35.85/11.24	WORST_CASE(Omega(n^1), O(n^1))
35.85/11.25	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
35.85/11.25	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.85/11.25	
35.85/11.25	
35.85/11.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
35.85/11.25	
35.85/11.25	(0) CpxTRS
35.85/11.25	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
35.85/11.25	(2) CpxWeightedTrs
35.85/11.25	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
35.85/11.25	    (4) CpxTypedWeightedTrs
35.85/11.25	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
35.85/11.25	    (6) CpxTypedWeightedCompleteTrs
35.85/11.25	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms]
35.85/11.25	    (8) CpxRNTS
35.85/11.25	    (9) CompleteCoflocoProof [FINISHED, 600 ms]
35.85/11.25	    (10) BOUNDS(1, n^1)
35.85/11.25	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
35.85/11.25	(12) TRS for Loop Detection
35.85/11.25	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
35.85/11.25	    (14) BEST
35.85/11.25	        (15) proven lower bound
35.85/11.25	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
35.85/11.25	            (17) BOUNDS(n^1, INF)
35.85/11.25	        (18) TRS for Loop Detection
35.85/11.25	
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(0)
35.85/11.25	Obligation:
35.85/11.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
35.85/11.25	
35.85/11.25	
35.85/11.25	The TRS R consists of the following rules:
35.85/11.25	
35.85/11.25	   le(0, y) -> true
35.85/11.25	   le(s(x), 0) -> false
35.85/11.25	   le(s(x), s(y)) -> le(x, y)
35.85/11.25	   minus(x, 0) -> x
35.85/11.25	   minus(0, x) -> 0
35.85/11.25	   minus(s(x), s(y)) -> minus(x, y)
35.85/11.25	   gcd(0, y) -> y
35.85/11.25	   gcd(s(x), 0) -> s(x)
35.85/11.25	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
35.85/11.25	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
35.85/11.25	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
35.85/11.25	
35.85/11.25	S is empty.
35.85/11.25	Rewrite Strategy: INNERMOST
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
35.85/11.25	Transformed relative TRS to weighted TRS
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(2)
35.85/11.25	Obligation:
35.85/11.25	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
35.85/11.25	
35.85/11.25	
35.85/11.25	The TRS R consists of the following rules:
35.85/11.25	
35.85/11.25	   le(0, y) -> true [1]
35.85/11.25	   le(s(x), 0) -> false [1]
35.85/11.25	   le(s(x), s(y)) -> le(x, y) [1]
35.85/11.25	   minus(x, 0) -> x [1]
35.85/11.25	   minus(0, x) -> 0 [1]
35.85/11.25	   minus(s(x), s(y)) -> minus(x, y) [1]
35.85/11.25	   gcd(0, y) -> y [1]
35.85/11.25	   gcd(s(x), 0) -> s(x) [1]
35.85/11.25	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
35.85/11.25	   if_gcd(true, x, y) -> gcd(minus(x, y), y) [1]
35.85/11.25	   if_gcd(false, x, y) -> gcd(minus(y, x), x) [1]
35.85/11.25	
35.85/11.25	Rewrite Strategy: INNERMOST
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
35.85/11.25	Infered types.
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(4)
35.85/11.25	Obligation:
35.85/11.25	Runtime Complexity Weighted TRS with Types.
35.85/11.25	The TRS R consists of the following rules:
35.85/11.25	
35.85/11.25	   le(0, y) -> true [1]
35.85/11.25	   le(s(x), 0) -> false [1]
35.85/11.25	   le(s(x), s(y)) -> le(x, y) [1]
35.85/11.25	   minus(x, 0) -> x [1]
35.85/11.25	   minus(0, x) -> 0 [1]
35.85/11.25	   minus(s(x), s(y)) -> minus(x, y) [1]
35.85/11.25	   gcd(0, y) -> y [1]
35.85/11.25	   gcd(s(x), 0) -> s(x) [1]
35.85/11.25	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
35.85/11.25	   if_gcd(true, x, y) -> gcd(minus(x, y), y) [1]
35.85/11.25	   if_gcd(false, x, y) -> gcd(minus(y, x), x) [1]
35.85/11.25	
35.85/11.25	The TRS has the following type information:
35.85/11.25	le :: 0:s -> 0:s -> true:false
35.85/11.25	0 :: 0:s
35.85/11.25	true :: true:false
35.85/11.25	s :: 0:s -> 0:s
35.85/11.25	false :: true:false
35.85/11.25	minus :: 0:s -> 0:s -> 0:s
35.85/11.25	gcd :: 0:s -> 0:s -> 0:s
35.85/11.25	if_gcd :: true:false -> 0:s -> 0:s -> 0:s
35.85/11.25	
35.85/11.25	Rewrite Strategy: INNERMOST
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(5) CompletionProof (UPPER BOUND(ID))
35.85/11.25	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
35.85/11.25	none
35.85/11.25	
35.85/11.25	And the following fresh constants: none
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(6)
35.85/11.25	Obligation:
35.85/11.25	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
35.85/11.25	
35.85/11.25	Runtime Complexity Weighted TRS with Types.
