891.97/291.45	WORST_CASE(Omega(n^1), O(n^2))
892.27/291.47	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
892.27/291.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
892.27/291.47	
892.27/291.47	
892.27/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
892.27/291.47	
892.27/291.47	(0) CpxTRS
892.27/291.47	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
892.27/291.47	(2) CpxTRS
892.27/291.47	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
892.27/291.47	    (4) CpxWeightedTrs
892.27/291.47	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
892.27/291.47	    (6) CpxTypedWeightedTrs
892.27/291.47	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
892.27/291.47	    (8) CpxTypedWeightedCompleteTrs
892.27/291.47	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
892.27/291.47	    (10) CpxRNTS
892.27/291.47	    (11) CompleteCoflocoProof [FINISHED, 317 ms]
892.27/291.47	    (12) BOUNDS(1, n^2)
892.27/291.47	(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
892.27/291.47	(14) TRS for Loop Detection
892.27/291.47	    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
892.27/291.47	    (16) BEST
892.27/291.47	        (17) proven lower bound
892.27/291.47	            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
892.27/291.47	            (19) BOUNDS(n^1, INF)
892.27/291.47	        (20) TRS for Loop Detection
892.27/291.47	
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(0)
892.27/291.47	Obligation:
892.27/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
892.27/291.47	
892.27/291.47	
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true
892.27/291.47	   lt(x, 0) -> false
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y)
892.27/291.47	   fac(x) -> help(x, 0)
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c)
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c)))
892.27/291.47	   if(false, x, c) -> s(0)
892.27/291.47	
892.27/291.47	S is empty.
892.27/291.47	Rewrite Strategy: FULL
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
892.27/291.47	Converted rc-obligation to irc-obligation.
892.27/291.47	
892.27/291.47	The duplicating contexts are:
892.27/291.47	help(x, [])
892.27/291.47	help([], c)
892.27/291.47	if(true, x, [])
892.27/291.47	
892.27/291.47	
892.27/291.47	The defined contexts are:
892.27/291.47	if([], x1, x2)
892.27/291.47	
892.27/291.47	
892.27/291.47	[] just represents basic- or constructor-terms in the following defined contexts:
892.27/291.47	if([], x1, x2)
892.27/291.47	
892.27/291.47	
892.27/291.47	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(2)
892.27/291.47	Obligation:
892.27/291.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
892.27/291.47	
892.27/291.47	
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true
892.27/291.47	   lt(x, 0) -> false
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y)
892.27/291.47	   fac(x) -> help(x, 0)
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c)
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c)))
892.27/291.47	   if(false, x, c) -> s(0)
892.27/291.47	
892.27/291.47	S is empty.
892.27/291.47	Rewrite Strategy: INNERMOST
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
892.27/291.47	Transformed relative TRS to weighted TRS
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(4)
892.27/291.47	Obligation:
892.27/291.47	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
892.27/291.47	
892.27/291.47	
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true [1]
892.27/291.47	   lt(x, 0) -> false [1]
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y) [1]
892.27/291.47	   fac(x) -> help(x, 0) [1]
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c) [1]
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c))) [1]
892.27/291.47	   if(false, x, c) -> s(0) [1]
892.27/291.47	
892.27/291.47	Rewrite Strategy: INNERMOST
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
892.27/291.47	Infered types.
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(6)
892.27/291.47	Obligation:
892.27/291.47	Runtime Complexity Weighted TRS with Types.
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true [1]
892.27/291.47	   lt(x, 0) -> false [1]
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y) [1]
892.27/291.47	   fac(x) -> help(x, 0) [1]
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c) [1]
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c))) [1]
892.27/291.47	   if(false, x, c) -> s(0) [1]
892.27/291.47	
892.27/291.47	The TRS has the following type information:
892.27/291.47	lt :: 0:s:times -> 0:s:times -> true:false
892.27/291.47	0 :: 0:s:times
892.27/291.47	s :: 0:s:times -> 0:s:times
892.27/291.47	true :: true:false
892.27/291.47	false :: true:false
892.27/291.47	fac :: 0:s:times -> 0:s:times
892.27/291.47	help :: 0:s:times -> 0:s:times -> 0:s:times
892.27/291.47	if :: true:false -> 0:s:times -> 0:s:times -> 0:s:times
892.27/291.47	times :: 0:s:times -> 0:s:times -> 0:s:times
892.27/291.47	
892.27/291.47	Rewrite Strategy: INNERMOST
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(7) CompletionProof (UPPER BOUND(ID))
892.27/291.47	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
892.27/291.47	
892.27/291.47	   lt(v0, v1) -> null_lt [0]
892.27/291.47	   if(v0, v1, v2) -> null_if [0]
892.27/291.47	
892.27/291.47	And the following fresh constants:    null_lt, null_if
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(8)
892.27/291.47	Obligation:
892.27/291.47	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
892.27/291.47	
892.27/291.47	Runtime Complexity Weighted TRS with Types.
