower bound
1078.27/291.51	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1078.27/291.51	            (17) BOUNDS(n^1, INF)
1078.27/291.51	        (18) TRS for Loop Detection
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(0)
1078.27/291.51	Obligation:
1078.27/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	The TRS R consists of the following rules:
1078.27/291.51	
1078.27/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
1078.27/291.51	   round(0) -> 0
1078.27/291.51	   round(s(0)) -> s(s(0))
1078.27/291.51	   round(s(s(x))) -> s(s(round(x)))
1078.27/291.51	   gt(0, v) -> false
1078.27/291.51	   gt(s(u), 0) -> true
1078.27/291.51	   gt(s(u), s(v)) -> gt(u, v)
1078.27/291.51	
1078.27/291.51	S is empty.
1078.27/291.51	Rewrite Strategy: INNERMOST
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1078.27/291.51	Transformed relative TRS to weighted TRS
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(2)
1078.27/291.51	Obligation:
1078.27/291.51	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	The TRS R consists of the following rules:
1078.27/291.51	
1078.27/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y))) [1]
1078.27/291.51	   round(0) -> 0 [1]
1078.27/291.51	   round(s(0)) -> s(s(0)) [1]
1078.27/291.51	   round(s(s(x))) -> s(s(round(x))) [1]
1078.27/291.51	   gt(0, v) -> false [1]
1078.27/291.51	   gt(s(u), 0) -> true [1]
1078.27/291.51	   gt(s(u), s(v)) -> gt(u, v) [1]
1078.27/291.51	
1078.27/291.51	Rewrite Strategy: INNERMOST
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1078.27/291.51	Infered types.
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(4)
1078.27/291.51	Obligation:
1078.27/291.51	Runtime Complexity Weighted TRS with Types.
1078.27/291.51	The TRS R consists of the following rules:
1078.27/291.51	
1078.27/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y))) [1]
1078.27/291.51	   round(0) -> 0 [1]
1078.27/291.51	   round(s(0)) -> s(s(0)) [1]
1078.27/291.51	   round(s(s(x))) -> s(s(round(x))) [1]
1078.27/291.51	   gt(0, v) -> false [1]
1078.27/291.51	   gt(s(u), 0) -> true [1]
1078.27/291.51	   gt(s(u), s(v)) -> gt(u, v) [1]
1078.27/291.51	
1078.27/291.51	The TRS has the following type information:
1078.27/291.51	f :: true:false -> s:0 -> s:0 -> f
1078.27/291.51	true :: true:false
1078.27/291.51	gt :: s:0 -> s:0 -> true:false
1078.27/291.51	round :: s:0 -> s:0
1078.27/291.51	s :: s:0 -> s:0
1078.27/291.51	0 :: s:0
1078.27/291.51	false :: true:false
1078.27/291.51	
1078.27/291.51	Rewrite Strategy: INNERMOST
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(5) CompletionProof (UPPER BOUND(ID))
1078.27/291.51	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
1078.27/291.51	
1078.27/291.51	   f(v0, v1, v2) -> null_f [0]
1078.27/291.51	
1078.27/291.51	And the following fresh constants:    null_f
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(6)
1078.27/291.51	Obligation:
1078.27/291.51	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
1078.27/291.51	
1078.27/291.51	Runtime Complexity Weighted TRS with Types.
