15.18/5.09	WORST_CASE(Omega(n^1), O(n^1))
15.18/5.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
15.18/5.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.18/5.11	
15.18/5.11	
15.18/5.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.18/5.11	
15.18/5.11	(0) CpxTRS
15.18/5.11	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
15.18/5.11	(2) CpxWeightedTrs
15.18/5.11	    (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
15.18/5.11	    (4) CpxWeightedTrs
15.18/5.11	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
15.18/5.11	    (6) CpxTypedWeightedTrs
15.18/5.11	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
15.18/5.11	    (8) CpxTypedWeightedCompleteTrs
15.18/5.11	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
15.18/5.11	    (10) CpxRNTS
15.18/5.11	    (11) CompleteCoflocoProof [FINISHED, 179 ms]
15.18/5.11	    (12) BOUNDS(1, n^1)
15.18/5.11	(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
15.18/5.11	(14) TRS for Loop Detection
15.18/5.11	    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
15.18/5.11	    (16) BEST
15.18/5.11	        (17) proven lower bound
15.18/5.11	            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
15.18/5.11	            (19) BOUNDS(n^1, INF)
15.18/5.11	        (20) TRS for Loop Detection
15.18/5.11	
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(0)
15.18/5.11	Obligation:
15.18/5.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.18/5.11	
15.18/5.11	
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   +(X, 0) -> X
15.18/5.11	   +(X, s(Y)) -> s(+(X, Y))
15.18/5.11	   double(X) -> +(X, X)
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X)
15.18/5.11	   g(X, Y) -> X
15.18/5.11	   g(X, Y) -> Y
15.18/5.11	
15.18/5.11	S is empty.
15.18/5.11	Rewrite Strategy: INNERMOST
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
15.18/5.11	Transformed relative TRS to weighted TRS
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(2)
15.18/5.11	Obligation:
15.18/5.11	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
15.18/5.11	
15.18/5.11	
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   +(X, 0) -> X [1]
15.18/5.11	   +(X, s(Y)) -> s(+(X, Y)) [1]
15.18/5.11	   double(X) -> +(X, X) [1]
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X) [1]
15.18/5.11	   g(X, Y) -> X [1]
15.18/5.11	   g(X, Y) -> Y [1]
15.18/5.11	
15.18/5.11	Rewrite Strategy: INNERMOST
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
15.18/5.11	Renamed defined symbols to avoid conflicts with arithmetic symbols:
15.18/5.11	
15.18/5.11	+ => plus
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(4)
15.18/5.11	Obligation:
15.18/5.11	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
15.18/5.11	
15.18/5.11	
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   plus(X, 0) -> X [1]
15.18/5.11	   plus(X, s(Y)) -> s(plus(X, Y)) [1]
15.18/5.11	   double(X) -> plus(X, X) [1]
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X) [1]
15.18/5.11	   g(X, Y) -> X [1]
15.18/5.11	   g(X, Y) -> Y [1]
15.18/5.11	
15.18/5.11	Rewrite Strategy: INNERMOST
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
15.18/5.11	Infered types.
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(6)
15.18/5.11	Obligation:
15.18/5.11	Runtime Complexity Weighted TRS with Types.
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   plus(X, 0) -> X [1]
15.18/5.11	   plus(X, s(Y)) -> s(plus(X, Y)) [1]
15.18/5.11	   double(X) -> plus(X, X) [1]
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X) [1]
15.18/5.11	   g(X, Y) -> X [1]
15.18/5.11	   g(X, Y) -> Y [1]
15.18/5.11	
15.18/5.11	The TRS has the following type information:
15.18/5.11	plus :: 0:s -> 0:s -> 0:s
15.18/5.11	0 :: 0:s
15.18/5.11	s :: 0:s -> 0:s
15.18/5.11	double :: 0:s -> 0:s
15.18/5.11	f :: 0:s -> 0:s -> 0:s -> f
15.18/5.11	g :: g -> g -> g
15.18/5.11	
15.18/5.11	Rewrite Strategy: INNERMOST
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(7) CompletionProof (UPPER BOUND(ID))
15.18/5.11	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
15.18/5.11	
15.18/5.11	   f(v0, v1, v2) -> null_f [0]
15.18/5.11	
15.18/5.11	And the following fresh constants:    null_f, const
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(8)
15.18/5.11	Obligation:
15.18/5.11	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
15.18/5.11	
15.18/5.11	Runtime Complexity Weighted TRS with Types.
