1110.88/291.48	WORST_CASE(Omega(n^1), O(n^2))
1110.88/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1110.88/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1110.88/291.50	
1110.88/291.50	(0) CpxTRS
1110.88/291.50	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1110.88/291.50	(2) CpxWeightedTrs
1110.88/291.50	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1110.88/291.50	    (4) CpxTypedWeightedTrs
1110.88/291.50	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
1110.88/291.50	    (6) CpxTypedWeightedCompleteTrs
1110.88/291.50	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1110.88/291.50	    (8) CpxRNTS
1110.88/291.50	    (9) CompleteCoflocoProof [FINISHED, 414 ms]
1110.88/291.50	    (10) BOUNDS(1, n^2)
1110.88/291.50	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1110.88/291.50	(12) TRS for Loop Detection
1110.88/291.50	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1110.88/291.50	    (14) BEST
1110.88/291.50	        (15) proven lower bound
1110.88/291.50	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1110.88/291.50	            (17) BOUNDS(n^1, INF)
1110.88/291.50	        (18) TRS for Loop Detection
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(0)
1110.88/291.50	Obligation:
1110.88/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	The TRS R consists of the following rules:
1110.88/291.50	
1110.88/291.50	   gt(0, y) -> false
1110.88/291.50	   gt(s(x), 0) -> true
1110.88/291.50	   gt(s(x), s(y)) -> gt(x, y)
1110.88/291.50	   plus(0, y) -> y
1110.88/291.50	   plus(s(x), y) -> s(plus(x, y))
1110.88/291.50	   double(0) -> 0
1110.88/291.50	   double(s(x)) -> s(s(double(x)))
1110.88/291.50	   average(x, y) -> aver(plus(x, y), 0)
1110.88/291.50	   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
1110.88/291.50	   if(true, sum, z) -> aver(sum, s(z))
1110.88/291.50	   if(false, sum, z) -> z
1110.88/291.50	
1110.88/291.50	S is empty.
1110.88/291.50	Rewrite Strategy: INNERMOST
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1110.88/291.50	Transformed relative TRS to weighted TRS
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(2)
1110.88/291.50	Obligation:
1110.88/291.50	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	The TRS R consists of the following rules:
1110.88/291.50	
1110.88/291.50	   gt(0, y) -> false [1]
1110.88/291.50	   gt(s(x), 0) -> true [1]
1110.88/291.50	   gt(s(x), s(y)) -> gt(x, y) [1]
1110.88/291.50	   plus(0, y) -> y [1]
1110.88/291.50	   plus(s(x), y) -> s(plus(x, y)) [1]
1110.88/291.50	   double(0) -> 0 [1]
1110.88/291.50	   double(s(x)) -> s(s(double(x))) [1]
1110.88/291.50	   average(x, y) -> aver(plus(x, y), 0) [1]
1110.88/291.50	   aver(sum, z) -> if(gt(sum, double(z)), sum, z) [1]
1110.88/291.50	   if(true, sum, z) -> aver(sum, s(z)) [1]
1110.88/291.50	   if(false, sum, z) -> z [1]
1110.88/291.50	
1110.88/291.50	Rewrite Strategy: INNERMOST
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1110.88/291.50	Infered types.
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(4)
1110.88/291.50	Obligation:
1110.88/291.50	Runtime Complexity Weighted TRS with Types.
1110.88/291.50	The TRS R consists of the following rules:
1110.88/291.50	
1110.88/291.50	   gt(0, y) -> false [1]
1110.88/291.50	   gt(s(x), 0) -> true [1]
1110.88/291.50	   gt(s(x), s(y)) -> gt(x, y) [1]
1110.88/291.50	   plus(0, y) -> y [1]
1110.88/291.50	   plus(s(x), y) -> s(plus(x, y)) [1]
1110.88/291.50	   double(0) -> 0 [1]
1110.88/291.50	   double(s(x)) -> s(s(double(x))) [1]
1110.88/291.50	   average(x, y) -> aver(plus(x, y), 0) [1]
1110.88/291.50	   aver(sum, z) -> if(gt(sum, double(z)), sum, z) [1]
1110.88/291.50	   if(true, sum, z) -> aver(sum, s(z)) [1]
1110.88/291.50	   if(false, sum, z) -> z [1]
1110.88/291.50	
1110.88/291.50	The TRS has the following type information:
1110.88/291.50	gt :: 0:s -> 0:s -> false:true
1110.88/291.50	0 :: 0:s
1110.88/291.50	false :: false:true
1110.88/291.50	s :: 0:s -> 0:s
1110.88/291.50	true :: false:true
1110.88/291.50	plus :: 0:s -> 0:s -> 0:s
1110.88/291.50	double :: 0:s -> 0:s
1110.88/291.50	average :: 0:s -> 0:s -> 0:s
1110.88/291.50	aver :: 0:s -> 0:s -> 0:s
1110.88/291.50	if :: false:true -> 0:s -> 0:s -> 0:s
1110.88/291.50	
1110.88/291.50	Rewrite Strategy: INNERMOST
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(5) CompletionProof (UPPER BOUND(ID))
1110.88/291.50	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
1110.88/291.50	none
1110.88/291.50	
1110.88/291.50	And the following fresh constants: none
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(6)
1110.88/291.50	Obligation:
1110.88/291.50	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
1110.88/291.50	
1110.88/291.50	Runtime Complexity Weighted TRS with Types.
