27.66/8.75	WORST_CASE(Omega(n^1), O(n^1))
27.66/8.76	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.66/8.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.66/8.76	
27.66/8.76	
27.66/8.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
27.66/8.76	
27.66/8.76	(0) CpxTRS
27.66/8.76	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
27.66/8.76	(2) CpxTRS
27.66/8.76	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
27.66/8.76	    (4) CpxWeightedTrs
27.66/8.76	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
27.66/8.76	    (6) CpxTypedWeightedTrs
27.66/8.76	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
27.66/8.76	    (8) CpxTypedWeightedCompleteTrs
27.66/8.76	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms]
27.66/8.76	    (10) CpxRNTS
27.66/8.76	    (11) CompleteCoflocoProof [FINISHED, 415 ms]
27.66/8.76	    (12) BOUNDS(1, n^1)
27.66/8.76	(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
27.66/8.76	(14) TRS for Loop Detection
27.66/8.76	    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
27.66/8.76	    (16) BEST
27.66/8.76	        (17) proven lower bound
27.66/8.76	            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
27.66/8.76	            (19) BOUNDS(n^1, INF)
27.66/8.76	        (20) TRS for Loop Detection
27.66/8.76	
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(0)
27.66/8.76	Obligation:
27.66/8.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
27.66/8.76	
27.66/8.76	
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y)
27.66/8.76	   quot(0, s(Y)) -> 0
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
27.66/8.76	   log(s(0)) -> 0
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))
27.66/8.76	
27.66/8.76	S is empty.
27.66/8.76	Rewrite Strategy: FULL
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
27.66/8.76	Converted rc-obligation to irc-obligation.
27.66/8.76	
27.66/8.76	The duplicating contexts are:
27.66/8.76	quot(s(X), s([]))
27.66/8.76	
27.66/8.76	
27.66/8.76	The defined contexts are:
27.66/8.76	log(s([]))
27.66/8.76	quot([], s(x1))
27.66/8.76	min([], x1)
27.66/8.76	quot([], s(s(0)))
27.66/8.76	
27.66/8.76	
27.66/8.76	[] just represents basic- or constructor-terms in the following defined contexts:
27.66/8.76	quot([], s(x1))
27.66/8.76	
27.66/8.76	
27.66/8.76	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(2)
27.66/8.76	Obligation:
27.66/8.76	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
27.66/8.76	
27.66/8.76	
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y)
27.66/8.76	   quot(0, s(Y)) -> 0
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
27.66/8.76	   log(s(0)) -> 0
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))
27.66/8.76	
27.66/8.76	S is empty.
27.66/8.76	Rewrite Strategy: INNERMOST
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
27.66/8.76	Transformed relative TRS to weighted TRS
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(4)
27.66/8.76	Obligation:
27.66/8.76	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
27.66/8.76	
27.66/8.76	
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X [1]
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y) [1]
27.66/8.76	   quot(0, s(Y)) -> 0 [1]
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) [1]
27.66/8.76	   log(s(0)) -> 0 [1]
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0)))))) [1]
27.66/8.76	
27.66/8.76	Rewrite Strategy: INNERMOST
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
27.66/8.76	Infered types.
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(6)
27.66/8.76	Obligation:
27.66/8.76	Runtime Complexity Weighted TRS with Types.
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X [1]
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y) [1]
27.66/8.76	   quot(0, s(Y)) -> 0 [1]
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) [1]
27.66/8.76	   log(s(0)) -> 0 [1]
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0)))))) [1]
27.66/8.76	
27.66/8.76	The TRS has the following type information:
27.66/8.76	min :: 0:s -> 0:s -> 0:s
27.66/8.76	0 :: 0:s
27.66/8.76	s :: 0:s -> 0:s
27.66/8.76	quot :: 0:s -> 0:s -> 0:s
27.66/8.76	log :: 0:s -> 0:s
27.66/8.76	
27.66/8.76	Rewrite Strategy: INNERMOST
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(7) CompletionProof (UPPER BOUND(ID))
27.66/8.76	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
27.66/8.76	
27.66/8.76	   min(v0, v1) -> null_min [0]
27.66/8.76	   quot(v0, v1) -> null_quot [0]
27.66/8.76	   log(v0) -> null_log [0]
27.66/8.76	
27.66/8.76	And the following fresh constants:    null_min, null_quot, null_log
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(8)
27.66/8.76	Obligation:
27.66/8.76	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
27.66/8.76	
27.66/8.76	Runtime Complexity Weighted TRS with Types.