35.85/11.25	The TRS R consists of the following rules:
35.85/11.25	
35.85/11.25	   le(0, y) -> true [1]
35.85/11.25	   le(s(x), 0) -> false [1]
35.85/11.25	   le(s(x), s(y)) -> le(x, y) [1]
35.85/11.25	   minus(x, 0) -> x [1]
35.85/11.25	   minus(0, x) -> 0 [1]
35.85/11.25	   minus(s(x), s(y)) -> minus(x, y) [1]
35.85/11.25	   gcd(0, y) -> y [1]
35.85/11.25	   gcd(s(x), 0) -> s(x) [1]
35.85/11.25	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
35.85/11.25	   if_gcd(true, x, y) -> gcd(minus(x, y), y) [1]
35.85/11.25	   if_gcd(false, x, y) -> gcd(minus(y, x), x) [1]
35.85/11.25	
35.85/11.25	The TRS has the following type information:
35.85/11.25	le :: 0:s -> 0:s -> true:false
35.85/11.25	0 :: 0:s
35.85/11.25	true :: true:false
35.85/11.25	s :: 0:s -> 0:s
35.85/11.25	false :: true:false
35.85/11.25	minus :: 0:s -> 0:s -> 0:s
35.85/11.25	gcd :: 0:s -> 0:s -> 0:s
35.85/11.25	if_gcd :: true:false -> 0:s -> 0:s -> 0:s
35.85/11.25	
35.85/11.25	Rewrite Strategy: INNERMOST
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
35.85/11.25	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
35.85/11.25	The constant constructors are abstracted as follows:
35.85/11.25	
35.85/11.25	   0 => 0
35.85/11.25	   true => 1
35.85/11.25	   false => 0
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(8)
35.85/11.25	Obligation:
35.85/11.25	Complexity RNTS consisting of the following rules:
35.85/11.25	
35.85/11.25	   gcd(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
35.85/11.25	   gcd(z, z') -{ 1 }-> if_gcd(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
35.85/11.25	   gcd(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0
35.85/11.25	   if_gcd(z, z', z'') -{ 1 }-> gcd(minus(x, y), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
35.85/11.25	   if_gcd(z, z', z'') -{ 1 }-> gcd(minus(y, x), x) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
35.85/11.25	   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
35.85/11.25	   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
35.85/11.25	   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
35.85/11.25	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
35.85/11.25	   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
35.85/11.25	   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
35.85/11.25	
35.85/11.25	Only complete derivations are relevant for the runtime complexity.
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(9) CompleteCoflocoProof (FINISHED)
35.85/11.25	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
35.85/11.25	
35.85/11.25	eq(start(V1, V, V15),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
35.85/11.25	eq(start(V1, V, V15),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
35.85/11.25	eq(start(V1, V, V15),0,[gcd(V1, V, Out)],[V1 >= 0,V >= 0]).
35.85/11.25	eq(start(V1, V, V15),0,[fun(V1, V, V15, Out)],[V1 >= 0,V >= 0,V15 >= 0]).
35.85/11.25	eq(le(V1, V, Out),1,[],[Out = 1,V2 >= 0,V1 = 0,V = V2]).
35.85/11.25	eq(le(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = 1 + V3,V = 0]).
35.85/11.25	eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
35.85/11.25	eq(minus(V1, V, Out),1,[],[Out = V6,V6 >= 0,V1 = V6,V = 0]).
35.85/11.25	eq(minus(V1, V, Out),1,[],[Out = 0,V = V7,V7 >= 0,V1 = 0]).
35.85/11.25	eq(minus(V1, V, Out),1,[minus(V8, V9, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = 1 + V8]).
35.85/11.25	eq(gcd(V1, V, Out),1,[],[Out = V10,V10 >= 0,V1 = 0,V = V10]).
35.85/11.25	eq(gcd(V1, V, Out),1,[],[Out = 1 + V11,V11 >= 0,V1 = 1 + V11,V = 0]).
35.85/11.25	eq(gcd(V1, V, Out),1,[le(V12, V13, Ret0),fun(Ret0, 1 + V13, 1 + V12, Ret2)],[Out = Ret2,V = 1 + V12,V13 >= 0,V12 >= 0,V1 = 1 + V13]).