1078.27/291.51	The TRS R consists of the following rules:
1078.27/291.51	
1078.27/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y))) [1]
1078.27/291.51	   round(0) -> 0 [1]
1078.27/291.51	   round(s(0)) -> s(s(0)) [1]
1078.27/291.51	   round(s(s(x))) -> s(s(round(x))) [1]
1078.27/291.51	   gt(0, v) -> false [1]
1078.27/291.51	   gt(s(u), 0) -> true [1]
1078.27/291.51	   gt(s(u), s(v)) -> gt(u, v) [1]
1078.27/291.51	   f(v0, v1, v2) -> null_f [0]
1078.27/291.51	
1078.27/291.51	The TRS has the following type information:
1078.27/291.51	f :: true:false -> s:0 -> s:0 -> null_f
1078.27/291.51	true :: true:false
1078.27/291.51	gt :: s:0 -> s:0 -> true:false
1078.27/291.51	round :: s:0 -> s:0
1078.27/291.51	s :: s:0 -> s:0
1078.27/291.51	0 :: s:0
1078.27/291.51	false :: true:false
1078.27/291.51	null_f :: null_f
1078.27/291.51	
1078.27/291.51	Rewrite Strategy: INNERMOST
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1078.27/291.51	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1078.27/291.51	The constant constructors are abstracted as follows:
1078.27/291.51	
1078.27/291.51	   true => 1
1078.27/291.51	   0 => 0
1078.27/291.51	   false => 0
1078.27/291.51	   null_f => 0
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(8)
1078.27/291.51	Obligation:
1078.27/291.51	Complexity RNTS consisting of the following rules:
1078.27/291.51	
1078.27/291.51	   f(z, z', z'') -{ 1 }-> f(gt(x, y), x, round(1 + y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
1078.27/291.51	   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
1078.27/291.51	   gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
1078.27/291.51	   gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0
1078.27/291.51	   gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0
1078.27/291.51	   round(z) -{ 1 }-> 0 :|: z = 0
1078.27/291.51	   round(z) -{ 1 }-> 1 + (1 + round(x)) :|: x >= 0, z = 1 + (1 + x)
1078.27/291.51	   round(z) -{ 1 }-> 1 + (1 + 0) :|: z = 1 + 0
1078.27/291.51	
1078.27/291.51	Only complete derivations are relevant for the runtime complexity.
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(9) CompleteCoflocoProof (FINISHED)
1078.27/291.51	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
1078.27/291.51	
1078.27/291.51	eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
1078.27/291.51	eq(start(V1, V, V2),0,[round(V1, Out)],[V1 >= 0]).
1078.27/291.51	eq(start(V1, V, V2),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
1078.27/291.51	eq(f(V1, V, V2, Out),1,[gt(V4, V3, Ret0),round(1 + V3, Ret2),f(Ret0, V4, Ret2, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
1078.27/291.51	eq(round(V1, Out),1,[],[Out = 0,V1 = 0]).
1078.27/291.51	eq(round(V1, Out),1,[],[Out = 2,V1 = 1]).
1078.27/291.51	eq(round(V1, Out),1,[round(V5, Ret11)],[Out = 2 + Ret11,V5 >= 0,V1 = 2 + V5]).
1078.27/291.51	eq(gt(V1, V, Out),1,[],[Out = 0,V6 >= 0,V = V6,V1 = 0]).
1078.27/291.51	eq(gt(V1, V, Out),1,[],[Out = 1,V1 = 1 + V7,V = 0,V7 >= 0]).
1078.27/291.51	eq(gt(V1, V, Out),1,[gt(V8, V9, Ret1)],[Out = Ret1,V9 >= 0,V = 1 + V9,V1 = 1 + V8,V8 >= 0]).
1078.27/291.51	eq(f(V1, V, V2, Out),0,[],[Out = 0,V11 >= 0,V2 = V12,V10 >= 0,V1 = V11,V = V10,V12 >= 0]).
1078.27/291.51	input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
1078.27/291.51	input_output_vars(round(V1,Out),[V1],[Out]).
1078.27/291.51	input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	CoFloCo proof output:
1078.27/291.51	Preprocessing Cost Relations
1078.27/291.51	=====================================