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X [1]
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y) [1]
27.66/8.76	   quot(0, s(Y)) -> 0 [1]
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) [1]
27.66/8.76	   log(s(0)) -> 0 [1]
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0)))))) [1]
27.66/8.76	   min(v0, v1) -> null_min [0]
27.66/8.76	   quot(v0, v1) -> null_quot [0]
27.66/8.76	   log(v0) -> null_log [0]
27.66/8.76	
27.66/8.76	The TRS has the following type information:
27.66/8.76	min :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log
27.66/8.76	0 :: 0:s:null_min:null_quot:null_log
27.66/8.76	s :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log
27.66/8.76	quot :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log
27.66/8.76	log :: 0:s:null_min:null_quot:null_log -> 0:s:null_min:null_quot:null_log
27.66/8.76	null_min :: 0:s:null_min:null_quot:null_log
27.66/8.76	null_quot :: 0:s:null_min:null_quot:null_log
27.66/8.76	null_log :: 0:s:null_min:null_quot:null_log
27.66/8.76	
27.66/8.76	Rewrite Strategy: INNERMOST
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
27.66/8.76	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
27.66/8.76	The constant constructors are abstracted as follows:
27.66/8.76	
27.66/8.76	   0 => 0
27.66/8.76	   null_min => 0
27.66/8.76	   null_quot => 0
27.66/8.76	   null_log => 0
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(10)
27.66/8.76	Obligation:
27.66/8.76	Complexity RNTS consisting of the following rules:
27.66/8.76	
27.66/8.76	   log(z) -{ 1 }-> 0 :|: z = 1 + 0
27.66/8.76	   log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
27.66/8.76	   log(z) -{ 1 }-> 1 + log(1 + quot(X, 1 + (1 + 0))) :|: z = 1 + (1 + X), X >= 0
27.66/8.76	   min(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
27.66/8.76	   min(z, z') -{ 1 }-> min(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
27.66/8.76	   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
27.66/8.76	   quot(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
27.66/8.76	   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
27.66/8.76	   quot(z, z') -{ 1 }-> 1 + quot(min(X, Y), 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
27.66/8.76	
27.66/8.76	Only complete derivations are relevant for the runtime complexity.
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(11) CompleteCoflocoProof (FINISHED)
27.66/8.76	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
27.66/8.76	
27.66/8.76	eq(start(V1, V),0,[min(V1, V, Out)],[V1 >= 0,V >= 0]).
27.66/8.76	eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]).
27.66/8.76	eq(start(V1, V),0,[log(V1, Out)],[V1 >= 0]).
27.66/8.76	eq(min(V1, V, Out),1,[],[Out = X1,X1 >= 0,V1 = X1,V = 0]).
27.66/8.76	eq(min(V1, V, Out),1,[min(X2, Y1, Ret)],[Out = Ret,V1 = 1 + X2,Y1 >= 0,V = 1 + Y1,X2 >= 0]).
27.66/8.76	eq(quot(V1, V, Out),1,[],[Out = 0,Y2 >= 0,V = 1 + Y2,V1 = 0]).
27.66/8.76	eq(quot(V1, V, Out),1,[min(X3, Y3, Ret10),quot(Ret10, 1 + Y3, Ret1)],[Out = 1 + Ret1,V1 = 1 + X3,Y3 >= 0,V = 1 + Y3,X3 >= 0]).
27.66/8.76	eq(log(V1, Out),1,[],[Out = 0,V1 = 1]).
27.66/8.76	eq(log(V1, Out),1,[quot(X4, 1 + (1 + 0), Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V1 = 2 + X4,X4 >= 0]).
