471.81/291.54	WORST_CASE(Omega(n^1), O(n^2))
471.81/291.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
471.81/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
471.81/291.55	
471.81/291.55	
471.81/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
471.81/291.55	
471.81/291.55	(0) CpxTRS
471.81/291.55	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
471.81/291.55	(2) CpxWeightedTrs
471.81/291.55	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
471.81/291.55	    (4) CpxTypedWeightedTrs
471.81/291.55	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
471.81/291.55	    (6) CpxTypedWeightedCompleteTrs
471.81/291.55	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
471.81/291.55	    (8) CpxRNTS
471.81/291.55	    (9) CompleteCoflocoProof [FINISHED, 1076 ms]
471.81/291.55	    (10) BOUNDS(1, n^2)
471.81/291.55	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
471.81/291.55	(12) TRS for Loop Detection
471.81/291.55	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
471.81/291.55	    (14) BEST
471.81/291.55	        (15) proven lower bound
471.81/291.55	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
471.81/291.55	            (17) BOUNDS(n^1, INF)
471.81/291.55	        (18) TRS for Loop Detection
471.81/291.55	
471.81/291.55	
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(0)
471.81/291.55	Obligation:
471.81/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
471.81/291.55	
471.81/291.55	
471.81/291.55	The TRS R consists of the following rules:
471.81/291.55	
471.81/291.55	   cond1(true, x) -> cond2(even(x), x)
471.81/291.55	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
471.81/291.55	   cond2(false, x) -> cond1(neq(x, 0), p(x))
471.81/291.55	   neq(0, 0) -> false
471.81/291.55	   neq(0, s(x)) -> true
471.81/291.55	   neq(s(x), 0) -> true
471.81/291.55	   neq(s(x), s(y)) -> neq(x, y)
471.81/291.55	   even(0) -> true
471.81/291.55	   even(s(0)) -> false
471.81/291.55	   even(s(s(x))) -> even(x)
471.81/291.55	   div2(0) -> 0
471.81/291.55	   div2(s(0)) -> 0
471.81/291.55	   div2(s(s(x))) -> s(div2(x))
471.81/291.55	   p(0) -> 0
471.81/291.55	   p(s(x)) -> x
471.81/291.55	
471.81/291.55	S is empty.
471.81/291.55	Rewrite Strategy: INNERMOST
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
471.81/291.55	Transformed relative TRS to weighted TRS
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(2)
471.81/291.55	Obligation:
471.81/291.55	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
471.81/291.55	
471.81/291.55	
471.81/291.55	The TRS R consists of the following rules:
471.81/291.55	
471.81/291.55	   cond1(true, x) -> cond2(even(x), x) [1]
471.81/291.55	   cond2(true, x) -> cond1(neq(x, 0), div2(x)) [1]
471.81/291.55	   cond2(false, x) -> cond1(neq(x, 0), p(x)) [1]
471.81/291.55	   neq(0, 0) -> false [1]
471.81/291.55	   neq(0, s(x)) -> true [1]
471.81/291.55	   neq(s(x), 0) -> true [1]
471.81/291.55	   neq(s(x), s(y)) -> neq(x, y) [1]
471.81/291.55	   even(0) -> true [1]
471.81/291.55	   even(s(0)) -> false [1]
471.81/291.55	   even(s(s(x))) -> even(x) [1]
471.81/291.55	   div2(0) -> 0 [1]
471.81/291.55	   div2(s(0)) -> 0 [1]
471.81/291.55	   div2(s(s(x))) -> s(div2(x)) [1]
471.81/291.55	   p(0) -> 0 [1]
471.81/291.55	   p(s(x)) -> x [1]
471.81/291.55	
471.81/291.55	Rewrite Strategy: INNERMOST
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
471.81/291.55	Infered types.
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(4)
471.81/291.55	Obligation:
471.81/291.55	Runtime Complexity Weighted TRS with Types.
471.81/291.55	The TRS R consists of the following rules:
471.81/291.55	
471.81/291.55	   cond1(true, x) -> cond2(even(x), x) [1]
471.81/291.55	   cond2(true, x) -> cond1(neq(x, 0), div2(x)) [1]
471.81/291.55	   cond2(false, x) -> cond1(neq(x, 0), p(x)) [1]
471.81/291.55	   neq(0, 0) -> false [1]
471.81/291.55	   neq(0, s(x)) -> true [1]
471.81/291.55	   neq(s(x), 0) -> true [1]
471.81/291.55	   neq(s(x), s(y)) -> neq(x, y) [1]
471.81/291.55	   even(0) -> true [1]
471.81/291.55	   even(s(0)) -> false [1]
471.81/291.55	   even(s(s(x))) -> even(x) [1]
471.81/291.55	   div2(0) -> 0 [1]
471.81/291.55	   div2(s(0)) -> 0 [1]
471.81/291.55	   div2(s(s(x))) -> s(div2(x)) [1]
471.81/291.55	   p(0) -> 0 [1]
471.81/291.55	   p(s(x)) -> x [1]
471.81/291.55	
471.81/291.55	The TRS has the following type information:
471.81/291.55	cond1 :: true:false -> 0:s:y -> cond1:cond2
471.81/291.55	true :: true:false
471.81/291.55	cond2 :: true:false -> 0:s:y -> cond1:cond2
471.81/291.55	even :: 0:s:y -> true:false
471.81/291.55	neq :: 0:s:y -> 0:s:y -> true:false
471.81/291.55	0 :: 0:s:y
471.81/291.55	div2 :: 0:s:y -> 0:s:y
471.81/291.55	false :: true:false
471.81/291.55	p :: 0:s:y -> 0:s:y
471.81/291.55	s :: 0:s:y -> 0:s:y
471.81/291.55	y :: 0:s:y
471.81/291.55	
471.81/291.55	Rewrite Strategy: INNERMOST
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(5) CompletionProof (UPPER BOUND(ID))
471.81/291.55	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
471.81/291.55	
471.81/291.55	   cond1(v0, v1) -> null_cond1 [0]
471.81/291.55	   neq(v0, v1) -> null_neq [0]
471.81/291.55	   even(v0) -> null_even [0]
471.81/291.55	   div2(v0) -> null_div2 [0]
471.81/291.55	   p(v0) -> null_p [0]
471.81/291.55	   cond2(v0, v1) -> null_cond2 [0]
471.81/291.55	
471.81/291.55	And the following fresh constants:    null_cond1, null_neq, null_even, null_div2, null_p, null_cond2
471.81/291.55	
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(6)
471.81/291.55	Obligation:
471.81/291.55	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
471.81/291.55	
471.81/291.55	Runtime Complexity Weighted TRS with Types.