1078.27/291.51	
1078.27/291.51	#### Computed strongly connected components 
1078.27/291.51	0. recursive  : [gt/3]
1078.27/291.51	1. recursive  : [round/2]
1078.27/291.51	2. recursive  : [f/4]
1078.27/291.51	3. non_recursive  : [start/3]
1078.27/291.51	
1078.27/291.51	#### Obtained direct recursion through partial evaluation 
1078.27/291.51	0. SCC is partially evaluated into gt/3
1078.27/291.51	1. SCC is partially evaluated into round/2
1078.27/291.51	2. SCC is partially evaluated into f/4
1078.27/291.51	3. SCC is partially evaluated into start/3
1078.27/291.51	
1078.27/291.51	Control-Flow Refinement of Cost Relations
1078.27/291.51	=====================================
1078.27/291.51	
1078.27/291.51	### Specialization of cost equations gt/3 
1078.27/291.51	* CE 11 is refined into CE [12] 
1078.27/291.51	* CE 10 is refined into CE [13] 
1078.27/291.51	* CE 9 is refined into CE [14] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	### Cost equations --> "Loop" of gt/3 
1078.27/291.51	* CEs [13] --> Loop 10 
1078.27/291.51	* CEs [14] --> Loop 11 
1078.27/291.51	* CEs [12] --> Loop 12 
1078.27/291.51	
1078.27/291.51	### Ranking functions of CR gt(V1,V,Out) 
1078.27/291.51	* RF of phase [12]: [V,V1]
1078.27/291.51	
1078.27/291.51	#### Partial ranking functions of CR gt(V1,V,Out) 
1078.27/291.51	* Partial RF of phase [12]:
1078.27/291.51	  - RF of loop [12:1]:
1078.27/291.51	    V
1078.27/291.51	    V1
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	### Specialization of cost equations round/2 
1078.27/291.51	* CE 8 is refined into CE [15] 
1078.27/291.51	* CE 7 is refined into CE [16] 
1078.27/291.51	* CE 6 is refined into CE [17] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	### Cost equations --> "Loop" of round/2 
1078.27/291.51	* CEs [16] --> Loop 13 
1078.27/291.51	* CEs [17] --> Loop 14 
1078.27/291.51	* CEs [15] --> Loop 15 
1078.27/291.51	
1078.27/291.51	### Ranking functions of CR round(V1,Out) 
1078.27/291.51	* RF of phase [15]: [V1-1]
1078.27/291.51	
1078.27/291.51	#### Partial ranking functions of CR round(V1,Out) 
1078.27/291.51	* Partial RF of phase [15]:
1078.27/291.51	  - RF of loop [15:1]:
1078.27/291.51	    V1-1
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	### Specialization of cost equations f/4 
1078.27/291.51	* CE 5 is refined into CE [18] 
1078.27/291.51	* CE 4 is refined into CE [19,20,21,22,23,24,25,26] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	### Cost equations --> "Loop" of f/4 
1078.27/291.51	* CEs [25] --> Loop 16 
1078.27/291.51	* CEs [26] --> Loop 17 
1078.27/291.51	* CEs [24] --> Loop 18 
1078.27/291.51	* CEs [23] --> Loop 19 
1078.27/291.51	* CEs [22] --> Loop 20 
1078.27/291.51	* CEs [21] --> Loop 21 
1078.27/291.51	* CEs [20] --> Loop 22 
1078.27/291.51	* CEs [19] --> Loop 23 
1078.27/291.51	* CEs [18] --> Loop 24 
1078.27/291.51	
1078.27/291.51	### Ranking functions of CR f(V1,V,V2,Out) 
1078.27/291.51	* RF of phase [16,17]: [V-V2]
1078.27/291.51	
1078.27/291.51	#### Partial ranking functions of CR f(V1,V,V2,Out) 
1078.27/291.51	* Partial RF of phase [16,17]:
1078.27/291.51	  - RF of loop [16:1]:
1078.27/291.51	    V/2-V2/2
1078.27/291.51	  - RF of loop [17:1]:
1078.27/291.51	    V-V2
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	### Specialization of cost equations start/3 
1078.27/291.51	* CE 1 is refined into CE [27,28,29,30,31] 
1078.27/291.51	* CE 2 is refined into CE [32,33,34,35] 
1078.27/291.51	* CE 3 is refined into CE [36,37,38,39] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	### Cost equations --> "Loop" of start/3 
