938.43/291.50	WORST_CASE(Omega(n^1), O(n^2))
938.59/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
938.59/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
938.59/291.52	
938.59/291.52	
938.59/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
938.59/291.52	
938.59/291.52	(0) CpxTRS
938.59/291.52	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
938.59/291.52	(2) CpxWeightedTrs
938.59/291.52	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
938.59/291.52	    (4) CpxTypedWeightedTrs
938.59/291.52	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
938.59/291.52	    (6) CpxTypedWeightedCompleteTrs
938.59/291.52	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
938.59/291.52	    (8) CpxRNTS
938.59/291.52	    (9) CompleteCoflocoProof [FINISHED, 322 ms]
938.59/291.52	    (10) BOUNDS(1, n^2)
938.59/291.52	(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
938.59/291.52	(12) CpxTRS
938.59/291.52	    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
938.59/291.52	    (14) typed CpxTrs
938.59/291.52	    (15) OrderProof [LOWER BOUND(ID), 0 ms]
938.59/291.52	    (16) typed CpxTrs
938.59/291.52	    (17) RewriteLemmaProof [LOWER BOUND(ID), 277 ms]
938.59/291.52	    (18) BEST
938.59/291.52	        (19) proven lower bound
938.59/291.52	            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
938.59/291.52	            (21) BOUNDS(n^1, INF)
938.59/291.52	        (22) typed CpxTrs
938.59/291.52	            (23) RewriteLemmaProof [LOWER BOUND(ID), 1298 ms]
938.59/291.52	            (24) BOUNDS(1, INF)
938.59/291.52	
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(0)
938.59/291.52	Obligation:
938.59/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
938.59/291.52	
938.59/291.52	
938.59/291.52	The TRS R consists of the following rules:
938.59/291.52	
938.59/291.52	   cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
938.59/291.52	   cond2(true, x, y) -> cond1(y, y)
938.59/291.52	   cond2(false, x, y) -> cond1(p(x), y)
938.59/291.52	   gr(0, x) -> false
938.59/291.52	   gr(s(x), 0) -> true
938.59/291.52	   gr(s(x), s(y)) -> gr(x, y)
938.59/291.52	   neq(0, 0) -> false
938.59/291.52	   neq(0, s(x)) -> true
938.59/291.52	   neq(s(x), 0) -> true
938.59/291.52	   neq(s(x), s(y)) -> neq(x, y)
938.59/291.52	   p(0) -> 0
938.59/291.52	   p(s(x)) -> x
938.59/291.52	
938.59/291.52	S is empty.
938.59/291.52	Rewrite Strategy: INNERMOST
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
938.59/291.52	Transformed relative TRS to weighted TRS
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(2)
938.59/291.52	Obligation:
938.59/291.52	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
938.59/291.52	
938.59/291.52	
938.59/291.52	The TRS R consists of the following rules:
938.59/291.52	
938.59/291.52	   cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y) [1]
938.59/291.52	   cond2(true, x, y) -> cond1(y, y) [1]
938.59/291.52	   cond2(false, x, y) -> cond1(p(x), y) [1]
938.59/291.52	   gr(0, x) -> false [1]
938.59/291.52	   gr(s(x), 0) -> true [1]
938.59/291.52	   gr(s(x), s(y)) -> gr(x, y) [1]
938.59/291.52	   neq(0, 0) -> false [1]
938.59/291.52	   neq(0, s(x)) -> true [1]
938.59/291.52	   neq(s(x), 0) -> true [1]
938.59/291.52	   neq(s(x), s(y)) -> neq(x, y) [1]
938.59/291.52	   p(0) -> 0 [1]
938.59/291.52	   p(s(x)) -> x [1]
938.59/291.52	
938.59/291.52	Rewrite Strategy: INNERMOST
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
938.59/291.52	Infered types.
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(4)
938.59/291.52	Obligation:
938.59/291.52	Runtime Complexity Weighted TRS with Types.