471.81/291.55	The TRS R consists of the following rules:
471.81/291.55	
471.81/291.55	   cond1(true, x) -> cond2(even(x), x) [1]
471.81/291.55	   cond2(true, x) -> cond1(neq(x, 0), div2(x)) [1]
471.81/291.55	   cond2(false, x) -> cond1(neq(x, 0), p(x)) [1]
471.81/291.55	   neq(0, 0) -> false [1]
471.81/291.55	   neq(0, s(x)) -> true [1]
471.81/291.55	   neq(s(x), 0) -> true [1]
471.81/291.55	   neq(s(x), s(y)) -> neq(x, y) [1]
471.81/291.55	   even(0) -> true [1]
471.81/291.55	   even(s(0)) -> false [1]
471.81/291.55	   even(s(s(x))) -> even(x) [1]
471.81/291.55	   div2(0) -> 0 [1]
471.81/291.55	   div2(s(0)) -> 0 [1]
471.81/291.55	   div2(s(s(x))) -> s(div2(x)) [1]
471.81/291.55	   p(0) -> 0 [1]
471.81/291.55	   p(s(x)) -> x [1]
471.81/291.55	   cond1(v0, v1) -> null_cond1 [0]
471.81/291.55	   neq(v0, v1) -> null_neq [0]
471.81/291.55	   even(v0) -> null_even [0]
471.81/291.55	   div2(v0) -> null_div2 [0]
471.81/291.55	   p(v0) -> null_p [0]
471.81/291.55	   cond2(v0, v1) -> null_cond2 [0]
471.81/291.55	
471.81/291.55	The TRS has the following type information:
471.81/291.55	cond1 :: true:false:null_neq:null_even -> 0:s:y:null_div2:null_p -> null_cond1:null_cond2
471.81/291.55	true :: true:false:null_neq:null_even
471.81/291.55	cond2 :: true:false:null_neq:null_even -> 0:s:y:null_div2:null_p -> null_cond1:null_cond2
471.81/291.55	even :: 0:s:y:null_div2:null_p -> true:false:null_neq:null_even
471.81/291.55	neq :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p -> true:false:null_neq:null_even
471.81/291.55	0 :: 0:s:y:null_div2:null_p
471.81/291.55	div2 :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
471.81/291.55	false :: true:false:null_neq:null_even
471.81/291.55	p :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
471.81/291.55	s :: 0:s:y:null_div2:null_p -> 0:s:y:null_div2:null_p
471.81/291.55	y :: 0:s:y:null_div2:null_p
471.81/291.55	null_cond1 :: null_cond1:null_cond2
471.81/291.55	null_neq :: true:false:null_neq:null_even
471.81/291.55	null_even :: true:false:null_neq:null_even
471.81/291.55	null_div2 :: 0:s:y:null_div2:null_p
471.81/291.55	null_p :: 0:s:y:null_div2:null_p
471.81/291.55	null_cond2 :: null_cond1:null_cond2
471.81/291.55	
471.81/291.55	Rewrite Strategy: INNERMOST
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
471.81/291.55	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
471.81/291.55	The constant constructors are abstracted as follows:
471.81/291.55	
471.81/291.55	   true => 2
471.81/291.55	   0 => 0
471.81/291.55	   false => 1
471.81/291.55	   y => 1
471.81/291.55	   null_cond1 => 0
471.81/291.55	   null_neq => 0
471.81/291.55	   null_even => 0
471.81/291.55	   null_div2 => 0
471.81/291.55	   null_p => 0
471.81/291.55	   null_cond2 => 0
471.81/291.55	
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(8)
471.81/291.55	Obligation:
471.81/291.55	Complexity RNTS consisting of the following rules:
471.81/291.55	
471.81/291.55	   cond1(z, z') -{ 1 }-> cond2(even(x), x) :|: z = 2, z' = x, x >= 0
471.81/291.55	   cond1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
471.81/291.55	   cond2(z, z') -{ 1 }-> cond1(neq(x, 0), p(x)) :|: z' = x, z = 1, x >= 0
471.81/291.55	   cond2(z, z') -{ 1 }-> cond1(neq(x, 0), div2(x)) :|: z = 2, z' = x, x >= 0
471.81/291.55	   cond2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
471.81/291.55	   div2(z) -{ 1 }-> 0 :|: z = 0
471.81/291.55	   div2(z) -{ 1 }-> 0 :|: z = 1 + 0
471.81/291.55	   div2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
471.81/291.55	   div2(z) -{ 1 }-> 1 + div2(x) :|: x >= 0, z = 1 + (1 + x)
471.81/291.55	   even(z) -{ 1 }-> even(x) :|: x >= 0, z = 1 + (1 + x)
471.81/291.55	   even(z) -{ 1 }-> 2 :|: z = 0
471.81/291.55	   even(z) -{ 1 }-> 1 :|: z = 1 + 0
471.81/291.55	   even(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
471.81/291.55	   neq(z, z') -{ 1 }-> neq(x, 1) :|: z' = 1 + 1, x >= 0, z = 1 + x
471.81/291.55	   neq(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0
471.81/291.55	   neq(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
471.81/291.55	   neq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
471.81/291.55	   neq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
471.81/291.55	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
471.81/291.55	   p(z) -{ 1 }-> 0 :|: z = 0
471.81/291.55	   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
471.81/291.55	
471.81/291.55	Only complete derivations are relevant for the runtime complexity.
471.81/291.55	
471.81/291.55	----------------------------------------
471.81/291.55	
471.81/291.55	(9) CompleteCoflocoProof (FINISHED)
471.81/291.55	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
471.81/291.55	
471.81/291.55	eq(start(V1, V),0,[cond1(V1, V, Out)],[V1 >= 0,V >= 0]).
471.81/291.55	eq(start(V1, V),0,[cond2(V1, V, Out)],[V1 >= 0,V >= 0]).
471.81/291.55	eq(start(V1, V),0,[neq(V1, V, Out)],[V1 >= 0,V >= 0]).
471.81/291.55	eq(start(V1, V),0,[even(V1, Out)],[V1 >= 0]).
471.81/291.55	eq(start(V1, V),0,[div2(V1, Out)],[V1 >= 0]).
471.81/291.55	eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
471.81/291.55	eq(cond1(V1, V, Out),1,[even(V2, Ret0),cond2(Ret0, V2, Ret)],[Out = Ret,V1 = 2,V = V2,V2 >= 0]).
471.81/291.55	eq(cond2(V1, V, Out),1,[neq(V3, 0, Ret01),div2(V3, Ret1),cond1(Ret01, Ret1, Ret2)],[Out = Ret2,V1 = 2,V = V3,V3 >= 0]).
