11.97/4.65	WORST_CASE(Omega(n^1), O(n^1))
11.97/4.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.97/4.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.97/4.67	
11.97/4.67	
11.97/4.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
11.97/4.67	
11.97/4.67	(0) CpxTRS
11.97/4.67	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
11.97/4.67	(2) CpxWeightedTrs
11.97/4.67	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
11.97/4.67	    (4) CpxTypedWeightedTrs
11.97/4.67	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
11.97/4.67	    (6) CpxTypedWeightedCompleteTrs
11.97/4.67	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
11.97/4.67	    (8) CpxRNTS
11.97/4.67	    (9) CompleteCoflocoProof [FINISHED, 40 ms]
11.97/4.67	    (10) BOUNDS(1, n^1)
11.97/4.67	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
11.97/4.67	(12) TRS for Loop Detection
11.97/4.67	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
11.97/4.67	    (14) BEST
11.97/4.67	        (15) proven lower bound
11.97/4.67	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
11.97/4.67	            (17) BOUNDS(n^1, INF)
11.97/4.67	        (18) TRS for Loop Detection
11.97/4.67	
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(0)
11.97/4.67	Obligation:
11.97/4.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
11.97/4.67	
11.97/4.67	
11.97/4.67	The TRS R consists of the following rules:
11.97/4.67	
11.97/4.67	   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
11.97/4.67	   u21(ackout(X), Y) -> u22(ackin(Y, X))
11.97/4.67	
11.97/4.67	S is empty.
11.97/4.67	Rewrite Strategy: INNERMOST
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
11.97/4.67	Transformed relative TRS to weighted TRS
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(2)
11.97/4.67	Obligation:
11.97/4.67	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
11.97/4.67	
11.97/4.67	
11.97/4.67	The TRS R consists of the following rules:
11.97/4.67	
11.97/4.67	   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
11.97/4.67	   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]
11.97/4.67	
11.97/4.67	Rewrite Strategy: INNERMOST
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
11.97/4.67	Infered types.
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(4)
11.97/4.67	Obligation:
11.97/4.67	Runtime Complexity Weighted TRS with Types.
11.97/4.67	The TRS R consists of the following rules:
11.97/4.67	
11.97/4.67	   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
11.97/4.67	   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]
11.97/4.67	
11.97/4.67	The TRS has the following type information:
11.97/4.67	ackin :: s -> s -> ackout:u22
11.97/4.67	s :: s -> s
11.97/4.67	u21 :: ackout:u22 -> s -> ackout:u22
11.97/4.67	ackout :: s -> ackout:u22
11.97/4.67	u22 :: ackout:u22 -> ackout:u22
11.97/4.67	
11.97/4.67	Rewrite Strategy: INNERMOST
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(5) CompletionProof (UPPER BOUND(ID))
11.97/4.67	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
11.97/4.67	
11.97/4.67	   ackin(v0, v1) -> null_ackin [0]
11.97/4.67	   u21(v0, v1) -> null_u21 [0]
11.97/4.67	
11.97/4.67	And the following fresh constants:    null_ackin, null_u21, const
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(6)
11.97/4.67	Obligation:
11.97/4.67	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
11.97/4.67	
11.97/4.67	Runtime Complexity Weighted TRS with Types.
11.97/4.67	The TRS R consists of the following rules:
11.97/4.67	
11.97/4.67	   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
11.97/4.67	   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]
11.97/4.67	   ackin(v0, v1) -> null_ackin [0]
11.97/4.67	   u21(v0, v1) -> null_u21 [0]
11.97/4.67	
11.97/4.67	The TRS has the following type information:
11.97/4.67	ackin :: s -> s -> ackout:u22:null_ackin:null_u21
11.97/4.67	s :: s -> s
11.97/4.67	u21 :: ackout:u22:null_ackin:null_u21 -> s -> ackout:u22:null_ackin:null_u21
11.97/4.67	ackout :: s -> ackout:u22:null_ackin:null_u21
11.97/4.67	u22 :: ackout:u22:null_ackin:null_u21 -> ackout:u22:null_ackin:null_u21
11.97/4.67	null_ackin :: ackout:u22:null_ackin:null_u21
11.97/4.67	null_u21 :: ackout:u22:null_ackin:null_u21
11.97/4.67	const :: s
11.97/4.67	
11.97/4.67	Rewrite Strategy: INNERMOST
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
11.97/4.67	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
11.97/4.67	The constant constructors are abstracted as follows:
11.97/4.67	
11.97/4.67	   null_ackin => 0
11.97/4.67	   null_u21 => 0
11.97/4.67	   const => 0
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(8)
11.97/4.67	Obligation:
11.97/4.67	Complexity RNTS consisting of the following rules:
11.97/4.67	
11.97/4.67	   ackin(z, z') -{ 1 }-> u21(ackin(1 + X, Y), X) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
11.97/4.67	   ackin(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
11.97/4.67	   u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
11.97/4.67	   u21(z, z') -{ 1 }-> 1 + ackin(Y, X) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
11.97/4.67	
11.97/4.67	Only complete derivations are relevant for the runtime complexity.