1110.88/291.50	The TRS R consists of the following rules:
1110.88/291.50	
1110.88/291.50	   gt(0, y) -> false [1]
1110.88/291.50	   gt(s(x), 0) -> true [1]
1110.88/291.50	   gt(s(x), s(y)) -> gt(x, y) [1]
1110.88/291.50	   plus(0, y) -> y [1]
1110.88/291.50	   plus(s(x), y) -> s(plus(x, y)) [1]
1110.88/291.50	   double(0) -> 0 [1]
1110.88/291.50	   double(s(x)) -> s(s(double(x))) [1]
1110.88/291.50	   average(x, y) -> aver(plus(x, y), 0) [1]
1110.88/291.50	   aver(sum, z) -> if(gt(sum, double(z)), sum, z) [1]
1110.88/291.50	   if(true, sum, z) -> aver(sum, s(z)) [1]
1110.88/291.50	   if(false, sum, z) -> z [1]
1110.88/291.50	
1110.88/291.50	The TRS has the following type information:
1110.88/291.50	gt :: 0:s -> 0:s -> false:true
1110.88/291.50	0 :: 0:s
1110.88/291.50	false :: false:true
1110.88/291.50	s :: 0:s -> 0:s
1110.88/291.50	true :: false:true
1110.88/291.50	plus :: 0:s -> 0:s -> 0:s
1110.88/291.50	double :: 0:s -> 0:s
1110.88/291.50	average :: 0:s -> 0:s -> 0:s
1110.88/291.50	aver :: 0:s -> 0:s -> 0:s
1110.88/291.50	if :: false:true -> 0:s -> 0:s -> 0:s
1110.88/291.50	
1110.88/291.50	Rewrite Strategy: INNERMOST
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1110.88/291.50	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1110.88/291.50	The constant constructors are abstracted as follows:
1110.88/291.50	
1110.88/291.50	   0 => 0
1110.88/291.50	   false => 0
1110.88/291.50	   true => 1
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(8)
1110.88/291.50	Obligation:
1110.88/291.50	Complexity RNTS consisting of the following rules:
1110.88/291.50	
1110.88/291.50	   aver(z', z'') -{ 1 }-> if(gt(sum, double(z)), sum, z) :|: z'' = z, z >= 0, sum >= 0, z' = sum
1110.88/291.50	   average(z', z'') -{ 1 }-> aver(plus(x, y), 0) :|: z' = x, z'' = y, x >= 0, y >= 0
1110.88/291.50	   double(z') -{ 1 }-> 0 :|: z' = 0
1110.88/291.50	   double(z') -{ 1 }-> 1 + (1 + double(x)) :|: z' = 1 + x, x >= 0
1110.88/291.50	   gt(z', z'') -{ 1 }-> gt(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
1110.88/291.50	   gt(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
1110.88/291.50	   gt(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
1110.88/291.50	   if(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, sum >= 0, z'' = sum, z' = 0
1110.88/291.50	   if(z', z'', z1) -{ 1 }-> aver(sum, 1 + z) :|: z1 = z, z >= 0, z' = 1, sum >= 0, z'' = sum
1110.88/291.50	   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
1110.88/291.50	   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
1110.88/291.50	
1110.88/291.50	Only complete derivations are relevant for the runtime complexity.
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(9) CompleteCoflocoProof (FINISHED)
1110.88/291.50	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
1110.88/291.50	
1110.88/291.50	eq(start(V, V1, V16),0,[gt(V, V1, Out)],[V >= 0,V1 >= 0]).
1110.88/291.50	eq(start(V, V1, V16),0,[plus(V, V1, Out)],[V >= 0,V1 >= 0]).
