1148.12/291.58	WORST_CASE(Omega(n^1), O(n^2))
1148.34/291.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1148.34/291.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1148.34/291.61	
1148.34/291.61	(0) CpxTRS
1148.34/291.61	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1148.34/291.61	(2) CpxWeightedTrs
1148.34/291.61	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1148.34/291.61	    (4) CpxTypedWeightedTrs
1148.34/291.61	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
1148.34/291.61	    (6) CpxTypedWeightedCompleteTrs
1148.34/291.61	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1148.34/291.61	    (8) CpxRNTS
1148.34/291.61	    (9) CompleteCoflocoProof [FINISHED, 1105 ms]
1148.34/291.61	    (10) BOUNDS(1, n^2)
1148.34/291.61	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1148.34/291.61	(12) TRS for Loop Detection
1148.34/291.61	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1148.34/291.61	    (14) BEST
1148.34/291.61	        (15) proven lower bound
1148.34/291.61	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1148.34/291.61	            (17) BOUNDS(n^1, INF)
1148.34/291.61	        (18) TRS for Loop Detection
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(0)
1148.34/291.61	Obligation:
1148.34/291.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	The TRS R consists of the following rules:
1148.34/291.61	
1148.34/291.61	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
1148.34/291.61	   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond2(false, x, y) -> cond3(eq(x, y), x, y)
1148.34/291.61	   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
1148.34/291.61	   gr(0, x) -> false
1148.34/291.61	   gr(s(x), 0) -> true
1148.34/291.61	   gr(s(x), s(y)) -> gr(x, y)
1148.34/291.61	   add(0, x) -> x
1148.34/291.61	   add(s(x), y) -> s(add(x, y))
1148.34/291.61	   eq(0, 0) -> true
1148.34/291.61	   eq(0, s(x)) -> false
1148.34/291.61	   eq(s(x), 0) -> false
1148.34/291.61	   eq(s(x), s(y)) -> eq(x, y)
1148.34/291.61	   p(0) -> 0
1148.34/291.61	   p(s(x)) -> x
1148.34/291.61	
1148.34/291.61	S is empty.
1148.34/291.61	Rewrite Strategy: INNERMOST
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1148.34/291.61	Transformed relative TRS to weighted TRS
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(2)
1148.34/291.61	Obligation:
1148.34/291.61	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	The TRS R consists of the following rules:
1148.34/291.61	
1148.34/291.61	   cond1(true, x, y) -> cond2(gr(x, y), x, y) [1]
1148.34/291.61	   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
1148.34/291.61	   cond2(false, x, y) -> cond3(eq(x, y), x, y) [1]
1148.34/291.61	   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
1148.34/291.61	   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1]
1148.34/291.61	   gr(0, x) -> false [1]
1148.34/291.61	   gr(s(x), 0) -> true [1]
1148.34/291.61	   gr(s(x), s(y)) -> gr(x, y) [1]
1148.34/291.61	   add(0, x) -> x [1]
1148.34/291.61	   add(s(x), y) -> s(add(x, y)) [1]
1148.34/291.61	   eq(0, 0) -> true [1]
1148.34/291.61	   eq(0, s(x)) -> false [1]
1148.34/291.61	   eq(s(x), 0) -> false [1]
1148.34/291.61	   eq(s(x), s(y)) -> eq(x, y) [1]
1148.34/291.61	   p(0) -> 0 [1]
1148.34/291.61	   p(s(x)) -> x [1]
1148.34/291.61	
1148.34/291.61	Rewrite Strategy: INNERMOST
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1148.34/291.61	Infered types.
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(4)
1148.34/291.61	Obligation:
1148.34/291.61	Runtime Complexity Weighted TRS with Types.
1148.34/291.61	The TRS R consists of the following rules:
1148.34/291.61	
1148.34/291.61	   cond1(true, x, y) -> cond2(gr(x, y), x, y) [1]
1148.34/291.61	   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
1148.34/291.61	   cond2(false, x, y) -> cond3(eq(x, y), x, y) [1]
1148.34/291.61	   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
1148.34/291.61	   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1]
1148.34/291.61	   gr(0, x) -> false [1]
1148.34/291.61	   gr(s(x), 0) -> true [1]
1148.34/291.61	   gr(s(x), s(y)) -> gr(x, y) [1]
1148.34/291.61	   add(0, x) -> x [1]
1148.34/291.61	   add(s(x), y) -> s(add(x, y)) [1]
1148.34/291.61	   eq(0, 0) -> true [1]
1148.34/291.61	   eq(0, s(x)) -> false [1]
1148.34/291.61	   eq(s(x), 0) -> false [1]
1148.34/291.61	   eq(s(x), s(y)) -> eq(x, y) [1]
1148.34/291.61	   p(0) -> 0 [1]
1148.34/291.61	   p(s(x)) -> x [1]
1148.34/291.61	
1148.34/291.61	The TRS has the following type information:
1148.34/291.61	cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
1148.34/291.61	true :: true:false
1148.34/291.61	cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
1148.34/291.61	gr :: 0:s -> 0:s -> true:false
1148.34/291.61	add :: 0:s -> 0:s -> 0:s
1148.34/291.61	0 :: 0:s
1148.34/291.61	p :: 0:s -> 0:s
1148.34/291.61	false :: true:false
1148.34/291.61	cond3 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3
1148.34/291.61	eq :: 0:s -> 0:s -> true:false
1148.34/291.61	s :: 0:s -> 0:s
1148.34/291.61	
1148.34/291.61	Rewrite Strategy: INNERMOST
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(5) CompletionProof (UPPER BOUND(ID))
1148.34/291.61	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
1148.34/291.61	
1148.34/291.61	   cond1(v0, v1, v2) -> null_cond1 [0]
1148.34/291.61	
1148.34/291.61	And the following fresh constants:    null_cond1
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(6)
1148.34/291.61	Obligation:
1148.34/291.61	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
1148.34/291.61	
1148.34/291.61	Runtime Complexity Weighted TRS with Types.