938.59/291.52	The TRS R consists of the following rules:
938.59/291.52	
938.59/291.52	   cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y) [1]
938.59/291.52	   cond2(true, x, y) -> cond1(y, y) [1]
938.59/291.52	   cond2(false, x, y) -> cond1(p(x), y) [1]
938.59/291.52	   gr(0, x) -> false [1]
938.59/291.52	   gr(s(x), 0) -> true [1]
938.59/291.52	   gr(s(x), s(y)) -> gr(x, y) [1]
938.59/291.52	   neq(0, 0) -> false [1]
938.59/291.52	   neq(0, s(x)) -> true [1]
938.59/291.52	   neq(s(x), 0) -> true [1]
938.59/291.52	   neq(s(x), s(y)) -> neq(x, y) [1]
938.59/291.52	   p(0) -> 0 [1]
938.59/291.52	   p(s(x)) -> x [1]
938.59/291.52	
938.59/291.52	The TRS has the following type information:
938.59/291.52	cond1 :: s:0 -> s:0 -> cond1:cond2
938.59/291.52	s :: s:0 -> s:0
938.59/291.52	cond2 :: true:false -> s:0 -> s:0 -> cond1:cond2
938.59/291.52	gr :: s:0 -> s:0 -> true:false
938.59/291.52	true :: true:false
938.59/291.52	false :: true:false
938.59/291.52	p :: s:0 -> s:0
938.59/291.52	0 :: s:0
938.59/291.52	neq :: s:0 -> s:0 -> true:false
938.59/291.52	
938.59/291.52	Rewrite Strategy: INNERMOST
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(5) CompletionProof (UPPER BOUND(ID))
938.59/291.52	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
938.59/291.52	
938.59/291.52	   cond1(v0, v1) -> null_cond1 [0]
938.59/291.52	
938.59/291.52	And the following fresh constants:    null_cond1
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(6)
938.59/291.52	Obligation:
938.59/291.52	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
938.59/291.52	
938.59/291.52	Runtime Complexity Weighted TRS with Types.
938.59/291.52	The TRS R consists of the following rules:
938.59/291.52	
938.59/291.52	   cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y) [1]
938.59/291.52	   cond2(true, x, y) -> cond1(y, y) [1]
938.59/291.52	   cond2(false, x, y) -> cond1(p(x), y) [1]
938.59/291.52	   gr(0, x) -> false [1]
938.59/291.52	   gr(s(x), 0) -> true [1]
938.59/291.52	   gr(s(x), s(y)) -> gr(x, y) [1]
938.59/291.52	   neq(0, 0) -> false [1]
938.59/291.52	   neq(0, s(x)) -> true [1]
938.59/291.52	   neq(s(x), 0) -> true [1]
938.59/291.52	   neq(s(x), s(y)) -> neq(x, y) [1]
938.59/291.52	   p(0) -> 0 [1]
938.59/291.52	   p(s(x)) -> x [1]
938.59/291.52	   cond1(v0, v1) -> null_cond1 [0]
938.59/291.52	
938.59/291.52	The TRS has the following type information:
938.59/291.52	cond1 :: s:0 -> s:0 -> null_cond1
938.59/291.52	s :: s:0 -> s:0
938.59/291.52	cond2 :: true:false -> s:0 -> s:0 -> null_cond1
938.59/291.52	gr :: s:0 -> s:0 -> true:false
938.59/291.52	true :: true:false
938.59/291.52	false :: true:false
938.59/291.52	p :: s:0 -> s:0
938.59/291.52	0 :: s:0
938.59/291.52	neq :: s:0 -> s:0 -> true:false
938.59/291.52	null_cond1 :: null_cond1
938.59/291.52	
938.59/291.52	Rewrite Strategy: INNERMOST
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
938.59/291.52	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
938.59/291.52	The constant constructors are abstracted as follows:
938.59/291.52	
938.59/291.52	   true => 1
938.59/291.52	   false => 0
938.59/291.52	   0 => 0
938.59/291.52	   null_cond1 => 0
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(8)
938.59/291.52	Obligation:
938.59/291.52	Complexity RNTS consisting of the following rules:
938.59/291.52	
938.59/291.52	   cond1(z, z') -{ 1 }-> cond2(gr(1 + x, y), 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
938.59/291.52	   cond1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
938.59/291.52	   cond2(z, z', z'') -{ 1 }-> cond1(y, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
938.59/291.52	   cond2(z, z', z'') -{ 1 }-> cond1(p(x), y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
938.59/291.52	   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
938.59/291.52	   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
938.59/291.52	   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
938.59/291.52	   neq(z, z') -{ 1 }-> neq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
938.59/291.52	   neq(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
938.59/291.52	   neq(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
938.59/291.52	   neq(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
938.59/291.52	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
938.59/291.52	   p(z) -{ 1 }-> 0 :|: z = 0
938.59/291.52	
938.59/291.52	Only complete derivations are relevant for the runtime complexity.
