15.88/4.83	WORST_CASE(Omega(n^1), O(n^1))
16.24/4.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
16.24/4.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.24/4.84	
16.24/4.84	
16.24/4.84	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.24/4.84	
16.24/4.84	(0) CpxTRS
16.24/4.84	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
16.24/4.84	(2) CpxWeightedTrs
16.24/4.84	    (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
16.24/4.84	    (4) CpxWeightedTrs
16.24/4.84	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
16.24/4.84	    (6) CpxTypedWeightedTrs
16.24/4.84	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
16.24/4.84	    (8) CpxTypedWeightedCompleteTrs
16.24/4.84	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
16.24/4.84	    (10) CpxRNTS
16.24/4.84	    (11) CompleteCoflocoProof [FINISHED, 169 ms]
16.24/4.84	    (12) BOUNDS(1, n^1)
16.24/4.84	(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
16.24/4.84	(14) TRS for Loop Detection
16.24/4.84	    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
16.24/4.84	    (16) BEST
16.24/4.84	        (17) proven lower bound
16.24/4.84	            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
16.24/4.84	            (19) BOUNDS(n^1, INF)
16.24/4.84	        (20) TRS for Loop Detection
16.24/4.84	
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(0)
16.24/4.84	Obligation:
16.24/4.84	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.24/4.84	
16.24/4.84	
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   +(X, 0) -> X
16.24/4.84	   +(X, s(Y)) -> s(+(X, Y))
16.24/4.84	   f(0, s(0), X) -> f(X, +(X, X), X)
16.24/4.84	   g(X, Y) -> X
16.24/4.84	   g(X, Y) -> Y
16.24/4.84	
16.24/4.84	S is empty.
16.24/4.84	Rewrite Strategy: INNERMOST
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
16.24/4.84	Transformed relative TRS to weighted TRS
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(2)
16.24/4.84	Obligation:
16.24/4.84	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
16.24/4.84	
16.24/4.84	
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   +(X, 0) -> X [1]
16.24/4.84	   +(X, s(Y)) -> s(+(X, Y)) [1]
16.24/4.84	   f(0, s(0), X) -> f(X, +(X, X), X) [1]
16.24/4.84	   g(X, Y) -> X [1]
16.24/4.84	   g(X, Y) -> Y [1]
16.24/4.84	
16.24/4.84	Rewrite Strategy: INNERMOST
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
16.24/4.84	Renamed defined symbols to avoid conflicts with arithmetic symbols:
16.24/4.84	
16.24/4.84	+ => plus
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(4)
16.24/4.84	Obligation:
16.24/4.84	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
16.24/4.84	
16.24/4.84	
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   plus(X, 0) -> X [1]
16.24/4.84	   plus(X, s(Y)) -> s(plus(X, Y)) [1]
16.24/4.84	   f(0, s(0), X) -> f(X, plus(X, X), X) [1]
16.24/4.84	   g(X, Y) -> X [1]
16.24/4.84	   g(X, Y) -> Y [1]
16.24/4.84	
16.24/4.84	Rewrite Strategy: INNERMOST
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
16.24/4.84	Infered types.
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(6)
16.24/4.84	Obligation:
16.24/4.84	Runtime Complexity Weighted TRS with Types.
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   plus(X, 0) -> X [1]
16.24/4.84	   plus(X, s(Y)) -> s(plus(X, Y)) [1]
16.24/4.84	   f(0, s(0), X) -> f(X, plus(X, X), X) [1]
16.24/4.84	   g(X, Y) -> X [1]
16.24/4.84	   g(X, Y) -> Y [1]
16.24/4.84	
16.24/4.84	The TRS has the following type information:
16.24/4.84	plus :: 0:s -> 0:s -> 0:s
16.24/4.84	0 :: 0:s
16.24/4.84	s :: 0:s -> 0:s
16.24/4.84	f :: 0:s -> 0:s -> 0:s -> f
16.24/4.84	g :: g -> g -> g
16.24/4.84	
16.24/4.84	Rewrite Strategy: INNERMOST
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(7) CompletionProof (UPPER BOUND(ID))
16.24/4.84	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
16.24/4.84	
16.24/4.84	   f(v0, v1, v2) -> null_f [0]
16.24/4.84	
16.24/4.84	And the following fresh constants:    null_f, const
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(8)
16.24/4.84	Obligation:
16.24/4.84	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
16.24/4.84	
16.24/4.84	Runtime Complexity Weighted TRS with Types.