1110.88/291.50	eq(start(V, V1, V16),0,[double(V, Out)],[V >= 0]).
1110.88/291.50	eq(start(V, V1, V16),0,[average(V, V1, Out)],[V >= 0,V1 >= 0]).
1110.88/291.50	eq(start(V, V1, V16),0,[aver(V, V1, Out)],[V >= 0,V1 >= 0]).
1110.88/291.50	eq(start(V, V1, V16),0,[if(V, V1, V16, Out)],[V >= 0,V1 >= 0,V16 >= 0]).
1110.88/291.50	eq(gt(V, V1, Out),1,[],[Out = 0,V1 = V2,V2 >= 0,V = 0]).
1110.88/291.50	eq(gt(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V3,V3 >= 0]).
1110.88/291.50	eq(gt(V, V1, Out),1,[gt(V4, V5, Ret)],[Out = Ret,V = 1 + V4,V4 >= 0,V5 >= 0,V1 = 1 + V5]).
1110.88/291.50	eq(plus(V, V1, Out),1,[],[Out = V6,V1 = V6,V6 >= 0,V = 0]).
1110.88/291.50	eq(plus(V, V1, Out),1,[plus(V7, V8, Ret1)],[Out = 1 + Ret1,V = 1 + V7,V1 = V8,V7 >= 0,V8 >= 0]).
1110.88/291.50	eq(double(V, Out),1,[],[Out = 0,V = 0]).
1110.88/291.50	eq(double(V, Out),1,[double(V9, Ret11)],[Out = 2 + Ret11,V = 1 + V9,V9 >= 0]).
1110.88/291.50	eq(average(V, V1, Out),1,[plus(V11, V10, Ret0),aver(Ret0, 0, Ret2)],[Out = Ret2,V = V11,V1 = V10,V11 >= 0,V10 >= 0]).
1110.88/291.50	eq(aver(V, V1, Out),1,[double(V13, Ret01),gt(V12, Ret01, Ret02),if(Ret02, V12, V13, Ret3)],[Out = Ret3,V1 = V13,V13 >= 0,V12 >= 0,V = V12]).
1110.88/291.50	eq(if(V, V1, V16, Out),1,[aver(V14, 1 + V15, Ret4)],[Out = Ret4,V16 = V15,V15 >= 0,V = 1,V14 >= 0,V1 = V14]).
1110.88/291.50	eq(if(V, V1, V16, Out),1,[],[Out = V17,V16 = V17,V17 >= 0,V18 >= 0,V1 = V18,V = 0]).
1110.88/291.50	input_output_vars(gt(V,V1,Out),[V,V1],[Out]).
1110.88/291.50	input_output_vars(plus(V,V1,Out),[V,V1],[Out]).
1110.88/291.50	input_output_vars(double(V,Out),[V],[Out]).
1110.88/291.50	input_output_vars(average(V,V1,Out),[V,V1],[Out]).
1110.88/291.50	input_output_vars(aver(V,V1,Out),[V,V1],[Out]).
1110.88/291.50	input_output_vars(if(V,V1,V16,Out),[V,V1,V16],[Out]).
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	CoFloCo proof output:
1110.88/291.50	Preprocessing Cost Relations
1110.88/291.50	=====================================
1110.88/291.50	
1110.88/291.50	#### Computed strongly connected components 
1110.88/291.50	0. recursive  : [double/2]
1110.88/291.50	1. recursive  : [gt/3]
1110.88/291.50	2. recursive  : [aver/3,if/4]
1110.88/291.50	3. recursive  : [plus/3]
1110.88/291.50	4. non_recursive  : [average/3]
1110.88/291.50	5. non_recursive  : [start/3]
1110.88/291.50	
1110.88/291.50	#### Obtained direct recursion through partial evaluation 
1110.88/291.50	0. SCC is partially evaluated into double/2
1110.88/291.50	1. SCC is partially evaluated into gt/3
1110.88/291.50	2. SCC is partially evaluated into aver/3
1110.88/291.50	3. SCC is partially evaluated into plus/3
1110.88/291.50	4. SCC is partially evaluated into average/3
1110.88/291.50	5. SCC is partially evaluated into start/3
1110.88/291.50	
1110.88/291.50	Control-Flow Refinement of Cost Relations
1110.88/291.50	=====================================
1110.88/291.50	
1110.88/291.50	### Specialization of cost equations double/2 
1110.88/291.50	* CE 16 is refined into CE [18] 
1110.88/291.50	* CE 15 is refined into CE [19] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Cost equations --> "Loop" of double/2 