1148.34/291.61	The TRS R consists of the following rules:
1148.34/291.61	
1148.34/291.61	   cond1(true, x, y) -> cond2(gr(x, y), x, y) [1]
1148.34/291.61	   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
1148.34/291.61	   cond2(false, x, y) -> cond3(eq(x, y), x, y) [1]
1148.34/291.61	   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y) [1]
1148.34/291.61	   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y)) [1]
1148.34/291.61	   gr(0, x) -> false [1]
1148.34/291.61	   gr(s(x), 0) -> true [1]
1148.34/291.61	   gr(s(x), s(y)) -> gr(x, y) [1]
1148.34/291.61	   add(0, x) -> x [1]
1148.34/291.61	   add(s(x), y) -> s(add(x, y)) [1]
1148.34/291.61	   eq(0, 0) -> true [1]
1148.34/291.61	   eq(0, s(x)) -> false [1]
1148.34/291.61	   eq(s(x), 0) -> false [1]
1148.34/291.61	   eq(s(x), s(y)) -> eq(x, y) [1]
1148.34/291.61	   p(0) -> 0 [1]
1148.34/291.61	   p(s(x)) -> x [1]
1148.34/291.61	   cond1(v0, v1, v2) -> null_cond1 [0]
1148.34/291.61	
1148.34/291.61	The TRS has the following type information:
1148.34/291.61	cond1 :: true:false -> 0:s -> 0:s -> null_cond1
1148.34/291.61	true :: true:false
1148.34/291.61	cond2 :: true:false -> 0:s -> 0:s -> null_cond1
1148.34/291.61	gr :: 0:s -> 0:s -> true:false
1148.34/291.61	add :: 0:s -> 0:s -> 0:s
1148.34/291.61	0 :: 0:s
1148.34/291.61	p :: 0:s -> 0:s
1148.34/291.61	false :: true:false
1148.34/291.61	cond3 :: true:false -> 0:s -> 0:s -> null_cond1
1148.34/291.61	eq :: 0:s -> 0:s -> true:false
1148.34/291.61	s :: 0:s -> 0:s
1148.34/291.61	null_cond1 :: null_cond1
1148.34/291.61	
1148.34/291.61	Rewrite Strategy: INNERMOST
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1148.34/291.61	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1148.34/291.61	The constant constructors are abstracted as follows:
1148.34/291.61	
1148.34/291.61	   true => 1
1148.34/291.61	   0 => 0
1148.34/291.61	   false => 0
1148.34/291.61	   null_cond1 => 0
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(8)
1148.34/291.61	Obligation:
1148.34/291.61	Complexity RNTS consisting of the following rules:
1148.34/291.61	
1148.34/291.61	   add(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
1148.34/291.61	   add(z, z') -{ 1 }-> 1 + add(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
1148.34/291.61	   cond1(z, z', z'') -{ 1 }-> cond2(gr(x, y), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
1148.34/291.61	   cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
1148.34/291.61	   cond2(z, z', z'') -{ 1 }-> cond3(eq(x, y), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
1148.34/291.61	   cond2(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
1148.34/291.61	   cond3(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), x, p(y)) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
1148.34/291.61	   cond3(z, z', z'') -{ 1 }-> cond1(gr(add(x, y), 0), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
1148.34/291.61	   eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1148.34/291.61	   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
1148.34/291.61	   eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
1148.34/291.61	   eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
1148.34/291.61	   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1148.34/291.61	   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
1148.34/291.61	   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
1148.34/291.61	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
1148.34/291.61	   p(z) -{ 1 }-> 0 :|: z = 0
1148.34/291.61	
1148.34/291.61	Only complete derivations are relevant for the runtime complexity.
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(9) CompleteCoflocoProof (FINISHED)
1148.34/291.61	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
1148.34/291.61	
1148.34/291.61	eq(start(V1, V, V2),0,[cond1(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
1148.34/291.61	eq(start(V1, V, V2),0,[cond2(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
1148.34/291.61	eq(start(V1, V, V2),0,[cond3(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
1148.34/291.61	eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
1148.34/291.61	eq(start(V1, V, V2),0,[add(V1, V, Out)],[V1 >= 0,V >= 0]).
1148.34/291.61	eq(start(V1, V, V2),0,[eq(V1, V, Out)],[V1 >= 0,V >= 0]).
1148.34/291.61	eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]).
1148.34/291.61	eq(cond1(V1, V, V2, Out),1,[gr(V4, V3, Ret0),cond2(Ret0, V4, V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
1148.34/291.61	eq(cond2(V1, V, V2, Out),1,[add(V5, V6, Ret00),gr(Ret00, 0, Ret01),p(V5, Ret1),cond1(Ret01, Ret1, V6, Ret2)],[Out = Ret2,V = V5,V2 = V6,V1 = 1,V5 >= 0,V6 >= 0]).
1148.34/291.61	eq(cond2(V1, V, V2, Out),1,[eq(V8, V7, Ret02),cond3(Ret02, V8, V7, Ret3)],[Out = Ret3,V = V8,V2 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
1148.34/291.61	eq(cond3(V1, V, V2, Out),1,[add(V9, V10, Ret001),gr(Ret001, 0, Ret03),p(V9, Ret11),cond1(Ret03, Ret11, V10, Ret4)],[Out = Ret4,V = V9,V2 = V10,V1 = 1,V9 >= 0,V10 >= 0]).