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true [1]
892.27/291.47	   lt(x, 0) -> false [1]
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y) [1]
892.27/291.47	   fac(x) -> help(x, 0) [1]
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c) [1]
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c))) [1]
892.27/291.47	   if(false, x, c) -> s(0) [1]
892.27/291.47	   lt(v0, v1) -> null_lt [0]
892.27/291.47	   if(v0, v1, v2) -> null_if [0]
892.27/291.47	
892.27/291.47	The TRS has the following type information:
892.27/291.47	lt :: 0:s:times:null_if -> 0:s:times:null_if -> true:false:null_lt
892.27/291.47	0 :: 0:s:times:null_if
892.27/291.47	s :: 0:s:times:null_if -> 0:s:times:null_if
892.27/291.47	true :: true:false:null_lt
892.27/291.47	false :: true:false:null_lt
892.27/291.47	fac :: 0:s:times:null_if -> 0:s:times:null_if
892.27/291.47	help :: 0:s:times:null_if -> 0:s:times:null_if -> 0:s:times:null_if
892.27/291.47	if :: true:false:null_lt -> 0:s:times:null_if -> 0:s:times:null_if -> 0:s:times:null_if
892.27/291.47	times :: 0:s:times:null_if -> 0:s:times:null_if -> 0:s:times:null_if
892.27/291.47	null_lt :: true:false:null_lt
892.27/291.47	null_if :: 0:s:times:null_if
892.27/291.47	
892.27/291.47	Rewrite Strategy: INNERMOST
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
892.27/291.47	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
892.27/291.47	The constant constructors are abstracted as follows:
892.27/291.47	
892.27/291.47	   0 => 0
892.27/291.47	   true => 2
892.27/291.47	   false => 1
892.27/291.47	   null_lt => 0
892.27/291.47	   null_if => 0
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(10)
892.27/291.47	Obligation:
892.27/291.47	Complexity RNTS consisting of the following rules:
892.27/291.47	
892.27/291.47	   fac(z) -{ 1 }-> help(x, 0) :|: x >= 0, z = x
892.27/291.47	   help(z, z') -{ 1 }-> if(lt(c, x), x, c) :|: c >= 0, x >= 0, z' = c, z = x
892.27/291.47	   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
892.27/291.47	   if(z, z', z'') -{ 1 }-> 1 + 0 :|: z' = x, c >= 0, z = 1, x >= 0, z'' = c
892.27/291.47	   if(z, z', z'') -{ 1 }-> 1 + (1 + c) + help(x, 1 + c) :|: z = 2, z' = x, c >= 0, x >= 0, z'' = c
892.27/291.47	   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
892.27/291.47	   lt(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0
892.27/291.47	   lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
892.27/291.47	   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
892.27/291.47	
892.27/291.47	Only complete derivations are relevant for the runtime complexity.
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(11) CompleteCoflocoProof (FINISHED)
892.27/291.47	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
892.27/291.47	
892.27/291.47	eq(start(V1, V, V9),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]).
892.27/291.47	eq(start(V1, V, V9),0,[fac(V1, Out)],[V1 >= 0]).
892.27/291.47	eq(start(V1, V, V9),0,[help(V1, V, Out)],[V1 >= 0,V >= 0]).
892.27/291.47	eq(start(V1, V, V9),0,[if(V1, V, V9, Out)],[V1 >= 0,V >= 0,V9 >= 0]).
892.27/291.47	eq(lt(V1, V, Out),1,[],[Out = 2,V = 1 + V2,V2 >= 0,V1 = 0]).
892.27/291.47	eq(lt(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = V3,V = 0]).
892.27/291.47	eq(lt(V1, V, Out),1,[lt(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
892.27/291.47	eq(fac(V1, Out),1,[help(V6, 0, Ret1)],[Out = Ret1,V6 >= 0,V1 = V6]).