1110.88/291.50	* CEs [19] --> Loop 13 
1110.88/291.50	* CEs [18] --> Loop 14 
1110.88/291.50	
1110.88/291.50	### Ranking functions of CR double(V,Out) 
1110.88/291.50	* RF of phase [14]: [V]
1110.88/291.50	
1110.88/291.50	#### Partial ranking functions of CR double(V,Out) 
1110.88/291.50	* Partial RF of phase [14]:
1110.88/291.50	  - RF of loop [14:1]:
1110.88/291.50	    V
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Specialization of cost equations gt/3 
1110.88/291.50	* CE 12 is refined into CE [20] 
1110.88/291.50	* CE 11 is refined into CE [21] 
1110.88/291.50	* CE 10 is refined into CE [22] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Cost equations --> "Loop" of gt/3 
1110.88/291.50	* CEs [21] --> Loop 15 
1110.88/291.50	* CEs [22] --> Loop 16 
1110.88/291.50	* CEs [20] --> Loop 17 
1110.88/291.50	
1110.88/291.50	### Ranking functions of CR gt(V,V1,Out) 
1110.88/291.50	* RF of phase [17]: [V,V1]
1110.88/291.50	
1110.88/291.50	#### Partial ranking functions of CR gt(V,V1,Out) 
1110.88/291.50	* Partial RF of phase [17]:
1110.88/291.50	  - RF of loop [17:1]:
1110.88/291.50	    V
1110.88/291.50	    V1
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Specialization of cost equations aver/3 
1110.88/291.50	* CE 9 is refined into CE [23,24] 
1110.88/291.50	* CE 8 is refined into CE [25,26,27] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Cost equations --> "Loop" of aver/3 
1110.88/291.50	* CEs [27] --> Loop 18 
1110.88/291.50	* CEs [26] --> Loop 19 
1110.88/291.50	* CEs [25] --> Loop 20 
1110.88/291.50	* CEs [24] --> Loop 21 
1110.88/291.50	* CEs [23] --> Loop 22 
1110.88/291.50	
1110.88/291.50	### Ranking functions of CR aver(V,V1,Out) 
1110.88/291.50	* RF of phase [21]: [V/2-V1]
1110.88/291.50	
1110.88/291.50	#### Partial ranking functions of CR aver(V,V1,Out) 
1110.88/291.50	* Partial RF of phase [21]:
1110.88/291.50	  - RF of loop [21:1]:
1110.88/291.50	    V/2-V1
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Specialization of cost equations plus/3 
1110.88/291.50	* CE 14 is refined into CE [28] 
1110.88/291.50	* CE 13 is refined into CE [29] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Cost equations --> "Loop" of plus/3 
1110.88/291.50	* CEs [29] --> Loop 23 
1110.88/291.50	* CEs [28] --> Loop 24 
1110.88/291.50	
1110.88/291.50	### Ranking functions of CR plus(V,V1,Out) 
1110.88/291.50	* RF of phase [24]: [V]
1110.88/291.50	
1110.88/291.50	#### Partial ranking functions of CR plus(V,V1,Out) 
1110.88/291.50	* Partial RF of phase [24]:
1110.88/291.50	  - RF of loop [24:1]:
1110.88/291.50	    V
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Specialization of cost equations average/3 
1110.88/291.50	* CE 17 is refined into CE [30,31,32,33,34] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Cost equations --> "Loop" of average/3 
1110.88/291.50	* CEs [34] --> Loop 25 
1110.88/291.50	* CEs [33] --> Loop 26 
1110.88/291.50	* CEs [32] --> Loop 27 
1110.88/291.50	* CEs [31] --> Loop 28 
1110.88/291.50	* CEs [30] --> Loop 29 
1110.88/291.50	
