1134.97/291.49	WORST_CASE(Omega(n^1), O(n^3))
1134.97/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1134.97/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1134.97/291.53	
1134.97/291.53	(0) CpxTRS
1134.97/291.53	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1134.97/291.53	(2) CpxWeightedTrs
1134.97/291.53	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1134.97/291.53	    (4) CpxTypedWeightedTrs
1134.97/291.53	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
1134.97/291.53	    (6) CpxTypedWeightedCompleteTrs
1134.97/291.53	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1134.97/291.53	    (8) CpxRNTS
1134.97/291.53	    (9) CompleteCoflocoProof [FINISHED, 616 ms]
1134.97/291.53	    (10) BOUNDS(1, n^3)
1134.97/291.53	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1134.97/291.53	(12) TRS for Loop Detection
1134.97/291.53	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1134.97/291.53	    (14) BEST
1134.97/291.53	        (15) proven lower bound
1134.97/291.53	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1134.97/291.53	            (17) BOUNDS(n^1, INF)
1134.97/291.53	        (18) TRS for Loop Detection
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(0)
1134.97/291.53	Obligation:
1134.97/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	The TRS R consists of the following rules:
1134.97/291.53	
1134.97/291.53	   minus(x, y) -> if(gt(x, y), x, y)
1134.97/291.53	   if(true, x, y) -> s(minus(p(x), y))
1134.97/291.53	   if(false, x, y) -> 0
1134.97/291.53	   p(0) -> 0
1134.97/291.53	   p(s(x)) -> x
1134.97/291.53	   ge(x, 0) -> true
1134.97/291.53	   ge(0, s(x)) -> false
1134.97/291.53	   ge(s(x), s(y)) -> ge(x, y)
1134.97/291.53	   gt(0, y) -> false
1134.97/291.53	   gt(s(x), 0) -> true
1134.97/291.53	   gt(s(x), s(y)) -> gt(x, y)
1134.97/291.53	   div(x, y) -> if1(ge(x, y), x, y)
1134.97/291.53	   if1(true, x, y) -> if2(gt(y, 0), x, y)
1134.97/291.53	   if1(false, x, y) -> 0
1134.97/291.53	   if2(true, x, y) -> s(div(minus(x, y), y))
1134.97/291.53	   if2(false, x, y) -> 0
1134.97/291.53	
1134.97/291.53	S is empty.
1134.97/291.53	Rewrite Strategy: INNERMOST
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1134.97/291.53	Transformed relative TRS to weighted TRS
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(2)
1134.97/291.53	Obligation:
1134.97/291.53	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	The TRS R consists of the following rules:
1134.97/291.53	
1134.97/291.53	   minus(x, y) -> if(gt(x, y), x, y) [1]
1134.97/291.53	   if(true, x, y) -> s(minus(p(x), y)) [1]
1134.97/291.53	   if(false, x, y) -> 0 [1]
1134.97/291.53	   p(0) -> 0 [1]
1134.97/291.53	   p(s(x)) -> x [1]
1134.97/291.53	   ge(x, 0) -> true [1]
1134.97/291.53	   ge(0, s(x)) -> false [1]
1134.97/291.53	   ge(s(x), s(y)) -> ge(x, y) [1]
1134.97/291.53	   gt(0, y) -> false [1]
1134.97/291.53	   gt(s(x), 0) -> true [1]
1134.97/291.53	   gt(s(x), s(y)) -> gt(x, y) [1]
1134.97/291.53	   div(x, y) -> if1(ge(x, y), x, y) [1]
1134.97/291.53	   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
1134.97/291.53	   if1(false, x, y) -> 0 [1]
1134.97/291.53	   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
1134.97/291.53	   if2(false, x, y) -> 0 [1]
1134.97/291.53	
1134.97/291.53	Rewrite Strategy: INNERMOST
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1134.97/291.53	Infered types.
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(4)
1134.97/291.53	Obligation:
1134.97/291.53	Runtime Complexity Weighted TRS with Types.
