51.31/14.23	WORST_CASE(Omega(n^1), O(n^1))
51.31/14.24	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
51.31/14.24	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
51.31/14.24	
51.31/14.24	
51.31/14.24	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
51.31/14.24	
51.31/14.24	(0) CpxTRS
51.31/14.24	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
51.31/14.24	(2) CpxWeightedTrs
51.31/14.24	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
51.31/14.24	    (4) CpxTypedWeightedTrs
51.31/14.24	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
51.31/14.24	    (6) CpxTypedWeightedCompleteTrs
51.31/14.24	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms]
51.31/14.24	    (8) CpxRNTS
51.31/14.24	    (9) CompleteCoflocoProof [FINISHED, 643 ms]
51.31/14.24	    (10) BOUNDS(1, n^1)
51.31/14.24	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
51.31/14.24	(12) TRS for Loop Detection
51.31/14.24	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
51.31/14.24	    (14) BEST
51.31/14.24	        (15) proven lower bound
51.31/14.24	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
51.31/14.24	            (17) BOUNDS(n^1, INF)
51.31/14.24	        (18) TRS for Loop Detection
51.31/14.24	
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(0)
51.31/14.24	Obligation:
51.31/14.24	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
51.31/14.24	
51.31/14.24	
51.31/14.24	The TRS R consists of the following rules:
51.31/14.24	
51.31/14.24	   ge(x, 0) -> true
51.31/14.24	   ge(0, s(y)) -> false
51.31/14.24	   ge(s(x), s(y)) -> ge(x, y)
51.31/14.24	   minus(x, 0) -> x
51.31/14.24	   minus(0, y) -> 0
51.31/14.24	   minus(s(x), s(y)) -> minus(x, y)
51.31/14.24	   id_inc(x) -> x
51.31/14.24	   id_inc(x) -> s(x)
51.31/14.24	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y)
51.31/14.24	   if(false, b, x, y) -> div_by_zero
51.31/14.24	   if(true, false, x, y) -> 0
51.31/14.24	   if(true, true, x, y) -> id_inc(div(minus(x, y), y))
51.31/14.24	
51.31/14.24	S is empty.
51.31/14.24	Rewrite Strategy: INNERMOST
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
51.31/14.24	Transformed relative TRS to weighted TRS
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(2)
51.31/14.24	Obligation:
51.31/14.24	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
51.31/14.24	
51.31/14.24	
51.31/14.24	The TRS R consists of the following rules:
51.31/14.24	
51.31/14.24	   ge(x, 0) -> true [1]
51.31/14.24	   ge(0, s(y)) -> false [1]
51.31/14.24	   ge(s(x), s(y)) -> ge(x, y) [1]
51.31/14.24	   minus(x, 0) -> x [1]
51.31/14.24	   minus(0, y) -> 0 [1]
51.31/14.24	   minus(s(x), s(y)) -> minus(x, y) [1]
51.31/14.24	   id_inc(x) -> x [1]
51.31/14.24	   id_inc(x) -> s(x) [1]
51.31/14.24	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y) [1]
51.31/14.24	   if(false, b, x, y) -> div_by_zero [1]
51.31/14.24	   if(true, false, x, y) -> 0 [1]
51.31/14.24	   if(true, true, x, y) -> id_inc(div(minus(x, y), y)) [1]
51.31/14.24	
51.31/14.24	Rewrite Strategy: INNERMOST
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
51.31/14.24	Infered types.
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(4)
51.31/14.24	Obligation:
51.31/14.24	Runtime Complexity Weighted TRS with Types.