35.85/11.25	eq(fun(V1, V, V15, Out),1,[minus(V16, V14, Ret01),gcd(Ret01, V14, Ret3)],[Out = Ret3,V = V16,V15 = V14,V1 = 1,V16 >= 0,V14 >= 0]).
35.85/11.25	eq(fun(V1, V, V15, Out),1,[minus(V18, V17, Ret02),gcd(Ret02, V17, Ret4)],[Out = Ret4,V = V17,V15 = V18,V17 >= 0,V18 >= 0,V1 = 0]).
35.85/11.25	input_output_vars(le(V1,V,Out),[V1,V],[Out]).
35.85/11.25	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
35.85/11.25	input_output_vars(gcd(V1,V,Out),[V1,V],[Out]).
35.85/11.25	input_output_vars(fun(V1,V,V15,Out),[V1,V,V15],[Out]).
35.85/11.25	
35.85/11.25	
35.85/11.25	CoFloCo proof output:
35.85/11.25	Preprocessing Cost Relations
35.85/11.25	=====================================
35.85/11.25	
35.85/11.25	#### Computed strongly connected components 
35.85/11.25	0. recursive  : [le/3]
35.85/11.25	1. recursive  : [minus/3]
35.85/11.25	2. recursive  : [fun/4,gcd/3]
35.85/11.25	3. non_recursive  : [start/3]
35.85/11.25	
35.85/11.25	#### Obtained direct recursion through partial evaluation 
35.85/11.25	0. SCC is partially evaluated into le/3
35.85/11.25	1. SCC is partially evaluated into minus/3
35.85/11.25	2. SCC is partially evaluated into gcd/3
35.85/11.25	3. SCC is partially evaluated into start/3
35.85/11.25	
35.85/11.25	Control-Flow Refinement of Cost Relations
35.85/11.25	=====================================
35.85/11.25	
35.85/11.25	### Specialization of cost equations le/3 
35.85/11.25	* CE 15 is refined into CE [16] 
35.85/11.25	* CE 14 is refined into CE [17] 
35.85/11.25	* CE 13 is refined into CE [18] 
35.85/11.25	
35.85/11.25	
35.85/11.25	### Cost equations --> "Loop" of le/3 
35.85/11.25	* CEs [17] --> Loop 13 
35.85/11.25	* CEs [18] --> Loop 14 
35.85/11.25	* CEs [16] --> Loop 15 
35.85/11.25	
35.85/11.25	### Ranking functions of CR le(V1,V,Out) 
35.85/11.25	* RF of phase [15]: [V,V1]
35.85/11.25	
35.85/11.25	#### Partial ranking functions of CR le(V1,V,Out) 
35.85/11.25	* Partial RF of phase [15]:
35.85/11.25	  - RF of loop [15:1]:
35.85/11.25	    V
35.85/11.25	    V1
35.85/11.25	
35.85/11.25	
35.85/11.25	### Specialization of cost equations minus/3 
35.85/11.25	* CE 8 is refined into CE [19] 
35.85/11.25	* CE 6 is refined into CE [20] 
35.85/11.25	* CE 7 is refined into CE [21] 
35.85/11.25	
35.85/11.25	
35.85/11.25	### Cost equations --> "Loop" of minus/3 
35.85/11.25	* CEs [20] --> Loop 16 
35.85/11.25	* CEs [21] --> Loop 17 
35.85/11.25	* CEs [19] --> Loop 18 
35.85/11.25	
35.85/11.25	### Ranking functions of CR minus(V1,V,Out) 
35.85/11.25	* RF of phase [18]: [V,V1]
35.85/11.25	
35.85/11.25	#### Partial ranking functions of CR minus(V1,V,Out) 
35.85/11.25	* Partial RF of phase [18]:
35.85/11.25	  - RF of loop [18:1]:
35.85/11.25	    V
35.85/11.25	    V1
35.85/11.25	
35.85/11.25	
35.85/11.25	### Specialization of cost equations gcd/3 
35.85/11.25	* CE 12 is refined into CE [22] 
35.85/11.25	* CE 11 is refined into CE [23] 
35.85/11.25	* CE 10 is refined into CE [24,25,26,27] 
35.85/11.25	* CE 9 is refined into CE [28,29] 
35.85/11.25	
35.85/11.25	