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   plus(X, 0) -> X [1]
16.24/4.84	   plus(X, s(Y)) -> s(plus(X, Y)) [1]
16.24/4.84	   f(0, s(0), X) -> f(X, plus(X, X), X) [1]
16.24/4.84	   g(X, Y) -> X [1]
16.24/4.84	   g(X, Y) -> Y [1]
16.24/4.84	   f(v0, v1, v2) -> null_f [0]
16.24/4.84	
16.24/4.84	The TRS has the following type information:
16.24/4.84	plus :: 0:s -> 0:s -> 0:s
16.24/4.84	0 :: 0:s
16.24/4.84	s :: 0:s -> 0:s
16.24/4.84	f :: 0:s -> 0:s -> 0:s -> null_f
16.24/4.84	g :: g -> g -> g
16.24/4.84	null_f :: null_f
16.24/4.84	const :: g
16.24/4.84	
16.24/4.84	Rewrite Strategy: INNERMOST
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
16.24/4.84	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
16.24/4.84	The constant constructors are abstracted as follows:
16.24/4.84	
16.24/4.84	   0 => 0
16.24/4.84	   null_f => 0
16.24/4.84	   const => 0
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(10)
16.24/4.84	Obligation:
16.24/4.84	Complexity RNTS consisting of the following rules:
16.24/4.84	
16.24/4.84	   f(z, z', z'') -{ 1 }-> f(X, plus(X, X), X) :|: z'' = X, X >= 0, z' = 1 + 0, z = 0
16.24/4.84	   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
16.24/4.84	   g(z, z') -{ 1 }-> X :|: z' = Y, Y >= 0, X >= 0, z = X
16.24/4.84	   g(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, X >= 0, z = X
16.24/4.84	   plus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
16.24/4.84	   plus(z, z') -{ 1 }-> 1 + plus(X, Y) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X
16.24/4.84	
16.24/4.84	Only complete derivations are relevant for the runtime complexity.
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(11) CompleteCoflocoProof (FINISHED)
16.24/4.84	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
16.24/4.84	
16.24/4.84	eq(start(V1, V, V2),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
16.24/4.84	eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
16.24/4.84	eq(start(V1, V, V2),0,[g(V1, V, Out)],[V1 >= 0,V >= 0]).
16.24/4.84	eq(plus(V1, V, Out),1,[],[Out = X1,X1 >= 0,V1 = X1,V = 0]).
16.24/4.84	eq(plus(V1, V, Out),1,[plus(X2, Y1, Ret1)],[Out = 1 + Ret1,Y1 >= 0,V = 1 + Y1,X2 >= 0,V1 = X2]).
16.24/4.84	eq(f(V1, V, V2, Out),1,[plus(X3, X3, Ret11),f(X3, Ret11, X3, Ret)],[Out = Ret,V2 = X3,X3 >= 0,V = 1,V1 = 0]).
16.24/4.84	eq(g(V1, V, Out),1,[],[Out = X4,V = Y2,Y2 >= 0,X4 >= 0,V1 = X4]).
16.24/4.84	eq(g(V1, V, Out),1,[],[Out = Y3,V = Y3,Y3 >= 0,X5 >= 0,V1 = X5]).
16.24/4.84	eq(f(V1, V, V2, Out),0,[],[Out = 0,V4 >= 0,V2 = V5,V3 >= 0,V1 = V4,V = V3,V5 >= 0]).
16.24/4.84	input_output_vars(plus(V1,V,Out),[V1,V],[Out]).
16.24/4.84	input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
16.24/4.84	input_output_vars(g(V1,V,Out),[V1,V],[Out]).