1148.34/291.61	eq(cond3(V1, V, V2, Out),1,[add(V12, V11, Ret002),gr(Ret002, 0, Ret04),p(V11, Ret21),cond1(Ret04, V12, Ret21, Ret5)],[Out = Ret5,V = V12,V2 = V11,V12 >= 0,V11 >= 0,V1 = 0]).
1148.34/291.61	eq(gr(V1, V, Out),1,[],[Out = 0,V = V13,V13 >= 0,V1 = 0]).
1148.34/291.61	eq(gr(V1, V, Out),1,[],[Out = 1,V14 >= 0,V1 = 1 + V14,V = 0]).
1148.34/291.61	eq(gr(V1, V, Out),1,[gr(V16, V15, Ret6)],[Out = Ret6,V = 1 + V15,V16 >= 0,V15 >= 0,V1 = 1 + V16]).
1148.34/291.61	eq(add(V1, V, Out),1,[],[Out = V17,V = V17,V17 >= 0,V1 = 0]).
1148.34/291.61	eq(add(V1, V, Out),1,[add(V19, V18, Ret12)],[Out = 1 + Ret12,V19 >= 0,V18 >= 0,V1 = 1 + V19,V = V18]).
1148.34/291.61	eq(eq(V1, V, Out),1,[],[Out = 1,V1 = 0,V = 0]).
1148.34/291.61	eq(eq(V1, V, Out),1,[],[Out = 0,V = 1 + V20,V20 >= 0,V1 = 0]).
1148.34/291.61	eq(eq(V1, V, Out),1,[],[Out = 0,V21 >= 0,V1 = 1 + V21,V = 0]).
1148.34/291.61	eq(eq(V1, V, Out),1,[eq(V22, V23, Ret7)],[Out = Ret7,V = 1 + V23,V22 >= 0,V23 >= 0,V1 = 1 + V22]).
1148.34/291.61	eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
1148.34/291.61	eq(p(V1, Out),1,[],[Out = V24,V24 >= 0,V1 = 1 + V24]).
1148.34/291.61	eq(cond1(V1, V, V2, Out),0,[],[Out = 0,V26 >= 0,V2 = V27,V25 >= 0,V1 = V26,V = V25,V27 >= 0]).
1148.34/291.61	input_output_vars(cond1(V1,V,V2,Out),[V1,V,V2],[Out]).
1148.34/291.61	input_output_vars(cond2(V1,V,V2,Out),[V1,V,V2],[Out]).
1148.34/291.61	input_output_vars(cond3(V1,V,V2,Out),[V1,V,V2],[Out]).
1148.34/291.61	input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
1148.34/291.61	input_output_vars(add(V1,V,Out),[V1,V],[Out]).
1148.34/291.61	input_output_vars(eq(V1,V,Out),[V1,V],[Out]).
1148.34/291.61	input_output_vars(p(V1,Out),[V1],[Out]).
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	CoFloCo proof output:
1148.34/291.61	Preprocessing Cost Relations
1148.34/291.61	=====================================
1148.34/291.61	
1148.34/291.61	#### Computed strongly connected components 
1148.34/291.61	0. recursive  : [add/3]
1148.34/291.61	1. recursive  : [gr/3]
1148.34/291.61	2. non_recursive  : [p/2]
1148.34/291.61	3. recursive  : [eq/3]
1148.34/291.61	4. recursive  : [cond1/4,cond2/4,cond3/4]
1148.34/291.61	5. non_recursive  : [start/3]
1148.34/291.61	
1148.34/291.61	#### Obtained direct recursion through partial evaluation 
1148.34/291.61	0. SCC is partially evaluated into add/3
1148.34/291.61	1. SCC is partially evaluated into gr/3
1148.34/291.61	2. SCC is partially evaluated into p/2
1148.34/291.61	3. SCC is partially evaluated into eq/3
1148.34/291.61	4. SCC is partially evaluated into cond1/4
1148.34/291.61	5. SCC is partially evaluated into start/3
1148.34/291.61	
1148.34/291.61	Control-Flow Refinement of Cost Relations
1148.34/291.61	=====================================
1148.34/291.61	
1148.34/291.61	### Specialization of cost equations add/3 
1148.34/291.61	* CE 11 is refined into CE [25] 
1148.34/291.61	* CE 10 is refined into CE [26] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Cost equations --> "Loop" of add/3 
1148.34/291.61	* CEs [26] --> Loop 17 
1148.34/291.61	* CEs [25] --> Loop 18 
1148.34/291.61	
1148.34/291.61	### Ranking functions of CR add(V1,V,Out) 
1148.34/291.61	* RF of phase [18]: [V1]
1148.34/291.61	
1148.34/291.61	#### Partial ranking functions of CR add(V1,V,Out) 
1148.34/291.61	* Partial RF of phase [18]:
1148.34/291.61	  - RF of loop [18:1]:
1148.34/291.61	    V1
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Specialization of cost equations gr/3 
1148.34/291.61	* CE 14 is refined into CE [27] 
1148.34/291.61	* CE 13 is refined into CE [28] 
1148.34/291.61	* CE 12 is refined into CE [29] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Cost equations --> "Loop" of gr/3 
1148.34/291.61	* CEs [28] --> Loop 19 
1148.34/291.61	* CEs [29] --> Loop 20 
1148.34/291.61	* CEs [27] --> Loop 21 
1148.34/291.61	
1148.34/291.61	### Ranking functions of CR gr(V1,V,Out) 
1148.34/291.61	* RF of phase [21]: [V,V1]
1148.34/291.61	
1148.34/291.61	#### Partial ranking functions of CR gr(V1,V,Out) 
1148.34/291.61	* Partial RF of phase [21]:
1148.34/291.61	  - RF of loop [21:1]:
1148.34/291.61	    V
1148.34/291.61	    V1
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Specialization of cost equations p/2 