1134.97/291.53	The TRS R consists of the following rules:
1134.97/291.53	
1134.97/291.53	   minus(x, y) -> if(gt(x, y), x, y) [1]
1134.97/291.53	   if(true, x, y) -> s(minus(p(x), y)) [1]
1134.97/291.53	   if(false, x, y) -> 0 [1]
1134.97/291.53	   p(0) -> 0 [1]
1134.97/291.53	   p(s(x)) -> x [1]
1134.97/291.53	   ge(x, 0) -> true [1]
1134.97/291.53	   ge(0, s(x)) -> false [1]
1134.97/291.53	   ge(s(x), s(y)) -> ge(x, y) [1]
1134.97/291.53	   gt(0, y) -> false [1]
1134.97/291.53	   gt(s(x), 0) -> true [1]
1134.97/291.53	   gt(s(x), s(y)) -> gt(x, y) [1]
1134.97/291.53	   div(x, y) -> if1(ge(x, y), x, y) [1]
1134.97/291.53	   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
1134.97/291.53	   if1(false, x, y) -> 0 [1]
1134.97/291.53	   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
1134.97/291.53	   if2(false, x, y) -> 0 [1]
1134.97/291.53	
1134.97/291.53	The TRS has the following type information:
1134.97/291.53	minus :: s:0 -> s:0 -> s:0
1134.97/291.53	if :: true:false -> s:0 -> s:0 -> s:0
1134.97/291.53	gt :: s:0 -> s:0 -> true:false
1134.97/291.53	true :: true:false
1134.97/291.53	s :: s:0 -> s:0
1134.97/291.53	p :: s:0 -> s:0
1134.97/291.53	false :: true:false
1134.97/291.53	0 :: s:0
1134.97/291.53	ge :: s:0 -> s:0 -> true:false
1134.97/291.53	div :: s:0 -> s:0 -> s:0
1134.97/291.53	if1 :: true:false -> s:0 -> s:0 -> s:0
1134.97/291.53	if2 :: true:false -> s:0 -> s:0 -> s:0
1134.97/291.53	
1134.97/291.53	Rewrite Strategy: INNERMOST
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(5) CompletionProof (UPPER BOUND(ID))
1134.97/291.53	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
1134.97/291.53	none
1134.97/291.53	
1134.97/291.53	And the following fresh constants: none
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(6)
1134.97/291.53	Obligation:
1134.97/291.53	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
1134.97/291.53	
1134.97/291.53	Runtime Complexity Weighted TRS with Types.
1134.97/291.53	The TRS R consists of the following rules:
1134.97/291.53	
1134.97/291.53	   minus(x, y) -> if(gt(x, y), x, y) [1]
1134.97/291.53	   if(true, x, y) -> s(minus(p(x), y)) [1]
1134.97/291.53	   if(false, x, y) -> 0 [1]
1134.97/291.53	   p(0) -> 0 [1]
1134.97/291.53	   p(s(x)) -> x [1]
1134.97/291.53	   ge(x, 0) -> true [1]
1134.97/291.53	   ge(0, s(x)) -> false [1]
1134.97/291.53	   ge(s(x), s(y)) -> ge(x, y) [1]
1134.97/291.53	   gt(0, y) -> false [1]
1134.97/291.53	   gt(s(x), 0) -> true [1]
1134.97/291.53	   gt(s(x), s(y)) -> gt(x, y) [1]
1134.97/291.53	   div(x, y) -> if1(ge(x, y), x, y) [1]
1134.97/291.53	   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
1134.97/291.53	   if1(false, x, y) -> 0 [1]
1134.97/291.53	   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
1134.97/291.53	   if2(false, x, y) -> 0 [1]
1134.97/291.53	
1134.97/291.53	The TRS has the following type information:
1134.97/291.53	minus :: s:0 -> s:0 -> s:0
1134.97/291.53	if :: true:false -> s:0 -> s:0 -> s:0
1134.97/291.53	gt :: s:0 -> s:0 -> true:false
1134.97/291.53	true :: true:false
1134.97/291.53	s :: s:0 -> s:0
1134.97/291.53	p :: s:0 -> s:0
1134.97/291.53	false :: true:false
1134.97/291.53	0 :: s:0
1134.97/291.53	ge :: s:0 -> s:0 -> true:false
1134.97/291.53	div :: s:0 -> s:0 -> s:0
1134.97/291.53	if1 :: true:false -> s:0 -> s:0 -> s:0
1134.97/291.53	if2 :: true:false -> s:0 -> s:0 -> s:0
1134.97/291.53	
1134.97/291.53	Rewrite Strategy: INNERMOST
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1134.97/291.53	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1134.97/291.53	The constant constructors are abstracted as follows:
1134.97/291.53	
1134.97/291.53	   true => 1
1134.97/291.53	   false => 0
1134.97/291.53	   0 => 0
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(8)
1134.97/291.53	Obligation:
1134.97/291.53	Complexity RNTS consisting of the following rules:
1134.97/291.53	
1134.97/291.53	   div(z, z') -{ 1 }-> if1(ge(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
1134.97/291.53	   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1134.97/291.53	   ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
1134.97/291.53	   ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
1134.97/291.53	   gt(z, z') -{ 1 }-> gt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1134.97/291.53	   gt(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
1134.97/291.53	   gt(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
1134.97/291.53	   if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
1134.97/291.53	   if(z, z', z'') -{ 1 }-> 1 + minus(p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
1134.97/291.53	   if1(z, z', z'') -{ 1 }-> if2(gt(y, 0), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
1134.97/291.53	   if1(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
1134.97/291.53	   if2(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
1134.97/291.53	   if2(z, z', z'') -{ 1 }-> 1 + div(minus(x, y), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
1134.97/291.53	   minus(z, z') -{ 1 }-> if(gt(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
1134.97/291.53	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
1134.97/291.53	   p(z) -{ 1 }-> 0 :|: z = 0
1134.97/291.53	
1134.97/291.53	Only complete derivations are relevant for the runtime complexity.
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(9) CompleteCoflocoProof (FINISHED)
1134.97/291.53	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
1134.97/291.53	
1134.97/291.53	eq(start(V1, V, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
1134.97/291.53	eq(start(V1, V, V5),0,[if(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
1134.97/291.53	eq(start(V1, V, V5),0,[p(V1, Out)],[V1 >= 0]).