1078.27/291.51	* CEs [34,35,39] --> Loop 25 
1078.27/291.51	* CEs [38] --> Loop 26 
1078.27/291.51	* CEs [27] --> Loop 27 
1078.27/291.51	* CEs [37] --> Loop 28 
1078.27/291.51	* CEs [28,29,30,31,33] --> Loop 29 
1078.27/291.51	* CEs [32,36] --> Loop 30 
1078.27/291.51	
1078.27/291.51	### Ranking functions of CR start(V1,V,V2) 
1078.27/291.51	
1078.27/291.51	#### Partial ranking functions of CR start(V1,V,V2) 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	Computing Bounds
1078.27/291.51	=====================================
1078.27/291.51	
1078.27/291.51	#### Cost of chains of gt(V1,V,Out):
1078.27/291.51	* Chain [[12],11]: 1*it(12)+1
1078.27/291.51	  Such that:it(12) =< V1
1078.27/291.51	
1078.27/291.51	  with precondition: [Out=0,V1>=1,V>=V1] 
1078.27/291.51	
1078.27/291.51	* Chain [[12],10]: 1*it(12)+1
1078.27/291.51	  Such that:it(12) =< V
1078.27/291.51	
1078.27/291.51	  with precondition: [Out=1,V>=1,V1>=V+1] 
1078.27/291.51	
1078.27/291.51	* Chain [11]: 1
1078.27/291.51	  with precondition: [V1=0,Out=0,V>=0] 
1078.27/291.51	
1078.27/291.51	* Chain [10]: 1
1078.27/291.51	  with precondition: [V=0,Out=1,V1>=1] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	#### Cost of chains of round(V1,Out):
1078.27/291.51	* Chain [[15],14]: 1*it(15)+1
1078.27/291.51	  Such that:it(15) =< Out
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=Out,V1>=2] 
1078.27/291.51	
1078.27/291.51	* Chain [[15],13]: 1*it(15)+1
1078.27/291.51	  Such that:it(15) =< Out
1078.27/291.51	
1078.27/291.51	  with precondition: [V1+1=Out,V1>=3] 
1078.27/291.51	
1078.27/291.51	* Chain [14]: 1
1078.27/291.51	  with precondition: [V1=0,Out=0] 
1078.27/291.51	
1078.27/291.51	* Chain [13]: 1
1078.27/291.51	  with precondition: [V1=1,Out=2] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	#### Cost of chains of f(V1,V,V2,Out):
1078.27/291.51	* Chain [[16,17],24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+0
1078.27/291.51	  Such that:aux(3) =< V+1
1078.27/291.51	aux(5) =< V-V2
1078.27/291.51	aux(6) =< V-V2+1
1078.27/291.51	it(16) =< V/2-V2/2
1078.27/291.51	it(16) =< aux(5)
1078.27/291.51	it(17) =< aux(5)
1078.27/291.51	it(16) =< aux(6)
1078.27/291.51	it(17) =< aux(6)
1078.27/291.51	aux(4) =< aux(3)-1
1078.27/291.51	s(10) =< it(16)*aux(3)
1078.27/291.51	s(12) =< it(17)*aux(4)
1078.27/291.51	s(11) =< s(12)
1078.27/291.51	s(9) =< s(10)
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
1078.27/291.51	
1078.27/291.51	* Chain [[16,17],19,24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+1*s(13)+1*s(14)+3
1078.27/291.51	  Such that:s(13) =< V
1078.27/291.51	aux(3) =< V+1
1078.27/291.51	s(14) =< V+3
1078.27/291.51	aux(5) =< V-V2
1078.27/291.51	aux(6) =< V-V2+1
1078.27/291.51	it(16) =< V/2-V2/2
1078.27/291.51	it(16) =< aux(5)
1078.27/291.51	it(17) =< aux(5)
1078.27/291.51	it(16) =< aux(6)
1078.27/291.51	it(17) =< aux(6)
1078.27/291.51	aux(4) =< aux(3)-1
1078.27/291.51	s(10) =< it(16)*aux(3)
1078.27/291.51	s(12) =< it(17)*aux(4)
1078.27/291.51	s(11) =< s(12)
1078.27/291.51	s(9) =< s(10)
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
1078.27/291.51	
1078.27/291.51	* Chain [[16,17],18,24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+1*s(15)+1*s(16)+3
1078.27/291.51	  Such that:s(15) =< V
1078.27/291.51	aux(3) =< V+1
1078.27/291.51	s(16) =< V+2
1078.27/291.51	aux(5) =< V-V2
1078.27/291.51	aux(6) =< V-V2+1
1078.27/291.51	it(16) =< V/2-V2/2
1078.27/291.51	it(16) =< aux(5)