51.31/14.24	The TRS R consists of the following rules:
51.31/14.24	
51.31/14.24	   ge(x, 0) -> true [1]
51.31/14.24	   ge(0, s(y)) -> false [1]
51.31/14.24	   ge(s(x), s(y)) -> ge(x, y) [1]
51.31/14.24	   minus(x, 0) -> x [1]
51.31/14.24	   minus(0, y) -> 0 [1]
51.31/14.24	   minus(s(x), s(y)) -> minus(x, y) [1]
51.31/14.24	   id_inc(x) -> x [1]
51.31/14.24	   id_inc(x) -> s(x) [1]
51.31/14.24	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y) [1]
51.31/14.24	   if(false, b, x, y) -> div_by_zero [1]
51.31/14.24	   if(true, false, x, y) -> 0 [1]
51.31/14.24	   if(true, true, x, y) -> id_inc(div(minus(x, y), y)) [1]
51.31/14.24	
51.31/14.24	The TRS has the following type information:
51.31/14.24	ge :: 0:s:div_by_zero -> 0:s:div_by_zero -> true:false
51.31/14.24	0 :: 0:s:div_by_zero
51.31/14.24	true :: true:false
51.31/14.24	s :: 0:s:div_by_zero -> 0:s:div_by_zero
51.31/14.24	false :: true:false
51.31/14.24	minus :: 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
51.31/14.24	id_inc :: 0:s:div_by_zero -> 0:s:div_by_zero
51.31/14.24	div :: 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
51.31/14.24	if :: true:false -> true:false -> 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
51.31/14.24	div_by_zero :: 0:s:div_by_zero
51.31/14.24	
51.31/14.24	Rewrite Strategy: INNERMOST
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(5) CompletionProof (UPPER BOUND(ID))
51.31/14.24	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
51.31/14.24	
51.31/14.24	   ge(v0, v1) -> null_ge [0]
51.31/14.24	   minus(v0, v1) -> null_minus [0]
51.31/14.24	   if(v0, v1, v2, v3) -> null_if [0]
51.31/14.24	
51.31/14.24	And the following fresh constants:    null_ge, null_minus, null_if
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(6)
51.31/14.24	Obligation:
51.31/14.24	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
51.31/14.24	
51.31/14.24	Runtime Complexity Weighted TRS with Types.
51.31/14.24	The TRS R consists of the following rules:
51.31/14.24	
51.31/14.24	   ge(x, 0) -> true [1]
51.31/14.24	   ge(0, s(y)) -> false [1]
51.31/14.24	   ge(s(x), s(y)) -> ge(x, y) [1]
51.31/14.24	   minus(x, 0) -> x [1]
51.31/14.24	   minus(0, y) -> 0 [1]
51.31/14.24	   minus(s(x), s(y)) -> minus(x, y) [1]
51.31/14.24	   id_inc(x) -> x [1]
51.31/14.24	   id_inc(x) -> s(x) [1]
51.31/14.24	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y) [1]
51.31/14.24	   if(false, b, x, y) -> div_by_zero [1]
51.31/14.24	   if(true, false, x, y) -> 0 [1]
51.31/14.24	   if(true, true, x, y) -> id_inc(div(minus(x, y), y)) [1]
51.31/14.24	   ge(v0, v1) -> null_ge [0]
51.31/14.24	   minus(v0, v1) -> null_minus [0]
51.31/14.24	   if(v0, v1, v2, v3) -> null_if [0]
51.31/14.24	
51.31/14.24	The TRS has the following type information:
51.31/14.24	ge :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> true:false:null_ge
51.31/14.24	0 :: 0:s:div_by_zero:null_minus:null_if
51.31/14.24	true :: true:false:null_ge
51.31/14.24	s :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
51.31/14.24	false :: true:false:null_ge
51.31/14.24	minus :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
51.31/14.24	id_inc :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
51.31/14.24	div :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
51.31/14.24	if :: true:false:null_ge -> true:false:null_ge -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
51.31/14.24	div_by_zero :: 0:s:div_by_zero:null_minus:null_if
51.31/14.24	null_ge :: true:false:null_ge
51.31/14.24	null_minus :: 0:s:div_by_zero:null_minus:null_if
51.31/14.24	null_if :: 0:s:div_by_zero:null_minus:null_if
51.31/14.24	
51.31/14.24	Rewrite Strategy: INNERMOST
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
51.31/14.24	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
51.31/14.24	The constant constructors are abstracted as follows:
51.31/14.24	
51.31/14.24	   0 => 0
51.31/14.24	   true => 2
51.31/14.24	   false => 1
51.31/14.24	   div_by_zero => 1
51.31/14.24	   null_ge => 0
51.31/14.24	   null_minus => 0
51.31/14.24	   null_if => 0
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(8)
51.31/14.24	Obligation:
51.31/14.24	Complexity RNTS consisting of the following rules:
51.31/14.24	
51.31/14.24	   div(z, z') -{ 1 }-> if(ge(y, 1 + 0), ge(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
51.31/14.24	   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
51.31/14.24	   ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0
51.31/14.24	   ge(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
51.31/14.24	   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
51.31/14.24	   id_inc(z) -{ 1 }-> x :|: x >= 0, z = x
51.31/14.24	   id_inc(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
51.31/14.24	   if(z, z', z'', z1) -{ 1 }-> id_inc(div(minus(x, y), y)) :|: z = 2, z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
51.31/14.24	   if(z, z', z'', z1) -{ 1 }-> 1 :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x
51.31/14.24	   if(z, z', z'', z1) -{ 1 }-> 0 :|: z = 2, z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
51.31/14.24	   if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
51.31/14.24	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
51.31/14.24	   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
51.31/14.24	   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
51.31/14.24	   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
51.31/14.24	
51.31/14.24	Only complete derivations are relevant for the runtime complexity.