1110.88/291.50	### Ranking functions of CR average(V,V1,Out) 
1110.88/291.50	
1110.88/291.50	#### Partial ranking functions of CR average(V,V1,Out) 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Specialization of cost equations start/3 
1110.88/291.50	* CE 2 is refined into CE [35,36,37] 
1110.88/291.50	* CE 1 is refined into CE [38] 
1110.88/291.50	* CE 3 is refined into CE [39,40,41,42] 
1110.88/291.50	* CE 4 is refined into CE [43,44] 
1110.88/291.50	* CE 5 is refined into CE [45,46] 
1110.88/291.50	* CE 6 is refined into CE [47,48,49,50,51] 
1110.88/291.50	* CE 7 is refined into CE [52,53,54,55,56,57] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	### Cost equations --> "Loop" of start/3 
1110.88/291.50	* CEs [37] --> Loop 30 
1110.88/291.50	* CEs [36] --> Loop 31 
1110.88/291.50	* CEs [35,40,41,42,44,46,50,51,54,55,56,57] --> Loop 32 
1110.88/291.50	* CEs [38,39,43,45,47,48,49,52,53] --> Loop 33 
1110.88/291.50	
1110.88/291.50	### Ranking functions of CR start(V,V1,V16) 
1110.88/291.50	
1110.88/291.50	#### Partial ranking functions of CR start(V,V1,V16) 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	Computing Bounds
1110.88/291.50	=====================================
1110.88/291.50	
1110.88/291.50	#### Cost of chains of double(V,Out):
1110.88/291.50	* Chain [[14],13]: 1*it(14)+1
1110.88/291.50	  Such that:it(14) =< Out/2
1110.88/291.50	
1110.88/291.50	  with precondition: [2*V=Out,V>=1] 
1110.88/291.50	
1110.88/291.50	* Chain [13]: 1
1110.88/291.50	  with precondition: [V=0,Out=0] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	#### Cost of chains of gt(V,V1,Out):
1110.88/291.50	* Chain [[17],16]: 1*it(17)+1
1110.88/291.50	  Such that:it(17) =< V
1110.88/291.50	
1110.88/291.50	  with precondition: [Out=0,V>=1,V1>=V] 
1110.88/291.50	
1110.88/291.50	* Chain [[17],15]: 1*it(17)+1
1110.88/291.50	  Such that:it(17) =< V1
1110.88/291.50	
1110.88/291.50	  with precondition: [Out=1,V1>=1,V>=V1+1] 
1110.88/291.50	
1110.88/291.50	* Chain [16]: 1
1110.88/291.50	  with precondition: [V=0,Out=0,V1>=0] 
1110.88/291.50	
1110.88/291.50	* Chain [15]: 1
1110.88/291.50	  with precondition: [V1=0,Out=1,V>=1] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	#### Cost of chains of aver(V,V1,Out):
1110.88/291.50	* Chain [[21],18]: 4*it(21)+1*s(1)+1*s(2)+1*s(7)+1*s(8)+4
1110.88/291.50	  Such that:s(2) =< V
1110.88/291.50	aux(1) =< V/2+1/2
1110.88/291.50	it(21) =< V/2-V1
1110.88/291.50	s(1) =< Out
1110.88/291.50	aux(2) =< it(21)*aux(1)
1110.88/291.50	s(7) =< it(21)*aux(1)
1110.88/291.50	s(8) =< aux(2)*2
1110.88/291.50	
1110.88/291.50	  with precondition: [V1>=1,V+1>=2*Out,2*Out>=V,Out>=V1+1] 
1110.88/291.50	
1110.88/291.50	* Chain [22,[21],18]: 4*it(21)+1*s(1)+1*s(2)+1*s(7)+1*s(8)+8
1110.88/291.50	  Such that:s(2) =< V
1110.88/291.50	it(21) =< V/2
1110.88/291.50	aux(1) =< V/2+1/2
1110.88/291.50	s(1) =< Out
1110.88/291.50	aux(2) =< it(21)*aux(1)
1110.88/291.50	s(7) =< it(21)*aux(1)
1110.88/291.50	s(8) =< aux(2)*2
1110.88/291.50	
1110.88/291.50	  with precondition: [V1=0,Out>=2,V+1>=2*Out,2*Out>=V] 
1110.88/291.50	
1110.88/291.50	* Chain [22,18]: 1*s(1)+1*s(2)+8