1134.97/291.53	eq(start(V1, V, V5),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
1134.97/291.53	eq(start(V1, V, V5),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
1134.97/291.53	eq(start(V1, V, V5),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
1134.97/291.53	eq(start(V1, V, V5),0,[if1(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
1134.97/291.53	eq(start(V1, V, V5),0,[if2(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
1134.97/291.53	eq(minus(V1, V, Out),1,[gt(V3, V2, Ret0),if(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
1134.97/291.53	eq(if(V1, V, V5, Out),1,[p(V4, Ret10),minus(Ret10, V6, Ret1)],[Out = 1 + Ret1,V = V4,V5 = V6,V1 = 1,V4 >= 0,V6 >= 0]).
1134.97/291.53	eq(if(V1, V, V5, Out),1,[],[Out = 0,V = V8,V5 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
1134.97/291.53	eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
1134.97/291.53	eq(p(V1, Out),1,[],[Out = V9,V9 >= 0,V1 = 1 + V9]).
1134.97/291.53	eq(ge(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = V10,V = 0]).
1134.97/291.53	eq(ge(V1, V, Out),1,[],[Out = 0,V = 1 + V11,V11 >= 0,V1 = 0]).
1134.97/291.53	eq(ge(V1, V, Out),1,[ge(V13, V12, Ret2)],[Out = Ret2,V = 1 + V12,V13 >= 0,V12 >= 0,V1 = 1 + V13]).
1134.97/291.53	eq(gt(V1, V, Out),1,[],[Out = 0,V14 >= 0,V1 = 0,V = V14]).
1134.97/291.53	eq(gt(V1, V, Out),1,[],[Out = 1,V15 >= 0,V1 = 1 + V15,V = 0]).
1134.97/291.53	eq(gt(V1, V, Out),1,[gt(V17, V16, Ret3)],[Out = Ret3,V = 1 + V16,V17 >= 0,V16 >= 0,V1 = 1 + V17]).
1134.97/291.53	eq(div(V1, V, Out),1,[ge(V19, V18, Ret01),if1(Ret01, V19, V18, Ret4)],[Out = Ret4,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
1134.97/291.53	eq(if1(V1, V, V5, Out),1,[gt(V20, 0, Ret02),if2(Ret02, V21, V20, Ret5)],[Out = Ret5,V = V21,V5 = V20,V1 = 1,V21 >= 0,V20 >= 0]).
1134.97/291.53	eq(if1(V1, V, V5, Out),1,[],[Out = 0,V = V23,V5 = V22,V23 >= 0,V22 >= 0,V1 = 0]).
1134.97/291.53	eq(if2(V1, V, V5, Out),1,[minus(V24, V25, Ret101),div(Ret101, V25, Ret11)],[Out = 1 + Ret11,V = V24,V5 = V25,V1 = 1,V24 >= 0,V25 >= 0]).
1134.97/291.53	eq(if2(V1, V, V5, Out),1,[],[Out = 0,V = V26,V5 = V27,V26 >= 0,V27 >= 0,V1 = 0]).
1134.97/291.53	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
1134.97/291.53	input_output_vars(if(V1,V,V5,Out),[V1,V,V5],[Out]).
1134.97/291.53	input_output_vars(p(V1,Out),[V1],[Out]).
1134.97/291.53	input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
1134.97/291.53	input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
1134.97/291.53	input_output_vars(div(V1,V,Out),[V1,V],[Out]).
1134.97/291.53	input_output_vars(if1(V1,V,V5,Out),[V1,V,V5],[Out]).
1134.97/291.53	input_output_vars(if2(V1,V,V5,Out),[V1,V,V5],[Out]).