1148.34/291.61	* CE 16 is refined into CE [30] 
1148.34/291.61	* CE 15 is refined into CE [31] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Cost equations --> "Loop" of p/2 
1148.34/291.61	* CEs [30] --> Loop 22 
1148.34/291.61	* CEs [31] --> Loop 23 
1148.34/291.61	
1148.34/291.61	### Ranking functions of CR p(V1,Out) 
1148.34/291.61	
1148.34/291.61	#### Partial ranking functions of CR p(V1,Out) 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Specialization of cost equations eq/3 
1148.34/291.61	* CE 24 is refined into CE [32] 
1148.34/291.61	* CE 23 is refined into CE [33] 
1148.34/291.61	* CE 22 is refined into CE [34] 
1148.34/291.61	* CE 21 is refined into CE [35] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Cost equations --> "Loop" of eq/3 
1148.34/291.61	* CEs [33] --> Loop 24 
1148.34/291.61	* CEs [34] --> Loop 25 
1148.34/291.61	* CEs [35] --> Loop 26 
1148.34/291.61	* CEs [32] --> Loop 27 
1148.34/291.61	
1148.34/291.61	### Ranking functions of CR eq(V1,V,Out) 
1148.34/291.61	* RF of phase [27]: [V,V1]
1148.34/291.61	
1148.34/291.61	#### Partial ranking functions of CR eq(V1,V,Out) 
1148.34/291.61	* Partial RF of phase [27]:
1148.34/291.61	  - RF of loop [27:1]:
1148.34/291.61	    V
1148.34/291.61	    V1
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Specialization of cost equations cond1/4 
1148.34/291.61	* CE 20 is refined into CE [36] 
1148.34/291.61	* CE 17 is refined into CE [37,38] 
1148.34/291.61	* CE 18 is refined into CE [39,40] 
1148.34/291.61	* CE 19 is refined into CE [41,42] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Cost equations --> "Loop" of cond1/4 
1148.34/291.61	* CEs [42] --> Loop 28 
1148.34/291.61	* CEs [38] --> Loop 29 
1148.34/291.61	* CEs [40] --> Loop 30 
1148.34/291.61	* CEs [37] --> Loop 31 
1148.34/291.61	* CEs [41] --> Loop 32 
1148.34/291.61	* CEs [39] --> Loop 33 
1148.34/291.61	* CEs [36] --> Loop 34 
1148.34/291.61	
1148.34/291.61	### Ranking functions of CR cond1(V1,V,V2,Out) 
1148.34/291.61	* RF of phase [28,30]: [V+V2-1]
1148.34/291.61	* RF of phase [29]: [V-1,V-V2]
1148.34/291.61	* RF of phase [31]: [V]
1148.34/291.61	* RF of phase [32]: [V2]
1148.34/291.61	
1148.34/291.61	#### Partial ranking functions of CR cond1(V1,V,V2,Out) 
1148.34/291.61	* Partial RF of phase [28,30]:
1148.34/291.61	  - RF of loop [28:1]:
1148.34/291.61	    -V+V2 depends on loops [30:1] 
1148.34/291.61	    V2-1
1148.34/291.61	  - RF of loop [30:1]:
1148.34/291.61	    V
1148.34/291.61	    V-V2+1 depends on loops [28:1] 
1148.34/291.61	* Partial RF of phase [29]:
1148.34/291.61	  - RF of loop [29:1]:
1148.34/291.61	    V-1
1148.34/291.61	    V-V2
1148.34/291.61	* Partial RF of phase [31]:
1148.34/291.61	  - RF of loop [31:1]:
1148.34/291.61	    V
1148.34/291.61	* Partial RF of phase [32]:
1148.34/291.61	  - RF of loop [32:1]:
1148.34/291.61	    V2
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Specialization of cost equations start/3 
1148.34/291.61	* CE 1 is refined into CE [43,44,45,46,47,48,49,50] 
1148.34/291.61	* CE 2 is refined into CE [51,52,53,54] 
1148.34/291.61	* CE 3 is refined into CE [55,56,57,58,59,60,61,62,63] 
1148.34/291.61	* CE 4 is refined into CE [64,65,66,67,68,69,70,71,72] 
1148.34/291.61	* CE 5 is refined into CE [73,74,75,76,77] 
1148.34/291.61	* CE 6 is refined into CE [78,79,80,81] 
1148.34/291.61	* CE 7 is refined into CE [82,83] 
1148.34/291.61	* CE 8 is refined into CE [84,85,86,87,88,89] 
1148.34/291.61	* CE 9 is refined into CE [90,91] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	### Cost equations --> "Loop" of start/3 
1148.34/291.61	* CEs [89] --> Loop 35 
1148.34/291.61	* CEs [48,75] --> Loop 36 
1148.34/291.61	* CEs [46,47,49,50,76,77] --> Loop 37 
1148.34/291.61	* CEs [44,45,73,74] --> Loop 38 
1148.34/291.61	* CEs [43,79,80,81,83,86,87,88,91] --> Loop 39 
1148.34/291.61	* CEs [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,78,82,84,85,90] --> Loop 40 
1148.34/291.61	
1148.34/291.61	### Ranking functions of CR start(V1,V,V2) 
1148.34/291.61	
1148.34/291.61	#### Partial ranking functions of CR start(V1,V,V2) 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	Computing Bounds