1110.88/291.50	  Such that:s(1) =< 1
1110.88/291.50	s(2) =< V
1110.88/291.50	
1110.88/291.50	  with precondition: [V1=0,Out=1,2>=V,V>=1] 
1110.88/291.50	
1110.88/291.50	* Chain [20]: 4
1110.88/291.50	  with precondition: [V=0,V1=0,Out=0] 
1110.88/291.50	
1110.88/291.50	* Chain [19]: 1*s(9)+4
1110.88/291.50	  Such that:s(9) =< V1
1110.88/291.50	
1110.88/291.50	  with precondition: [V=0,V1=Out,V1>=1] 
1110.88/291.50	
1110.88/291.50	* Chain [18]: 1*s(1)+1*s(2)+4
1110.88/291.50	  Such that:s(2) =< V
1110.88/291.50	s(1) =< V1
1110.88/291.50	
1110.88/291.50	  with precondition: [V1=Out,V>=1,V1>=1,2*V1>=V] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	#### Cost of chains of plus(V,V1,Out):
1110.88/291.50	* Chain [[24],23]: 1*it(24)+1
1110.88/291.50	  Such that:it(24) =< -V1+Out
1110.88/291.50	
1110.88/291.50	  with precondition: [V+V1=Out,V>=1,V1>=0] 
1110.88/291.50	
1110.88/291.50	* Chain [23]: 1
1110.88/291.50	  with precondition: [V=0,V1=Out,V1>=0] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	#### Cost of chains of average(V,V1,Out):
1110.88/291.50	* Chain [29]: 6
1110.88/291.50	  with precondition: [V=0,V1=0,Out=0] 
1110.88/291.50	
1110.88/291.50	* Chain [28]: 1*s(10)+1*s(11)+10
1110.88/291.50	  Such that:s(10) =< 1
1110.88/291.50	s(11) =< V1
1110.88/291.50	
1110.88/291.50	  with precondition: [V=0,Out=1,2>=V1,V1>=1] 
1110.88/291.50	
1110.88/291.50	* Chain [27]: 1*s(12)+4*s(13)+1*s(15)+1*s(17)+1*s(18)+10
1110.88/291.50	  Such that:s(12) =< V1
1110.88/291.50	s(13) =< V1/2
1110.88/291.50	aux(3) =< V1/2+1/2
1110.88/291.50	s(15) =< aux(3)
1110.88/291.50	s(16) =< s(13)*aux(3)
1110.88/291.50	s(17) =< s(13)*aux(3)
1110.88/291.50	s(18) =< s(16)*2
1110.88/291.50	
1110.88/291.50	  with precondition: [V=0,Out>=2,V1+1>=2*Out,2*Out>=V1] 
1110.88/291.50	
1110.88/291.50	* Chain [26]: 1*s(19)+1*s(20)+1*s(21)+10
1110.88/291.50	  Such that:s(20) =< 1
1110.88/291.50	s(19) =< V
1110.88/291.50	s(21) =< V+V1
1110.88/291.50	
1110.88/291.50	  with precondition: [Out=1,V>=1,V1>=0,2>=V+V1] 
1110.88/291.50	
1110.88/291.50	* Chain [25]: 1*s(22)+1*s(23)+4*s(24)+1*s(26)+1*s(28)+1*s(29)+10
1110.88/291.50	  Such that:s(22) =< V
1110.88/291.50	s(23) =< V+V1
1110.88/291.50	s(24) =< V/2+V1/2
1110.88/291.50	aux(4) =< V/2+V1/2+1/2
1110.88/291.50	s(26) =< aux(4)
1110.88/291.50	s(27) =< s(24)*aux(4)
1110.88/291.50	s(28) =< s(24)*aux(4)
1110.88/291.50	s(29) =< s(27)*2
1110.88/291.50	
1110.88/291.50	  with precondition: [V>=1,V1>=0,Out>=2,V+V1+1>=2*Out,2*Out>=V+V1] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	#### Cost of chains of start(V,V1,V16):
1110.88/291.50	* Chain [33]: 1*s(30)+3*s(31)+4*s(33)+1*s(35)+1*s(37)+1*s(38)+10
1110.88/291.50	  Such that:s(30) =< 1
1110.88/291.50	s(33) =< V1/2
1110.88/291.50	s(34) =< V1/2+1/2
1110.88/291.50	aux(5) =< V1
1110.88/291.50	s(31) =< aux(5)
1110.88/291.50	s(35) =< s(34)
1110.88/291.50	s(36) =< s(33)*s(34)
1110.88/291.50	s(37) =< s(33)*s(34)
1110.88/291.50	s(38) =< s(36)*2
1110.88/291.50	
1110.88/291.50	  with precondition: [V=0] 
1110.88/291.50	