1078.27/291.51	it(17) =< aux(5)
1078.27/291.51	it(16) =< aux(6)
1078.27/291.51	it(17) =< aux(6)
1078.27/291.51	aux(4) =< aux(3)-1
1078.27/291.51	s(10) =< it(16)*aux(3)
1078.27/291.51	s(12) =< it(17)*aux(4)
1078.27/291.51	s(11) =< s(12)
1078.27/291.51	s(9) =< s(10)
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
1078.27/291.51	
1078.27/291.51	* Chain [24]: 0
1078.27/291.51	  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 
1078.27/291.51	
1078.27/291.51	* Chain [23,24]: 3
1078.27/291.51	  with precondition: [V1=1,V=0,V2=0,Out=0] 
1078.27/291.51	
1078.27/291.51	* Chain [22,24]: 1*s(17)+3
1078.27/291.51	  Such that:s(17) =< V2+2
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,V=0,Out=0,V2>=2] 
1078.27/291.51	
1078.27/291.51	* Chain [21,24]: 1*s(18)+3
1078.27/291.51	  Such that:s(18) =< V2+1
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,V=0,Out=0,V2>=1] 
1078.27/291.51	
1078.27/291.51	* Chain [20,[16,17],24]: 3*it(16)+3*it(17)+2*s(9)+2*s(11)+3
1078.27/291.51	  Such that:aux(3) =< V+1
1078.27/291.51	it(16) =< V/2
1078.27/291.51	aux(7) =< V
1078.27/291.51	it(16) =< aux(7)
1078.27/291.51	it(17) =< aux(7)
1078.27/291.51	aux(4) =< aux(3)-1
1078.27/291.51	s(10) =< it(16)*aux(3)
1078.27/291.51	s(12) =< it(17)*aux(4)
1078.27/291.51	s(11) =< s(12)
1078.27/291.51	s(9) =< s(10)
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,V2=0,Out=0,V>=3] 
1078.27/291.51	
1078.27/291.51	* Chain [20,[16,17],19,24]: 3*it(16)+4*it(17)+2*s(9)+2*s(11)+1*s(14)+6
1078.27/291.51	  Such that:aux(3) =< V+1
1078.27/291.51	s(14) =< V+3
1078.27/291.51	it(16) =< V/2
1078.27/291.51	aux(8) =< V
1078.27/291.51	it(17) =< aux(8)
1078.27/291.51	it(16) =< aux(8)
1078.27/291.51	aux(4) =< aux(3)-1
1078.27/291.51	s(10) =< it(16)*aux(3)
1078.27/291.51	s(12) =< it(17)*aux(4)
1078.27/291.51	s(11) =< s(12)
1078.27/291.51	s(9) =< s(10)
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,V2=0,Out=0,V>=3] 
1078.27/291.51	
1078.27/291.51	* Chain [20,[16,17],18,24]: 3*it(16)+4*it(17)+2*s(9)+2*s(11)+1*s(16)+6
1078.27/291.51	  Such that:aux(3) =< V+1
1078.27/291.51	s(16) =< V+2
1078.27/291.51	it(16) =< V/2
1078.27/291.51	aux(9) =< V
1078.27/291.51	it(17) =< aux(9)
1078.27/291.51	it(16) =< aux(9)
1078.27/291.51	aux(4) =< aux(3)-1
1078.27/291.51	s(10) =< it(16)*aux(3)
1078.27/291.51	s(12) =< it(17)*aux(4)
1078.27/291.51	s(11) =< s(12)
1078.27/291.51	s(9) =< s(10)
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,V2=0,Out=0,V>=3] 
1078.27/291.51	
1078.27/291.51	* Chain [20,24]: 3
1078.27/291.51	  with precondition: [V1=1,V2=0,Out=0,V>=1] 
1078.27/291.51	
1078.27/291.51	* Chain [20,19,24]: 1*s(13)+1*s(14)+6
1078.27/291.51	  Such that:s(14) =< 4
1078.27/291.51	s(13) =< V
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,V2=0,Out=0,2>=V,V>=1] 
1078.27/291.51	
1078.27/291.51	* Chain [20,18,24]: 1*s(15)+1*s(16)+6
1078.27/291.51	  Such that:s(16) =< 3
1078.27/291.51	s(15) =< V
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,V2=0,Out=0,2>=V,V>=1] 
1078.27/291.51	
1078.27/291.51	* Chain [19,24]: 1*s(13)+1*s(14)+3
1078.27/291.51	  Such that:s(13) =< V
1078.27/291.51	s(14) =< V2+2
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,Out=0,V>=1,V2>=2,V2>=V] 
1078.27/291.51	
1078.27/291.51	* Chain [18,24]: 1*s(15)+1*s(16)+3
1078.27/291.51	  Such that:s(15) =< V
1078.27/291.51	s(16) =< V2+1
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1,Out=0,V>=1,V2>=V] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	#### Cost of chains of start(V1,V,V2):
1078.27/291.51	* Chain [30]: 1