27.66/8.76	eq(min(V1, V, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
27.66/8.76	eq(quot(V1, V, Out),0,[],[Out = 0,V5 >= 0,V4 >= 0,V1 = V5,V = V4]).
27.66/8.76	eq(log(V1, Out),0,[],[Out = 0,V6 >= 0,V1 = V6]).
27.66/8.76	input_output_vars(min(V1,V,Out),[V1,V],[Out]).
27.66/8.76	input_output_vars(quot(V1,V,Out),[V1,V],[Out]).
27.66/8.76	input_output_vars(log(V1,Out),[V1],[Out]).
27.66/8.76	
27.66/8.76	
27.66/8.76	CoFloCo proof output:
27.66/8.76	Preprocessing Cost Relations
27.66/8.76	=====================================
27.66/8.76	
27.66/8.76	#### Computed strongly connected components 
27.66/8.76	0. recursive  : [min/3]
27.66/8.76	1. recursive  : [quot/3]
27.66/8.76	2. recursive  : [log/2]
27.66/8.76	3. non_recursive  : [start/2]
27.66/8.76	
27.66/8.76	#### Obtained direct recursion through partial evaluation 
27.66/8.76	0. SCC is partially evaluated into min/3
27.66/8.76	1. SCC is partially evaluated into quot/3
27.66/8.76	2. SCC is partially evaluated into log/2
27.66/8.76	3. SCC is partially evaluated into start/2
27.66/8.76	
27.66/8.76	Control-Flow Refinement of Cost Relations
27.66/8.76	=====================================
27.66/8.76	
27.66/8.76	### Specialization of cost equations min/3 
27.66/8.76	* CE 6 is refined into CE [13] 
27.66/8.76	* CE 4 is refined into CE [14] 
27.66/8.76	* CE 5 is refined into CE [15] 
27.66/8.76	
27.66/8.76	
27.66/8.76	### Cost equations --> "Loop" of min/3 
27.66/8.76	* CEs [15] --> Loop 9 
27.66/8.76	* CEs [13] --> Loop 10 
27.66/8.76	* CEs [14] --> Loop 11 
27.66/8.76	
27.66/8.76	### Ranking functions of CR min(V1,V,Out) 
27.66/8.76	* RF of phase [9]: [V,V1]
27.66/8.76	
27.66/8.76	#### Partial ranking functions of CR min(V1,V,Out) 
27.66/8.76	* Partial RF of phase [9]:
27.66/8.76	  - RF of loop [9:1]:
27.66/8.76	    V
27.66/8.76	    V1
27.66/8.76	
27.66/8.76	
27.66/8.76	### Specialization of cost equations quot/3 
27.66/8.76	* CE 7 is refined into CE [16] 
27.66/8.76	* CE 9 is refined into CE [17] 
27.66/8.76	* CE 8 is refined into CE [18,19,20] 
27.66/8.76	
27.66/8.76	
27.66/8.76	### Cost equations --> "Loop" of quot/3 
27.66/8.76	* CEs [20] --> Loop 12 
27.66/8.76	* CEs [19] --> Loop 13 
27.66/8.76	* CEs [18] --> Loop 14 
27.66/8.76	* CEs [16,17] --> Loop 15 
27.66/8.76	
27.66/8.76	### Ranking functions of CR quot(V1,V,Out) 
27.66/8.76	* RF of phase [12]: [V1-1,V1-V+1]
27.66/8.76	* RF of phase [14]: [V1]
27.66/8.76	
27.66/8.76	#### Partial ranking functions of CR quot(V1,V,Out) 
27.66/8.76	* Partial RF of phase [12]:
27.66/8.76	  - RF of loop [12:1]:
27.66/8.76	    V1-1
27.66/8.76	    V1-V+1
27.66/8.76	* Partial RF of phase [14]:
27.66/8.76	  - RF of loop [14:1]:
27.66/8.76	    V1
27.66/8.76	
27.66/8.76	
27.66/8.76	### Specialization of cost equations log/2 