1110.88/291.50	* Chain [32]: 1*s(40)+9*s(41)+2*s(42)+2*s(45)+2*s(47)+4*s(50)+1*s(52)+1*s(54)+1*s(55)+4*s(59)+2*s(61)+1*s(63)+1*s(64)+4*s(69)+1*s(72)+1*s(73)+10
1110.88/291.50	  Such that:s(59) =< V/2
1110.88/291.50	s(69) =< V/2-V1
1110.88/291.50	s(50) =< V/2+V1/2
1110.88/291.50	s(51) =< V/2+V1/2+1/2
1110.88/291.50	s(40) =< V16+1
1110.88/291.50	aux(8) =< 1
1110.88/291.50	aux(9) =< V
1110.88/291.50	aux(10) =< V+V1
1110.88/291.50	aux(11) =< V/2+1/2
1110.88/291.50	aux(12) =< V1
1110.88/291.50	s(45) =< aux(8)
1110.88/291.50	s(41) =< aux(9)
1110.88/291.50	s(47) =< aux(10)
1110.88/291.50	s(42) =< aux(12)
1110.88/291.50	s(52) =< s(51)
1110.88/291.50	s(53) =< s(50)*s(51)
1110.88/291.50	s(54) =< s(50)*s(51)
1110.88/291.50	s(55) =< s(53)*2
1110.88/291.50	s(61) =< aux(11)
1110.88/291.50	s(62) =< s(59)*aux(11)
1110.88/291.50	s(63) =< s(59)*aux(11)
1110.88/291.50	s(64) =< s(62)*2
1110.88/291.50	s(71) =< s(69)*aux(11)
1110.88/291.50	s(72) =< s(69)*aux(11)
1110.88/291.50	s(73) =< s(71)*2
1110.88/291.50	
1110.88/291.50	  with precondition: [V>=1] 
1110.88/291.50	
1110.88/291.50	* Chain [31]: 1*s(74)+1*s(75)+5
1110.88/291.50	  Such that:s(74) =< V1
1110.88/291.50	s(75) =< V16+1
1110.88/291.50	
1110.88/291.50	  with precondition: [V=1,V1>=1,V16>=0,2*V16+2>=V1] 
1110.88/291.50	
1110.88/291.50	* Chain [30]: 1*s(76)+4*s(78)+1*s(79)+1*s(81)+1*s(82)+5
1110.88/291.50	  Such that:s(76) =< V1
1110.88/291.50	s(78) =< V1/2-V16
1110.88/291.50	aux(13) =< V1/2+1/2
1110.88/291.50	s(79) =< aux(13)
1110.88/291.50	s(80) =< s(78)*aux(13)
1110.88/291.50	s(81) =< s(78)*aux(13)
1110.88/291.50	s(82) =< s(80)*2
1110.88/291.50	
1110.88/291.50	  with precondition: [V=1,V16>=0,V1>=2*V16+3] 
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	Closed-form bounds of start(V,V1,V16): 
1110.88/291.50	-------------------------------------
1110.88/291.50	* Chain [33] with precondition: [V=0] 
1110.88/291.50	    - Upper bound: nat(V1)*3+11+nat(V1/2+1/2)+nat(V1/2+1/2)*3*nat(V1/2)+nat(V1/2)*4 
1110.88/291.50	    - Complexity: n^2 
1110.88/291.50	* Chain [32] with precondition: [V>=1] 
1110.88/291.50	    - Upper bound: 9*V+12+nat(V1)*2+nat(V+V1)*2+nat(V16+1)+nat(V/2+V1/2+1/2)+nat(V/2+V1/2+1/2)*3*nat(V/2+V1/2)+nat(V/2+V1/2)*4+(V+1)+(3/2*V+3/2)*nat(V/2-V1)+V/2*(3/2*V+3/2)+nat(V/2-V1)*4+2*V 
1110.88/291.50	    - Complexity: n^2 
1110.88/291.50	* Chain [31] with precondition: [V=1,V1>=1,V16>=0,2*V16+2>=V1] 
1110.88/291.50	    - Upper bound: V1+V16+6 
1110.88/291.50	    - Complexity: n 
1110.88/291.50	* Chain [30] with precondition: [V=1,V16>=0,V1>=2*V16+3] 
1110.88/291.50	    - Upper bound: 2*V1-4*V16+(3/2*V1+11/2+(V1/2-V16)*(3/2*V1+3/2)) 
1110.88/291.50	    - Complexity: n^2 
1110.88/291.50	
1110.88/291.50	### Maximum cost of start(V,V1,V16): nat(V1)+5+max([max([nat(V16+1),nat(V1/2+1/2)*3*nat(V1/2-V16)+nat(V1/2+1/2)+nat(V1/2-V16)*4]),nat(V1)+6+max([nat(V1/2+1/2)+nat(V1)+nat(V1/2+1/2)*3*nat(V1/2)+nat(V1/2)*4,9*V+1+nat(V+V1)*2+nat(V16+1)+nat(V/2+V1/2+1/2)+nat(V/2+V1/2+1/2)*3*nat(V/2+V1/2)+nat(V/2+V1/2)*4+(V+1)+(3/2*V+3/2)*nat(V/2-V1)+V/2*(3/2*V+3/2)+nat(V/2-V1)*4+2*V])]) 
1110.88/291.50	Asymptotic class: n^2 
1110.88/291.50	* Total analysis performed in 330 ms.