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(9) CompleteCoflocoProof (FINISHED)
51.31/14.24	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
51.31/14.24	
51.31/14.24	eq(start(V1, V, V15, V18),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
51.31/14.24	eq(start(V1, V, V15, V18),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
51.31/14.24	eq(start(V1, V, V15, V18),0,[fun(V1, Out)],[V1 >= 0]).
51.31/14.24	eq(start(V1, V, V15, V18),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
51.31/14.24	eq(start(V1, V, V15, V18),0,[if(V1, V, V15, V18, Out)],[V1 >= 0,V >= 0,V15 >= 0,V18 >= 0]).
51.31/14.24	eq(ge(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = V2,V = 0]).
51.31/14.24	eq(ge(V1, V, Out),1,[],[Out = 1,V = 1 + V3,V3 >= 0,V1 = 0]).
51.31/14.24	eq(ge(V1, V, Out),1,[ge(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
51.31/14.24	eq(minus(V1, V, Out),1,[],[Out = V6,V6 >= 0,V1 = V6,V = 0]).
51.31/14.24	eq(minus(V1, V, Out),1,[],[Out = 0,V7 >= 0,V1 = 0,V = V7]).
51.31/14.24	eq(minus(V1, V, Out),1,[minus(V8, V9, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = 1 + V8]).
51.31/14.24	eq(fun(V1, Out),1,[],[Out = V10,V10 >= 0,V1 = V10]).
51.31/14.24	eq(fun(V1, Out),1,[],[Out = 1 + V11,V11 >= 0,V1 = V11]).
51.31/14.24	eq(div(V1, V, Out),1,[ge(V12, 1 + 0, Ret0),ge(V13, V12, Ret11),if(Ret0, Ret11, V13, V12, Ret2)],[Out = Ret2,V13 >= 0,V12 >= 0,V1 = V13,V = V12]).
51.31/14.24	eq(if(V1, V, V15, V18, Out),1,[],[Out = 1,V17 >= 0,V18 = V14,V1 = 1,V16 >= 0,V14 >= 0,V = V17,V15 = V16]).
51.31/14.24	eq(if(V1, V, V15, V18, Out),1,[],[Out = 0,V1 = 2,V18 = V20,V19 >= 0,V20 >= 0,V15 = V19,V = 1]).
51.31/14.24	eq(if(V1, V, V15, V18, Out),1,[minus(V22, V21, Ret00),div(Ret00, V21, Ret01),fun(Ret01, Ret3)],[Out = Ret3,V1 = 2,V18 = V21,V = 2,V22 >= 0,V21 >= 0,V15 = V22]).
51.31/14.24	eq(ge(V1, V, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V1 = V24,V = V23]).
51.31/14.24	eq(minus(V1, V, Out),0,[],[Out = 0,V26 >= 0,V25 >= 0,V1 = V26,V = V25]).
51.31/14.24	eq(if(V1, V, V15, V18, Out),0,[],[Out = 0,V18 = V29,V28 >= 0,V15 = V30,V27 >= 0,V1 = V28,V = V27,V30 >= 0,V29 >= 0]).
51.31/14.24	input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
51.31/14.24	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
51.31/14.24	input_output_vars(fun(V1,Out),[V1],[Out]).
51.31/14.24	input_output_vars(div(V1,V,Out),[V1,V],[Out]).
51.31/14.24	input_output_vars(if(V1,V,V15,V18,Out),[V1,V,V15,V18],[Out]).