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	CoFloCo proof output:
1134.97/291.53	Preprocessing Cost Relations
1134.97/291.53	=====================================
1134.97/291.53	
1134.97/291.53	#### Computed strongly connected components 
1134.97/291.53	0. recursive  : [ge/3]
1134.97/291.53	1. recursive  : [gt/3]
1134.97/291.53	2. non_recursive  : [p/2]
1134.97/291.53	3. recursive  : [if/4,minus/3]
1134.97/291.53	4. recursive  : [(div)/3,if1/4,if2/4]
1134.97/291.53	5. non_recursive  : [start/3]
1134.97/291.53	
1134.97/291.53	#### Obtained direct recursion through partial evaluation 
1134.97/291.53	0. SCC is partially evaluated into ge/3
1134.97/291.53	1. SCC is partially evaluated into gt/3
1134.97/291.53	2. SCC is partially evaluated into p/2
1134.97/291.53	3. SCC is partially evaluated into minus/3
1134.97/291.53	4. SCC is partially evaluated into (div)/3
1134.97/291.53	5. SCC is partially evaluated into start/3
1134.97/291.53	
1134.97/291.53	Control-Flow Refinement of Cost Relations
1134.97/291.53	=====================================
1134.97/291.53	
1134.97/291.53	### Specialization of cost equations ge/3 
1134.97/291.53	* CE 23 is refined into CE [24] 
1134.97/291.53	* CE 21 is refined into CE [25] 
1134.97/291.53	* CE 22 is refined into CE [26] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Cost equations --> "Loop" of ge/3 
1134.97/291.53	* CEs [25] --> Loop 15 
1134.97/291.53	* CEs [26] --> Loop 16 
1134.97/291.53	* CEs [24] --> Loop 17 
1134.97/291.53	
1134.97/291.53	### Ranking functions of CR ge(V1,V,Out) 
1134.97/291.53	* RF of phase [17]: [V,V1]
1134.97/291.53	
1134.97/291.53	#### Partial ranking functions of CR ge(V1,V,Out) 
1134.97/291.53	* Partial RF of phase [17]:
1134.97/291.53	  - RF of loop [17:1]:
1134.97/291.53	    V
1134.97/291.53	    V1
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Specialization of cost equations gt/3 
1134.97/291.53	* CE 13 is refined into CE [27] 
1134.97/291.53	* CE 12 is refined into CE [28] 
1134.97/291.53	* CE 11 is refined into CE [29] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Cost equations --> "Loop" of gt/3 
1134.97/291.53	* CEs [28] --> Loop 18 
1134.97/291.53	* CEs [29] --> Loop 19 
1134.97/291.53	* CEs [27] --> Loop 20 
1134.97/291.53	
1134.97/291.53	### Ranking functions of CR gt(V1,V,Out) 
1134.97/291.53	* RF of phase [20]: [V,V1]
1134.97/291.53	
1134.97/291.53	#### Partial ranking functions of CR gt(V1,V,Out) 
1134.97/291.53	* Partial RF of phase [20]:
1134.97/291.53	  - RF of loop [20:1]:
1134.97/291.53	    V
1134.97/291.53	    V1
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Specialization of cost equations p/2 
1134.97/291.53	* CE 20 is refined into CE [30] 
1134.97/291.53	* CE 19 is refined into CE [31] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Cost equations --> "Loop" of p/2 
1134.97/291.53	* CEs [30] --> Loop 21 
1134.97/291.53	* CEs [31] --> Loop 22 
1134.97/291.53	
1134.97/291.53	### Ranking functions of CR p(V1,Out) 
1134.97/291.53	
1134.97/291.53	#### Partial ranking functions of CR p(V1,Out) 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Specialization of cost equations minus/3 
1134.97/291.53	* CE 15 is refined into CE [32,33] 
1134.97/291.53	* CE 14 is refined into CE [34,35] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Cost equations --> "Loop" of minus/3 
1134.97/291.53	* CEs [35] --> Loop 23 
1134.97/291.53	* CEs [34] --> Loop 24 
1134.97/291.53	* CEs [33] --> Loop 25 
1134.97/291.53	* CEs [32] --> Loop 26 
1134.97/291.53	
1134.97/291.53	### Ranking functions of CR minus(V1,V,Out) 
1134.97/291.53	* RF of phase [25]: [V1-1,V1-V]
1134.97/291.53	* RF of phase [26]: [V1]
1134.97/291.53	
1134.97/291.53	#### Partial ranking functions of CR minus(V1,V,Out) 
1134.97/291.53	* Partial RF of phase [25]:
1134.97/291.53	  - RF of loop [25:1]:
1134.97/291.53	    V1-1
1134.97/291.53	    V1-V
1134.97/291.53	* Partial RF of phase [26]:
1134.97/291.53	  - RF of loop [26:1]:
1134.97/291.53	    V1
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Specialization of cost equations (div)/3 
1134.97/291.53	* CE 16 is refined into CE [36,37] 
1134.97/291.53	* CE 18 is refined into CE [38] 
1134.97/291.53	* CE 17 is refined into CE [39,40] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Cost equations --> "Loop" of (div)/3 
1134.97/291.53	* CEs [40] --> Loop 27 
1134.97/291.53	* CEs [39] --> Loop 28 
1134.97/291.53	* CEs [37] --> Loop 29 
1134.97/291.53	* CEs [38] --> Loop 30 
1134.97/291.53	* CEs [36] --> Loop 31 
1134.97/291.53	
1134.97/291.53	### Ranking functions of CR div(V1,V,Out) 
1134.97/291.53	* RF of phase [27]: [V1-1,V1-V]
1134.97/291.53	
1134.97/291.53	#### Partial ranking functions of CR div(V1,V,Out) 
1134.97/291.53	* Partial RF of phase [27]:
1134.97/291.53	  - RF of loop [27:1]:
1134.97/291.53	    V1-1
1134.97/291.53	    V1-V
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Specialization of cost equations start/3 
1134.97/291.53	* CE 2 is refined into CE [41,42,43,44,45,46] 
1134.97/291.53	* CE 3 is refined into CE [47] 
1134.97/291.53	* CE 4 is refined into CE [48,49,50,51,52,53,54,55] 
1134.97/291.53	* CE 5 is refined into CE [56,57,58,59,60] 
1134.97/291.53	* CE 1 is refined into CE [61] 
1134.97/291.53	* CE 6 is refined into CE [62,63,64,65] 
1134.97/291.53	* CE 7 is refined into CE [66,67] 
1134.97/291.53	* CE 8 is refined into CE [68,69,70,71] 
1134.97/291.53	* CE 9 is refined into CE [72,73,74,75] 
1134.97/291.53	* CE 10 is refined into CE [76,77,78,79,80,81] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	### Cost equations --> "Loop" of start/3 
1134.97/291.53	* CEs [43,52] --> Loop 32 
1134.97/291.53	* CEs [45,46,54,55,60] --> Loop 33 
1134.97/291.53	* CEs [59] --> Loop 34 
1134.97/291.53	* CEs [42,51] --> Loop 35 
1134.97/291.53	* CEs [44,53] --> Loop 36 
1134.97/291.53	* CEs [47,50,58] --> Loop 37 
1134.97/291.53	* CEs [57,64,65,67,70,71,74,75,78,79,80,81] --> Loop 38 
1134.97/291.53	* CEs [41,48,49,56,63,69,73,77] --> Loop 39 
1134.97/291.53	* CEs [61,62,66,68,72,76] --> Loop 40 
1134.97/291.53	
1134.97/291.53	### Ranking functions of CR start(V1,V,V5) 
1134.97/291.53	
1134.97/291.53	#### Partial ranking functions of CR start(V1,V,V5) 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	Computing Bounds
1134.97/291.53	=====================================
1134.97/291.53	
1134.97/291.53	#### Cost of chains of ge(V1,V,Out):
1134.97/291.53	* Chain [[17],16]: 1*it(17)+1
1134.97/291.53	  Such that:it(17) =< V1
1134.97/291.53	
1134.97/291.53	  with precondition: [Out=0,V1>=1,V>=V1+1] 
1134.97/291.53	
1134.97/291.53	* Chain [[17],15]: 1*it(17)+1
1134.97/291.53	  Such that:it(17) =< V
1134.97/291.53	
1134.97/291.53	  with precondition: [Out=1,V>=1,V1>=V] 
1134.97/291.53	
1134.97/291.53	* Chain [16]: 1
1134.97/291.53	  with precondition: [V1=0,Out=0,V>=1] 
1134.97/291.53	
1134.97/291.53	* Chain [15]: 1
1134.97/291.53	  with precondition: [V=0,Out=1,V1>=0] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	#### Cost of chains of gt(V1,V,Out):
1134.97/291.53	* Chain [[20],19]: 1*it(20)+1
1134.97/291.53	  Such that:it(20) =< V1
1134.97/291.53	
1134.97/291.53	  with precondition: [Out=0,V1>=1,V>=V1] 
1134.97/291.53	
1134.97/291.53	* Chain [[20],18]: 1*it(20)+1
1134.97/291.53	  Such that:it(20) =< V
1134.97/291.53	
1134.97/291.53	  with precondition: [Out=1,V>=1,V1>=V+1] 
1134.97/291.53	
1134.97/291.53	* Chain [19]: 1
1134.97/291.53	  with precondition: [V1=0,Out=0,V>=0] 
1134.97/291.53	
1134.97/291.53	* Chain [18]: 1
1134.97/291.53	  with precondition: [V=0,Out=1,V1>=1] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	#### Cost of chains of p(V1,Out):