471.81/291.55	eq(cond2(V1, V, Out),1,[neq(V4, 0, Ret02),p(V4, Ret11),cond1(Ret02, Ret11, Ret3)],[Out = Ret3,V = V4,V1 = 1,V4 >= 0]).
471.81/291.55	eq(neq(V1, V, Out),1,[],[Out = 1,V1 = 0,V = 0]).
471.81/291.55	eq(neq(V1, V, Out),1,[],[Out = 2,V = 1 + V5,V5 >= 0,V1 = 0]).
471.81/291.55	eq(neq(V1, V, Out),1,[],[Out = 2,V6 >= 0,V1 = 1 + V6,V = 0]).
471.81/291.55	eq(neq(V1, V, Out),1,[neq(V7, 1, Ret4)],[Out = Ret4,V = 2,V7 >= 0,V1 = 1 + V7]).
471.81/291.55	eq(even(V1, Out),1,[],[Out = 2,V1 = 0]).
471.81/291.55	eq(even(V1, Out),1,[],[Out = 1,V1 = 1]).
471.81/291.55	eq(even(V1, Out),1,[even(V8, Ret5)],[Out = Ret5,V8 >= 0,V1 = 2 + V8]).
471.81/291.55	eq(div2(V1, Out),1,[],[Out = 0,V1 = 0]).
471.81/291.55	eq(div2(V1, Out),1,[],[Out = 0,V1 = 1]).
471.81/291.55	eq(div2(V1, Out),1,[div2(V9, Ret12)],[Out = 1 + Ret12,V9 >= 0,V1 = 2 + V9]).
471.81/291.55	eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
471.81/291.55	eq(p(V1, Out),1,[],[Out = V10,V10 >= 0,V1 = 1 + V10]).
471.81/291.55	eq(cond1(V1, V, Out),0,[],[Out = 0,V12 >= 0,V11 >= 0,V1 = V12,V = V11]).
471.81/291.55	eq(neq(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
471.81/291.55	eq(even(V1, Out),0,[],[Out = 0,V15 >= 0,V1 = V15]).
471.81/291.55	eq(div2(V1, Out),0,[],[Out = 0,V16 >= 0,V1 = V16]).
471.81/291.55	eq(p(V1, Out),0,[],[Out = 0,V17 >= 0,V1 = V17]).
471.81/291.55	eq(cond2(V1, V, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V1 = V18,V = V19]).
471.81/291.55	input_output_vars(cond1(V1,V,Out),[V1,V],[Out]).
471.81/291.55	input_output_vars(cond2(V1,V,Out),[V1,V],[Out]).
471.81/291.55	input_output_vars(neq(V1,V,Out),[V1,V],[Out]).
471.81/291.55	input_output_vars(even(V1,Out),[V1],[Out]).
471.81/291.55	input_output_vars(div2(V1,Out),[V1],[Out]).
471.81/291.55	input_output_vars(p(V1,Out),[V1],[Out]).
471.81/291.55	
471.81/291.55	
471.81/291.55	CoFloCo proof output:
471.81/291.55	Preprocessing Cost Relations
471.81/291.55	=====================================
471.81/291.55	
471.81/291.55	#### Computed strongly connected components 
471.81/291.55	0. recursive  : [div2/2]
471.81/291.55	1. recursive  : [neq/3]
471.81/291.55	2. non_recursive  : [p/2]
471.81/291.55	3. recursive  : [even/2]
471.81/291.55	4. recursive  : [cond1/3,cond2/3]
471.81/291.55	5. non_recursive  : [start/2]
471.81/291.55	
471.81/291.55	#### Obtained direct recursion through partial evaluation 
471.81/291.55	0. SCC is partially evaluated into div2/2
471.81/291.55	1. SCC is partially evaluated into neq/3
471.81/291.55	2. SCC is partially evaluated into p/2
471.81/291.55	3. SCC is partially evaluated into even/2
471.81/291.55	4. SCC is partially evaluated into cond2/3
471.81/291.55	5. SCC is partially evaluated into start/2
471.81/291.55	
471.81/291.55	Control-Flow Refinement of Cost Relations
471.81/291.55	=====================================
471.81/291.56	
471.81/291.56	### Specialization of cost equations div2/2 
471.81/291.56	* CE 23 is refined into CE [29] 
471.81/291.56	* CE 22 is refined into CE [30] 
471.81/291.56	* CE 25 is refined into CE [31] 
471.81/291.56	* CE 24 is refined into CE [32] 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Cost equations --> "Loop" of div2/2 
471.81/291.56	* CEs [32] --> Loop 20 
471.81/291.56	* CEs [29] --> Loop 21 
471.81/291.56	* CEs [30,31] --> Loop 22 
471.81/291.56	
471.81/291.56	### Ranking functions of CR div2(V1,Out) 
471.81/291.56	* RF of phase [20]: [V1-1]
471.81/291.56	
471.81/291.56	#### Partial ranking functions of CR div2(V1,Out) 
471.81/291.56	* Partial RF of phase [20]:
471.81/291.56	  - RF of loop [20:1]:
471.81/291.56	    V1-1
471.81/291.56	
471.81/291.56	
471.81/291.56	### Specialization of cost equations neq/3 
471.81/291.56	* CE 21 is refined into CE [33] 
471.81/291.56	* CE 19 is refined into CE [34] 
471.81/291.56	* CE 18 is refined into CE [35] 
471.81/291.56	* CE 17 is refined into CE [36] 
471.81/291.56	* CE 20 is refined into CE [37] 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Cost equations --> "Loop" of neq/3 
471.81/291.56	* CEs [37] --> Loop 23 
471.81/291.56	* CEs [33] --> Loop 24 
471.81/291.56	* CEs [34] --> Loop 25 
471.81/291.56	* CEs [35] --> Loop 26 
471.81/291.56	* CEs [36] --> Loop 27 
471.81/291.56	
471.81/291.56	### Ranking functions of CR neq(V1,V,Out) 
471.81/291.56	
471.81/291.56	#### Partial ranking functions of CR neq(V1,V,Out) 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Specialization of cost equations p/2 
471.81/291.56	* CE 27 is refined into CE [38] 
471.81/291.56	* CE 26 is refined into CE [39] 
471.81/291.56	* CE 28 is refined into CE [40] 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Cost equations --> "Loop" of p/2 
471.81/291.56	* CEs [38] --> Loop 28 
471.81/291.56	* CEs [39,40] --> Loop 29 
471.81/291.56	
471.81/291.56	### Ranking functions of CR p(V1,Out) 
471.81/291.56	
471.81/291.56	#### Partial ranking functions of CR p(V1,Out) 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Specialization of cost equations even/2 
471.81/291.56	* CE 11 is refined into CE [41] 
471.81/291.56	* CE 9 is refined into CE [42] 
471.81/291.56	* CE 8 is refined into CE [43] 
471.81/291.56	* CE 10 is refined into CE [44] 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Cost equations --> "Loop" of even/2 
471.81/291.56	* CEs [44] --> Loop 30 
471.81/291.56	* CEs [41] --> Loop 31 
471.81/291.56	* CEs [42] --> Loop 32 
471.81/291.56	* CEs [43] --> Loop 33 
471.81/291.56	
471.81/291.56	### Ranking functions of CR even(V1,Out) 
471.81/291.56	* RF of phase [30]: [V1-1]
471.81/291.56	
471.81/291.56	#### Partial ranking functions of CR even(V1,Out) 