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(9) CompleteCoflocoProof (FINISHED)
11.97/4.67	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
11.97/4.67	
11.97/4.67	eq(start(V1, V),0,[ackin(V1, V, Out)],[V1 >= 0,V >= 0]).
11.97/4.67	eq(start(V1, V),0,[u21(V1, V, Out)],[V1 >= 0,V >= 0]).
11.97/4.67	eq(ackin(V1, V, Out),1,[ackin(1 + X1, Y1, Ret0),u21(Ret0, X1, Ret)],[Out = Ret,V1 = 1 + X1,Y1 >= 0,V = 1 + Y1,X1 >= 0]).
11.97/4.67	eq(u21(V1, V, Out),1,[ackin(Y2, X2, Ret1)],[Out = 1 + Ret1,V1 = 1 + X2,V = Y2,Y2 >= 0,X2 >= 0]).
11.97/4.67	eq(ackin(V1, V, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
11.97/4.67	eq(u21(V1, V, Out),0,[],[Out = 0,V5 >= 0,V4 >= 0,V1 = V5,V = V4]).
11.97/4.67	input_output_vars(ackin(V1,V,Out),[V1,V],[Out]).
11.97/4.67	input_output_vars(u21(V1,V,Out),[V1,V],[Out]).
11.97/4.67	
11.97/4.67	
11.97/4.67	CoFloCo proof output:
11.97/4.67	Preprocessing Cost Relations
11.97/4.67	=====================================
11.97/4.67	
11.97/4.67	#### Computed strongly connected components 
11.97/4.67	0. recursive [multiple] : [ackin/3,u21/3]
11.97/4.67	1. non_recursive  : [start/2]
11.97/4.67	
11.97/4.67	#### Obtained direct recursion through partial evaluation 
11.97/4.67	0. SCC is partially evaluated into ackin/3
11.97/4.67	1. SCC is partially evaluated into start/2
11.97/4.67	
11.97/4.67	Control-Flow Refinement of Cost Relations
11.97/4.67	=====================================
11.97/4.67	
11.97/4.67	### Specialization of cost equations ackin/3 
11.97/4.67	* CE 6 is refined into CE [7] 
11.97/4.67	* CE 5 is refined into CE [8] 
11.97/4.67	* CE 4 is refined into CE [9] 
11.97/4.67	
11.97/4.67	
11.97/4.67	### Cost equations --> "Loop" of ackin/3 
11.97/4.67	* CEs [9] --> Loop 5 
11.97/4.67	* CEs [8] --> Loop 6 
11.97/4.67	* CEs [7] --> Loop 7 
11.97/4.67	
11.97/4.67	### Ranking functions of CR ackin(V1,V,Out) 
11.97/4.67	
11.97/4.67	#### Partial ranking functions of CR ackin(V1,V,Out) 
11.97/4.67	* Partial RF of phase [5,6]:
11.97/4.67	  - RF of loop [5:1,6:1]:
11.97/4.67	    V depends on loops [6:2] 
11.97/4.67	  - RF of loop [6:2]:
11.97/4.67	    V1
11.97/4.67	
11.97/4.67	
11.97/4.67	### Specialization of cost equations start/2 
11.97/4.67	* CE 1 is refined into CE [10] 
11.97/4.67	* CE 2 is refined into CE [11] 
11.97/4.67	* CE 3 is refined into CE [12] 
11.97/4.67	
11.97/4.67	
11.97/4.67	### Cost equations --> "Loop" of start/2 
11.97/4.67	* CEs [10,11,12] --> Loop 8 
11.97/4.67	
11.97/4.67	### Ranking functions of CR start(V1,V) 
11.97/4.67	
11.97/4.67	#### Partial ranking functions of CR start(V1,V) 
11.97/4.67	
11.97/4.67	
11.97/4.67	Computing Bounds
11.97/4.67	=====================================
11.97/4.67	
11.97/4.67	#### Cost of chains of ackin(V1,V,Out):
11.97/4.67	* Chain [7]: 0
11.97/4.67	  with precondition: [Out=0,V1>=0,V>=0] 
11.97/4.67	
11.97/4.67	* Chain [multiple([5,6],[[7]])]: 1*it(5)+0
11.97/4.67	  Such that:it(5) =< V
11.97/4.67	
11.97/4.67	  with precondition: [Out=0,V1>=1,V>=1] 
11.97/4.67	
11.97/4.67	
11.97/4.67	#### Cost of chains of start(V1,V):
11.97/4.67	* Chain [8]: 1*s(2)+1*s(3)+1