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(10)
1110.88/291.50	BOUNDS(1, n^2)
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1110.88/291.50	Transformed a relative TRS into a decreasing-loop problem.
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(12)
1110.88/291.50	Obligation:
1110.88/291.50	Analyzing the following TRS for decreasing loops:
1110.88/291.50	
1110.88/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	The TRS R consists of the following rules:
1110.88/291.50	
1110.88/291.50	   gt(0, y) -> false
1110.88/291.50	   gt(s(x), 0) -> true
1110.88/291.50	   gt(s(x), s(y)) -> gt(x, y)
1110.88/291.50	   plus(0, y) -> y
1110.88/291.50	   plus(s(x), y) -> s(plus(x, y))
1110.88/291.50	   double(0) -> 0
1110.88/291.50	   double(s(x)) -> s(s(double(x)))
1110.88/291.50	   average(x, y) -> aver(plus(x, y), 0)
1110.88/291.50	   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
1110.88/291.50	   if(true, sum, z) -> aver(sum, s(z))
1110.88/291.50	   if(false, sum, z) -> z
1110.88/291.50	
1110.88/291.50	S is empty.
1110.88/291.50	Rewrite Strategy: INNERMOST
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(13) DecreasingLoopProof (LOWER BOUND(ID))
1110.88/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1110.88/291.50	
1110.88/291.50	The rewrite sequence
1110.88/291.50	
1110.88/291.50	gt(s(x), s(y)) ->^+ gt(x, y)
1110.88/291.50	
1110.88/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1110.88/291.50	
1110.88/291.50	The pumping substitution is [x / s(x), y / s(y)].
1110.88/291.50	
1110.88/291.50	The result substitution is [ ].
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(14)
1110.88/291.50	Complex Obligation (BEST)
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(15)
1110.88/291.50	Obligation:
1110.88/291.50	Proved the lower bound n^1 for the following obligation:
1110.88/291.50	
1110.88/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	The TRS R consists of the following rules:
1110.88/291.50	
1110.88/291.50	   gt(0, y) -> false
1110.88/291.50	   gt(s(x), 0) -> true
1110.88/291.50	   gt(s(x), s(y)) -> gt(x, y)
1110.88/291.50	   plus(0, y) -> y
1110.88/291.50	   plus(s(x), y) -> s(plus(x, y))
1110.88/291.50	   double(0) -> 0
1110.88/291.50	   double(s(x)) -> s(s(double(x)))
1110.88/291.50	   average(x, y) -> aver(plus(x, y), 0)
1110.88/291.50	   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
1110.88/291.50	   if(true, sum, z) -> aver(sum, s(z))
1110.88/291.50	   if(false, sum, z) -> z
1110.88/291.50	
1110.88/291.50	S is empty.
1110.88/291.50	Rewrite Strategy: INNERMOST
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(16) LowerBoundPropagationProof (FINISHED)
1110.88/291.50	Propagated lower bound.
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(17)
1110.88/291.50	BOUNDS(n^1, INF)
1110.88/291.50	
1110.88/291.50	----------------------------------------
1110.88/291.50	
1110.88/291.50	(18)
1110.88/291.50	Obligation:
1110.88/291.50	Analyzing the following TRS for decreasing loops:
1110.88/291.50	
1110.88/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1110.88/291.50	
1110.88/291.50	
1110.88/291.50	The TRS R consists of the following rules:
1110.88/291.50	
1110.88/291.50	   gt(0, y) -> false
1110.88/291.50	   gt(s(x), 0) -> true
1110.88/291.50	   gt(s(x), s(y)) -> gt(x, y)
1110.88/291.50	   plus(0, y) -> y
1110.88/291.50	   plus(s(x), y) -> s(plus(x, y))
1110.88/291.50	   double(0) -> 0
1110.88/291.50	   double(s(x)) -> s(s(double(x)))
1110.88/291.50	   average(x, y) -> aver(plus(x, y), 0)
1110.88/291.50	   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
1110.88/291.50	   if(true, sum, z) -> aver(sum, s(z))
1110.88/291.50	   if(false, sum, z) -> z
1110.88/291.50	
1110.88/291.50	S is empty.
1110.88/291.50	Rewrite Strategy: INNERMOST
1111.09/291.56	EOF