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   plus(X, 0) -> X [1]
15.18/5.11	   plus(X, s(Y)) -> s(plus(X, Y)) [1]
15.18/5.11	   double(X) -> plus(X, X) [1]
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X) [1]
15.18/5.11	   g(X, Y) -> X [1]
15.18/5.11	   g(X, Y) -> Y [1]
15.18/5.11	   f(v0, v1, v2) -> null_f [0]
15.18/5.11	
15.18/5.11	The TRS has the following type information:
15.18/5.11	plus :: 0:s -> 0:s -> 0:s
15.18/5.11	0 :: 0:s
15.18/5.11	s :: 0:s -> 0:s
15.18/5.11	double :: 0:s -> 0:s
15.18/5.11	f :: 0:s -> 0:s -> 0:s -> null_f
15.18/5.11	g :: g -> g -> g
15.18/5.11	null_f :: null_f
15.18/5.11	const :: g
15.18/5.11	
15.18/5.11	Rewrite Strategy: INNERMOST
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
15.18/5.11	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
15.18/5.11	The constant constructors are abstracted as follows:
15.18/5.11	
15.18/5.11	   0 => 0
15.18/5.11	   null_f => 0
15.18/5.11	   const => 0
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(10)
15.18/5.11	Obligation:
15.18/5.11	Complexity RNTS consisting of the following rules:
15.18/5.11	
15.18/5.11	   double(z) -{ 1 }-> plus(X, X) :|: X >= 0, z = X
15.18/5.11	   f(z, z', z'') -{ 1 }-> f(X, double(X), X) :|: z'' = X, X >= 0, z' = 1 + 0, z = 0
15.18/5.11	   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
15.18/5.11	   g(z, z') -{ 1 }-> X :|: z' = Y, Y >= 0, X >= 0, z = X
15.18/5.11	   g(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, X >= 0, z = X
15.18/5.11	   plus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
15.18/5.11	   plus(z, z') -{ 1 }-> 1 + plus(X, Y) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X
15.18/5.11	
15.18/5.11	Only complete derivations are relevant for the runtime complexity.
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(11) CompleteCoflocoProof (FINISHED)
15.18/5.11	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
15.18/5.11	
15.18/5.11	eq(start(V1, V, V2),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
15.18/5.11	eq(start(V1, V, V2),0,[double(V1, Out)],[V1 >= 0]).
15.18/5.11	eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
15.18/5.11	eq(start(V1, V, V2),0,[g(V1, V, Out)],[V1 >= 0,V >= 0]).
15.18/5.11	eq(plus(V1, V, Out),1,[],[Out = X1,X1 >= 0,V1 = X1,V = 0]).
15.18/5.11	eq(plus(V1, V, Out),1,[plus(X2, Y1, Ret1)],[Out = 1 + Ret1,Y1 >= 0,V = 1 + Y1,X2 >= 0,V1 = X2]).
15.18/5.11	eq(double(V1, Out),1,[plus(X3, X3, Ret)],[Out = Ret,X3 >= 0,V1 = X3]).
15.18/5.11	eq(f(V1, V, V2, Out),1,[double(X4, Ret11),f(X4, Ret11, X4, Ret2)],[Out = Ret2,V2 = X4,X4 >= 0,V = 1,V1 = 0]).
15.18/5.11	eq(g(V1, V, Out),1,[],[Out = X5,V = Y2,Y2 >= 0,X5 >= 0,V1 = X5]).
15.18/5.11	eq(g(V1, V, Out),1,[],[Out = Y3,V = Y3,Y3 >= 0,X6 >= 0,V1 = X6]).
15.18/5.11	eq(f(V1, V, V2, Out),0,[],[Out = 0,V4 >= 0,V2 = V5,V3 >= 0,V1 = V4,V = V3,V5 >= 0]).
15.18/5.11	input_output_vars(plus(V1,V,Out),[V1,V],[Out]).
15.18/5.11	input_output_vars(double(V1,Out),[V1],[Out]).
15.18/5.11	input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
15.18/5.11	input_output_vars(g(V1,V,Out),[V1,V],[Out]).