1134.97/291.53	* Chain [22]: 1
1134.97/291.53	  with precondition: [V1=0,Out=0] 
1134.97/291.53	
1134.97/291.53	* Chain [21]: 1
1134.97/291.53	  with precondition: [V1=Out+1,V1>=1] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	#### Cost of chains of minus(V1,V,Out):
1134.97/291.53	* Chain [[26],24]: 4*it(26)+3
1134.97/291.53	  Such that:it(26) =< Out
1134.97/291.53	
1134.97/291.53	  with precondition: [V=0,V1=Out,V1>=1] 
1134.97/291.53	
1134.97/291.53	* Chain [[25],23]: 4*it(25)+1*s(1)+1*s(4)+3
1134.97/291.53	  Such that:it(25) =< Out
1134.97/291.53	aux(2) =< V1-Out
1134.97/291.53	s(1) =< aux(2)
1134.97/291.53	s(4) =< it(25)*aux(2)
1134.97/291.53	
1134.97/291.53	  with precondition: [V1=Out+V,V>=1,V1>=V+1] 
1134.97/291.53	
1134.97/291.53	* Chain [24]: 3
1134.97/291.53	  with precondition: [V1=0,Out=0,V>=0] 
1134.97/291.53	
1134.97/291.53	* Chain [23]: 1*s(1)+3
1134.97/291.53	  Such that:s(1) =< V1
1134.97/291.53	
1134.97/291.53	  with precondition: [Out=0,V1>=1,V>=V1] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	#### Cost of chains of div(V1,V,Out):
1134.97/291.53	* Chain [[27],29]: 8*it(27)+3*s(5)+4*s(16)+1*s(17)+3
1134.97/291.53	  Such that:aux(7) =< V1-V
1134.97/291.53	s(12) =< V
1134.97/291.53	aux(9) =< V1
1134.97/291.53	it(27) =< aux(9)
1134.97/291.53	s(5) =< aux(9)
1134.97/291.53	it(27) =< aux(7)
1134.97/291.53	s(16) =< it(27)*aux(7)
1134.97/291.53	s(17) =< s(16)*s(12)
1134.97/291.53	
1134.97/291.53	  with precondition: [V>=1,Out>=1,V1>=Out+V] 
1134.97/291.53	
1134.97/291.53	* Chain [[27],28,31]: 8*it(27)+2*s(15)+4*s(16)+1*s(17)+2*s(19)+11
1134.97/291.53	  Such that:aux(6) =< V1
1134.97/291.53	aux(11) =< V1-V
1134.97/291.53	aux(12) =< V
1134.97/291.53	it(27) =< aux(11)
1134.97/291.53	s(19) =< aux(12)
1134.97/291.53	it(27) =< aux(6)
1134.97/291.53	s(18) =< aux(6)
1134.97/291.53	s(18) =< aux(11)
1134.97/291.53	s(16) =< it(27)*aux(11)
1134.97/291.53	s(15) =< s(18)
1134.97/291.53	s(17) =< s(16)*aux(12)
1134.97/291.53	
1134.97/291.53	  with precondition: [V>=1,Out>=2,V1+2>=2*V+Out] 
1134.97/291.53	
1134.97/291.53	* Chain [31]: 3
1134.97/291.53	  with precondition: [V1=0,Out=0,V>=1] 
1134.97/291.53	
1134.97/291.53	* Chain [30]: 5
1134.97/291.53	  with precondition: [V=0,Out=0,V1>=0] 
1134.97/291.53	
1134.97/291.53	* Chain [29]: 1*s(5)+3
1134.97/291.53	  Such that:s(5) =< V1
1134.97/291.53	
1134.97/291.53	  with precondition: [Out=0,V1>=1,V>=V1+1] 
1134.97/291.53	
1134.97/291.53	* Chain [28,31]: 2*s(19)+11
1134.97/291.53	  Such that:aux(10) =< V1
1134.97/291.53	s(19) =< aux(10)
1134.97/291.53	
1134.97/291.53	  with precondition: [Out=1,V1=V,V1>=1] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	#### Cost of chains of start(V1,V,V5):
1134.97/291.53	* Chain [40]: 3
1134.97/291.53	  with precondition: [V1=0] 
1134.97/291.53	
1134.97/291.53	* Chain [39]: 4*s(21)+9
1134.97/291.53	  Such that:s(21) =< V1
1134.97/291.53	
1134.97/291.53	  with precondition: [V=0,V1>=0] 
1134.97/291.53	
1134.97/291.53	* Chain [38]: 7*s(22)+4*s(23)+7*s(25)+1*s(26)+16*s(37)+8*s(39)+2*s(40)+2*s(48)+11
1134.97/291.53	  Such that:aux(13) =< V1
1134.97/291.53	aux(14) =< V1-V
1134.97/291.53	aux(15) =< V
1134.97/291.53	s(22) =< aux(13)
1134.97/291.53	s(23) =< aux(14)
1134.97/291.53	s(25) =< aux(15)
1134.97/291.53	s(37) =< aux(13)
1134.97/291.53	s(37) =< aux(14)
1134.97/291.53	s(39) =< s(37)*aux(14)
1134.97/291.53	s(40) =< s(39)*aux(15)
1134.97/291.53	s(46) =< aux(13)
1134.97/291.53	s(46) =< aux(14)
1134.97/291.53	s(48) =< s(46)
1134.97/291.53	s(26) =< s(23)*aux(15)
1134.97/291.53	
1134.97/291.53	  with precondition: [V1>=1] 
1134.97/291.53	
1134.97/291.53	* Chain [37]: 8*s(50)+9
1134.97/291.53	  Such that:aux(16) =< V
1134.97/291.53	s(50) =< aux(16)
1134.97/291.53	
1134.97/291.53	  with precondition: [V1=1,V5=0,V>=0] 
1134.97/291.53	
1134.97/291.53	* Chain [36]: 14*s(52)+2*s(55)+17