892.27/291.47	eq(help(V1, V, Out),1,[lt(V8, V7, Ret0),if(Ret0, V7, V8, Ret2)],[Out = Ret2,V8 >= 0,V7 >= 0,V = V8,V1 = V7]).
892.27/291.47	eq(if(V1, V, V9, Out),1,[help(V10, 1 + V11, Ret11)],[Out = 2 + Ret11 + V11,V1 = 2,V = V10,V11 >= 0,V10 >= 0,V9 = V11]).
892.27/291.47	eq(if(V1, V, V9, Out),1,[],[Out = 1,V = V13,V12 >= 0,V1 = 1,V13 >= 0,V9 = V12]).
892.27/291.47	eq(lt(V1, V, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V1 = V15,V = V14]).
892.27/291.47	eq(if(V1, V, V9, Out),0,[],[Out = 0,V17 >= 0,V9 = V18,V16 >= 0,V1 = V17,V = V16,V18 >= 0]).
892.27/291.47	input_output_vars(lt(V1,V,Out),[V1,V],[Out]).
892.27/291.47	input_output_vars(fac(V1,Out),[V1],[Out]).
892.27/291.47	input_output_vars(help(V1,V,Out),[V1,V],[Out]).
892.27/291.47	input_output_vars(if(V1,V,V9,Out),[V1,V,V9],[Out]).
892.27/291.47	
892.27/291.47	
892.27/291.47	CoFloCo proof output:
892.27/291.47	Preprocessing Cost Relations
892.27/291.47	=====================================
892.27/291.47	
892.27/291.47	#### Computed strongly connected components 
892.27/291.47	0. recursive  : [lt/3]
892.27/291.47	1. recursive  : [help/3,if/4]
892.27/291.47	2. non_recursive  : [fac/2]
892.27/291.47	3. non_recursive  : [start/3]
892.27/291.47	
892.27/291.47	#### Obtained direct recursion through partial evaluation 
892.27/291.47	0. SCC is partially evaluated into lt/3
892.27/291.47	1. SCC is partially evaluated into help/3
892.27/291.47	2. SCC is completely evaluated into other SCCs
892.27/291.47	3. SCC is partially evaluated into start/3
892.27/291.47	
892.27/291.47	Control-Flow Refinement of Cost Relations
892.27/291.47	=====================================
892.27/291.47	
892.27/291.47	### Specialization of cost equations lt/3 
892.27/291.47	* CE 13 is refined into CE [14] 
892.27/291.47	* CE 11 is refined into CE [15] 
892.27/291.47	* CE 10 is refined into CE [16] 
892.27/291.47	* CE 12 is refined into CE [17] 
892.27/291.47	
892.27/291.47	
892.27/291.47	### Cost equations --> "Loop" of lt/3 
892.27/291.47	* CEs [17] --> Loop 10 
892.27/291.47	* CEs [14] --> Loop 11 
892.27/291.47	* CEs [15] --> Loop 12 
892.27/291.47	* CEs [16] --> Loop 13 
892.27/291.47	
892.27/291.47	### Ranking functions of CR lt(V1,V,Out) 
892.27/291.47	* RF of phase [10]: [V,V1]
892.27/291.47	
892.27/291.47	#### Partial ranking functions of CR lt(V1,V,Out) 
892.27/291.47	* Partial RF of phase [10]:
892.27/291.47	  - RF of loop [10:1]:
892.27/291.47	    V
892.27/291.47	    V1
892.27/291.47	
892.27/291.47	
892.27/291.47	### Specialization of cost equations help/3 
892.27/291.47	* CE 9 is refined into CE [18,19] 
892.27/291.47	* CE 8 is refined into CE [20,21] 
892.27/291.47	* CE 7 is refined into CE [22,23,24,25,26] 
892.27/291.47	
892.27/291.47	
892.27/291.47	### Cost equations --> "Loop" of help/3 
892.27/291.47	* CEs [21] --> Loop 14 
892.27/291.47	* CEs [22] --> Loop 15 
892.27/291.47	* CEs [20] --> Loop 16 
892.27/291.47	* CEs [23,24,25,26] --> Loop 17 
892.27/291.47	* CEs [19] --> Loop 18 
892.27/291.47	* CEs [18] --> Loop 19 
892.27/291.47	
892.27/291.47	### Ranking functions of CR help(V1,V,Out) 
892.27/291.47	* RF of phase [18]: [V1-V]
892.27/291.47	