15.18/5.11	
15.18/5.11	
15.18/5.11	CoFloCo proof output:
15.18/5.11	Preprocessing Cost Relations
15.18/5.11	=====================================
15.18/5.11	
15.18/5.11	#### Computed strongly connected components 
15.18/5.11	0. recursive  : [plus/3]
15.18/5.11	1. non_recursive  : [double/2]
15.18/5.11	2. recursive  : [f/4]
15.18/5.11	3. non_recursive  : [g/3]
15.18/5.11	4. non_recursive  : [start/3]
15.18/5.11	
15.18/5.11	#### Obtained direct recursion through partial evaluation 
15.18/5.11	0. SCC is partially evaluated into plus/3
15.18/5.11	1. SCC is completely evaluated into other SCCs
15.18/5.11	2. SCC is partially evaluated into f/4
15.18/5.11	3. SCC is partially evaluated into g/3
15.18/5.11	4. SCC is partially evaluated into start/3
15.18/5.11	
15.18/5.11	Control-Flow Refinement of Cost Relations
15.18/5.11	=====================================
15.18/5.11	
15.18/5.11	### Specialization of cost equations plus/3 
15.18/5.11	* CE 6 is refined into CE [11] 
15.18/5.11	* CE 5 is refined into CE [12] 
15.18/5.11	
15.18/5.11	
15.18/5.11	### Cost equations --> "Loop" of plus/3 
15.18/5.11	* CEs [12] --> Loop 8 
15.18/5.11	* CEs [11] --> Loop 9 
15.18/5.11	
15.18/5.11	### Ranking functions of CR plus(V1,V,Out) 
15.18/5.11	* RF of phase [9]: [V]
15.18/5.11	
15.18/5.11	#### Partial ranking functions of CR plus(V1,V,Out) 
15.18/5.11	* Partial RF of phase [9]:
15.18/5.11	  - RF of loop [9:1]:
15.18/5.11	    V
15.18/5.11	
15.18/5.11	
15.18/5.11	### Specialization of cost equations f/4 
15.18/5.11	* CE 8 is refined into CE [13] 
15.18/5.11	* CE 7 is refined into CE [14,15] 
15.18/5.11	
15.18/5.11	
15.18/5.11	### Cost equations --> "Loop" of f/4 
15.18/5.11	* CEs [15] --> Loop 10 
15.18/5.11	* CEs [14] --> Loop 11 
15.18/5.11	* CEs [13] --> Loop 12 
15.18/5.11	
15.18/5.11	### Ranking functions of CR f(V1,V,V2,Out) 
15.18/5.11	
15.18/5.11	#### Partial ranking functions of CR f(V1,V,V2,Out) 
15.18/5.11	
15.18/5.11	
15.18/5.11	### Specialization of cost equations g/3 
15.18/5.11	* CE 10 is refined into CE [16] 
15.18/5.11	* CE 9 is refined into CE [17] 
15.18/5.11	
15.18/5.11	
15.18/5.11	### Cost equations --> "Loop" of g/3 
15.18/5.11	* CEs [16] --> Loop 13 
15.18/5.11	* CEs [17] --> Loop 14 
15.18/5.11	
15.18/5.11	### Ranking functions of CR g(V1,V,Out) 
15.18/5.11	
15.18/5.11	#### Partial ranking functions of CR g(V1,V,Out) 
15.18/5.11	
15.18/5.11	
15.18/5.11	### Specialization of cost equations start/3 
15.18/5.11	* CE 1 is refined into CE [18,19] 
15.18/5.11	* CE 2 is refined into CE [20,21] 
15.18/5.11	* CE 3 is refined into CE [22,23] 
15.18/5.11	* CE 4 is refined into CE [24,25] 
15.18/5.11	
15.18/5.11	
15.18/5.11	### Cost equations --> "Loop" of start/3 
15.18/5.11	* CEs [21] --> Loop 15 
15.18/5.11	* CEs [18,19,22,24,25] --> Loop 16 
15.18/5.11	* CEs [20,23] --> Loop 17 
15.18/5.11	
15.18/5.11	### Ranking functions of CR start(V1,V,V2) 
15.18/5.11	
15.18/5.11	#### Partial ranking functions of CR start(V1,V,V2) 
15.18/5.11	
15.18/5.11	
15.18/5.11	Computing Bounds
15.18/5.11	=====================================
15.18/5.11	
15.18/5.11	#### Cost of chains of plus(V1,V,Out):
15.18/5.11	* Chain [[9],8]: 1*it(9)+1
15.18/5.11	  Such that:it(9) =< V
15.18/5.11	
15.18/5.11	  with precondition: [V+V1=Out,V1>=0,V>=1] 
15.18/5.11	
15.18/5.11	* Chain [8]: 1
15.18/5.11	  with precondition: [V=0,V1=Out,V1>=0] 
15.18/5.11	
15.18/5.11	
15.18/5.11	#### Cost of chains of f(V1,V,V2,Out):
15.18/5.11	* Chain [12]: 0
15.18/5.11	  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 
15.18/5.11	
15.18/5.11	* Chain [11,12]: 3
15.18/5.11	  with precondition: [V1=0,V=1,V2=0,Out=0] 
15.18/5.11	
15.18/5.11	* Chain [10,12]: 1*s(1)+3
15.18/5.11	  Such that:s(1) =< V2
15.18/5.11	
15.18/5.11	  with precondition: [V1=0,V=1,Out=0,V2>=1] 
15.18/5.11	
15.18/5.11	
15.18/5.11	#### Cost of chains of g(V1,V,Out):
15.18/5.11	* Chain [14]: 1
15.18/5.11	  with precondition: [V1=Out,V1>=0,V>=0] 
15.18/5.11	
15.18/5.11	* Chain [13]: 1