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(9) CompleteCoflocoProof (FINISHED)
938.59/291.52	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
938.59/291.52	
938.59/291.52	eq(start(V1, V, V5),0,[cond1(V1, V, Out)],[V1 >= 0,V >= 0]).
938.59/291.52	eq(start(V1, V, V5),0,[cond2(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
938.59/291.52	eq(start(V1, V, V5),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
938.59/291.52	eq(start(V1, V, V5),0,[neq(V1, V, Out)],[V1 >= 0,V >= 0]).
938.59/291.52	eq(start(V1, V, V5),0,[p(V1, Out)],[V1 >= 0]).
938.59/291.52	eq(cond1(V1, V, Out),1,[gr(1 + V3, V2, Ret0),cond2(Ret0, 1 + V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = 1 + V3,V = V2]).
938.59/291.52	eq(cond2(V1, V, V5, Out),1,[cond1(V6, V6, Ret1)],[Out = Ret1,V = V4,V5 = V6,V1 = 1,V4 >= 0,V6 >= 0]).
938.59/291.52	eq(cond2(V1, V, V5, Out),1,[p(V8, Ret01),cond1(Ret01, V7, Ret2)],[Out = Ret2,V = V8,V5 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
938.59/291.52	eq(gr(V1, V, Out),1,[],[Out = 0,V = V9,V9 >= 0,V1 = 0]).
938.59/291.52	eq(gr(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = 1 + V10,V = 0]).
938.59/291.52	eq(gr(V1, V, Out),1,[gr(V12, V11, Ret3)],[Out = Ret3,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]).
938.59/291.52	eq(neq(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]).
938.59/291.52	eq(neq(V1, V, Out),1,[],[Out = 1,V = 1 + V13,V13 >= 0,V1 = 0]).
938.59/291.52	eq(neq(V1, V, Out),1,[],[Out = 1,V14 >= 0,V1 = 1 + V14,V = 0]).
938.59/291.52	eq(neq(V1, V, Out),1,[neq(V16, V15, Ret4)],[Out = Ret4,V = 1 + V15,V16 >= 0,V15 >= 0,V1 = 1 + V16]).
938.59/291.52	eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
938.59/291.52	eq(p(V1, Out),1,[],[Out = V17,V17 >= 0,V1 = 1 + V17]).
938.59/291.52	eq(cond1(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
938.59/291.52	input_output_vars(cond1(V1,V,Out),[V1,V],[Out]).
938.59/291.52	input_output_vars(cond2(V1,V,V5,Out),[V1,V,V5],[Out]).
938.59/291.52	input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
938.59/291.52	input_output_vars(neq(V1,V,Out),[V1,V],[Out]).
938.59/291.52	input_output_vars(p(V1,Out),[V1],[Out]).
938.59/291.52	
938.59/291.52	
938.59/291.52	CoFloCo proof output:
938.59/291.52	Preprocessing Cost Relations
938.59/291.52	=====================================
938.59/291.52	
938.59/291.52	#### Computed strongly connected components 
938.59/291.52	0. non_recursive  : [p/2]
938.59/291.52	1. recursive  : [gr/3]
938.59/291.52	2. recursive  : [cond1/3,cond2/4]
938.59/291.52	3. recursive  : [neq/3]
938.59/291.52	4. non_recursive  : [start/3]
938.59/291.52	
938.59/291.52	#### Obtained direct recursion through partial evaluation 
938.59/291.52	0. SCC is partially evaluated into p/2
938.59/291.52	1. SCC is partially evaluated into gr/3
938.59/291.52	2. SCC is partially evaluated into cond1/3
938.59/291.52	3. SCC is partially evaluated into neq/3
938.59/291.52	4. SCC is partially evaluated into start/3
938.59/291.52	
938.59/291.52	Control-Flow Refinement of Cost Relations
938.59/291.52	=====================================