892.27/291.47	#### Partial ranking functions of CR help(V1,V,Out) 
892.27/291.47	* Partial RF of phase [18]:
892.27/291.47	  - RF of loop [18:1]:
892.27/291.47	    V1-V
892.27/291.47	
892.27/291.47	
892.27/291.47	### Specialization of cost equations start/3 
892.27/291.47	* CE 3 is refined into CE [27,28,29,30] 
892.27/291.47	* CE 1 is refined into CE [31] 
892.27/291.47	* CE 2 is refined into CE [32] 
892.27/291.47	* CE 4 is refined into CE [33,34,35,36,37] 
892.27/291.47	* CE 5 is refined into CE [38,39,40,41,42] 
892.27/291.47	* CE 6 is refined into CE [43,44,45,46,47,48,49] 
892.27/291.47	
892.27/291.47	
892.27/291.47	### Cost equations --> "Loop" of start/3 
892.27/291.47	* CEs [34,46,47] --> Loop 20 
892.27/291.47	* CEs [27,28,29,30] --> Loop 21 
892.27/291.47	* CEs [32,39,44] --> Loop 22 
892.27/291.47	* CEs [31,33,35,36,37,38,40,41,42,43,45,48,49] --> Loop 23 
892.27/291.47	
892.27/291.47	### Ranking functions of CR start(V1,V,V9) 
892.27/291.47	
892.27/291.47	#### Partial ranking functions of CR start(V1,V,V9) 
892.27/291.47	
892.27/291.47	
892.27/291.47	Computing Bounds
892.27/291.47	=====================================
892.27/291.47	
892.27/291.47	#### Cost of chains of lt(V1,V,Out):
892.27/291.47	* Chain [[10],13]: 1*it(10)+1
892.27/291.47	  Such that:it(10) =< V1
892.27/291.47	
892.27/291.47	  with precondition: [Out=2,V1>=1,V>=V1+1] 
892.27/291.47	
892.27/291.47	* Chain [[10],12]: 1*it(10)+1
892.27/291.47	  Such that:it(10) =< V
892.27/291.47	
892.27/291.47	  with precondition: [Out=1,V>=1,V1>=V] 
892.27/291.47	
892.27/291.47	* Chain [[10],11]: 1*it(10)+0
892.27/291.47	  Such that:it(10) =< V
892.27/291.47	
892.27/291.47	  with precondition: [Out=0,V1>=1,V>=1] 
892.27/291.47	
892.27/291.47	* Chain [13]: 1
892.27/291.47	  with precondition: [V1=0,Out=2,V>=1] 
892.27/291.47	
892.27/291.47	* Chain [12]: 1
892.27/291.47	  with precondition: [V=0,Out=1,V1>=0] 
892.27/291.47	
892.27/291.47	* Chain [11]: 0
892.27/291.47	  with precondition: [Out=0,V1>=0,V>=0] 
892.27/291.47	
892.27/291.47	
892.27/291.47	#### Cost of chains of help(V1,V,Out):
892.27/291.47	* Chain [[18],17]: 3*it(18)+3*s(2)+1*s(7)+2
892.27/291.47	  Such that:it(18) =< V1-V
892.27/291.47	aux(3) =< V1
892.27/291.47	s(2) =< aux(3)
892.27/291.47	s(7) =< it(18)*aux(3)
892.27/291.47	
892.27/291.47	  with precondition: [V>=1,V1>=V+1,Out>=V+2] 
892.27/291.47	
892.27/291.47	* Chain [[18],14]: 3*it(18)+1*s(7)+1*s(8)+3
892.27/291.47	  Such that:it(18) =< V1-V
892.27/291.47	aux(4) =< V1
892.27/291.47	s(8) =< aux(4)
892.27/291.47	s(7) =< it(18)*aux(4)
892.27/291.47	
892.27/291.47	  with precondition: [V>=1,V1>=V+1,Out+3*V+1>=4*V1] 
892.27/291.47	
892.27/291.47	* Chain [19,[18],17]: 6*it(18)+1*s(7)+5
892.27/291.47	  Such that:aux(5) =< V1
892.27/291.47	it(18) =< aux(5)
892.27/291.47	s(7) =< it(18)*aux(5)
892.27/291.47	
892.27/291.47	  with precondition: [V=0,V1>=2,Out>=5] 
892.27/291.47	
892.27/291.47	* Chain [19,[18],14]: 4*it(18)+1*s(7)+6
892.27/291.47	  Such that:aux(6) =< V1
892.27/291.47	it(18) =< aux(6)
892.27/291.47	s(7) =< it(18)*aux(6)
892.27/291.47	
892.27/291.47	  with precondition: [V=0,V1>=2,Out+2>=4*V1] 
892.27/291.47	
892.27/291.47	* Chain [19,17]: 2*s(2)+1*s(4)+5