1134.97/291.53	  Such that:aux(19) =< V5
1134.97/291.53	s(52) =< aux(19)
1134.97/291.53	s(55) =< s(52)*aux(19)
1134.97/291.53	
1134.97/291.53	  with precondition: [V1=1,V=2*V5,V>=2] 
1134.97/291.53	
1134.97/291.53	* Chain [35]: 2*s(64)+9
1134.97/291.53	  Such that:aux(20) =< V
1134.97/291.53	s(64) =< aux(20)
1134.97/291.53	
1134.97/291.53	  with precondition: [V1=1,V>=1,V5>=V] 
1134.97/291.53	
1134.97/291.53	* Chain [34]: 1*s(66)+5
1134.97/291.53	  Such that:s(66) =< V
1134.97/291.53	
1134.97/291.53	  with precondition: [V1=1,V>=2,V5+1>=V] 
1134.97/291.53	
1134.97/291.53	* Chain [33]: 26*s(67)+9*s(69)+5*s(70)+32*s(74)+16*s(76)+4*s(77)+4*s(89)+17
1134.97/291.53	  Such that:aux(29) =< V-2*V5
1134.97/291.53	aux(30) =< V-V5
1134.97/291.53	aux(31) =< V5
1134.97/291.53	s(67) =< aux(30)
1134.97/291.53	s(74) =< aux(30)
1134.97/291.53	s(74) =< aux(29)
1134.97/291.53	s(76) =< s(74)*aux(29)
1134.97/291.53	s(77) =< s(76)*aux(31)
1134.97/291.53	s(69) =< aux(31)
1134.97/291.53	s(70) =< s(67)*aux(31)
1134.97/291.53	s(87) =< aux(30)
1134.97/291.53	s(87) =< aux(29)
1134.97/291.53	s(89) =< s(87)
1134.97/291.53	
1134.97/291.53	  with precondition: [V1=1,V5>=1,V>=V5+2] 
1134.97/291.53	
1134.97/291.53	* Chain [32]: 10*s(119)+2*s(121)+2*s(122)+9
1134.97/291.53	  Such that:aux(34) =< V-V5
1134.97/291.53	aux(35) =< V5
1134.97/291.53	s(119) =< aux(34)
1134.97/291.53	s(121) =< aux(35)
1134.97/291.53	s(122) =< s(119)*aux(35)
1134.97/291.53	
1134.97/291.53	  with precondition: [V1=1,2*V5>=V+1,V>=V5+1] 
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	Closed-form bounds of start(V1,V,V5): 
1134.97/291.53	-------------------------------------
1134.97/291.53	* Chain [40] with precondition: [V1=0] 
1134.97/291.53	    - Upper bound: 3 
1134.97/291.53	    - Complexity: constant 
1134.97/291.53	* Chain [39] with precondition: [V=0,V1>=0] 
1134.97/291.53	    - Upper bound: 4*V1+9 
1134.97/291.53	    - Complexity: n 
1134.97/291.53	* Chain [38] with precondition: [V1>=1] 
1134.97/291.53	    - Upper bound: 25*V1+11+2*V1*nat(V)*nat(V1-V)+8*V1*nat(V1-V)+nat(V)*7+nat(V1-V)*nat(V)+nat(V1-V)*4 
1134.97/291.53	    - Complexity: n^3 
1134.97/291.53	* Chain [37] with precondition: [V1=1,V5=0,V>=0] 
1134.97/291.53	    - Upper bound: 8*V+9 
1134.97/291.53	    - Complexity: n 
1134.97/291.53	* Chain [36] with precondition: [V1=1,V=2*V5,V>=2] 
1134.97/291.53	    - Upper bound: 14*V5+17+2*V5*V5 
1134.97/291.53	    - Complexity: n^2 
1134.97/291.53	* Chain [35] with precondition: [V1=1,V>=1,V5>=V] 
1134.97/291.53	    - Upper bound: 2*V+9 
1134.97/291.53	    - Complexity: n 
1134.97/291.53	* Chain [34] with precondition: [V1=1,V>=2,V5+1>=V] 
1134.97/291.53	    - Upper bound: V+5 
1134.97/291.53	    - Complexity: n 
1134.97/291.53	* Chain [33] with precondition: [V1=1,V5>=1,V>=V5+2] 
1134.97/291.53	    - Upper bound: 62*V-62*V5+(9*V5+17+(V-V5)*(5*V5)+(V-V5)*(4*V5)*nat(V-2*V5))+(16*V-16*V5)*nat(V-2*V5) 
1134.97/291.53	    - Complexity: n^3 
1134.97/291.53	* Chain [32] with precondition: [V1=1,2*V5>=V+1,V>=V5+1] 
1134.97/291.53	    - Upper bound: 10*V-10*V5+(2*V5+9+(V-V5)*(2*V5)) 
1134.97/291.53	    - Complexity: n^2 
1134.97/291.53	
1134.97/291.53	### Maximum cost of start(V1,V,V5): max([max([4*V1+6,nat(V5)*2+6+max([nat(V5)*2*nat(V-V5)+nat(V-V5)*10,nat(V5)*7+8+max([nat(V5)*2*nat(V5)+nat(V5)*5,nat(V5)*4*nat(V-V5)*nat(V-2*V5)+nat(V5)*5*nat(V-V5)+nat(V-V5)*62+nat(V-V5)*16*nat(V-2*V5)])])]),nat(V)*5+max([nat(V),25*V1+2+2*V1*nat(V)*nat(V1-V)+8*V1*nat(V1-V)+nat(V1-V)*nat(V)+nat(V1-V)*4])+(nat(V)+4)+(nat(V)+2)])+3 
1134.97/291.53	Asymptotic class: n^3 
1134.97/291.53	* Total analysis performed in 510 ms.
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(10)
1134.97/291.53	BOUNDS(1, n^3)
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1134.97/291.53	Transformed a relative TRS into a decreasing-loop problem.