51.31/14.24	
51.31/14.24	
51.31/14.24	CoFloCo proof output:
51.31/14.24	Preprocessing Cost Relations
51.31/14.24	=====================================
51.31/14.24	
51.31/14.24	#### Computed strongly connected components 
51.31/14.24	0. recursive  : [ge/3]
51.31/14.24	1. non_recursive  : [fun/2]
51.31/14.24	2. recursive  : [minus/3]
51.31/14.24	3. recursive [non_tail] : [(div)/3,if/5]
51.31/14.24	4. non_recursive  : [start/4]
51.31/14.24	
51.31/14.24	#### Obtained direct recursion through partial evaluation 
51.31/14.24	0. SCC is partially evaluated into ge/3
51.31/14.24	1. SCC is partially evaluated into fun/2
51.31/14.24	2. SCC is partially evaluated into minus/3
51.31/14.24	3. SCC is partially evaluated into (div)/3
51.31/14.24	4. SCC is partially evaluated into start/4
51.31/14.24	
51.31/14.24	Control-Flow Refinement of Cost Relations
51.31/14.24	=====================================
51.31/14.24	
51.31/14.24	### Specialization of cost equations ge/3 
51.31/14.24	* CE 22 is refined into CE [23] 
51.31/14.24	* CE 19 is refined into CE [24] 
51.31/14.24	* CE 20 is refined into CE [25] 
51.31/14.24	* CE 21 is refined into CE [26] 
51.31/14.24	
51.31/14.24	
51.31/14.24	### Cost equations --> "Loop" of ge/3 
51.31/14.24	* CEs [26] --> Loop 16 
51.31/14.24	* CEs [23] --> Loop 17 
51.31/14.24	* CEs [24] --> Loop 18 
51.31/14.24	* CEs [25] --> Loop 19 
51.31/14.24	
51.31/14.24	### Ranking functions of CR ge(V1,V,Out) 
51.31/14.24	* RF of phase [16]: [V,V1]
51.31/14.24	
51.31/14.24	#### Partial ranking functions of CR ge(V1,V,Out) 
51.31/14.24	* Partial RF of phase [16]:
51.31/14.24	  - RF of loop [16:1]:
51.31/14.24	    V
51.31/14.24	    V1
51.31/14.24	
51.31/14.24	
51.31/14.24	### Specialization of cost equations fun/2 
51.31/14.24	* CE 17 is refined into CE [27] 
51.31/14.24	* CE 18 is refined into CE [28] 
51.31/14.24	
51.31/14.24	
51.31/14.24	### Cost equations --> "Loop" of fun/2 
51.31/14.24	* CEs [27] --> Loop 20 
51.31/14.24	* CEs [28] --> Loop 21 
51.31/14.24	
51.31/14.24	### Ranking functions of CR fun(V1,Out) 
51.31/14.24	
51.31/14.24	#### Partial ranking functions of CR fun(V1,Out) 
51.31/14.24	
51.31/14.24	
51.31/14.24	### Specialization of cost equations minus/3 
51.31/14.24	* CE 9 is refined into CE [29] 
51.31/14.24	* CE 10 is refined into CE [30] 
51.31/14.24	* CE 12 is refined into CE [31] 
51.31/14.24	* CE 11 is refined into CE [32] 
51.31/14.24	
51.31/14.24	
51.31/14.24	### Cost equations --> "Loop" of minus/3 
51.31/14.24	* CEs [32] --> Loop 22 
51.31/14.24	* CEs [29] --> Loop 23 
51.31/14.24	* CEs [30,31] --> Loop 24 
51.31/14.24	
51.31/14.24	### Ranking functions of CR minus(V1,V,Out) 
51.31/14.24	* RF of phase [22]: [V,V1]
51.31/14.24	
51.31/14.24	#### Partial ranking functions of CR minus(V1,V,Out) 
51.31/14.24	* Partial RF of phase [22]:
51.31/14.24	  - RF of loop [22:1]:
51.31/14.24	    V
51.31/14.24	    V1
51.31/14.24	
51.31/14.24	
51.31/14.24	### Specialization of cost equations (div)/3 
51.31/14.24	* CE 16 is refined into CE [33,34] 
51.31/14.24	* CE 13 is refined into CE [35,36,37,38,39,40,41,42,43,44,45] 
51.31/14.24	* CE 15 is refined into CE [46,47] 
51.31/14.24	* CE 14 is refined into CE [48,49,50,51] 