27.66/8.76	* CE 10 is refined into CE [21] 
27.66/8.76	* CE 12 is refined into CE [22] 
27.66/8.76	* CE 11 is refined into CE [23,24,25,26] 
27.66/8.76	
27.66/8.76	
27.66/8.76	### Cost equations --> "Loop" of log/2 
27.66/8.76	* CEs [26] --> Loop 16 
27.66/8.76	* CEs [25] --> Loop 17 
27.66/8.76	* CEs [24] --> Loop 18 
27.66/8.76	* CEs [23] --> Loop 19 
27.66/8.76	* CEs [21,22] --> Loop 20 
27.66/8.76	
27.66/8.76	### Ranking functions of CR log(V1,Out) 
27.66/8.76	* RF of phase [16,17]: [V1-3,V1/2-3/2]
27.66/8.76	
27.66/8.76	#### Partial ranking functions of CR log(V1,Out) 
27.66/8.76	* Partial RF of phase [16,17]:
27.66/8.76	  - RF of loop [16:1]:
27.66/8.76	    V1/2-2
27.66/8.76	  - RF of loop [17:1]:
27.66/8.76	    V1-3
27.66/8.76	
27.66/8.76	
27.66/8.76	### Specialization of cost equations start/2 
27.66/8.76	* CE 1 is refined into CE [27,28,29] 
27.66/8.76	* CE 2 is refined into CE [30,31,32,33,34] 
27.66/8.76	* CE 3 is refined into CE [35,36,37,38,39,40] 
27.66/8.76	
27.66/8.76	
27.66/8.76	### Cost equations --> "Loop" of start/2 
27.66/8.76	* CEs [30] --> Loop 21 
27.66/8.76	* CEs [27,28,29,31,32,33,34,35,36,37,38,39,40] --> Loop 22 
27.66/8.76	
27.66/8.76	### Ranking functions of CR start(V1,V) 
27.66/8.76	
27.66/8.76	#### Partial ranking functions of CR start(V1,V) 
27.66/8.76	
27.66/8.76	
27.66/8.76	Computing Bounds
27.66/8.76	=====================================
27.66/8.76	
27.66/8.76	#### Cost of chains of min(V1,V,Out):
27.66/8.76	* Chain [[9],11]: 1*it(9)+1
27.66/8.76	  Such that:it(9) =< V
27.66/8.76	
27.66/8.76	  with precondition: [V1=Out+V,V>=1,V1>=V] 
27.66/8.76	
27.66/8.76	* Chain [[9],10]: 1*it(9)+0
27.66/8.76	  Such that:it(9) =< V
27.66/8.76	
27.66/8.76	  with precondition: [Out=0,V1>=1,V>=1] 
27.66/8.76	
27.66/8.76	* Chain [11]: 1
27.66/8.76	  with precondition: [V=0,V1=Out,V1>=0] 
27.66/8.76	
27.66/8.76	* Chain [10]: 0
27.66/8.76	  with precondition: [Out=0,V1>=0,V>=0] 
27.66/8.76	
27.66/8.76	
27.66/8.76	#### Cost of chains of quot(V1,V,Out):
27.66/8.76	* Chain [[14],15]: 2*it(14)+1
27.66/8.76	  Such that:it(14) =< Out
27.66/8.76	
27.66/8.76	  with precondition: [V=1,Out>=1,V1>=Out] 
27.66/8.76	
27.66/8.76	* Chain [[14],13,15]: 2*it(14)+1*s(2)+2
27.66/8.76	  Such that:s(2) =< 1
27.66/8.76	it(14) =< Out
27.66/8.76	
27.66/8.76	  with precondition: [V=1,Out>=2,V1>=Out] 
27.66/8.76	
27.66/8.76	* Chain [[12],15]: 2*it(12)+1*s(5)+1
27.66/8.76	  Such that:it(12) =< V1-V+1
27.66/8.76	aux(3) =< V1
27.66/8.76	it(12) =< aux(3)
27.66/8.76	s(5) =< aux(3)
27.66/8.76	
27.66/8.76	  with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] 
27.66/8.76	
27.66/8.76	* Chain [[12],13,15]: 2*it(12)+1*s(2)+1*s(5)+2
27.66/8.76	  Such that:it(12) =< V1-V+1
27.66/8.76	s(2) =< V
27.66/8.76	aux(4) =< V1