1148.34/291.61	=====================================
1148.34/291.61	
1148.34/291.61	#### Cost of chains of add(V1,V,Out):
1148.34/291.61	* Chain [[18],17]: 1*it(18)+1
1148.34/291.61	  Such that:it(18) =< -V+Out
1148.34/291.61	
1148.34/291.61	  with precondition: [V+V1=Out,V1>=1,V>=0] 
1148.34/291.61	
1148.34/291.61	* Chain [17]: 1
1148.34/291.61	  with precondition: [V1=0,V=Out,V>=0] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	#### Cost of chains of gr(V1,V,Out):
1148.34/291.61	* Chain [[21],20]: 1*it(21)+1
1148.34/291.61	  Such that:it(21) =< V1
1148.34/291.61	
1148.34/291.61	  with precondition: [Out=0,V1>=1,V>=V1] 
1148.34/291.61	
1148.34/291.61	* Chain [[21],19]: 1*it(21)+1
1148.34/291.61	  Such that:it(21) =< V
1148.34/291.61	
1148.34/291.61	  with precondition: [Out=1,V>=1,V1>=V+1] 
1148.34/291.61	
1148.34/291.61	* Chain [20]: 1
1148.34/291.61	  with precondition: [V1=0,Out=0,V>=0] 
1148.34/291.61	
1148.34/291.61	* Chain [19]: 1
1148.34/291.61	  with precondition: [V=0,Out=1,V1>=1] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	#### Cost of chains of p(V1,Out):
1148.34/291.61	* Chain [23]: 1
1148.34/291.61	  with precondition: [V1=0,Out=0] 
1148.34/291.61	
1148.34/291.61	* Chain [22]: 1
1148.34/291.61	  with precondition: [V1=Out+1,V1>=1] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	#### Cost of chains of eq(V1,V,Out):
1148.34/291.61	* Chain [[27],26]: 1*it(27)+1
1148.34/291.61	  Such that:it(27) =< V1
1148.34/291.61	
1148.34/291.61	  with precondition: [Out=1,V1=V,V1>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [[27],25]: 1*it(27)+1
1148.34/291.61	  Such that:it(27) =< V1
1148.34/291.61	
1148.34/291.61	  with precondition: [Out=0,V1>=1,V>=V1+1] 
1148.34/291.61	
1148.34/291.61	* Chain [[27],24]: 1*it(27)+1
1148.34/291.61	  Such that:it(27) =< V
1148.34/291.61	
1148.34/291.61	  with precondition: [Out=0,V>=1,V1>=V+1] 
1148.34/291.61	
1148.34/291.61	* Chain [26]: 1
1148.34/291.61	  with precondition: [V1=0,V=0,Out=1] 
1148.34/291.61	
1148.34/291.61	* Chain [25]: 1
1148.34/291.61	  with precondition: [V1=0,Out=0,V>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [24]: 1
1148.34/291.61	  with precondition: [V=0,Out=0,V1>=1] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	#### Cost of chains of cond1(V1,V,V2,Out):
1148.34/291.61	* Chain [[32],34]: 8*it(32)+0
1148.34/291.61	  Such that:it(32) =< V2
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,V=0,Out=0,V2>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [[32],33,34]: 8*it(32)+8
1148.34/291.61	  Such that:it(32) =< V2
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,V=0,Out=0,V2>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [[31],34]: 6*it(31)+1*s(3)+0
1148.34/291.61	  Such that:aux(3) =< V
1148.34/291.61	it(31) =< aux(3)
1148.34/291.61	s(3) =< it(31)*aux(3)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,V2=0,Out=0,V>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [[31],33,34]: 6*it(31)+1*s(3)+8
1148.34/291.61	  Such that:aux(4) =< V
1148.34/291.61	it(31) =< aux(4)
1148.34/291.61	s(3) =< it(31)*aux(4)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,V2=0,Out=0,V>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [[29],[28,30],[32],34]: 16*it(28)+6*it(29)+8*it(32)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+0
1148.34/291.61	  Such that:aux(20) =< V
1148.34/291.61	it(29) =< V-V2
1148.34/291.61	aux(17) =< 2*V2
1148.34/291.61	aux(21) =< V2
1148.34/291.61	it(28) =< aux(21)
1148.34/291.61	it(32) =< aux(17)
1148.34/291.61	it(28) =< aux(17)
1148.34/291.61	s(17) =< it(28)*aux(17)
1148.34/291.61	s(15) =< it(28)*aux(21)
1148.34/291.61	s(16) =< s(17)
1148.34/291.61	s(14) =< s(15)
1148.34/291.61	it(29) =< aux(20)
1148.34/291.61	s(22) =< it(29)*aux(21)
1148.34/291.61	s(23) =< it(29)*aux(20)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
1148.34/291.61	
1148.34/291.61	* Chain [[29],[28,30],[32],33,34]: 16*it(28)+6*it(29)+8*it(32)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+8
1148.34/291.61	  Such that:aux(20) =< V
1148.34/291.61	it(29) =< V-V2
1148.34/291.61	aux(23) =< 2*V2
1148.34/291.61	aux(24) =< V2
1148.34/291.61	it(28) =< aux(24)
1148.34/291.61	it(32) =< aux(23)