471.81/291.56	* Partial RF of phase [30]:
471.81/291.56	  - RF of loop [30:1]:
471.81/291.56	    V1-1
471.81/291.56	
471.81/291.56	
471.81/291.56	### Specialization of cost equations cond2/3 
471.81/291.56	* CE 13 is refined into CE [45,46,47,48,49] 
471.81/291.56	* CE 12 is refined into CE [50,51,52,53,54] 
471.81/291.56	* CE 16 is refined into CE [55] 
471.81/291.56	* CE 15 is refined into CE [56,57,58,59,60,61] 
471.81/291.56	* CE 14 is refined into CE [62,63,64,65,66,67,68] 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Cost equations --> "Loop" of cond2/3 
471.81/291.56	* CEs [61] --> Loop 34 
471.81/291.56	* CEs [56] --> Loop 35 
471.81/291.56	* CEs [60] --> Loop 36 
471.81/291.56	* CEs [58] --> Loop 37 
471.81/291.56	* CEs [59] --> Loop 38 
471.81/291.56	* CEs [57] --> Loop 39 
471.81/291.56	* CEs [68] --> Loop 40 
471.81/291.56	* CEs [67] --> Loop 41 
471.81/291.56	* CEs [66] --> Loop 42 
471.81/291.56	* CEs [63] --> Loop 43 
471.81/291.56	* CEs [65] --> Loop 44 
471.81/291.56	* CEs [62,64] --> Loop 45 
471.81/291.56	* CEs [45,46,47,48,49] --> Loop 46 
471.81/291.56	* CEs [50,51,52,53,54,55] --> Loop 47 
471.81/291.56	
471.81/291.56	### Ranking functions of CR cond2(V1,V,Out) 
471.81/291.56	* RF of phase [34,36,40,41]: [V-2,-2*V1+V+1,-V1+V-1]
471.81/291.56	
471.81/291.56	#### Partial ranking functions of CR cond2(V1,V,Out) 
471.81/291.56	* Partial RF of phase [34,36,40,41]:
471.81/291.56	  - RF of loop [34:1,41:1]:
471.81/291.56	    V-3
471.81/291.56	  - RF of loop [36:1]:
471.81/291.56	    V/3-5/3
471.81/291.56	    V1-1 depends on loops [40:1] 
471.81/291.56	  - RF of loop [40:1]:
471.81/291.56	    V-2
471.81/291.56	    -V1+2 depends on loops [36:1] 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Specialization of cost equations start/2 
471.81/291.56	* CE 1 is refined into CE [69] 
471.81/291.56	* CE 2 is refined into CE [70,71,72,73,74,75,76,77,78] 
471.81/291.56	* CE 3 is refined into CE [79,80,81] 
471.81/291.56	* CE 4 is refined into CE [82,83,84,85,86] 
471.81/291.56	* CE 5 is refined into CE [87,88,89,90,91] 
471.81/291.56	* CE 6 is refined into CE [92,93] 
471.81/291.56	* CE 7 is refined into CE [94,95] 
471.81/291.56	
471.81/291.56	
471.81/291.56	### Cost equations --> "Loop" of start/2 
471.81/291.56	* CEs [81] --> Loop 48 
471.81/291.56	* CEs [85] --> Loop 49 
471.81/291.56	* CEs [72] --> Loop 50 
471.81/291.56	* CEs [70,71,73,74,75,76,77,78,80] --> Loop 51 
471.81/291.56	* CEs [84,88] --> Loop 52 
471.81/291.56	* CEs [69,79,82,83,86,87,89,90,91,92,93,94,95] --> Loop 53 
471.81/291.56	
471.81/291.56	### Ranking functions of CR start(V1,V) 
471.81/291.56	
471.81/291.56	#### Partial ranking functions of CR start(V1,V) 
471.81/291.56	
471.81/291.56	
471.81/291.56	Computing Bounds
471.81/291.56	=====================================
471.81/291.56	
471.81/291.56	#### Cost of chains of div2(V1,Out):
471.81/291.56	* Chain [[20],22]: 1*it(20)+1
471.81/291.56	  Such that:it(20) =< 2*Out
471.81/291.56	
471.81/291.56	  with precondition: [Out>=1,V1>=2*Out] 
471.81/291.56	
471.81/291.56	* Chain [[20],21]: 1*it(20)+1
471.81/291.56	  Such that:it(20) =< 2*Out
471.81/291.56	
471.81/291.56	  with precondition: [V1=2*Out+1,V1>=3] 
471.81/291.56	
471.81/291.56	* Chain [22]: 1
471.81/291.56	  with precondition: [Out=0,V1>=0] 
471.81/291.56	
471.81/291.56	* Chain [21]: 1
471.81/291.56	  with precondition: [V1=1,Out=0] 
471.81/291.56	
471.81/291.56	
471.81/291.56	#### Cost of chains of neq(V1,V,Out):
471.81/291.56	* Chain [27]: 1
471.81/291.56	  with precondition: [V1=0,V=0,Out=1] 
471.81/291.56	
471.81/291.56	* Chain [26]: 1
471.81/291.56	  with precondition: [V1=0,Out=2,V>=1] 
471.81/291.56	
471.81/291.56	* Chain [25]: 1
471.81/291.56	  with precondition: [V=0,Out=2,V1>=1] 
471.81/291.56	
471.81/291.56	* Chain [24]: 0
471.81/291.56	  with precondition: [Out=0,V1>=0,V>=0] 
471.81/291.56	
471.81/291.56	* Chain [23,26]: 2
471.81/291.56	  with precondition: [V1=1,V=2,Out=2] 
471.81/291.56	
471.81/291.56	* Chain [23,24]: 1
471.81/291.56	  with precondition: [V=2,Out=0,V1>=1] 
471.81/291.56	
471.81/291.56	
471.81/291.56	#### Cost of chains of p(V1,Out):
471.81/291.56	* Chain [29]: 1
471.81/291.56	  with precondition: [Out=0,V1>=0] 
471.81/291.56	
471.81/291.56	* Chain [28]: 1
471.81/291.56	  with precondition: [V1=Out+1,V1>=1] 
471.81/291.56	
471.81/291.56	
471.81/291.56	#### Cost of chains of even(V1,Out):
471.81/291.56	* Chain [[30],33]: 1*it(30)+1
471.81/291.56	  Such that:it(30) =< V1
471.81/291.56	
471.81/291.56	  with precondition: [Out=2,V1>=2] 
471.81/291.56	
471.81/291.56	* Chain [[30],32]: 1*it(30)+1
471.81/291.56	  Such that:it(30) =< V1
471.81/291.56	
471.81/291.56	  with precondition: [Out=1,V1>=3] 
471.81/291.56	
471.81/291.56	* Chain [[30],31]: 1*it(30)+0
471.81/291.56	  Such that:it(30) =< V1
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,V1>=2] 
471.81/291.56	
471.81/291.56	* Chain [33]: 1
471.81/291.56	  with precondition: [V1=0,Out=2] 
471.81/291.56	
471.81/291.56	* Chain [32]: 1
471.81/291.56	  with precondition: [V1=1,Out=1] 
471.81/291.56	
471.81/291.56	* Chain [31]: 0
471.81/291.56	  with precondition: [Out=0,V1>=0] 
471.81/291.56	
471.81/291.56	
471.81/291.56	#### Cost of chains of cond2(V1,V,Out):
471.81/291.56	* Chain [[34,36,40,41],47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+3
471.81/291.56	  Such that:aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(16) =< -V1+V
471.81/291.56	aux(17) =< V
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(16)
471.81/291.56	it(36) =< aux(16)