35.85/11.25	### Cost equations --> "Loop" of gcd/3 
35.85/11.25	* CEs [29] --> Loop 19 
35.85/11.25	* CEs [26,27] --> Loop 20 
35.85/11.25	* CEs [28] --> Loop 21 
35.85/11.25	* CEs [24,25] --> Loop 22 
35.85/11.25	* CEs [22] --> Loop 23 
35.85/11.25	* CEs [23] --> Loop 24 
35.85/11.25	
35.85/11.25	### Ranking functions of CR gcd(V1,V,Out) 
35.85/11.25	* RF of phase [19,20]: [V1+V-3]
35.85/11.25	* RF of phase [22]: [V1]
35.85/11.25	
35.85/11.25	#### Partial ranking functions of CR gcd(V1,V,Out) 
35.85/11.25	* Partial RF of phase [19,20]:
35.85/11.25	  - RF of loop [19:1]:
35.85/11.25	    V-2
35.85/11.25	    V1/2+V/2-2
35.85/11.25	  - RF of loop [20:1]:
35.85/11.25	    V1-1 depends on loops [19:1] 
35.85/11.25	    V1-V+1 depends on loops [19:1] 
35.85/11.25	* Partial RF of phase [22]:
35.85/11.25	  - RF of loop [22:1]:
35.85/11.25	    V1
35.85/11.25	
35.85/11.25	
35.85/11.25	### Specialization of cost equations start/3 
35.85/11.25	* CE 2 is refined into CE [30,31,32,33,34,35,36,37,38] 
35.85/11.25	* CE 1 is refined into CE [39,40,41,42,43,44,45,46,47] 
35.85/11.25	* CE 3 is refined into CE [48,49,50,51] 
35.85/11.25	* CE 4 is refined into CE [52,53,54,55] 
35.85/11.25	* CE 5 is refined into CE [56,57,58,59,60,61] 
35.85/11.25	
35.85/11.25	
35.85/11.25	### Cost equations --> "Loop" of start/3 
35.85/11.25	* CEs [60,61] --> Loop 25 
35.85/11.25	* CEs [50,55,59] --> Loop 26 
35.85/11.25	* CEs [35] --> Loop 27 
35.85/11.25	* CEs [33,34] --> Loop 28 
35.85/11.25	* CEs [36,37,38,57] --> Loop 29 
35.85/11.25	* CEs [32,51,54] --> Loop 30 
35.85/11.25	* CEs [30,31] --> Loop 31 
35.85/11.25	* CEs [43] --> Loop 32 
35.85/11.25	* CEs [44] --> Loop 33 
35.85/11.25	* CEs [45] --> Loop 34 
35.85/11.25	* CEs [41,49,53,58] --> Loop 35 
35.85/11.25	* CEs [39,40,42,46,47,48,52,56] --> Loop 36 
35.85/11.25	
35.85/11.25	### Ranking functions of CR start(V1,V,V15) 
35.85/11.25	
35.85/11.25	#### Partial ranking functions of CR start(V1,V,V15) 
35.85/11.25	
35.85/11.25	
35.85/11.25	Computing Bounds
35.85/11.25	=====================================
35.85/11.25	
35.85/11.25	#### Cost of chains of le(V1,V,Out):
35.85/11.25	* Chain [[15],14]: 1*it(15)+1
35.85/11.25	  Such that:it(15) =< V1
35.85/11.25	
35.85/11.25	  with precondition: [Out=1,V1>=1,V>=V1] 
35.85/11.25	
35.85/11.25	* Chain [[15],13]: 1*it(15)+1
35.85/11.25	  Such that:it(15) =< V
35.85/11.25	
35.85/11.25	  with precondition: [Out=0,V>=1,V1>=V+1] 
35.85/11.25	
35.85/11.25	* Chain [14]: 1
35.85/11.25	  with precondition: [V1=0,Out=1,V>=0] 
35.85/11.25	
35.85/11.25	* Chain [13]: 1
35.85/11.25	  with precondition: [V=0,Out=0,V1>=1] 
35.85/11.25	
35.85/11.25	
35.85/11.25	#### Cost of chains of minus(V1,V,Out):
35.85/11.25	* Chain [[18],17]: 1*it(18)+1
35.85/11.25	  Such that:it(18) =< V1
35.85/11.25	
35.85/11.25	  with precondition: [Out=0,V1>=1,V>=V1] 
35.85/11.25	
35.85/11.25	* Chain [[18],16]: 1*it(18)+1
35.85/11.25	  Such that:it(18) =< V
35.85/11.25	
35.85/11.25	  with precondition: [V1=Out+V,V>=1,V1>=V] 
35.85/11.25	
35.85/11.25	* Chain [17]: 1