1078.27/291.51	  with precondition: [V1=0] 
1078.27/291.51	
1078.27/291.51	* Chain [29]: 2*s(92)+2*s(93)+1*s(94)+1*s(95)+2*s(96)+2*s(97)+17*s(101)+9*s(102)+6*s(106)+6*s(107)+9*s(120)+9*s(121)+6*s(125)+6*s(126)+6
1078.27/291.51	  Such that:s(94) =< 3
1078.27/291.51	s(95) =< 4
1078.27/291.51	s(116) =< V-V2
1078.27/291.51	s(117) =< V-V2+1
1078.27/291.51	s(100) =< V/2
1078.27/291.51	s(118) =< V/2-V2/2
1078.27/291.51	aux(19) =< V
1078.27/291.51	aux(20) =< V+1
1078.27/291.51	aux(21) =< V+2
1078.27/291.51	aux(22) =< V+3
1078.27/291.51	aux(23) =< V2+1
1078.27/291.51	aux(24) =< V2+2
1078.27/291.51	s(96) =< aux(21)
1078.27/291.51	s(97) =< aux(22)
1078.27/291.51	s(92) =< aux(23)
1078.27/291.51	s(93) =< aux(24)
1078.27/291.51	s(101) =< aux(19)
1078.27/291.51	s(102) =< s(100)
1078.27/291.51	s(102) =< aux(19)
1078.27/291.51	s(103) =< aux(20)-1
1078.27/291.51	s(104) =< s(102)*aux(20)
1078.27/291.51	s(105) =< s(101)*s(103)
1078.27/291.51	s(106) =< s(105)
1078.27/291.51	s(107) =< s(104)
1078.27/291.51	s(120) =< s(118)
1078.27/291.51	s(120) =< s(116)
1078.27/291.51	s(121) =< s(116)
1078.27/291.51	s(120) =< s(117)
1078.27/291.51	s(121) =< s(117)
1078.27/291.51	s(123) =< s(120)*aux(20)
1078.27/291.51	s(124) =< s(121)*s(103)
1078.27/291.51	s(125) =< s(124)
1078.27/291.51	s(126) =< s(123)
1078.27/291.51	
1078.27/291.51	  with precondition: [V1=1] 
1078.27/291.51	
1078.27/291.51	* Chain [28]: 1
1078.27/291.51	  with precondition: [V=0,V1>=1] 
1078.27/291.51	
1078.27/291.51	* Chain [27]: 3
1078.27/291.51	  with precondition: [V1>=0,V>=0,V2>=0] 
1078.27/291.51	
1078.27/291.51	* Chain [26]: 1*s(127)+1
1078.27/291.51	  Such that:s(127) =< V1
1078.27/291.51	
1078.27/291.51	  with precondition: [V1>=1,V>=V1] 
1078.27/291.51	
1078.27/291.51	* Chain [25]: 1*s(128)+1*s(129)+1*s(130)+1
1078.27/291.51	  Such that:s(129) =< V1
1078.27/291.51	s(128) =< V1+1
1078.27/291.51	s(130) =< V
1078.27/291.51	
1078.27/291.51	  with precondition: [V1>=2] 
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	Closed-form bounds of start(V1,V,V2): 
1078.27/291.51	-------------------------------------
1078.27/291.51	* Chain [30] with precondition: [V1=0] 
1078.27/291.51	    - Upper bound: 1 
1078.27/291.51	    - Complexity: constant 
1078.27/291.51	* Chain [29] with precondition: [V1=1] 
1078.27/291.51	    - Upper bound: nat(V)*17+13+nat(V)*6*nat(nat(V+1)+ -1)+nat(nat(V+1)+ -1)*6*nat(V-V2)+nat(V+1)*6*nat(V/2-V2/2)+nat(V+1)*6*nat(V/2)+nat(V+2)*2+nat(V+3)*2+nat(V2+1)*2+nat(V2+2)*2+nat(V-V2)*9+nat(V/2-V2/2)*9+nat(V/2)*9 
1078.27/291.51	    - Complexity: n^2 
1078.27/291.51	* Chain [28] with precondition: [V=0,V1>=1] 
1078.27/291.51	    - Upper bound: 1 
1078.27/291.51	    - Complexity: constant 
1078.27/291.51	* Chain [27] with precondition: [V1>=0,V>=0,V2>=0] 
1078.27/291.51	    - Upper bound: 3 
1078.27/291.51	    - Complexity: constant 
1078.27/291.51	* Chain [26] with precondition: [V1>=1,V>=V1] 
1078.27/291.51	    - Upper bound: V1+1 
1078.27/291.51	    - Complexity: n 
1078.27/291.51	* Chain [25] with precondition: [V1>=2] 
1078.27/291.51	    - Upper bound: V1+1+nat(V)+(V1+1) 
1078.27/291.51	    - Complexity: n 
1078.27/291.51	
1078.27/291.51	### Maximum cost of start(V1,V,V2): max([max([2,nat(V)*17+12+nat(V)*6*nat(nat(V+1)+ -1)+nat(nat(V+1)+ -1)*6*nat(V-V2)+nat(V+1)*6*nat(V/2-V2/2)+nat(V+1)*6*nat(V/2)+nat(V+2)*2+nat(V+3)*2+nat(V2+1)*2+nat(V2+2)*2+nat(V-V2)*9+nat(V/2-V2/2)*9+nat(V/2)*9]),V1+1+nat(V)+V1])+1 
1078.27/291.51	Asymptotic class: n^2 
1078.27/291.51	* Total analysis performed in 404 ms.