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(12)
1134.97/291.53	Obligation:
1134.97/291.53	Analyzing the following TRS for decreasing loops:
1134.97/291.53	
1134.97/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	The TRS R consists of the following rules:
1134.97/291.53	
1134.97/291.53	   minus(x, y) -> if(gt(x, y), x, y)
1134.97/291.53	   if(true, x, y) -> s(minus(p(x), y))
1134.97/291.53	   if(false, x, y) -> 0
1134.97/291.53	   p(0) -> 0
1134.97/291.53	   p(s(x)) -> x
1134.97/291.53	   ge(x, 0) -> true
1134.97/291.53	   ge(0, s(x)) -> false
1134.97/291.53	   ge(s(x), s(y)) -> ge(x, y)
1134.97/291.53	   gt(0, y) -> false
1134.97/291.53	   gt(s(x), 0) -> true
1134.97/291.53	   gt(s(x), s(y)) -> gt(x, y)
1134.97/291.53	   div(x, y) -> if1(ge(x, y), x, y)
1134.97/291.53	   if1(true, x, y) -> if2(gt(y, 0), x, y)
1134.97/291.53	   if1(false, x, y) -> 0
1134.97/291.53	   if2(true, x, y) -> s(div(minus(x, y), y))
1134.97/291.53	   if2(false, x, y) -> 0
1134.97/291.53	
1134.97/291.53	S is empty.
1134.97/291.53	Rewrite Strategy: INNERMOST
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(13) DecreasingLoopProof (LOWER BOUND(ID))
1134.97/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1134.97/291.53	
1134.97/291.53	The rewrite sequence
1134.97/291.53	
1134.97/291.53	gt(s(x), s(y)) ->^+ gt(x, y)
1134.97/291.53	
1134.97/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1134.97/291.53	
1134.97/291.53	The pumping substitution is [x / s(x), y / s(y)].
1134.97/291.53	
1134.97/291.53	The result substitution is [ ].
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(14)
1134.97/291.53	Complex Obligation (BEST)
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(15)
1134.97/291.53	Obligation:
1134.97/291.53	Proved the lower bound n^1 for the following obligation:
1134.97/291.53	
1134.97/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	The TRS R consists of the following rules:
1134.97/291.53	
1134.97/291.53	   minus(x, y) -> if(gt(x, y), x, y)
1134.97/291.53	   if(true, x, y) -> s(minus(p(x), y))
1134.97/291.53	   if(false, x, y) -> 0
1134.97/291.53	   p(0) -> 0
1134.97/291.53	   p(s(x)) -> x
1134.97/291.53	   ge(x, 0) -> true
1134.97/291.53	   ge(0, s(x)) -> false
1134.97/291.53	   ge(s(x), s(y)) -> ge(x, y)
1134.97/291.53	   gt(0, y) -> false
1134.97/291.53	   gt(s(x), 0) -> true
1134.97/291.53	   gt(s(x), s(y)) -> gt(x, y)
1134.97/291.53	   div(x, y) -> if1(ge(x, y), x, y)
1134.97/291.53	   if1(true, x, y) -> if2(gt(y, 0), x, y)
1134.97/291.53	   if1(false, x, y) -> 0
1134.97/291.53	   if2(true, x, y) -> s(div(minus(x, y), y))
1134.97/291.53	   if2(false, x, y) -> 0
1134.97/291.53	
1134.97/291.53	S is empty.
1134.97/291.53	Rewrite Strategy: INNERMOST
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(16) LowerBoundPropagationProof (FINISHED)
1134.97/291.53	Propagated lower bound.
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(17)
1134.97/291.53	BOUNDS(n^1, INF)
1134.97/291.53	
1134.97/291.53	----------------------------------------
1134.97/291.53	
1134.97/291.53	(18)
1134.97/291.53	Obligation:
1134.97/291.53	Analyzing the following TRS for decreasing loops:
1134.97/291.53	
1134.97/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1134.97/291.53	
1134.97/291.53	
1134.97/291.53	The TRS R consists of the following rules:
1134.97/291.53	
1134.97/291.53	   minus(x, y) -> if(gt(x, y), x, y)
1134.97/291.53	   if(true, x, y) -> s(minus(p(x), y))
1134.97/291.53	   if(false, x, y) -> 0
1134.97/291.53	   p(0) -> 0
1134.97/291.53	   p(s(x)) -> x
1134.97/291.53	   ge(x, 0) -> true
1134.97/291.53	   ge(0, s(x)) -> false
1134.97/291.53	   ge(s(x), s(y)) -> ge(x, y)
1134.97/291.53	   gt(0, y) -> false
1134.97/291.53	   gt(s(x), 0) -> true
1134.97/291.53	   gt(s(x), s(y)) -> gt(x, y)
1134.97/291.53	   div(x, y) -> if1(ge(x, y), x, y)
1134.97/291.53	   if1(true, x, y) -> if2(gt(y, 0), x, y)
1134.97/291.53	   if1(false, x, y) -> 0
1134.97/291.53	   if2(true, x, y) -> s(div(minus(x, y), y))
1134.97/291.53	   if2(false, x, y) -> 0
1134.97/291.53	
1134.97/291.53	S is empty.
1134.97/291.53	Rewrite Strategy: INNERMOST
1135.41/291.61	EOF