35.85/11.25	  with precondition: [V1=0,Out=0,V>=0] 
35.85/11.25	
35.85/11.25	* Chain [16]: 1
35.85/11.25	  with precondition: [V=0,V1=Out,V1>=0] 
35.85/11.25	
35.85/11.25	
35.85/11.25	#### Cost of chains of gcd(V1,V,Out):
35.85/11.25	* Chain [[22],24]: 6*it(22)+1
35.85/11.25	  Such that:it(22) =< V1
35.85/11.25	
35.85/11.25	  with precondition: [V=1,Out=1,V1>=1] 
35.85/11.25	
35.85/11.25	* Chain [[19,20],24]: 4*it(19)+4*it(20)+2*s(13)+4*s(15)+1
35.85/11.25	  Such that:aux(9) =< V1-V+1
35.85/11.25	aux(21) =< V1+V
35.85/11.25	aux(22) =< V1+V-Out
35.85/11.25	it(19) =< V1/2+V/2
35.85/11.25	aux(24) =< V1/2+V/2-Out/2
35.85/11.25	aux(25) =< V
35.85/11.25	aux(26) =< V-Out
35.85/11.25	aux(8) =< 2*V-2*Out
35.85/11.25	aux(27) =< V1
35.85/11.25	it(19) =< aux(21)
35.85/11.25	it(20) =< aux(21)
35.85/11.25	s(14) =< aux(21)
35.85/11.25	it(19) =< aux(22)
35.85/11.25	it(20) =< aux(22)
35.85/11.25	s(14) =< aux(22)
35.85/11.25	it(19) =< aux(24)
35.85/11.25	it(20) =< aux(24)
35.85/11.25	aux(6) =< aux(25)
35.85/11.25	it(19) =< aux(25)
35.85/11.25	aux(6) =< aux(26)
35.85/11.25	it(19) =< aux(26)
35.85/11.25	it(20) =< aux(8)+aux(9)
35.85/11.25	it(20) =< aux(6)+aux(27)
35.85/11.25	s(16) =< aux(6)+aux(27)
35.85/11.25	s(16) =< it(20)*aux(25)
35.85/11.25	s(15) =< s(16)
35.85/11.25	s(13) =< s(14)
35.85/11.25	
35.85/11.25	  with precondition: [Out>=2,V1>=Out,V>=Out] 
35.85/11.25	
35.85/11.25	* Chain [[19,20],21,[22],24]: 4*it(19)+4*it(20)+6*it(22)+2*s(13)+4*s(15)+1*s(17)+5
35.85/11.25	  Such that:s(17) =< 1
35.85/11.25	aux(9) =< V1-V+1
35.85/11.25	it(19) =< V1/2+V/2
35.85/11.25	aux(8) =< 2*V
35.85/11.25	aux(28) =< V1
35.85/11.25	aux(29) =< V1+V
35.85/11.25	aux(30) =< V
35.85/11.25	it(22) =< aux(30)
35.85/11.25	it(19) =< aux(29)
35.85/11.25	it(20) =< aux(29)
35.85/11.25	it(19) =< aux(30)
35.85/11.25	it(20) =< aux(8)+aux(9)
35.85/11.25	it(20) =< aux(30)+aux(28)
35.85/11.25	s(16) =< aux(30)+aux(28)
35.85/11.25	s(16) =< it(20)*aux(30)
35.85/11.25	s(15) =< s(16)
35.85/11.25	s(13) =< aux(29)
35.85/11.25	
35.85/11.25	  with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 
35.85/11.25	
35.85/11.25	* Chain [24]: 1
35.85/11.25	  with precondition: [V1=0,V=Out,V>=0] 
35.85/11.25	
35.85/11.25	* Chain [23]: 1
35.85/11.25	  with precondition: [V=0,V1=Out,V1>=1] 
35.85/11.25	
35.85/11.25	* Chain [21,[22],24]: 6*it(22)+1*s(17)+5
35.85/11.25	  Such that:s(17) =< 1
35.85/11.25	it(22) =< V
35.85/11.25	
35.85/11.25	  with precondition: [V1=1,Out=1,V>=2] 
35.85/11.25	
35.85/11.25	
35.85/11.25	#### Cost of chains of start(V1,V,V15):
35.85/11.25	* Chain [36]: 5*s(18)+8*s(19)+1*s(20)+8*s(22)+4*s(28)+4*s(30)+4*s(42)+4*s(46)+7
35.85/11.25	  Such that:s(20) =< 1
35.85/11.25	aux(35) =< -2*V+V15+1
35.85/11.25	aux(36) =< -V+V15
35.85/11.25	aux(37) =< V
35.85/11.25	aux(38) =< 2*V
35.85/11.25	aux(39) =< V15
35.85/11.25	aux(40) =< V15/2
35.85/11.25	s(18) =< aux(39)
35.85/11.25	s(22) =< aux(40)
35.85/11.25	s(19) =< aux(37)
35.85/11.25	s(22) =< aux(39)
35.85/11.25	s(28) =< aux(39)
35.85/11.25	s(22) =< aux(37)