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(10)
1078.27/291.51	BOUNDS(1, n^2)
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1078.27/291.51	Transformed a relative TRS into a decreasing-loop problem.
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(12)
1078.27/291.51	Obligation:
1078.27/291.51	Analyzing the following TRS for decreasing loops:
1078.27/291.51	
1078.27/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	The TRS R consists of the following rules:
1078.27/291.51	
1078.27/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
1078.27/291.51	   round(0) -> 0
1078.27/291.51	   round(s(0)) -> s(s(0))
1078.27/291.51	   round(s(s(x))) -> s(s(round(x)))
1078.27/291.51	   gt(0, v) -> false
1078.27/291.51	   gt(s(u), 0) -> true
1078.27/291.51	   gt(s(u), s(v)) -> gt(u, v)
1078.27/291.51	
1078.27/291.51	S is empty.
1078.27/291.51	Rewrite Strategy: INNERMOST
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(13) DecreasingLoopProof (LOWER BOUND(ID))
1078.27/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1078.27/291.51	
1078.27/291.51	The rewrite sequence
1078.27/291.51	
1078.27/291.51	gt(s(u), s(v)) ->^+ gt(u, v)
1078.27/291.51	
1078.27/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1078.27/291.51	
1078.27/291.51	The pumping substitution is [u / s(u), v / s(v)].
1078.27/291.51	
1078.27/291.51	The result substitution is [ ].
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(14)
1078.27/291.51	Complex Obligation (BEST)
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(15)
1078.27/291.51	Obligation:
1078.27/291.51	Proved the lower bound n^1 for the following obligation:
1078.27/291.51	
1078.27/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	The TRS R consists of the following rules:
1078.27/291.51	
1078.27/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
1078.27/291.51	   round(0) -> 0
1078.27/291.51	   round(s(0)) -> s(s(0))
1078.27/291.51	   round(s(s(x))) -> s(s(round(x)))
1078.27/291.51	   gt(0, v) -> false
1078.27/291.51	   gt(s(u), 0) -> true
1078.27/291.51	   gt(s(u), s(v)) -> gt(u, v)
1078.27/291.51	
1078.27/291.51	S is empty.
1078.27/291.51	Rewrite Strategy: INNERMOST
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(16) LowerBoundPropagationProof (FINISHED)
1078.27/291.51	Propagated lower bound.
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(17)
1078.27/291.51	BOUNDS(n^1, INF)
1078.27/291.51	
1078.27/291.51	----------------------------------------
1078.27/291.51	
1078.27/291.51	(18)
1078.27/291.51	Obligation:
1078.27/291.51	Analyzing the following TRS for decreasing loops:
1078.27/291.51	
1078.27/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1078.27/291.51	
1078.27/291.51	
1078.27/291.51	The TRS R consists of the following rules:
1078.27/291.51	
1078.27/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
1078.27/291.51	   round(0) -> 0
1078.27/291.51	   round(s(0)) -> s(s(0))
1078.27/291.51	   round(s(s(x))) -> s(s(round(x)))
1078.27/291.51	   gt(0, v) -> false
1078.27/291.51	   gt(s(u), 0) -> true
1078.27/291.51	   gt(s(u), s(v)) -> gt(u, v)
1078.27/291.51	
1078.27/291.51	S is empty.
1078.27/291.51	Rewrite Strategy: INNERMOST
1078.49/291.58	EOF