51.31/14.24	
51.31/14.24	
51.31/14.24	### Cost equations --> "Loop" of (div)/3 
51.31/14.24	* CEs [51] --> Loop 25 
51.31/14.24	* CEs [50] --> Loop 26 
51.31/14.24	* CEs [49] --> Loop 27 
51.31/14.24	* CEs [48] --> Loop 28 
51.31/14.24	* CEs [33,34] --> Loop 29 
51.31/14.24	* CEs [35,36,38] --> Loop 30 
51.31/14.24	* CEs [37,39,40,41,42,43,44,45,46,47] --> Loop 31 
51.31/14.24	
51.31/14.24	### Ranking functions of CR div(V1,V,Out) 
51.31/14.24	* RF of phase [25,26]: [V1,V1-V+1]
51.31/14.24	
51.31/14.24	#### Partial ranking functions of CR div(V1,V,Out) 
51.31/14.24	* Partial RF of phase [25,26]:
51.31/14.24	  - RF of loop [25:1,26:1]:
51.31/14.24	    V1
51.31/14.24	    V1-V+1
51.31/14.24	
51.31/14.24	
51.31/14.24	### Specialization of cost equations start/4 
51.31/14.24	* CE 2 is refined into CE [52,53,54,55,56,57,58,59,60,61,62,63,64,65] 
51.31/14.24	* CE 3 is refined into CE [66] 
51.31/14.24	* CE 1 is refined into CE [67] 
51.31/14.24	* CE 4 is refined into CE [68] 
51.31/14.24	* CE 5 is refined into CE [69,70,71,72,73] 
51.31/14.24	* CE 6 is refined into CE [74,75,76] 
51.31/14.24	* CE 7 is refined into CE [77,78] 
51.31/14.24	* CE 8 is refined into CE [79,80,81,82] 
51.31/14.24	
51.31/14.24	
51.31/14.24	### Cost equations --> "Loop" of start/4 
51.31/14.24	* CEs [70,74,80] --> Loop 32 
51.31/14.24	* CEs [52,53,54,55,56,57,58,59,60,61,62,63,64,65] --> Loop 33 
51.31/14.24	* CEs [66] --> Loop 34 
51.31/14.24	* CEs [68] --> Loop 35 
51.31/14.24	* CEs [67,69,71,72,73,75,76,77,78,79,81,82] --> Loop 36 
51.31/14.24	
51.31/14.24	### Ranking functions of CR start(V1,V,V15,V18) 
51.31/14.24	
51.31/14.24	#### Partial ranking functions of CR start(V1,V,V15,V18) 
51.31/14.24	
51.31/14.24	
51.31/14.24	Computing Bounds
51.31/14.24	=====================================
51.31/14.24	
51.31/14.24	#### Cost of chains of ge(V1,V,Out):
51.31/14.24	* Chain [[16],19]: 1*it(16)+1
51.31/14.24	  Such that:it(16) =< V1
51.31/14.24	
51.31/14.24	  with precondition: [Out=1,V1>=1,V>=V1+1] 
51.31/14.24	
51.31/14.24	* Chain [[16],18]: 1*it(16)+1
51.31/14.24	  Such that:it(16) =< V
51.31/14.24	
51.31/14.24	  with precondition: [Out=2,V>=1,V1>=V] 
51.31/14.24	
51.31/14.24	* Chain [[16],17]: 1*it(16)+0
51.31/14.24	  Such that:it(16) =< V
51.31/14.24	
51.31/14.24	  with precondition: [Out=0,V1>=1,V>=1] 
51.31/14.24	
51.31/14.24	* Chain [19]: 1
51.31/14.24	  with precondition: [V1=0,Out=1,V>=1] 
51.31/14.24	
51.31/14.24	* Chain [18]: 1
51.31/14.24	  with precondition: [V=0,Out=2,V1>=0] 
51.31/14.24	
51.31/14.24	* Chain [17]: 0
51.31/14.24	  with precondition: [Out=0,V1>=0,V>=0] 
51.31/14.24	
51.31/14.24	
51.31/14.24	#### Cost of chains of fun(V1,Out):
51.31/14.24	* Chain [21]: 1
51.31/14.24	  with precondition: [V1+1=Out,V1>=0] 
51.31/14.24	
51.31/14.24	* Chain [20]: 1
51.31/14.24	  with precondition: [V1=Out,V1>=0] 
51.31/14.24	
51.31/14.24	
51.31/14.24	#### Cost of chains of minus(V1,V,Out):
51.31/14.24	* Chain [[22],24]: 1*it(22)+1
51.31/14.24	  Such that:it(22) =< V
51.31/14.24	
51.31/14.24	  with precondition: [Out=0,V1>=1,V>=1] 
51.31/14.24	
51.31/14.24	* Chain [[22],23]: 1*it(22)+1
51.31/14.24	  Such that:it(22) =< V
51.31/14.24	