35.85/11.25	s(28) =< aux(38)+aux(35)
35.85/11.25	s(28) =< aux(37)+aux(36)
35.85/11.25	s(29) =< aux(37)+aux(36)
35.85/11.25	s(29) =< s(28)*aux(37)
35.85/11.25	s(30) =< s(29)
35.85/11.25	s(42) =< aux(39)
35.85/11.25	s(42) =< aux(40)
35.85/11.25	s(42) =< aux(38)+aux(35)
35.85/11.25	s(42) =< aux(37)+aux(36)
35.85/11.25	s(45) =< aux(37)+aux(36)
35.85/11.25	s(45) =< s(42)*aux(37)
35.85/11.25	s(46) =< s(45)
35.85/11.25	
35.85/11.25	  with precondition: [V1=0,V>=0] 
35.85/11.25	
35.85/11.25	* Chain [35]: 3
35.85/11.25	  with precondition: [V=0,V1>=0] 
35.85/11.25	
35.85/11.25	* Chain [34]: 1*s(48)+6*s(49)+3
35.85/11.25	  Such that:s(48) =< 1
35.85/11.25	s(49) =< V15
35.85/11.25	
35.85/11.25	  with precondition: [V1=0,V=1,V15>=2] 
35.85/11.25	
35.85/11.25	* Chain [33]: 7*s(50)+1*s(51)+7
35.85/11.25	  Such that:s(51) =< 1
35.85/11.25	aux(41) =< V
35.85/11.25	s(50) =< aux(41)
35.85/11.25	
35.85/11.25	  with precondition: [V1=0,V+1=V15,V>=2] 
35.85/11.25	
35.85/11.25	* Chain [32]: 1*s(53)+3
35.85/11.25	  Such that:s(53) =< V15
35.85/11.25	
35.85/11.25	  with precondition: [V1=0,V=V15,V>=1] 
35.85/11.25	
35.85/11.25	* Chain [31]: 3
35.85/11.25	  with precondition: [V1=1,V=0,V15>=0] 
35.85/11.25	
35.85/11.25	* Chain [30]: 2*s(54)+3
35.85/11.25	  Such that:aux(42) =< V1
35.85/11.25	s(54) =< aux(42)
35.85/11.25	
35.85/11.25	  with precondition: [V1>=1,V>=V1] 
35.85/11.25	
35.85/11.25	* Chain [29]: 3*s(56)+16*s(57)+8*s(58)+8*s(61)+4*s(67)+4*s(69)+4*s(81)+4*s(85)+7
35.85/11.25	  Such that:aux(47) =< 1
35.85/11.25	aux(48) =< V
35.85/11.25	aux(49) =< V-2*V15+1
35.85/11.25	aux(50) =< V-V15
35.85/11.25	aux(51) =< V/2
35.85/11.25	aux(52) =< V15
35.85/11.25	aux(53) =< 2*V15
35.85/11.25	s(56) =< aux(47)
35.85/11.25	s(57) =< aux(48)
35.85/11.25	s(61) =< aux(51)
35.85/11.25	s(58) =< aux(52)
35.85/11.25	s(61) =< aux(48)
35.85/11.25	s(67) =< aux(48)
35.85/11.25	s(61) =< aux(52)
35.85/11.25	s(67) =< aux(53)+aux(49)
35.85/11.25	s(67) =< aux(52)+aux(50)
35.85/11.25	s(68) =< aux(52)+aux(50)
35.85/11.25	s(68) =< s(67)*aux(52)
35.85/11.25	s(69) =< s(68)
35.85/11.25	s(81) =< aux(48)
35.85/11.25	s(81) =< aux(51)
35.85/11.25	s(81) =< aux(53)+aux(49)
35.85/11.25	s(81) =< aux(52)+aux(50)
35.85/11.25	s(84) =< aux(52)+aux(50)
35.85/11.25	s(84) =< s(81)*aux(52)
35.85/11.25	s(85) =< s(84)
35.85/11.25	
35.85/11.25	  with precondition: [V1=1,V>=2] 
35.85/11.25	
35.85/11.25	* Chain [28]: 1*s(89)+1*s(90)+3
35.85/11.25	  Such that:s(89) =< V
35.85/11.25	s(90) =< V15
35.85/11.25	
35.85/11.25	  with precondition: [V1=1,V>=1,V15>=V] 
35.85/11.25	
35.85/11.25	* Chain [27]: 7*s(91)+1*s(92)+7
35.85/11.25	  Such that:s(92) =< 1
35.85/11.25	aux(54) =< V
35.85/11.25	s(91) =< aux(54)
35.85/11.25	
35.85/11.25	  with precondition: [V1=1,V=V15+1,V>=3] 
35.85/11.25	
35.85/11.25	* Chain [26]: 2*s(94)+6*s(96)+1
35.85/11.25	  Such that:s(96) =< V1
35.85/11.25	aux(55) =< V
35.85/11.25	s(94) =< aux(55)
35.85/11.25	
35.85/11.25	  with precondition: [V>=1,V1>=V] 
35.85/11.25	
35.85/11.25	* Chain [25]: 1*s(97)+8*s(99)+6*s(104)+4*s(105)+4*s(107)+4*s(108)+4*s(118)+4*s(122)+5