1148.34/291.61	it(28) =< aux(23)
1148.34/291.61	s(17) =< it(28)*aux(23)
1148.34/291.61	s(15) =< it(28)*aux(24)
1148.34/291.61	s(16) =< s(17)
1148.34/291.61	s(14) =< s(15)
1148.34/291.61	it(29) =< aux(20)
1148.34/291.61	s(22) =< it(29)*aux(24)
1148.34/291.61	s(23) =< it(29)*aux(20)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
1148.34/291.61	
1148.34/291.61	* Chain [[29],[28,30],34]: 16*it(28)+6*it(29)+3*s(14)+3*s(16)+1*s(22)+1*s(23)+0
1148.34/291.61	  Such that:aux(20) =< V
1148.34/291.61	it(29) =< V-V2
1148.34/291.61	aux(26) =< 2*V2
1148.34/291.61	aux(27) =< V2
1148.34/291.61	it(28) =< aux(27)
1148.34/291.61	it(28) =< aux(26)
1148.34/291.61	s(17) =< it(28)*aux(26)
1148.34/291.61	s(15) =< it(28)*aux(27)
1148.34/291.61	s(16) =< s(17)
1148.34/291.61	s(14) =< s(15)
1148.34/291.61	it(29) =< aux(20)
1148.34/291.61	s(22) =< it(29)*aux(27)
1148.34/291.61	s(23) =< it(29)*aux(20)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
1148.34/291.61	
1148.34/291.61	* Chain [[29],34]: 6*it(29)+1*s(22)+1*s(23)+0
1148.34/291.61	  Such that:aux(20) =< V
1148.34/291.61	it(29) =< V-V2
1148.34/291.61	aux(19) =< V2
1148.34/291.61	it(29) =< aux(20)
1148.34/291.61	s(22) =< it(29)*aux(19)
1148.34/291.61	s(23) =< it(29)*aux(20)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
1148.34/291.61	
1148.34/291.61	* Chain [[28,30],[32],34]: 8*it(28)+8*it(30)+8*it(32)+3*s(14)+3*s(16)+0
1148.34/291.61	  Such that:it(28) =< V2
1148.34/291.61	aux(16) =< V
1148.34/291.61	aux(17) =< V+V2
1148.34/291.61	it(30) =< aux(16)
1148.34/291.61	it(32) =< aux(17)
1148.34/291.61	it(28) =< aux(17)
1148.34/291.61	it(30) =< aux(17)
1148.34/291.61	s(17) =< it(30)*aux(17)
1148.34/291.61	s(15) =< it(28)*aux(16)
1148.34/291.61	s(16) =< s(17)
1148.34/291.61	s(14) =< s(15)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,Out=0,V>=1,V2>=V] 
1148.34/291.61	
1148.34/291.61	* Chain [[28,30],[32],33,34]: 8*it(28)+8*it(30)+8*it(32)+3*s(14)+3*s(16)+8
1148.34/291.61	  Such that:it(28) =< V2
1148.34/291.61	aux(22) =< V
1148.34/291.61	aux(23) =< V+V2
1148.34/291.61	it(30) =< aux(22)
1148.34/291.61	it(32) =< aux(23)
1148.34/291.61	it(28) =< aux(23)
1148.34/291.61	it(30) =< aux(23)
1148.34/291.61	s(17) =< it(30)*aux(23)
1148.34/291.61	s(15) =< it(28)*aux(22)
1148.34/291.61	s(16) =< s(17)
1148.34/291.61	s(14) =< s(15)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,Out=0,V>=1,V2>=V] 
1148.34/291.61	
1148.34/291.61	* Chain [[28,30],34]: 8*it(28)+8*it(30)+3*s(14)+3*s(16)+0
1148.34/291.61	  Such that:it(28) =< V2
1148.34/291.61	aux(25) =< V
1148.34/291.61	aux(26) =< V+V2
1148.34/291.61	it(30) =< aux(25)
1148.34/291.61	it(28) =< aux(26)
1148.34/291.61	it(30) =< aux(26)
1148.34/291.61	s(17) =< it(30)*aux(26)
1148.34/291.61	s(15) =< it(28)*aux(25)
1148.34/291.61	s(16) =< s(17)
1148.34/291.61	s(14) =< s(15)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,Out=0,V>=1,V2>=V] 
1148.34/291.61	
1148.34/291.61	* Chain [34]: 0
1148.34/291.61	  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 
1148.34/291.61	
1148.34/291.61	* Chain [33,34]: 8
1148.34/291.61	  with precondition: [V1=1,V=0,V2=0,Out=0] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	#### Cost of chains of start(V1,V,V2):
1148.34/291.61	* Chain [40]: 38*s(98)+19*s(100)+144*s(109)+27*s(113)+27*s(114)+48*s(115)+63*s(118)+8*s(122)+48*s(130)+48*s(131)+18*s(134)+18*s(135)+32*s(136)+48*s(150)+8*s(151)+8*s(152)+14
1148.34/291.61	  Such that:aux(50) =< 1
1148.34/291.61	aux(51) =< V
1148.34/291.61	aux(52) =< V-V2+1
1148.34/291.61	aux(53) =< V+V2
1148.34/291.61	aux(54) =< V2
1148.34/291.61	aux(55) =< 2*V2
1148.34/291.61	s(100) =< aux(50)
1148.34/291.61	s(118) =< aux(51)
1148.34/291.61	s(98) =< aux(54)
1148.34/291.61	s(122) =< s(118)*aux(51)
1148.34/291.61	s(150) =< aux(52)
1148.34/291.61	s(150) =< aux(51)
1148.34/291.61	s(151) =< s(150)*aux(54)
1148.34/291.61	s(152) =< s(150)*aux(51)
1148.34/291.61	s(109) =< aux(54)
1148.34/291.61	s(109) =< aux(55)
1148.34/291.61	s(111) =< s(109)*aux(55)
1148.34/291.61	s(112) =< s(109)*aux(54)
1148.34/291.61	s(113) =< s(111)
1148.34/291.61	s(114) =< s(112)
1148.34/291.61	s(115) =< aux(55)
1148.34/291.61	s(130) =< aux(54)
1148.34/291.61	s(131) =< aux(51)
1148.34/291.61	s(130) =< aux(53)