51.31/14.24	  with precondition: [V1=Out+V,V>=1,V1>=V] 
51.31/14.24	
51.31/14.24	* Chain [24]: 1
51.31/14.24	  with precondition: [Out=0,V1>=0,V>=0] 
51.31/14.24	
51.31/14.24	* Chain [23]: 1
51.31/14.24	  with precondition: [V=0,V1=Out,V1>=0] 
51.31/14.24	
51.31/14.24	
51.31/14.24	#### Cost of chains of div(V1,V,Out):
51.31/14.24	* Chain [[25,26],31]: 12*it(25)+10*s(3)+4*s(5)+7*s(7)+2*s(32)+4
51.31/14.24	  Such that:aux(1) =< 1
51.31/14.24	aux(9) =< V1-V+1
51.31/14.24	aux(3) =< V
51.31/14.24	aux(12) =< V1
51.31/14.24	s(3) =< aux(1)
51.31/14.24	s(7) =< aux(12)
51.31/14.24	s(5) =< aux(3)
51.31/14.24	aux(6) =< aux(12)
51.31/14.24	it(25) =< aux(12)
51.31/14.24	aux(6) =< aux(9)
51.31/14.24	it(25) =< aux(9)
51.31/14.24	s(32) =< aux(6)
51.31/14.24	
51.31/14.24	  with precondition: [V>=1,Out>=0,V1>=V,V1+1>=Out+V] 
51.31/14.24	
51.31/14.24	* Chain [[25,26],28,31]: 12*it(25)+11*s(3)+6*s(5)+2*s(32)+4*s(33)+10
51.31/14.24	  Such that:aux(14) =< 1
51.31/14.24	aux(8) =< V1
51.31/14.24	aux(9) =< V1-V+1
51.31/14.24	aux(15) =< V
51.31/14.24	aux(16) =< V1-V
51.31/14.24	s(3) =< aux(14)
51.31/14.24	s(5) =< aux(15)
51.31/14.24	aux(6) =< aux(8)
51.31/14.24	it(25) =< aux(8)
51.31/14.24	s(34) =< aux(8)
51.31/14.24	aux(6) =< aux(9)
51.31/14.24	it(25) =< aux(9)
51.31/14.24	aux(6) =< aux(16)
51.31/14.24	it(25) =< aux(16)
51.31/14.24	s(34) =< aux(16)
51.31/14.24	s(32) =< aux(6)
51.31/14.24	s(33) =< s(34)
51.31/14.24	
51.31/14.24	  with precondition: [V>=1,Out>=1,V1>=2*V,V1+2>=2*V+Out] 
51.31/14.24	
51.31/14.24	* Chain [[25,26],27,31]: 12*it(25)+11*s(3)+6*s(5)+2*s(32)+4*s(33)+10
51.31/14.24	  Such that:aux(18) =< 1
51.31/14.24	aux(8) =< V1
51.31/14.24	aux(9) =< V1-V+1
51.31/14.24	aux(19) =< V
51.31/14.24	aux(20) =< V1-V
51.31/14.24	s(3) =< aux(18)
51.31/14.24	s(5) =< aux(19)
51.31/14.24	aux(6) =< aux(8)
51.31/14.24	it(25) =< aux(8)
51.31/14.24	s(34) =< aux(8)
51.31/14.24	aux(6) =< aux(9)
51.31/14.24	it(25) =< aux(9)
51.31/14.24	aux(6) =< aux(20)
51.31/14.24	it(25) =< aux(20)
51.31/14.24	s(34) =< aux(20)
51.31/14.24	s(32) =< aux(6)
51.31/14.24	s(33) =< s(34)
51.31/14.24	
51.31/14.24	  with precondition: [V>=1,Out>=0,V1>=2*V,V1+1>=2*V+Out] 
51.31/14.24	
51.31/14.24	* Chain [31]: 10*s(3)+4*s(5)+3*s(7)+4
51.31/14.24	  Such that:aux(1) =< 1
51.31/14.24	aux(2) =< V1
51.31/14.24	aux(3) =< V
51.31/14.24	s(3) =< aux(1)
51.31/14.24	s(7) =< aux(2)
51.31/14.24	s(5) =< aux(3)
51.31/14.24	
51.31/14.24	  with precondition: [Out=0,V1>=0,V>=0] 
51.31/14.24	
51.31/14.24	* Chain [30]: 4
51.31/14.24	  with precondition: [V=0,Out=0,V1>=0] 
51.31/14.24	
51.31/14.24	* Chain [29]: 4
51.31/14.24	  with precondition: [V=0,Out=1,V1>=0] 
51.31/14.24	
51.31/14.24	* Chain [28,31]: 11*s(3)+6*s(5)+10
51.31/14.24	  Such that:aux(14) =< 1
51.31/14.24	aux(15) =< V
51.31/14.24	s(3) =< aux(14)
51.31/14.24	s(5) =< aux(15)
51.31/14.24	
51.31/14.24	  with precondition: [Out=1,V>=1,V1>=V] 
51.31/14.24	
51.31/14.24	* Chain [27,31]: 11*s(3)+6*s(5)+10
51.31/14.24	  Such that:aux(18) =< 1
51.31/14.24	aux(19) =< V
51.31/14.24	s(3) =< aux(18)
51.31/14.24	s(5) =< aux(19)
51.31/14.24	
51.31/14.24	  with precondition: [Out=0,V>=1,V1>=V] 
51.31/14.24	
51.31/14.24	
51.31/14.24	#### Cost of chains of start(V1,V,V15,V18):