471.81/291.56	it(34) =< aux(17)
471.81/291.56	it(36) =< aux(17)
471.81/291.56	s(21) =< aux(17)
471.81/291.56	aux(9) =< aux(17)
471.81/291.56	s(22) =< aux(17)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(17)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],46]: 20*it(34)+4*s(20)+6*s(21)+1*s(26)+1*s(27)+3
471.81/291.56	  Such that:aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+4
471.81/291.56	aux(12) =< -V1+V
471.81/291.56	aux(13) =< -V1+V+2
471.81/291.56	aux(19) =< V
471.81/291.56	it(34) =< aux(19)
471.81/291.56	s(21) =< aux(19)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(34) =< aux(12)
471.81/291.56	it(34) =< aux(13)
471.81/291.56	aux(9) =< aux(19)
471.81/291.56	s(22) =< aux(19)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(19)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],45,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+8
471.81/291.56	  Such that:aux(11) =< -2*V1+V
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(20) =< -V1+V
471.81/291.56	aux(21) =< V
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(20)
471.81/291.56	it(36) =< aux(20)
471.81/291.56	it(34) =< aux(21)
471.81/291.56	it(36) =< aux(21)
471.81/291.56	s(21) =< aux(21)
471.81/291.56	aux(9) =< aux(21)
471.81/291.56	s(22) =< aux(21)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(21)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=2*V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],45,46]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+8
471.81/291.56	  Such that:aux(11) =< -2*V1+V
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(22) =< -V1+V
471.81/291.56	aux(23) =< V
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(22)
471.81/291.56	it(36) =< aux(22)
471.81/291.56	it(34) =< aux(23)
471.81/291.56	it(36) =< aux(23)
471.81/291.56	s(21) =< aux(23)
471.81/291.56	aux(9) =< aux(23)
471.81/291.56	s(22) =< aux(23)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(23)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=2*V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],43,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+7
471.81/291.56	  Such that:aux(11) =< -2*V1+V
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(24) =< -V1+V
471.81/291.56	aux(25) =< V
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(24)
471.81/291.56	it(36) =< aux(24)
471.81/291.56	it(34) =< aux(25)
471.81/291.56	it(36) =< aux(25)
471.81/291.56	s(21) =< aux(25)
471.81/291.56	aux(9) =< aux(25)
471.81/291.56	s(22) =< aux(25)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(25)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=2*V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],42,47]: 20*it(34)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+1*s(33)+7
471.81/291.56	  Such that:aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	aux(12) =< -V1+V
471.81/291.56	aux(26) =< -V1+V+1
471.81/291.56	aux(27) =< V
471.81/291.56	s(33) =< aux(26)
471.81/291.56	it(34) =< aux(27)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(34) =< aux(12)
471.81/291.56	it(34) =< aux(26)
471.81/291.56	s(21) =< aux(27)
471.81/291.56	aux(9) =< aux(27)
471.81/291.56	s(22) =< aux(27)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(27)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=2*V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],39,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+7
471.81/291.56	  Such that:aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(28) =< -V1+V
471.81/291.56	aux(29) =< V
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(28)
471.81/291.56	it(36) =< aux(28)
471.81/291.56	it(34) =< aux(29)
471.81/291.56	it(36) =< aux(29)
471.81/291.56	s(21) =< aux(29)
471.81/291.56	aux(9) =< aux(29)
471.81/291.56	s(22) =< aux(29)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(29)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],38,47]: 20*it(34)+4*s(20)+4*s(21)+1*s(26)+1*s(27)+1*s(37)+7
471.81/291.56	  Such that:aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+4
471.81/291.56	aux(12) =< -V1+V
471.81/291.56	aux(30) =< -V1+V+2
471.81/291.56	aux(31) =< V
471.81/291.56	s(37) =< aux(30)
471.81/291.56	it(34) =< aux(31)
471.81/291.56	s(21) =< aux(31)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(34) =< aux(12)
471.81/291.56	it(34) =< aux(30)
471.81/291.56	aux(9) =< aux(31)
471.81/291.56	s(22) =< aux(31)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(31)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],37,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+2*s(39)+8
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(32) =< -V1+V
471.81/291.56	aux(33) =< V
471.81/291.56	s(39) =< s(38)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(32)
471.81/291.56	it(36) =< aux(32)
471.81/291.56	it(34) =< aux(33)
471.81/291.56	it(36) =< aux(33)
471.81/291.56	s(21) =< aux(33)
471.81/291.56	aux(9) =< aux(33)
471.81/291.56	s(22) =< aux(33)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(33)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],37,45,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+2*s(39)+13
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(34) =< -V1+V
471.81/291.56	aux(35) =< V
471.81/291.56	s(39) =< s(38)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(34)
471.81/291.56	it(36) =< aux(34)
471.81/291.56	it(34) =< aux(35)
471.81/291.56	it(36) =< aux(35)
471.81/291.56	s(21) =< aux(35)
471.81/291.56	aux(9) =< aux(35)
471.81/291.56	s(22) =< aux(35)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(35)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],37,45,46]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+2*s(39)+13