27.66/8.76	it(12) =< aux(4)
27.66/8.76	s(5) =< aux(4)
27.66/8.76	
27.66/8.76	  with precondition: [V>=2,Out>=2,V1+3>=2*Out+V] 
27.66/8.76	
27.66/8.76	* Chain [15]: 1
27.66/8.76	  with precondition: [Out=0,V1>=0,V>=0] 
27.66/8.76	
27.66/8.76	* Chain [13,15]: 1*s(2)+2
27.66/8.76	  Such that:s(2) =< V
27.66/8.76	
27.66/8.76	  with precondition: [Out=1,V1>=1,V>=1] 
27.66/8.76	
27.66/8.76	
27.66/8.76	#### Cost of chains of log(V1,Out):
27.66/8.76	* Chain [[16,17],20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+1
27.66/8.76	  Such that:s(25) =< 2*V1
27.66/8.76	aux(14) =< 5/2*V1
27.66/8.76	aux(13) =< 5/2*V1+27/2
27.66/8.76	aux(15) =< V1
27.66/8.76	aux(16) =< V1/2
27.66/8.76	aux(8) =< aux(15)
27.66/8.76	it(16) =< aux(15)
27.66/8.76	it(17) =< aux(15)
27.66/8.76	aux(8) =< aux(16)
27.66/8.76	it(16) =< aux(16)
27.66/8.76	it(17) =< aux(16)
27.66/8.76	it(17) =< aux(13)
27.66/8.76	s(23) =< aux(13)
27.66/8.76	it(17) =< aux(14)
27.66/8.76	s(23) =< aux(14)
27.66/8.76	s(22) =< aux(8)*2
27.66/8.76	s(24) =< s(25)
27.66/8.76	s(21) =< s(23)
27.66/8.76	
27.66/8.76	  with precondition: [Out>=1,V1>=3*Out+1] 
27.66/8.76	
27.66/8.76	* Chain [[16,17],19,20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+3
27.66/8.76	  Such that:s(25) =< 2*V1
27.66/8.76	aux(14) =< 5/2*V1
27.66/8.76	aux(13) =< 5/2*V1+27/2
27.66/8.76	aux(17) =< V1
27.66/8.76	aux(18) =< V1/2
27.66/8.76	aux(8) =< aux(17)
27.66/8.76	it(16) =< aux(17)
27.66/8.76	it(17) =< aux(17)
27.66/8.76	aux(8) =< aux(18)
27.66/8.76	it(16) =< aux(18)
27.66/8.76	it(17) =< aux(18)
27.66/8.76	it(17) =< aux(13)
27.66/8.76	s(23) =< aux(13)
27.66/8.76	it(17) =< aux(14)
27.66/8.76	s(23) =< aux(14)
27.66/8.76	s(22) =< aux(8)*2
27.66/8.76	s(24) =< s(25)
27.66/8.76	s(21) =< s(23)
27.66/8.76	
27.66/8.76	  with precondition: [Out>=2,V1+2>=3*Out] 
27.66/8.76	
27.66/8.76	* Chain [[16,17],18,20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+1*s(26)+4
27.66/8.76	  Such that:s(26) =< 2
27.66/8.76	s(25) =< 2*V1
27.66/8.76	aux(14) =< 5/2*V1
27.66/8.76	aux(13) =< 5/2*V1+27/2
27.66/8.76	aux(19) =< V1
27.66/8.76	aux(20) =< V1/2
27.66/8.76	aux(8) =< aux(19)
27.66/8.76	it(16) =< aux(19)
27.66/8.76	it(17) =< aux(19)
27.66/8.76	aux(8) =< aux(20)
27.66/8.76	it(16) =< aux(20)
27.66/8.76	it(17) =< aux(20)
27.66/8.76	it(17) =< aux(13)
27.66/8.76	s(23) =< aux(13)
27.66/8.76	it(17) =< aux(14)
27.66/8.76	s(23) =< aux(14)
27.66/8.76	s(22) =< aux(8)*2
27.66/8.76	s(24) =< s(25)
27.66/8.76	s(21) =< s(23)
27.66/8.76	
27.66/8.76	  with precondition: [Out>=2,V1+3>=4*Out] 
27.66/8.76	
27.66/8.76	* Chain [[16,17],18,19,20]: 3*it(16)+2*it(17)+3*s(21)+1*s(22)+3*s(24)+1*s(26)+6
27.66/8.76	  Such that:s(26) =< 2
27.66/8.76	s(25) =< 2*V1
27.66/8.76	aux(14) =< 5/2*V1
27.66/8.76	aux(13) =< 5/2*V1+27/2
27.66/8.76	aux(21) =< V1