51.31/14.24	* Chain [36]: 36*s(83)+11*s(84)+64*s(91)+24*s(102)+4*s(104)+8*s(105)+12*s(108)+2*s(109)+10
51.31/14.24	  Such that:aux(27) =< 1
51.31/14.24	aux(28) =< V1
51.31/14.24	aux(29) =< V1-V
51.31/14.24	aux(30) =< V1-V+1
51.31/14.24	aux(31) =< V
51.31/14.24	s(84) =< aux(28)
51.31/14.24	s(83) =< aux(31)
51.31/14.24	s(91) =< aux(27)
51.31/14.24	s(101) =< aux(28)
51.31/14.24	s(102) =< aux(28)
51.31/14.24	s(103) =< aux(28)
51.31/14.24	s(101) =< aux(30)
51.31/14.24	s(102) =< aux(30)
51.31/14.24	s(101) =< aux(29)
51.31/14.24	s(102) =< aux(29)
51.31/14.24	s(103) =< aux(29)
51.31/14.24	s(104) =< s(101)
51.31/14.24	s(105) =< s(103)
51.31/14.24	s(107) =< aux(28)
51.31/14.24	s(108) =< aux(28)
51.31/14.24	s(107) =< aux(30)
51.31/14.24	s(108) =< aux(30)
51.31/14.24	s(109) =< s(107)
51.31/14.24	
51.31/14.24	  with precondition: [V1>=0] 
51.31/14.24	
51.31/14.24	* Chain [35]: 1
51.31/14.24	  with precondition: [V1=1,V>=0,V15>=0,V18>=0] 
51.31/14.24	
51.31/14.24	* Chain [34]: 1
51.31/14.24	  with precondition: [V1=2,V=1,V15>=0,V18>=0] 
51.31/14.24	
51.31/14.24	* Chain [33]: 212*s(125)+6*s(127)+92*s(134)+20*s(156)+48*s(173)+8*s(175)+16*s(176)+24*s(179)+4*s(180)+13
51.31/14.24	  Such that:aux(40) =< 1
51.31/14.24	aux(41) =< V15
51.31/14.24	aux(42) =< V15-2*V18
51.31/14.24	aux(43) =< V15-2*V18+1
51.31/14.24	aux(44) =< V15-V18
51.31/14.24	aux(45) =< V18
51.31/14.24	s(125) =< aux(40)
51.31/14.24	s(127) =< aux(41)
51.31/14.24	s(134) =< aux(45)
51.31/14.24	s(156) =< aux(44)
51.31/14.24	s(172) =< aux(44)
51.31/14.24	s(173) =< aux(44)
51.31/14.24	s(174) =< aux(44)
51.31/14.24	s(172) =< aux(43)
51.31/14.24	s(173) =< aux(43)
51.31/14.24	s(172) =< aux(42)
51.31/14.24	s(173) =< aux(42)
51.31/14.24	s(174) =< aux(42)
51.31/14.24	s(175) =< s(172)
51.31/14.24	s(176) =< s(174)
51.31/14.24	s(178) =< aux(44)
51.31/14.24	s(179) =< aux(44)
51.31/14.24	s(178) =< aux(43)
51.31/14.24	s(179) =< aux(43)
51.31/14.24	s(180) =< s(178)
51.31/14.24	
51.31/14.24	  with precondition: [V1=2,V=2,V15>=0,V18>=0] 
51.31/14.24	
51.31/14.24	* Chain [32]: 4
51.31/14.24	  with precondition: [V=0,V1>=0] 
51.31/14.24	
51.31/14.24	
51.31/14.24	Closed-form bounds of start(V1,V,V15,V18): 
51.31/14.24	-------------------------------------
51.31/14.24	* Chain [36] with precondition: [V1>=0] 
51.31/14.24	    - Upper bound: 61*V1+74+nat(V)*36 
51.31/14.24	    - Complexity: n 
51.31/14.24	* Chain [35] with precondition: [V1=1,V>=0,V15>=0,V18>=0] 
51.31/14.24	    - Upper bound: 1 
51.31/14.24	    - Complexity: constant 
51.31/14.24	* Chain [34] with precondition: [V1=2,V=1,V15>=0,V18>=0] 
51.31/14.24	    - Upper bound: 1 
51.31/14.24	    - Complexity: constant 
51.31/14.24	* Chain [33] with precondition: [V1=2,V=2,V15>=0,V18>=0] 
51.31/14.24	    - Upper bound: 6*V15+92*V18+225+nat(V15-V18)*120 
51.31/14.24	    - Complexity: n 
51.31/14.24	* Chain [32] with precondition: [V=0,V1>=0] 
51.31/14.24	    - Upper bound: 4 
51.31/14.24	    - Complexity: constant 
51.31/14.24	
51.31/14.24	### Maximum cost of start(V1,V,V15,V18): max([3,61*V1+73+nat(V)*36,nat(V15)*6+224+nat(V18)*92+nat(V15-V18)*120])+1 
51.31/14.24	Asymptotic class: n 
51.31/14.24	* Total analysis performed in 542 ms.