892.27/291.47	  Such that:s(4) =< 1
892.27/291.47	aux(1) =< V1
892.27/291.47	s(2) =< aux(1)
892.27/291.47	
892.27/291.47	  with precondition: [V=0,Out=2,V1>=1] 
892.27/291.47	
892.27/291.47	* Chain [19,14]: 1*s(8)+6
892.27/291.47	  Such that:s(8) =< 1
892.27/291.47	
892.27/291.47	  with precondition: [V1=1,V=0,Out=3] 
892.27/291.47	
892.27/291.47	* Chain [17]: 2*s(2)+1*s(4)+2
892.27/291.47	  Such that:s(4) =< V
892.27/291.47	aux(1) =< V1
892.27/291.47	s(2) =< aux(1)
892.27/291.47	
892.27/291.47	  with precondition: [Out=0,V1>=0,V>=0] 
892.27/291.47	
892.27/291.47	* Chain [16]: 3
892.27/291.47	  with precondition: [V1=0,Out=1,V>=0] 
892.27/291.47	
892.27/291.47	* Chain [15]: 2
892.27/291.47	  with precondition: [V=0,Out=0,V1>=1] 
892.27/291.47	
892.27/291.47	* Chain [14]: 1*s(8)+3
892.27/291.47	  Such that:s(8) =< V1
892.27/291.47	
892.27/291.47	  with precondition: [Out=1,V1>=1,V>=V1] 
892.27/291.47	
892.27/291.47	
892.27/291.47	#### Cost of chains of start(V1,V,V9):
892.27/291.47	* Chain [23]: 3*s(26)+22*s(28)+1*s(32)+2*s(37)+6*s(44)+2*s(46)+7
892.27/291.47	  Such that:s(32) =< 1
892.27/291.47	s(43) =< V1-V
892.27/291.47	aux(10) =< V1
892.27/291.47	aux(11) =< V
892.27/291.47	s(28) =< aux(10)
892.27/291.47	s(26) =< aux(11)
892.27/291.47	s(37) =< s(28)*aux(10)
892.27/291.47	s(44) =< s(43)
892.27/291.47	s(46) =< s(44)*aux(10)
892.27/291.47	
892.27/291.47	  with precondition: [V1>=0] 
892.27/291.47	
892.27/291.47	* Chain [22]: 9
892.27/291.47	  with precondition: [V1=1] 
892.27/291.47	
892.27/291.47	* Chain [21]: 1*s(50)+7*s(51)+6*s(55)+2*s(57)+4
892.27/291.47	  Such that:s(54) =< V-V9
892.27/291.47	s(50) =< V9+1
892.27/291.47	aux(13) =< V
892.27/291.47	s(51) =< aux(13)
892.27/291.47	s(55) =< s(54)
892.27/291.47	s(57) =< s(55)*aux(13)
892.27/291.47	
892.27/291.47	  with precondition: [V1=2,V>=0,V9>=0] 
892.27/291.47	
892.27/291.47	* Chain [20]: 1*s(58)+12*s(60)+2*s(63)+6
892.27/291.47	  Such that:s(58) =< 1
892.27/291.47	aux(14) =< V1
892.27/291.47	s(60) =< aux(14)
892.27/291.47	s(63) =< s(60)*aux(14)
892.27/291.47	
892.27/291.47	  with precondition: [V=0,V1>=0] 
892.27/291.47	
892.27/291.47	
892.27/291.47	Closed-form bounds of start(V1,V,V9): 
892.27/291.47	-------------------------------------
892.27/291.47	* Chain [23] with precondition: [V1>=0] 
892.27/291.47	    - Upper bound: 22*V1+8+2*V1*V1+2*V1*nat(V1-V)+nat(V)*3+nat(V1-V)*6 
892.27/291.47	    - Complexity: n^2 
892.27/291.47	* Chain [22] with precondition: [V1=1] 
892.27/291.47	    - Upper bound: 9 
892.27/291.47	    - Complexity: constant 
892.27/291.47	* Chain [21] with precondition: [V1=2,V>=0,V9>=0] 
892.27/291.47	    - Upper bound: 7*V+4+2*V*nat(V-V9)+(V9+1)+nat(V-V9)*6 
892.27/291.47	    - Complexity: n^2 
892.27/291.47	* Chain [20] with precondition: [V=0,V1>=0] 
892.27/291.47	    - Upper bound: 12*V1+7+2*V1*V1 
892.27/291.47	    - Complexity: n^2 
892.27/291.47	
892.27/291.47	### Maximum cost of start(V1,V,V9): max([max([5,nat(V)*2*nat(V-V9)+nat(V)*7+nat(V9+1)+nat(V-V9)*6]),10*V1+1+2*V1*nat(V1-V)+nat(V)*3+nat(V1-V)*6+(12*V1+3+2*V1*V1)])+4 
892.27/291.47	Asymptotic class: n^2 
892.27/291.47	* Total analysis performed in 248 ms.