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(36) =< -V1+V
471.81/291.56	aux(37) =< V
471.81/291.56	s(39) =< s(38)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(36)
471.81/291.56	it(36) =< aux(36)
471.81/291.56	it(34) =< aux(37)
471.81/291.56	it(36) =< aux(37)
471.81/291.56	s(21) =< aux(37)
471.81/291.56	aux(9) =< aux(37)
471.81/291.56	s(22) =< aux(37)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(37)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],37,43,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+2*s(39)+12
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(38) =< -V1+V
471.81/291.56	aux(39) =< V
471.81/291.56	s(39) =< s(38)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(38)
471.81/291.56	it(36) =< aux(38)
471.81/291.56	it(34) =< aux(39)
471.81/291.56	it(36) =< aux(39)
471.81/291.56	s(21) =< aux(39)
471.81/291.56	aux(9) =< aux(39)
471.81/291.56	s(22) =< aux(39)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(39)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],37,42,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+1*s(33)+2*s(39)+12
471.81/291.56	  Such that:s(33) =< 1
471.81/291.56	s(38) =< 2
471.81/291.56	aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(40) =< -V1+V
471.81/291.56	aux(41) =< V
471.81/291.56	s(39) =< s(38)
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(40)
471.81/291.56	it(36) =< aux(40)
471.81/291.56	it(34) =< aux(41)
471.81/291.56	it(36) =< aux(41)
471.81/291.56	s(21) =< aux(41)
471.81/291.56	aux(9) =< aux(41)
471.81/291.56	s(22) =< aux(41)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(41)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],35,47]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+8
471.81/291.56	  Such that:aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(42) =< -V1+V
471.81/291.56	aux(43) =< V
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(42)
471.81/291.56	it(36) =< aux(42)
471.81/291.56	it(34) =< aux(43)
471.81/291.56	it(36) =< aux(43)
471.81/291.56	s(21) =< aux(43)
471.81/291.56	aux(9) =< aux(43)
471.81/291.56	s(22) =< aux(43)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(43)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [[34,36,40,41],35,46]: 15*it(34)+5*it(36)+4*s(20)+2*s(21)+1*s(26)+1*s(27)+8
471.81/291.56	  Such that:aux(10) =< -2*V1+V+1
471.81/291.56	aux(11) =< -2*V1+V+2
471.81/291.56	it(36) =< V/3
471.81/291.56	aux(44) =< -V1+V
471.81/291.56	aux(45) =< V
471.81/291.56	it(34) =< aux(10)
471.81/291.56	it(36) =< aux(10)
471.81/291.56	it(34) =< aux(11)
471.81/291.56	it(36) =< aux(11)
471.81/291.56	it(34) =< aux(44)
471.81/291.56	it(36) =< aux(44)
471.81/291.56	it(34) =< aux(45)
471.81/291.56	it(36) =< aux(45)
471.81/291.56	s(21) =< aux(45)
471.81/291.56	aux(9) =< aux(45)
471.81/291.56	s(22) =< aux(45)*2
471.81/291.56	s(27) =< it(34)*aux(9)
471.81/291.56	s(26) =< it(34)*aux(45)
471.81/291.56	s(20) =< s(22)
471.81/291.56	
471.81/291.56	  with precondition: [Out=0,2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	* Chain [47]: 3
471.81/291.56	  with precondition: [Out=0,V1>=0,V>=0] 
471.81/291.56	
471.81/291.56	* Chain [46]: 4*s(29)+3
471.81/291.56	  Such that:aux(18) =< V
471.81/291.56	s(29) =< aux(18)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,Out=0,V>=0] 
471.81/291.56	
471.81/291.56	* Chain [45,47]: 8
471.81/291.56	  with precondition: [V1=1,Out=0,V>=1] 
471.81/291.56	
471.81/291.56	* Chain [45,46]: 8
471.81/291.56	  with precondition: [V1=1,Out=0,V>=1] 
471.81/291.56	
471.81/291.56	* Chain [44,47]: 8
471.81/291.56	  with precondition: [V1=1,V=2,Out=0] 
471.81/291.56	
471.81/291.56	* Chain [44,45,47]: 13
471.81/291.56	  with precondition: [V1=1,V=2,Out=0] 
471.81/291.56	
471.81/291.56	* Chain [44,45,46]: 13
471.81/291.56	  with precondition: [V1=1,V=2,Out=0] 
471.81/291.56	
471.81/291.56	* Chain [44,43,47]: 12
471.81/291.56	  with precondition: [V1=1,V=2,Out=0] 
471.81/291.56	
471.81/291.56	* Chain [44,42,47]: 1*s(33)+12
471.81/291.56	  Such that:s(33) =< 1
471.81/291.56	
471.81/291.56	  with precondition: [V1=1,V=2,Out=0] 
471.81/291.56	
471.81/291.56	* Chain [43,47]: 7
471.81/291.56	  with precondition: [V1=1,Out=0,V>=1] 
471.81/291.56	
471.81/291.56	* Chain [42,47]: 1*s(33)+7
471.81/291.56	  Such that:s(33) =< V
471.81/291.56	
471.81/291.56	  with precondition: [V1=1,Out=0,V>=1] 
471.81/291.56	
471.81/291.56	* Chain [39,47]: 7
471.81/291.56	  with precondition: [V1=2,Out=0,V>=1] 
471.81/291.56	
471.81/291.56	* Chain [38,47]: 2*s(36)+1*s(37)+7
471.81/291.56	  Such that:s(35) =< V
471.81/291.56	s(37) =< V/2
471.81/291.56	s(36) =< s(35)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,Out=0,V>=2] 
471.81/291.56	
471.81/291.56	* Chain [37,47]: 2*s(39)+8
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	s(39) =< s(38)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,Out=0,V>=2] 
471.81/291.56	
471.81/291.56	* Chain [37,45,47]: 2*s(39)+13
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	s(39) =< s(38)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,Out=0,V>=2] 
471.81/291.56	
471.81/291.56	* Chain [37,45,46]: 2*s(39)+13
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	s(39) =< s(38)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,Out=0,V>=2] 
471.81/291.56	
471.81/291.56	* Chain [37,43,47]: 2*s(39)+12
471.81/291.56	  Such that:s(38) =< 2
471.81/291.56	s(39) =< s(38)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,Out=0,V>=2] 
471.81/291.56	
471.81/291.56	* Chain [37,42,47]: 1*s(33)+2*s(39)+12