51.31/14.24	
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(10)
51.31/14.24	BOUNDS(1, n^1)
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
51.31/14.24	Transformed a relative TRS into a decreasing-loop problem.
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(12)
51.31/14.24	Obligation:
51.31/14.24	Analyzing the following TRS for decreasing loops:
51.31/14.24	
51.31/14.24	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
51.31/14.24	
51.31/14.24	
51.31/14.24	The TRS R consists of the following rules:
51.31/14.24	
51.31/14.24	   ge(x, 0) -> true
51.31/14.24	   ge(0, s(y)) -> false
51.31/14.24	   ge(s(x), s(y)) -> ge(x, y)
51.31/14.24	   minus(x, 0) -> x
51.31/14.24	   minus(0, y) -> 0
51.31/14.24	   minus(s(x), s(y)) -> minus(x, y)
51.31/14.24	   id_inc(x) -> x
51.31/14.24	   id_inc(x) -> s(x)
51.31/14.24	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y)
51.31/14.24	   if(false, b, x, y) -> div_by_zero
51.31/14.24	   if(true, false, x, y) -> 0
51.31/14.24	   if(true, true, x, y) -> id_inc(div(minus(x, y), y))
51.31/14.24	
51.31/14.24	S is empty.
51.31/14.24	Rewrite Strategy: INNERMOST
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(13) DecreasingLoopProof (LOWER BOUND(ID))
51.31/14.24	The following loop(s) give(s) rise to the lower bound Omega(n^1):
51.31/14.24	
51.31/14.24	The rewrite sequence
51.31/14.24	
51.31/14.24	minus(s(x), s(y)) ->^+ minus(x, y)
51.31/14.24	
51.31/14.24	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
51.31/14.24	
51.31/14.24	The pumping substitution is [x / s(x), y / s(y)].
51.31/14.24	
51.31/14.24	The result substitution is [ ].
51.31/14.24	
51.31/14.24	
51.31/14.24	
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(14)
51.31/14.24	Complex Obligation (BEST)
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(15)
51.31/14.24	Obligation:
51.31/14.24	Proved the lower bound n^1 for the following obligation:
51.31/14.24	
51.31/14.24	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
51.31/14.24	
51.31/14.24	
51.31/14.24	The TRS R consists of the following rules:
51.31/14.24	
51.31/14.24	   ge(x, 0) -> true
51.31/14.24	   ge(0, s(y)) -> false
51.31/14.24	   ge(s(x), s(y)) -> ge(x, y)
51.31/14.24	   minus(x, 0) -> x
51.31/14.24	   minus(0, y) -> 0
51.31/14.24	   minus(s(x), s(y)) -> minus(x, y)
51.31/14.24	   id_inc(x) -> x
51.31/14.24	   id_inc(x) -> s(x)
51.31/14.24	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y)
51.31/14.24	   if(false, b, x, y) -> div_by_zero
51.31/14.24	   if(true, false, x, y) -> 0
51.31/14.24	   if(true, true, x, y) -> id_inc(div(minus(x, y), y))
51.31/14.24	
51.31/14.24	S is empty.
51.31/14.24	Rewrite Strategy: INNERMOST
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(16) LowerBoundPropagationProof (FINISHED)
51.31/14.24	Propagated lower bound.
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(17)
51.31/14.24	BOUNDS(n^1, INF)
51.31/14.24	
51.31/14.24	----------------------------------------
51.31/14.24	
51.31/14.24	(18)
51.31/14.24	Obligation:
51.31/14.24	Analyzing the following TRS for decreasing loops:
51.31/14.24	
51.31/14.24	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
51.31/14.24	
51.31/14.24	
51.31/14.24	The TRS R consists of the following rules:
51.31/14.24	
51.31/14.24	   ge(x, 0) -> true
51.31/14.24	   ge(0, s(y)) -> false
51.31/14.24	   ge(s(x), s(y)) -> ge(x, y)
51.31/14.24	   minus(x, 0) -> x
51.31/14.24	   minus(0, y) -> 0
51.31/14.24	   minus(s(x), s(y)) -> minus(x, y)
51.31/14.24	   id_inc(x) -> x
51.31/14.24	   id_inc(x) -> s(x)
51.31/14.24	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y)
51.31/14.24	   if(false, b, x, y) -> div_by_zero
51.31/14.24	   if(true, false, x, y) -> 0
51.31/14.24	   if(true, true, x, y) -> id_inc(div(minus(x, y), y))
51.31/14.24	
51.31/14.24	S is empty.
51.31/14.24	Rewrite Strategy: INNERMOST
51.42/14.29	EOF