892.27/291.47	
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(12)
892.27/291.47	BOUNDS(1, n^2)
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
892.27/291.47	Transformed a relative TRS into a decreasing-loop problem.
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(14)
892.27/291.47	Obligation:
892.27/291.47	Analyzing the following TRS for decreasing loops:
892.27/291.47	
892.27/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
892.27/291.47	
892.27/291.47	
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true
892.27/291.47	   lt(x, 0) -> false
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y)
892.27/291.47	   fac(x) -> help(x, 0)
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c)
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c)))
892.27/291.47	   if(false, x, c) -> s(0)
892.27/291.47	
892.27/291.47	S is empty.
892.27/291.47	Rewrite Strategy: FULL
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(15) DecreasingLoopProof (LOWER BOUND(ID))
892.27/291.47	The following loop(s) give(s) rise to the lower bound Omega(n^1):
892.27/291.47	
892.27/291.47	The rewrite sequence
892.27/291.47	
892.27/291.47	lt(s(x), s(y)) ->^+ lt(x, y)
892.27/291.47	
892.27/291.47	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
892.27/291.47	
892.27/291.47	The pumping substitution is [x / s(x), y / s(y)].
892.27/291.47	
892.27/291.47	The result substitution is [ ].
892.27/291.47	
892.27/291.47	
892.27/291.47	
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(16)
892.27/291.47	Complex Obligation (BEST)
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(17)
892.27/291.47	Obligation:
892.27/291.47	Proved the lower bound n^1 for the following obligation:
892.27/291.47	
892.27/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
892.27/291.47	
892.27/291.47	
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true
892.27/291.47	   lt(x, 0) -> false
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y)
892.27/291.47	   fac(x) -> help(x, 0)
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c)
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c)))
892.27/291.47	   if(false, x, c) -> s(0)
892.27/291.47	
892.27/291.47	S is empty.
892.27/291.47	Rewrite Strategy: FULL
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(18) LowerBoundPropagationProof (FINISHED)
892.27/291.47	Propagated lower bound.
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(19)
892.27/291.47	BOUNDS(n^1, INF)
892.27/291.47	
892.27/291.47	----------------------------------------
892.27/291.47	
892.27/291.47	(20)
892.27/291.47	Obligation:
892.27/291.47	Analyzing the following TRS for decreasing loops:
892.27/291.47	
892.27/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
892.27/291.47	
892.27/291.47	
892.27/291.47	The TRS R consists of the following rules:
892.27/291.47	
892.27/291.47	   lt(0, s(x)) -> true
892.27/291.47	   lt(x, 0) -> false
892.27/291.47	   lt(s(x), s(y)) -> lt(x, y)
892.27/291.47	   fac(x) -> help(x, 0)
892.27/291.47	   help(x, c) -> if(lt(c, x), x, c)
892.27/291.47	   if(true, x, c) -> times(s(c), help(x, s(c)))
892.27/291.47	   if(false, x, c) -> s(0)
892.27/291.47	
892.27/291.47	S is empty.
892.27/291.47	Rewrite Strategy: FULL
892.36/291.51	EOF