35.85/11.25	  Such that:s(97) =< 1
35.85/11.25	aux(59) =< V1
35.85/11.25	aux(60) =< V1-V+1
35.85/11.25	aux(61) =< V1+V
35.85/11.25	aux(62) =< V1/2+V/2
35.85/11.25	aux(63) =< V
35.85/11.25	aux(64) =< 2*V
35.85/11.25	s(99) =< aux(62)
35.85/11.25	s(104) =< aux(63)
35.85/11.25	s(99) =< aux(61)
35.85/11.25	s(105) =< aux(61)
35.85/11.25	s(99) =< aux(63)
35.85/11.25	s(105) =< aux(64)+aux(60)
35.85/11.25	s(105) =< aux(63)+aux(59)
35.85/11.25	s(106) =< aux(63)+aux(59)
35.85/11.25	s(106) =< s(105)*aux(63)
35.85/11.25	s(107) =< s(106)
35.85/11.25	s(108) =< aux(61)
35.85/11.25	s(118) =< aux(61)
35.85/11.25	s(118) =< aux(62)
35.85/11.25	s(118) =< aux(64)+aux(60)
35.85/11.25	s(118) =< aux(63)+aux(59)
35.85/11.25	s(121) =< aux(63)+aux(59)
35.85/11.25	s(121) =< s(118)*aux(63)
35.85/11.25	s(122) =< s(121)
35.85/11.25	
35.85/11.25	  with precondition: [V1>=2,V>=2] 
35.85/11.25	
35.85/11.25	
35.85/11.25	Closed-form bounds of start(V1,V,V15): 
35.85/11.25	-------------------------------------
35.85/11.25	* Chain [36] with precondition: [V1=0,V>=0] 
35.85/11.25	    - Upper bound: 16*V+8+nat(V15)*13+nat(-V+V15)*8+nat(V15/2)*8 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [35] with precondition: [V=0,V1>=0] 
35.85/11.25	    - Upper bound: 3 
35.85/11.25	    - Complexity: constant 
35.85/11.25	* Chain [34] with precondition: [V1=0,V=1,V15>=2] 
35.85/11.25	    - Upper bound: 6*V15+4 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [33] with precondition: [V1=0,V+1=V15,V>=2] 
35.85/11.25	    - Upper bound: 7*V+8 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [32] with precondition: [V1=0,V=V15,V>=1] 
35.85/11.25	    - Upper bound: V15+3 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [31] with precondition: [V1=1,V=0,V15>=0] 
35.85/11.25	    - Upper bound: 3 
35.85/11.25	    - Complexity: constant 
35.85/11.25	* Chain [30] with precondition: [V1>=1,V>=V1] 
35.85/11.25	    - Upper bound: 2*V1+3 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [29] with precondition: [V1=1,V>=2] 
35.85/11.25	    - Upper bound: 24*V+10+nat(V15)*16+nat(V-V15)*8+4*V 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [28] with precondition: [V1=1,V>=1,V15>=V] 
35.85/11.25	    - Upper bound: V+V15+3 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [27] with precondition: [V1=1,V=V15+1,V>=3] 
35.85/11.25	    - Upper bound: 7*V+8 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [26] with precondition: [V>=1,V1>=V] 
35.85/11.25	    - Upper bound: 6*V1+2*V+1 
35.85/11.25	    - Complexity: n 
35.85/11.25	* Chain [25] with precondition: [V1>=2,V>=2] 
35.85/11.25	    - Upper bound: 24*V1+30*V+6 
35.85/11.25	    - Complexity: n 
35.85/11.25	
35.85/11.25	### Maximum cost of start(V1,V,V15): max([max([2*V1,nat(V15)*5+1+nat(V15)])+2,max([max([6*V1,5*V+5+max([2,7*V+max([24*V1+16*V,2*V+2+nat(V15)*13+max([nat(V15/2)*8+nat(-V+V15)*8,8*V+2+nat(V15)*3+nat(V-V15)*8+4*V])])])])+V,nat(V15)+2])+V])+1 
35.85/11.25	Asymptotic class: n 
35.85/11.25	* Total analysis performed in 497 ms.