27.66/8.76	aux(22) =< V1/2
27.66/8.76	aux(8) =< aux(21)
27.66/8.76	it(16) =< aux(21)
27.66/8.76	it(17) =< aux(21)
27.66/8.76	aux(8) =< aux(22)
27.66/8.76	it(16) =< aux(22)
27.66/8.76	it(17) =< aux(22)
27.66/8.76	it(17) =< aux(13)
27.66/8.76	s(23) =< aux(13)
27.66/8.76	it(17) =< aux(14)
27.66/8.76	s(23) =< aux(14)
27.66/8.76	s(22) =< aux(8)*2
27.66/8.76	s(24) =< s(25)
27.66/8.76	s(21) =< s(23)
27.66/8.76	
27.66/8.76	  with precondition: [Out>=3,V1+7>=4*Out] 
27.66/8.76	
27.66/8.76	* Chain [20]: 1
27.66/8.76	  with precondition: [Out=0,V1>=0] 
27.66/8.76	
27.66/8.76	* Chain [19,20]: 3
27.66/8.76	  with precondition: [Out=1,V1>=2] 
27.66/8.76	
27.66/8.76	* Chain [18,20]: 1*s(26)+4
27.66/8.76	  Such that:s(26) =< 2
27.66/8.76	
27.66/8.76	  with precondition: [Out=1,V1>=3] 
27.66/8.76	
27.66/8.76	* Chain [18,19,20]: 1*s(26)+6
27.66/8.76	  Such that:s(26) =< 2
27.66/8.76	
27.66/8.76	  with precondition: [Out=2,V1>=3] 
27.66/8.76	
27.66/8.76	
27.66/8.76	#### Cost of chains of start(V1,V):
27.66/8.76	* Chain [22]: 4*s(53)+4*s(56)+2*s(58)+3*s(63)+12*s(70)+8*s(71)+4*s(73)+12*s(74)+12*s(75)+6
27.66/8.76	  Such that:aux(28) =< 2
27.66/8.76	aux(29) =< V1
27.66/8.76	aux(30) =< V1-V+1
27.66/8.76	aux(31) =< 2*V1
27.66/8.76	aux(32) =< V1/2
27.66/8.76	aux(33) =< 5/2*V1
27.66/8.76	aux(34) =< 5/2*V1+27/2
27.66/8.76	aux(35) =< V
27.66/8.76	s(63) =< aux(28)
27.66/8.76	s(56) =< aux(30)
27.66/8.76	s(53) =< aux(35)
27.66/8.76	s(69) =< aux(29)
27.66/8.76	s(70) =< aux(29)
27.66/8.76	s(71) =< aux(29)
27.66/8.76	s(69) =< aux(32)
27.66/8.76	s(70) =< aux(32)
27.66/8.76	s(71) =< aux(32)
27.66/8.76	s(71) =< aux(34)
27.66/8.76	s(72) =< aux(34)
27.66/8.76	s(71) =< aux(33)
27.66/8.76	s(72) =< aux(33)
27.66/8.76	s(73) =< s(69)*2
27.66/8.76	s(74) =< aux(31)
27.66/8.76	s(75) =< s(72)
27.66/8.76	s(56) =< aux(29)
27.66/8.76	s(58) =< aux(29)
27.66/8.76	
27.66/8.76	  with precondition: [V1>=0] 
27.66/8.76	
27.66/8.76	* Chain [21]: 1*s(102)+4*s(104)+2
27.66/8.76	  Such that:s(102) =< 1
27.66/8.76	s(103) =< V1
27.66/8.76	s(104) =< s(103)
27.66/8.76	
27.66/8.76	  with precondition: [V=1,V1>=1] 
27.66/8.76	
27.66/8.76	
27.66/8.76	Closed-form bounds of start(V1,V): 
27.66/8.76	-------------------------------------
27.66/8.76	* Chain [22] with precondition: [V1>=0] 
27.66/8.76	    - Upper bound: 30*V1+12+nat(V)*4+24*V1+(30*V1+162)+nat(V1-V+1)*4 
27.66/8.76	    - Complexity: n 
27.66/8.76	* Chain [21] with precondition: [V=1,V1>=1] 
27.66/8.76	    - Upper bound: 4*V1+3 
27.66/8.76	    - Complexity: n 
27.66/8.76	
27.66/8.76	### Maximum cost of start(V1,V): 26*V1+9+nat(V)*4+24*V1+(30*V1+162)+nat(V1-V+1)*4+(4*V1+3) 
27.66/8.76	Asymptotic class: n 
27.66/8.76	* Total analysis performed in 324 ms.
27.66/8.76	
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(12)