938.59/291.52	
938.59/291.52	### Specialization of cost equations p/2 
938.59/291.52	* CE 8 is refined into CE [19] 
938.59/291.52	* CE 7 is refined into CE [20] 
938.59/291.52	
938.59/291.52	
938.59/291.52	### Cost equations --> "Loop" of p/2 
938.59/291.52	* CEs [19] --> Loop 14 
938.59/291.52	* CEs [20] --> Loop 15 
938.59/291.52	
938.59/291.52	### Ranking functions of CR p(V1,Out) 
938.59/291.52	
938.59/291.52	#### Partial ranking functions of CR p(V1,Out) 
938.59/291.52	
938.59/291.52	
938.59/291.52	### Specialization of cost equations gr/3 
938.59/291.52	* CE 14 is refined into CE [21] 
938.59/291.52	* CE 13 is refined into CE [22] 
938.59/291.52	* CE 12 is refined into CE [23] 
938.59/291.52	
938.59/291.52	
938.59/291.52	### Cost equations --> "Loop" of gr/3 
938.59/291.52	* CEs [22] --> Loop 16 
938.59/291.52	* CEs [23] --> Loop 17 
938.59/291.52	* CEs [21] --> Loop 18 
938.59/291.52	
938.59/291.52	### Ranking functions of CR gr(V1,V,Out) 
938.59/291.52	* RF of phase [18]: [V,V1]
938.59/291.52	
938.59/291.52	#### Partial ranking functions of CR gr(V1,V,Out) 
938.59/291.52	* Partial RF of phase [18]:
938.59/291.52	  - RF of loop [18:1]:
938.59/291.52	    V
938.59/291.52	    V1
938.59/291.52	
938.59/291.52	
938.59/291.52	### Specialization of cost equations cond1/3 
938.59/291.52	* CE 11 is refined into CE [24] 
938.59/291.52	* CE 9 is refined into CE [25] 
938.59/291.52	* CE 10 is refined into CE [26,27] 
938.59/291.52	
938.59/291.52	
938.59/291.52	### Cost equations --> "Loop" of cond1/3 
938.59/291.52	* CEs [27] --> Loop 19 
938.59/291.52	* CEs [25] --> Loop 20 
938.59/291.52	* CEs [26] --> Loop 21 
938.59/291.52	* CEs [24] --> Loop 22 
938.59/291.52	
938.59/291.52	### Ranking functions of CR cond1(V1,V,Out) 
938.59/291.52	* RF of phase [20]: [V1]
938.59/291.52	
938.59/291.52	#### Partial ranking functions of CR cond1(V1,V,Out) 
938.59/291.52	* Partial RF of phase [20]:
938.59/291.52	  - RF of loop [20:1]:
938.59/291.52	    V1
938.59/291.52	
938.59/291.52	
938.59/291.52	### Specialization of cost equations neq/3 
938.59/291.52	* CE 18 is refined into CE [28] 
938.59/291.52	* CE 17 is refined into CE [29] 
938.59/291.52	* CE 16 is refined into CE [30] 
938.59/291.52	* CE 15 is refined into CE [31] 
938.59/291.52	
938.59/291.52	
938.59/291.52	### Cost equations --> "Loop" of neq/3 
938.59/291.52	* CEs [29] --> Loop 23 
938.59/291.52	* CEs [30] --> Loop 24 
938.59/291.52	* CEs [31] --> Loop 25 
938.59/291.52	* CEs [28] --> Loop 26 
938.59/291.52	
938.59/291.52	### Ranking functions of CR neq(V1,V,Out) 
938.59/291.52	* RF of phase [26]: [V,V1]
938.59/291.52	
938.59/291.52	#### Partial ranking functions of CR neq(V1,V,Out) 
938.59/291.52	* Partial RF of phase [26]:
938.59/291.52	  - RF of loop [26:1]:
938.59/291.52	    V
938.59/291.52	    V1
938.59/291.52	
938.59/291.52	
938.59/291.52	### Specialization of cost equations start/3 
938.59/291.52	* CE 2 is refined into CE [32] 
938.59/291.52	* CE 1 is refined into CE [33,34] 
938.59/291.52	* CE 3 is refined into CE [35] 