35.85/11.25	
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(10)
35.85/11.25	BOUNDS(1, n^1)
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
35.85/11.25	Transformed a relative TRS into a decreasing-loop problem.
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(12)
35.85/11.25	Obligation:
35.85/11.25	Analyzing the following TRS for decreasing loops:
35.85/11.25	
35.85/11.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
35.85/11.25	
35.85/11.25	
35.85/11.25	The TRS R consists of the following rules:
35.85/11.25	
35.85/11.25	   le(0, y) -> true
35.85/11.25	   le(s(x), 0) -> false
35.85/11.25	   le(s(x), s(y)) -> le(x, y)
35.85/11.25	   minus(x, 0) -> x
35.85/11.25	   minus(0, x) -> 0
35.85/11.25	   minus(s(x), s(y)) -> minus(x, y)
35.85/11.25	   gcd(0, y) -> y
35.85/11.25	   gcd(s(x), 0) -> s(x)
35.85/11.25	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
35.85/11.25	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
35.85/11.25	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
35.85/11.25	
35.85/11.25	S is empty.
35.85/11.25	Rewrite Strategy: INNERMOST
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(13) DecreasingLoopProof (LOWER BOUND(ID))
35.85/11.25	The following loop(s) give(s) rise to the lower bound Omega(n^1):
35.85/11.25	
35.85/11.25	The rewrite sequence
35.85/11.25	
35.85/11.25	minus(s(x), s(y)) ->^+ minus(x, y)
35.85/11.25	
35.85/11.25	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
35.85/11.25	
35.85/11.25	The pumping substitution is [x / s(x), y / s(y)].
35.85/11.25	
35.85/11.25	The result substitution is [ ].
35.85/11.25	
35.85/11.25	
35.85/11.25	
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(14)
35.85/11.25	Complex Obligation (BEST)
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(15)
35.85/11.25	Obligation:
35.85/11.25	Proved the lower bound n^1 for the following obligation:
35.85/11.25	
35.85/11.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
35.85/11.25	
35.85/11.25	
35.85/11.25	The TRS R consists of the following rules:
35.85/11.25	
35.85/11.25	   le(0, y) -> true
35.85/11.25	   le(s(x), 0) -> false
35.85/11.25	   le(s(x), s(y)) -> le(x, y)
35.85/11.25	   minus(x, 0) -> x
35.85/11.25	   minus(0, x) -> 0
35.85/11.25	   minus(s(x), s(y)) -> minus(x, y)
35.85/11.25	   gcd(0, y) -> y
35.85/11.25	   gcd(s(x), 0) -> s(x)
35.85/11.25	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
35.85/11.25	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
35.85/11.25	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
35.85/11.25	
35.85/11.25	S is empty.
35.85/11.25	Rewrite Strategy: INNERMOST
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(16) LowerBoundPropagationProof (FINISHED)
35.85/11.25	Propagated lower bound.
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(17)
35.85/11.25	BOUNDS(n^1, INF)
35.85/11.25	
35.85/11.25	----------------------------------------
35.85/11.25	
35.85/11.25	(18)
35.85/11.25	Obligation:
35.85/11.25	Analyzing the following TRS for decreasing loops:
35.85/11.25	
35.85/11.25	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
35.85/11.25	
35.85/11.25	
35.85/11.25	The TRS R consists of the following rules:
35.85/11.25	
35.85/11.25	   le(0, y) -> true
35.85/11.25	   le(s(x), 0) -> false
35.85/11.25	   le(s(x), s(y)) -> le(x, y)
35.85/11.25	   minus(x, 0) -> x
35.85/11.25	   minus(0, x) -> 0
35.85/11.25	   minus(s(x), s(y)) -> minus(x, y)
35.85/11.25	   gcd(0, y) -> y
35.85/11.25	   gcd(s(x), 0) -> s(x)
35.85/11.25	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
35.85/11.25	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
35.85/11.25	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
35.85/11.25	
35.85/11.25	S is empty.
35.85/11.25	Rewrite Strategy: INNERMOST
36.15/11.31	EOF