27.66/8.76	BOUNDS(1, n^1)
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
27.66/8.76	Transformed a relative TRS into a decreasing-loop problem.
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(14)
27.66/8.76	Obligation:
27.66/8.76	Analyzing the following TRS for decreasing loops:
27.66/8.76	
27.66/8.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
27.66/8.76	
27.66/8.76	
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y)
27.66/8.76	   quot(0, s(Y)) -> 0
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
27.66/8.76	   log(s(0)) -> 0
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))
27.66/8.76	
27.66/8.76	S is empty.
27.66/8.76	Rewrite Strategy: FULL
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(15) DecreasingLoopProof (LOWER BOUND(ID))
27.66/8.76	The following loop(s) give(s) rise to the lower bound Omega(n^1):
27.66/8.76	
27.66/8.76	The rewrite sequence
27.66/8.76	
27.66/8.76	min(s(X), s(Y)) ->^+ min(X, Y)
27.66/8.76	
27.66/8.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
27.66/8.76	
27.66/8.76	The pumping substitution is [X / s(X), Y / s(Y)].
27.66/8.76	
27.66/8.76	The result substitution is [ ].
27.66/8.76	
27.66/8.76	
27.66/8.76	
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(16)
27.66/8.76	Complex Obligation (BEST)
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(17)
27.66/8.76	Obligation:
27.66/8.76	Proved the lower bound n^1 for the following obligation:
27.66/8.76	
27.66/8.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
27.66/8.76	
27.66/8.76	
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y)
27.66/8.76	   quot(0, s(Y)) -> 0
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
27.66/8.76	   log(s(0)) -> 0
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))
27.66/8.76	
27.66/8.76	S is empty.
27.66/8.76	Rewrite Strategy: FULL
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(18) LowerBoundPropagationProof (FINISHED)
27.66/8.76	Propagated lower bound.
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(19)
27.66/8.76	BOUNDS(n^1, INF)
27.66/8.76	
27.66/8.76	----------------------------------------
27.66/8.76	
27.66/8.76	(20)
27.66/8.76	Obligation:
27.66/8.76	Analyzing the following TRS for decreasing loops:
27.66/8.76	
27.66/8.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
27.66/8.76	
27.66/8.76	
27.66/8.76	The TRS R consists of the following rules:
27.66/8.76	
27.66/8.76	   min(X, 0) -> X
27.66/8.76	   min(s(X), s(Y)) -> min(X, Y)
27.66/8.76	   quot(0, s(Y)) -> 0
27.66/8.76	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
27.66/8.76	   log(s(0)) -> 0
27.66/8.76	   log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))
27.66/8.76	
27.66/8.76	S is empty.
27.66/8.76	Rewrite Strategy: FULL
27.74/8.81	EOF