15.18/5.11	  with precondition: [V=Out,V1>=0,V>=0] 
15.18/5.11	
15.18/5.11	
15.18/5.11	#### Cost of chains of start(V1,V,V2):
15.18/5.11	* Chain [17]: 1*s(2)+3
15.18/5.11	  Such that:s(2) =< V2
15.18/5.11	
15.18/5.11	  with precondition: [V1=0] 
15.18/5.11	
15.18/5.11	* Chain [16]: 1*s(3)+3
15.18/5.11	  Such that:s(3) =< V
15.18/5.11	
15.18/5.11	  with precondition: [V1>=0,V>=0] 
15.18/5.11	
15.18/5.11	* Chain [15]: 1*s(4)+2
15.18/5.11	  Such that:s(4) =< V1
15.18/5.11	
15.18/5.11	  with precondition: [V1>=1] 
15.18/5.11	
15.18/5.11	
15.18/5.11	Closed-form bounds of start(V1,V,V2): 
15.18/5.11	-------------------------------------
15.18/5.11	* Chain [17] with precondition: [V1=0] 
15.18/5.11	    - Upper bound: nat(V2)+3 
15.18/5.11	    - Complexity: n 
15.18/5.11	* Chain [16] with precondition: [V1>=0,V>=0] 
15.18/5.11	    - Upper bound: V+3 
15.18/5.11	    - Complexity: n 
15.18/5.11	* Chain [15] with precondition: [V1>=1] 
15.18/5.11	    - Upper bound: V1+2 
15.18/5.11	    - Complexity: n 
15.18/5.11	
15.18/5.11	### Maximum cost of start(V1,V,V2): max([V1,nat(V)+1,nat(V2)+1])+2 
15.18/5.11	Asymptotic class: n 
15.18/5.11	* Total analysis performed in 109 ms.
15.18/5.11	
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(12)
15.18/5.11	BOUNDS(1, n^1)
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
15.18/5.11	Transformed a relative TRS into a decreasing-loop problem.
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(14)
15.18/5.11	Obligation:
15.18/5.11	Analyzing the following TRS for decreasing loops:
15.18/5.11	
15.18/5.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.18/5.11	
15.18/5.11	
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   +(X, 0) -> X
15.18/5.11	   +(X, s(Y)) -> s(+(X, Y))
15.18/5.11	   double(X) -> +(X, X)
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X)
15.18/5.11	   g(X, Y) -> X
15.18/5.11	   g(X, Y) -> Y
15.18/5.11	
15.18/5.11	S is empty.
15.18/5.11	Rewrite Strategy: INNERMOST
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(15) DecreasingLoopProof (LOWER BOUND(ID))
15.18/5.11	The following loop(s) give(s) rise to the lower bound Omega(n^1):
15.18/5.11	
15.18/5.11	The rewrite sequence
15.18/5.11	
15.18/5.11	+(X, s(Y)) ->^+ s(+(X, Y))
15.18/5.11	
15.18/5.11	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
15.18/5.11	
15.18/5.11	The pumping substitution is [Y / s(Y)].
15.18/5.11	
15.18/5.11	The result substitution is [ ].
15.18/5.11	
15.18/5.11	
15.18/5.11	
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(16)
15.18/5.11	Complex Obligation (BEST)
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(17)
15.18/5.11	Obligation:
15.18/5.11	Proved the lower bound n^1 for the following obligation:
15.18/5.11	
15.18/5.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.18/5.11	
15.18/5.11	
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   +(X, 0) -> X
15.18/5.11	   +(X, s(Y)) -> s(+(X, Y))
15.18/5.11	   double(X) -> +(X, X)
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X)
15.18/5.11	   g(X, Y) -> X
15.18/5.11	   g(X, Y) -> Y
15.18/5.11	
15.18/5.11	S is empty.
15.18/5.11	Rewrite Strategy: INNERMOST
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(18) LowerBoundPropagationProof (FINISHED)
15.18/5.11	Propagated lower bound.
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(19)
15.18/5.11	BOUNDS(n^1, INF)
15.18/5.11	
15.18/5.11	----------------------------------------
15.18/5.11	
15.18/5.11	(20)
15.18/5.11	Obligation:
15.18/5.11	Analyzing the following TRS for decreasing loops:
15.18/5.11	
15.18/5.11	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.18/5.11	
15.18/5.11	
15.18/5.11	The TRS R consists of the following rules:
15.18/5.11	
15.18/5.11	   +(X, 0) -> X
15.18/5.11	   +(X, s(Y)) -> s(+(X, Y))
15.18/5.11	   double(X) -> +(X, X)
15.18/5.11	   f(0, s(0), X) -> f(X, double(X), X)
15.18/5.11	   g(X, Y) -> X
15.18/5.11	   g(X, Y) -> Y
15.18/5.11	
15.18/5.11	S is empty.
15.18/5.11	Rewrite Strategy: INNERMOST
15.49/5.15	EOF