11.97/4.67	  Such that:s(2) =< V1
11.97/4.67	s(3) =< V
11.97/4.67	
11.97/4.67	  with precondition: [V1>=0,V>=0] 
11.97/4.67	
11.97/4.67	
11.97/4.67	Closed-form bounds of start(V1,V): 
11.97/4.67	-------------------------------------
11.97/4.67	* Chain [8] with precondition: [V1>=0,V>=0] 
11.97/4.67	    - Upper bound: V1+V+1 
11.97/4.67	    - Complexity: n 
11.97/4.67	
11.97/4.67	### Maximum cost of start(V1,V): V1+V+1 
11.97/4.67	Asymptotic class: n 
11.97/4.67	* Total analysis performed in 73 ms.
11.97/4.67	
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(10)
11.97/4.67	BOUNDS(1, n^1)
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
11.97/4.67	Transformed a relative TRS into a decreasing-loop problem.
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(12)
11.97/4.67	Obligation:
11.97/4.67	Analyzing the following TRS for decreasing loops:
11.97/4.67	
11.97/4.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
11.97/4.67	
11.97/4.67	
11.97/4.67	The TRS R consists of the following rules:
11.97/4.67	
11.97/4.67	   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
11.97/4.67	   u21(ackout(X), Y) -> u22(ackin(Y, X))
11.97/4.67	
11.97/4.67	S is empty.
11.97/4.67	Rewrite Strategy: INNERMOST
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(13) DecreasingLoopProof (LOWER BOUND(ID))
11.97/4.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
11.97/4.67	
11.97/4.67	The rewrite sequence
11.97/4.67	
11.97/4.67	ackin(s(X), s(Y)) ->^+ u21(ackin(s(X), Y), X)
11.97/4.67	
11.97/4.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
11.97/4.67	
11.97/4.67	The pumping substitution is [Y / s(Y)].
11.97/4.67	
11.97/4.67	The result substitution is [ ].
11.97/4.67	
11.97/4.67	
11.97/4.67	
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(14)
11.97/4.67	Complex Obligation (BEST)
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(15)
11.97/4.67	Obligation:
11.97/4.67	Proved the lower bound n^1 for the following obligation:
11.97/4.67	
11.97/4.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
11.97/4.67	
11.97/4.67	
11.97/4.67	The TRS R consists of the following rules:
11.97/4.67	
11.97/4.67	   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
11.97/4.67	   u21(ackout(X), Y) -> u22(ackin(Y, X))
11.97/4.67	
11.97/4.67	S is empty.
11.97/4.67	Rewrite Strategy: INNERMOST
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(16) LowerBoundPropagationProof (FINISHED)
11.97/4.67	Propagated lower bound.
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(17)
11.97/4.67	BOUNDS(n^1, INF)
11.97/4.67	
11.97/4.67	----------------------------------------
11.97/4.67	
11.97/4.67	(18)
11.97/4.67	Obligation:
11.97/4.67	Analyzing the following TRS for decreasing loops:
11.97/4.67	
11.97/4.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
11.97/4.67	
11.97/4.67	
11.97/4.67	The TRS R consists of the following rules:
11.97/4.67	
11.97/4.67	   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
11.97/4.67	   u21(ackout(X), Y) -> u22(ackin(Y, X))
11.97/4.67	
11.97/4.67	S is empty.
11.97/4.67	Rewrite Strategy: INNERMOST
12.11/4.72	EOF