16.24/4.84	
16.24/4.84	
16.24/4.84	CoFloCo proof output:
16.24/4.84	Preprocessing Cost Relations
16.24/4.84	=====================================
16.24/4.84	
16.24/4.84	#### Computed strongly connected components 
16.24/4.84	0. recursive  : [plus/3]
16.24/4.84	1. recursive  : [f/4]
16.24/4.84	2. non_recursive  : [g/3]
16.24/4.84	3. non_recursive  : [start/3]
16.24/4.84	
16.24/4.84	#### Obtained direct recursion through partial evaluation 
16.24/4.84	0. SCC is partially evaluated into plus/3
16.24/4.84	1. SCC is partially evaluated into f/4
16.24/4.84	2. SCC is partially evaluated into g/3
16.24/4.84	3. SCC is partially evaluated into start/3
16.24/4.84	
16.24/4.84	Control-Flow Refinement of Cost Relations
16.24/4.84	=====================================
16.24/4.84	
16.24/4.84	### Specialization of cost equations plus/3 
16.24/4.84	* CE 5 is refined into CE [10] 
16.24/4.84	* CE 4 is refined into CE [11] 
16.24/4.84	
16.24/4.84	
16.24/4.84	### Cost equations --> "Loop" of plus/3 
16.24/4.84	* CEs [11] --> Loop 8 
16.24/4.84	* CEs [10] --> Loop 9 
16.24/4.84	
16.24/4.84	### Ranking functions of CR plus(V1,V,Out) 
16.24/4.84	* RF of phase [9]: [V]
16.24/4.84	
16.24/4.84	#### Partial ranking functions of CR plus(V1,V,Out) 
16.24/4.84	* Partial RF of phase [9]:
16.24/4.84	  - RF of loop [9:1]:
16.24/4.84	    V
16.24/4.84	
16.24/4.84	
16.24/4.84	### Specialization of cost equations f/4 
16.24/4.84	* CE 7 is refined into CE [12] 
16.24/4.84	* CE 6 is refined into CE [13,14] 
16.24/4.84	
16.24/4.84	
16.24/4.84	### Cost equations --> "Loop" of f/4 
16.24/4.84	* CEs [14] --> Loop 10 
16.24/4.84	* CEs [13] --> Loop 11 
16.24/4.84	* CEs [12] --> Loop 12 
16.24/4.84	
16.24/4.84	### Ranking functions of CR f(V1,V,V2,Out) 
16.24/4.84	
16.24/4.84	#### Partial ranking functions of CR f(V1,V,V2,Out) 
16.24/4.84	
16.24/4.84	
16.24/4.84	### Specialization of cost equations g/3 
16.24/4.84	* CE 9 is refined into CE [15] 
16.24/4.84	* CE 8 is refined into CE [16] 
16.24/4.84	
16.24/4.84	
16.24/4.84	### Cost equations --> "Loop" of g/3 
16.24/4.84	* CEs [15] --> Loop 13 
16.24/4.84	* CEs [16] --> Loop 14 
16.24/4.84	
16.24/4.84	### Ranking functions of CR g(V1,V,Out) 
16.24/4.84	
16.24/4.84	#### Partial ranking functions of CR g(V1,V,Out) 
16.24/4.84	
16.24/4.84	
16.24/4.84	### Specialization of cost equations start/3 
16.24/4.84	* CE 1 is refined into CE [17,18] 
16.24/4.84	* CE 2 is refined into CE [19,20] 
16.24/4.84	* CE 3 is refined into CE [21,22] 
16.24/4.84	
16.24/4.84	
16.24/4.84	### Cost equations --> "Loop" of start/3 
16.24/4.84	* CEs [17] --> Loop 15 
16.24/4.84	* CEs [18,19,20,21,22] --> Loop 16 
16.24/4.84	
16.24/4.84	### Ranking functions of CR start(V1,V,V2) 
16.24/4.84	
16.24/4.84	#### Partial ranking functions of CR start(V1,V,V2) 
16.24/4.84	
16.24/4.84	
16.24/4.84	Computing Bounds
16.24/4.84	=====================================
16.24/4.84	
16.24/4.84	#### Cost of chains of plus(V1,V,Out):
16.24/4.84	* Chain [[9],8]: 1*it(9)+1
16.24/4.84	  Such that:it(9) =< V
16.24/4.84	
16.24/4.84	  with precondition: [V+V1=Out,V1>=0,V>=1] 
16.24/4.84	
16.24/4.84	* Chain [8]: 1
16.24/4.84	  with precondition: [V=0,V1=Out,V1>=0] 
16.24/4.84	
16.24/4.84	
16.24/4.84	#### Cost of chains of f(V1,V,V2,Out):
16.24/4.84	* Chain [12]: 0
16.24/4.84	  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 
16.24/4.84	
16.24/4.84	* Chain [11,12]: 2
16.24/4.84	  with precondition: [V1=0,V=1,V2=0,Out=0] 
16.24/4.84	
16.24/4.84	* Chain [10,12]: 1*s(1)+2
16.24/4.84	  Such that:s(1) =< V2
16.24/4.84	
16.24/4.84	  with precondition: [V1=0,V=1,Out=0,V2>=1] 
16.24/4.84	
16.24/4.84	
16.24/4.84	#### Cost of chains of g(V1,V,Out):
16.24/4.84	* Chain [14]: 1