1148.34/291.61	s(131) =< aux(53)
1148.34/291.61	s(132) =< s(131)*aux(53)
1148.34/291.61	s(133) =< s(130)*aux(51)
1148.34/291.61	s(134) =< s(132)
1148.34/291.61	s(135) =< s(133)
1148.34/291.61	s(136) =< aux(53)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=0] 
1148.34/291.61	
1148.34/291.61	* Chain [39]: 3*s(196)+2*s(197)+12
1148.34/291.61	  Such that:aux(56) =< V1
1148.34/291.61	aux(57) =< V
1148.34/291.61	s(196) =< aux(56)
1148.34/291.61	s(197) =< aux(57)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [38]: 32*s(202)+12
1148.34/291.61	  Such that:aux(58) =< V2
1148.34/291.61	s(202) =< aux(58)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1>=0,V>=0,V2>=0] 
1148.34/291.61	
1148.34/291.61	* Chain [37]: 3*s(205)+1*s(206)+16*s(208)+48*s(213)+48*s(214)+18*s(217)+18*s(218)+32*s(219)+48*s(225)+8*s(226)+8*s(227)+96*s(228)+18*s(231)+18*s(232)+32*s(233)+12
1148.34/291.61	  Such that:s(206) =< 1
1148.34/291.61	aux(61) =< V
1148.34/291.61	aux(62) =< V-V2
1148.34/291.61	aux(63) =< V+V2
1148.34/291.61	aux(64) =< V2
1148.34/291.61	aux(65) =< 2*V2
1148.34/291.61	s(205) =< aux(61)
1148.34/291.61	s(208) =< aux(64)
1148.34/291.61	s(213) =< aux(64)
1148.34/291.61	s(214) =< aux(61)
1148.34/291.61	s(213) =< aux(63)
1148.34/291.61	s(214) =< aux(63)
1148.34/291.61	s(215) =< s(214)*aux(63)
1148.34/291.61	s(216) =< s(213)*aux(61)
1148.34/291.61	s(217) =< s(215)
1148.34/291.61	s(218) =< s(216)
1148.34/291.61	s(219) =< aux(63)
1148.34/291.61	s(225) =< aux(62)
1148.34/291.61	s(225) =< aux(61)
1148.34/291.61	s(226) =< s(225)*aux(64)
1148.34/291.61	s(227) =< s(225)*aux(61)
1148.34/291.61	s(228) =< aux(64)
1148.34/291.61	s(228) =< aux(65)
1148.34/291.61	s(229) =< s(228)*aux(65)
1148.34/291.61	s(230) =< s(228)*aux(64)
1148.34/291.61	s(231) =< s(229)
1148.34/291.61	s(232) =< s(230)
1148.34/291.61	s(233) =< aux(65)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,V>=1,V2>=0] 
1148.34/291.61	
1148.34/291.61	* Chain [36]: 25*s(257)+4*s(260)+12
1148.34/291.61	  Such that:aux(67) =< V
1148.34/291.61	s(257) =< aux(67)
1148.34/291.61	s(260) =< s(257)*aux(67)
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=1,V2=0,V>=1] 
1148.34/291.61	
1148.34/291.61	* Chain [35]: 1*s(264)+1
1148.34/291.61	  Such that:s(264) =< V
1148.34/291.61	
1148.34/291.61	  with precondition: [V1=V,V1>=1] 
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	Closed-form bounds of start(V1,V,V2): 
1148.34/291.61	-------------------------------------
1148.34/291.61	* Chain [40] with precondition: [V1=0] 
1148.34/291.61	    - Upper bound: nat(V)*111+33+nat(V)*8*nat(V)+nat(V)*18*nat(V2)+nat(V)*18*nat(V+V2)+nat(V)*8*nat(V-V2+1)+nat(V2)*230+nat(V2)*27*nat(V2)+nat(V2)*27*nat(2*V2)+nat(V2)*8*nat(V-V2+1)+nat(2*V2)*48+nat(V+V2)*32+nat(V-V2+1)*48 
1148.34/291.61	    - Complexity: n^2 
1148.34/291.61	* Chain [39] with precondition: [V1>=1] 
1148.34/291.61	    - Upper bound: 3*V1+12+nat(V)*2 
1148.34/291.61	    - Complexity: n 
1148.34/291.61	* Chain [38] with precondition: [V1>=0,V>=0,V2>=0] 
1148.34/291.61	    - Upper bound: 32*V2+12 
1148.34/291.61	    - Complexity: n 
1148.34/291.61	* Chain [37] with precondition: [V1=1,V>=1,V2>=0] 
1148.34/291.61	    - Upper bound: 51*V+13+18*V*V2+(V+V2)*(18*V)+8*V*nat(V-V2)+160*V2+18*V2*V2+18*V2*(2*V2)+8*V2*nat(V-V2)+64*V2+(32*V+32*V2)+nat(V-V2)*48 
1148.34/291.61	    - Complexity: n^2 
1148.34/291.61	* Chain [36] with precondition: [V1=1,V2=0,V>=1] 
1148.34/291.61	    - Upper bound: 25*V+12+4*V*V 
1148.34/291.61	    - Complexity: n^2 
1148.34/291.61	* Chain [35] with precondition: [V1=V,V1>=1] 
1148.34/291.61	    - Upper bound: V+1 
1148.34/291.61	    - Complexity: n 
1148.34/291.61	
1148.34/291.61	### Maximum cost of start(V1,V,V2): max([nat(V2)*32+11,nat(V)+11+max([3*V1,nat(V)*23+max([nat(V)*4*nat(V),nat(V)*26+1+nat(V)*18*nat(V2)+nat(V)*18*nat(V+V2)+nat(V2)*160+nat(V2)*18*nat(V2)+nat(V2)*18*nat(2*V2)+nat(2*V2)*32+nat(V+V2)*32+max([nat(V2)*8*nat(V-V2)+nat(V)*8*nat(V-V2)+nat(V-V2)*48,nat(V)*60+20+nat(V)*8*nat(V)+nat(V)*8*nat(V-V2+1)+nat(V2)*70+nat(V2)*9*nat(V2)+nat(V2)*9*nat(2*V2)+nat(V2)*8*nat(V-V2+1)+nat(2*V2)*16+nat(V-V2+1)*48])])])+nat(V)])+1 
1148.34/291.61	Asymptotic class: n^2 
1148.34/291.61	* Total analysis performed in 960 ms.