471.81/291.56	  Such that:s(33) =< 1
471.81/291.56	s(38) =< 2
471.81/291.56	s(39) =< s(38)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,Out=0,V>=2] 
471.81/291.56	
471.81/291.56	* Chain [35,47]: 8
471.81/291.56	  with precondition: [V1=2,Out=0,V>=1] 
471.81/291.56	
471.81/291.56	* Chain [35,46]: 8
471.81/291.56	  with precondition: [V1=2,Out=0,V>=1] 
471.81/291.56	
471.81/291.56	
471.81/291.56	#### Cost of chains of start(V1,V):
471.81/291.56	* Chain [53]: 1*s(251)+1*s(252)+5*s(253)+13
471.81/291.56	  Such that:s(251) =< 1
471.81/291.56	s(252) =< V
471.81/291.56	aux(57) =< V1
471.81/291.56	s(253) =< aux(57)
471.81/291.56	
471.81/291.56	  with precondition: [V1>=0] 
471.81/291.56	
471.81/291.56	* Chain [52]: 2
471.81/291.56	  with precondition: [V1=1] 
471.81/291.56	
471.81/291.56	* Chain [51]: 9*s(258)+50*s(264)+536*s(266)+120*s(284)+28*s(291)+28*s(292)+120*s(293)+1*s(297)+40*s(298)+2*s(299)+2*s(300)+2*s(310)+15
471.81/291.56	  Such that:s(281) =< V+1
471.81/291.56	s(279) =< V+2
471.81/291.56	aux(65) =< 1
471.81/291.56	aux(66) =< 2
471.81/291.56	aux(67) =< V
471.81/291.56	aux(68) =< V/2
471.81/291.56	aux(69) =< V/3
471.81/291.56	s(258) =< aux(65)
471.81/291.56	s(310) =< aux(68)
471.81/291.56	s(264) =< aux(66)
471.81/291.56	s(266) =< aux(67)
471.81/291.56	s(284) =< aux(69)
471.81/291.56	s(284) =< aux(67)
471.81/291.56	s(289) =< aux(67)
471.81/291.56	s(290) =< aux(67)*2
471.81/291.56	s(291) =< s(266)*s(289)
471.81/291.56	s(292) =< s(266)*aux(67)
471.81/291.56	s(293) =< s(290)
471.81/291.56	s(297) =< s(281)
471.81/291.56	s(298) =< aux(67)
471.81/291.56	s(298) =< s(279)
471.81/291.56	s(298) =< s(281)
471.81/291.56	s(299) =< s(298)*s(289)
471.81/291.56	s(300) =< s(298)*aux(67)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,V>=0] 
471.81/291.56	
471.81/291.56	* Chain [50]: 2*s(354)+15
471.81/291.56	  Such that:aux(70) =< 1
471.81/291.56	s(354) =< aux(70)
471.81/291.56	
471.81/291.56	  with precondition: [V1=2,V=1] 
471.81/291.56	
471.81/291.56	* Chain [49]: 1
471.81/291.56	  with precondition: [V=0,V1>=1] 
471.81/291.56	
471.81/291.56	* Chain [48]: 1*s(356)+45*s(367)+15*s(368)+10*s(369)+135*s(370)+36*s(371)+9*s(374)+9*s(375)+60*s(376)+45*s(377)+3*s(378)+3*s(379)+1*s(380)+40*s(381)+2*s(382)+2*s(383)+1*s(384)+20*s(385)+1*s(386)+1*s(387)+13
471.81/291.56	  Such that:s(356) =< 1
471.81/291.56	s(358) =< 2
471.81/291.56	s(359) =< -2*V1+V
471.81/291.56	s(360) =< -2*V1+V+1
471.81/291.56	s(361) =< -2*V1+V+2
471.81/291.56	s(362) =< -2*V1+V+4
471.81/291.56	s(363) =< -V1+V
471.81/291.56	s(357) =< -V1+V+1
471.81/291.56	s(364) =< -V1+V+2
471.81/291.56	s(365) =< V
471.81/291.56	s(366) =< V/3
471.81/291.56	s(367) =< s(366)
471.81/291.56	s(368) =< s(366)
471.81/291.56	s(369) =< s(358)
471.81/291.56	s(370) =< s(360)
471.81/291.56	s(367) =< s(360)
471.81/291.56	s(370) =< s(361)
471.81/291.56	s(367) =< s(361)
471.81/291.56	s(370) =< s(363)
471.81/291.56	s(367) =< s(363)
471.81/291.56	s(370) =< s(365)
471.81/291.56	s(367) =< s(365)
471.81/291.56	s(371) =< s(365)
471.81/291.56	s(372) =< s(365)
471.81/291.56	s(373) =< s(365)*2
471.81/291.56	s(374) =< s(370)*s(372)
471.81/291.56	s(375) =< s(370)*s(365)
471.81/291.56	s(376) =< s(373)
471.81/291.56	s(377) =< s(360)
471.81/291.56	s(368) =< s(360)
471.81/291.56	s(377) =< s(359)
471.81/291.56	s(368) =< s(359)
471.81/291.56	s(377) =< s(363)
471.81/291.56	s(368) =< s(363)
471.81/291.56	s(377) =< s(365)
471.81/291.56	s(368) =< s(365)
471.81/291.56	s(378) =< s(377)*s(372)
471.81/291.56	s(379) =< s(377)*s(365)
471.81/291.56	s(380) =< s(364)
471.81/291.56	s(381) =< s(365)
471.81/291.56	s(381) =< s(360)
471.81/291.56	s(381) =< s(362)
471.81/291.56	s(381) =< s(363)
471.81/291.56	s(381) =< s(364)
471.81/291.56	s(382) =< s(381)*s(372)
471.81/291.56	s(383) =< s(381)*s(365)
471.81/291.56	s(384) =< s(357)
471.81/291.56	s(385) =< s(365)
471.81/291.56	s(385) =< s(360)
471.81/291.56	s(385) =< s(361)
471.81/291.56	s(385) =< s(363)
471.81/291.56	s(385) =< s(357)
471.81/291.56	s(386) =< s(385)*s(372)
471.81/291.56	s(387) =< s(385)*s(365)
471.81/291.56	
471.81/291.56	  with precondition: [2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	
471.81/291.56	
471.81/291.56	Closed-form bounds of start(V1,V): 
471.81/291.56	-------------------------------------
471.81/291.56	* Chain [53] with precondition: [V1>=0] 
471.81/291.56	    - Upper bound: 5*V1+14+nat(V) 
471.81/291.56	    - Complexity: n 
471.81/291.56	* Chain [52] with precondition: [V1=1] 
471.81/291.56	    - Upper bound: 2 
471.81/291.56	    - Complexity: constant 
471.81/291.56	* Chain [51] with precondition: [V1=2,V>=0] 
471.81/291.56	    - Upper bound: 816*V+124+60*V*V+(V+1)+V+40*V 
471.81/291.56	    - Complexity: n^2 
471.81/291.56	* Chain [50] with precondition: [V1=2,V=1] 
471.81/291.56	    - Upper bound: 17 
471.81/291.56	    - Complexity: constant 
471.81/291.56	* Chain [49] with precondition: [V=0,V1>=1] 
471.81/291.56	    - Upper bound: 1 
471.81/291.56	    - Complexity: constant 
471.81/291.56	* Chain [48] with precondition: [2>=V1,V1>=1,V>=V1+2] 
471.81/291.56	    - Upper bound: 216*V+34+6*V*V+(-2*V1+V+1)*(24*V)+(-V1+V+1)+(-V1+V+2)+(-360*V1+180*V+180)+20*V 
471.81/291.56	    - Complexity: n^2 
471.81/291.56	
471.81/291.56	### Maximum cost of start(V1,V): max([16,nat(V)+13+max([5*V1,nat(V)*215+20+nat(V)*6*nat(V)+nat(V/3)*60+max([nat(V)*24*nat(-2*V1+V+1)+nat(-V1+V+1)+nat(-V1+V+2)+nat(-2*V1+V+1)*180,nat(V)*600+90+nat(V)*54*nat(V)+nat(V+1)+nat(V/2)*2+nat(V/3)*60])])])+1 
471.81/291.56	Asymptotic class: n^2 
471.81/291.56	* Total analysis performed in 964 ms.