16.24/4.84	  with precondition: [V1=Out,V1>=0,V>=0] 
16.24/4.84	
16.24/4.84	* Chain [13]: 1
16.24/4.84	  with precondition: [V=Out,V1>=0,V>=0] 
16.24/4.84	
16.24/4.84	
16.24/4.84	#### Cost of chains of start(V1,V,V2):
16.24/4.84	* Chain [16]: 1*s(2)+1*s(3)+2
16.24/4.84	  Such that:s(2) =< V
16.24/4.84	s(3) =< V2
16.24/4.84	
16.24/4.84	  with precondition: [V1>=0,V>=0] 
16.24/4.84	
16.24/4.84	* Chain [15]: 1
16.24/4.84	  with precondition: [V=0,V1>=0] 
16.24/4.84	
16.24/4.84	
16.24/4.84	Closed-form bounds of start(V1,V,V2): 
16.24/4.84	-------------------------------------
16.24/4.84	* Chain [16] with precondition: [V1>=0,V>=0] 
16.24/4.84	    - Upper bound: V+2+nat(V2) 
16.24/4.84	    - Complexity: n 
16.24/4.84	* Chain [15] with precondition: [V=0,V1>=0] 
16.24/4.84	    - Upper bound: 1 
16.24/4.84	    - Complexity: constant 
16.24/4.84	
16.24/4.84	### Maximum cost of start(V1,V,V2): V+1+nat(V2)+1 
16.24/4.84	Asymptotic class: n 
16.24/4.84	* Total analysis performed in 101 ms.
16.24/4.84	
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(12)
16.24/4.84	BOUNDS(1, n^1)
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
16.24/4.84	Transformed a relative TRS into a decreasing-loop problem.
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(14)
16.24/4.84	Obligation:
16.24/4.84	Analyzing the following TRS for decreasing loops:
16.24/4.84	
16.24/4.84	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.24/4.84	
16.24/4.84	
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   +(X, 0) -> X
16.24/4.84	   +(X, s(Y)) -> s(+(X, Y))
16.24/4.84	   f(0, s(0), X) -> f(X, +(X, X), X)
16.24/4.84	   g(X, Y) -> X
16.24/4.84	   g(X, Y) -> Y
16.24/4.84	
16.24/4.84	S is empty.
16.24/4.84	Rewrite Strategy: INNERMOST
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(15) DecreasingLoopProof (LOWER BOUND(ID))
16.24/4.84	The following loop(s) give(s) rise to the lower bound Omega(n^1):
16.24/4.84	
16.24/4.84	The rewrite sequence
16.24/4.84	
16.24/4.84	+(X, s(Y)) ->^+ s(+(X, Y))
16.24/4.84	
16.24/4.84	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
16.24/4.84	
16.24/4.84	The pumping substitution is [Y / s(Y)].
16.24/4.84	
16.24/4.84	The result substitution is [ ].
16.24/4.84	
16.24/4.84	
16.24/4.84	
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(16)
16.24/4.84	Complex Obligation (BEST)
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(17)
16.24/4.84	Obligation:
16.24/4.84	Proved the lower bound n^1 for the following obligation:
16.24/4.84	
16.24/4.84	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.24/4.84	
16.24/4.84	
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   +(X, 0) -> X
16.24/4.84	   +(X, s(Y)) -> s(+(X, Y))
16.24/4.84	   f(0, s(0), X) -> f(X, +(X, X), X)
16.24/4.84	   g(X, Y) -> X
16.24/4.84	   g(X, Y) -> Y
16.24/4.84	
16.24/4.84	S is empty.
16.24/4.84	Rewrite Strategy: INNERMOST
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(18) LowerBoundPropagationProof (FINISHED)
16.24/4.84	Propagated lower bound.
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(19)
16.24/4.84	BOUNDS(n^1, INF)
16.24/4.84	
16.24/4.84	----------------------------------------
16.24/4.84	
16.24/4.84	(20)
16.24/4.84	Obligation:
16.24/4.84	Analyzing the following TRS for decreasing loops:
16.24/4.84	
16.24/4.84	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.24/4.84	
16.24/4.84	
16.24/4.84	The TRS R consists of the following rules:
16.24/4.84	
16.24/4.84	   +(X, 0) -> X
16.24/4.84	   +(X, s(Y)) -> s(+(X, Y))
16.24/4.84	   f(0, s(0), X) -> f(X, +(X, X), X)
16.24/4.84	   g(X, Y) -> X
16.24/4.84	   g(X, Y) -> Y
16.24/4.84	
16.24/4.84	S is empty.
16.24/4.84	Rewrite Strategy: INNERMOST
16.29/4.91	EOF