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(10)
1148.34/291.61	BOUNDS(1, n^2)
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1148.34/291.61	Transformed a relative TRS into a decreasing-loop problem.
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(12)
1148.34/291.61	Obligation:
1148.34/291.61	Analyzing the following TRS for decreasing loops:
1148.34/291.61	
1148.34/291.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	The TRS R consists of the following rules:
1148.34/291.61	
1148.34/291.61	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
1148.34/291.61	   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond2(false, x, y) -> cond3(eq(x, y), x, y)
1148.34/291.61	   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
1148.34/291.61	   gr(0, x) -> false
1148.34/291.61	   gr(s(x), 0) -> true
1148.34/291.61	   gr(s(x), s(y)) -> gr(x, y)
1148.34/291.61	   add(0, x) -> x
1148.34/291.61	   add(s(x), y) -> s(add(x, y))
1148.34/291.61	   eq(0, 0) -> true
1148.34/291.61	   eq(0, s(x)) -> false
1148.34/291.61	   eq(s(x), 0) -> false
1148.34/291.61	   eq(s(x), s(y)) -> eq(x, y)
1148.34/291.61	   p(0) -> 0
1148.34/291.61	   p(s(x)) -> x
1148.34/291.61	
1148.34/291.61	S is empty.
1148.34/291.61	Rewrite Strategy: INNERMOST
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(13) DecreasingLoopProof (LOWER BOUND(ID))
1148.34/291.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1148.34/291.61	
1148.34/291.61	The rewrite sequence
1148.34/291.61	
1148.34/291.61	add(s(x), y) ->^+ s(add(x, y))
1148.34/291.61	
1148.34/291.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1148.34/291.61	
1148.34/291.61	The pumping substitution is [x / s(x)].
1148.34/291.61	
1148.34/291.61	The result substitution is [ ].
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(14)
1148.34/291.61	Complex Obligation (BEST)
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(15)
1148.34/291.61	Obligation:
1148.34/291.61	Proved the lower bound n^1 for the following obligation:
1148.34/291.61	
1148.34/291.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	The TRS R consists of the following rules:
1148.34/291.61	
1148.34/291.61	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
1148.34/291.61	   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond2(false, x, y) -> cond3(eq(x, y), x, y)
1148.34/291.61	   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
1148.34/291.61	   gr(0, x) -> false
1148.34/291.61	   gr(s(x), 0) -> true
1148.34/291.61	   gr(s(x), s(y)) -> gr(x, y)
1148.34/291.61	   add(0, x) -> x
1148.34/291.61	   add(s(x), y) -> s(add(x, y))
1148.34/291.61	   eq(0, 0) -> true
1148.34/291.61	   eq(0, s(x)) -> false
1148.34/291.61	   eq(s(x), 0) -> false
1148.34/291.61	   eq(s(x), s(y)) -> eq(x, y)
1148.34/291.61	   p(0) -> 0
1148.34/291.61	   p(s(x)) -> x
1148.34/291.61	
1148.34/291.61	S is empty.
1148.34/291.61	Rewrite Strategy: INNERMOST
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(16) LowerBoundPropagationProof (FINISHED)
1148.34/291.61	Propagated lower bound.
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(17)
1148.34/291.61	BOUNDS(n^1, INF)
1148.34/291.61	
1148.34/291.61	----------------------------------------
1148.34/291.61	
1148.34/291.61	(18)
1148.34/291.61	Obligation:
1148.34/291.61	Analyzing the following TRS for decreasing loops:
1148.34/291.61	
1148.34/291.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1148.34/291.61	
1148.34/291.61	
1148.34/291.61	The TRS R consists of the following rules:
1148.34/291.61	
1148.34/291.61	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
1148.34/291.61	   cond2(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond2(false, x, y) -> cond3(eq(x, y), x, y)
1148.34/291.61	   cond3(true, x, y) -> cond1(gr(add(x, y), 0), p(x), y)
1148.34/291.61	   cond3(false, x, y) -> cond1(gr(add(x, y), 0), x, p(y))
1148.34/291.61	   gr(0, x) -> false
1148.34/291.61	   gr(s(x), 0) -> true
1148.34/291.61	   gr(s(x), s(y)) -> gr(x, y)
1148.34/291.61	   add(0, x) -> x
1148.34/291.61	   add(s(x), y) -> s(add(x, y))
1148.34/291.61	   eq(0, 0) -> true
1148.34/291.61	   eq(0, s(x)) -> false
1148.34/291.61	   eq(s(x), 0) -> false
1148.34/291.61	   eq(s(x), s(y)) -> eq(x, y)
1148.34/291.61	   p(0) -> 0
1148.34/291.61	   p(s(x)) -> x
1148.34/291.61	
1148.34/291.61	S is empty.
1148.34/291.61	Rewrite Strategy: INNERMOST
1148.68/291.73	EOF