471.81/291.56	
471.81/291.56	
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(10)
471.81/291.56	BOUNDS(1, n^2)
471.81/291.56	
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
471.81/291.56	Transformed a relative TRS into a decreasing-loop problem.
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(12)
471.81/291.56	Obligation:
471.81/291.56	Analyzing the following TRS for decreasing loops:
471.81/291.56	
471.81/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
471.81/291.56	
471.81/291.56	
471.81/291.56	The TRS R consists of the following rules:
471.81/291.56	
471.81/291.56	   cond1(true, x) -> cond2(even(x), x)
471.81/291.56	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
471.81/291.56	   cond2(false, x) -> cond1(neq(x, 0), p(x))
471.81/291.56	   neq(0, 0) -> false
471.81/291.56	   neq(0, s(x)) -> true
471.81/291.56	   neq(s(x), 0) -> true
471.81/291.56	   neq(s(x), s(y)) -> neq(x, y)
471.81/291.56	   even(0) -> true
471.81/291.56	   even(s(0)) -> false
471.81/291.56	   even(s(s(x))) -> even(x)
471.81/291.56	   div2(0) -> 0
471.81/291.56	   div2(s(0)) -> 0
471.81/291.56	   div2(s(s(x))) -> s(div2(x))
471.81/291.56	   p(0) -> 0
471.81/291.56	   p(s(x)) -> x
471.81/291.56	
471.81/291.56	S is empty.
471.81/291.56	Rewrite Strategy: INNERMOST
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(13) DecreasingLoopProof (LOWER BOUND(ID))
471.81/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
471.81/291.56	
471.81/291.56	The rewrite sequence
471.81/291.56	
471.81/291.56	even(s(s(x))) ->^+ even(x)
471.81/291.56	
471.81/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
471.81/291.56	
471.81/291.56	The pumping substitution is [x / s(s(x))].
471.81/291.56	
471.81/291.56	The result substitution is [ ].
471.81/291.56	
471.81/291.56	
471.81/291.56	
471.81/291.56	
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(14)
471.81/291.56	Complex Obligation (BEST)
471.81/291.56	
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(15)
471.81/291.56	Obligation:
471.81/291.56	Proved the lower bound n^1 for the following obligation:
471.81/291.56	
471.81/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
471.81/291.56	
471.81/291.56	
471.81/291.56	The TRS R consists of the following rules:
471.81/291.56	
471.81/291.56	   cond1(true, x) -> cond2(even(x), x)
471.81/291.56	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
471.81/291.56	   cond2(false, x) -> cond1(neq(x, 0), p(x))
471.81/291.56	   neq(0, 0) -> false
471.81/291.56	   neq(0, s(x)) -> true
471.81/291.56	   neq(s(x), 0) -> true
471.81/291.56	   neq(s(x), s(y)) -> neq(x, y)
471.81/291.56	   even(0) -> true
471.81/291.56	   even(s(0)) -> false
471.81/291.56	   even(s(s(x))) -> even(x)
471.81/291.56	   div2(0) -> 0
471.81/291.56	   div2(s(0)) -> 0
471.81/291.56	   div2(s(s(x))) -> s(div2(x))
471.81/291.56	   p(0) -> 0
471.81/291.56	   p(s(x)) -> x
471.81/291.56	
471.81/291.56	S is empty.
471.81/291.56	Rewrite Strategy: INNERMOST
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(16) LowerBoundPropagationProof (FINISHED)
471.81/291.56	Propagated lower bound.
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(17)
471.81/291.56	BOUNDS(n^1, INF)
471.81/291.56	
471.81/291.56	----------------------------------------
471.81/291.56	
471.81/291.56	(18)
471.81/291.56	Obligation:
471.81/291.56	Analyzing the following TRS for decreasing loops:
471.81/291.56	
471.81/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
471.81/291.56	
471.81/291.56	
471.81/291.56	The TRS R consists of the following rules:
471.81/291.56	
471.81/291.56	   cond1(true, x) -> cond2(even(x), x)
471.81/291.56	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
471.81/291.56	   cond2(false, x) -> cond1(neq(x, 0), p(x))
471.81/291.56	   neq(0, 0) -> false
471.81/291.56	   neq(0, s(x)) -> true
471.81/291.56	   neq(s(x), 0) -> true
471.81/291.56	   neq(s(x), s(y)) -> neq(x, y)
471.81/291.56	   even(0) -> true
471.81/291.56	   even(s(0)) -> false
471.81/291.56	   even(s(s(x))) -> even(x)
471.81/291.56	   div2(0) -> 0
471.81/291.56	   div2(s(0)) -> 0
471.81/291.56	   div2(s(s(x))) -> s(div2(x))
471.81/291.56	   p(0) -> 0
471.81/291.56	   p(s(x)) -> x
471.81/291.56	
471.81/291.56	S is empty.
471.81/291.56	Rewrite Strategy: INNERMOST
472.02/291.62	EOF