938.59/291.52	* CE 4 is refined into CE [36,37,38,39] 
938.59/291.52	* CE 5 is refined into CE [40,41,42,43,44,45] 
938.59/291.52	* CE 6 is refined into CE [46,47] 
938.59/291.52	
938.59/291.52	
938.59/291.52	### Cost equations --> "Loop" of start/3 
938.59/291.52	* CEs [43] --> Loop 27 
938.59/291.52	* CEs [37,42,47] --> Loop 28 
938.59/291.52	* CEs [32,35,38,39,44,45] --> Loop 29 
938.59/291.52	* CEs [33,34,36,40,41,46] --> Loop 30 
938.59/291.52	
938.59/291.52	### Ranking functions of CR start(V1,V,V5) 
938.59/291.52	
938.59/291.52	#### Partial ranking functions of CR start(V1,V,V5) 
938.59/291.52	
938.59/291.52	
938.59/291.52	Computing Bounds
938.59/291.52	=====================================
938.59/291.52	
938.59/291.52	#### Cost of chains of p(V1,Out):
938.59/291.52	* Chain [15]: 1
938.59/291.52	  with precondition: [V1=0,Out=0] 
938.59/291.52	
938.59/291.52	* Chain [14]: 1
938.59/291.52	  with precondition: [V1=Out+1,V1>=1] 
938.59/291.52	
938.59/291.52	
938.59/291.52	#### Cost of chains of gr(V1,V,Out):
938.59/291.52	* Chain [[18],17]: 1*it(18)+1
938.59/291.52	  Such that:it(18) =< V1
938.59/291.52	
938.59/291.52	  with precondition: [Out=0,V1>=1,V>=V1] 
938.59/291.52	
938.59/291.52	* Chain [[18],16]: 1*it(18)+1
938.59/291.52	  Such that:it(18) =< V
938.59/291.52	
938.59/291.52	  with precondition: [Out=1,V>=1,V1>=V+1] 
938.59/291.52	
938.59/291.52	* Chain [17]: 1
938.59/291.52	  with precondition: [V1=0,Out=0,V>=0] 
938.59/291.52	
938.59/291.52	* Chain [16]: 1
938.59/291.52	  with precondition: [V=0,Out=1,V1>=1] 
938.59/291.52	
938.59/291.52	
938.59/291.52	#### Cost of chains of cond1(V1,V,Out):
938.59/291.52	* Chain [[20],22]: 4*it(20)+1*s(3)+0
938.59/291.52	  Such that:aux(3) =< V1
938.59/291.52	it(20) =< aux(3)
938.59/291.52	s(3) =< it(20)*aux(3)
938.59/291.52	
938.59/291.52	  with precondition: [Out=0,V1>=1,V>=V1] 
938.59/291.52	
938.59/291.52	* Chain [22]: 0
938.59/291.52	  with precondition: [Out=0,V1>=0,V>=0] 
938.59/291.52	
938.59/291.52	* Chain [21,22]: 3
938.59/291.52	  with precondition: [V=0,Out=0,V1>=1] 
938.59/291.52	
938.59/291.52	* Chain [19,[20],22]: 5*it(20)+1*s(3)+3
938.59/291.52	  Such that:aux(4) =< V
938.59/291.52	it(20) =< aux(4)
938.59/291.52	s(3) =< it(20)*aux(4)
938.59/291.52	
938.59/291.52	  with precondition: [Out=0,V>=1,V1>=V+1] 
938.59/291.52	
938.59/291.52	* Chain [19,22]: 1*s(4)+3
938.59/291.52	  Such that:s(4) =< V
938.59/291.52	
938.59/291.52	  with precondition: [Out=0,V>=1,V1>=V+1] 
938.59/291.52	
938.59/291.52	
938.59/291.52	#### Cost of chains of neq(V1,V,Out):
938.59/291.52	* Chain [[26],25]: 1*it(26)+1
938.59/291.52	  Such that:it(26) =< V1
938.59/291.52	
938.59/291.52	  with precondition: [Out=0,V1=V,V1>=1] 
938.59/291.52	
938.59/291.52	* Chain [[26],24]: 1*it(26)+1
938.59/291.52	  Such that:it(26) =< V1
938.59/291.52	
938.59/291.52	  with precondition: [Out=1,V1>=1,V>=V1+1] 
938.59/291.52	
938.59/291.52	* Chain [[26],23]: 1*it(26)+1
938.59/291.52	  Such that:it(26) =< V
938.59/291.52	
938.59/291.52	  with precondition: [Out=1,V>=1,V1>=V+1] 
938.59/291.52	
938.59/291.52	* Chain [25]: 1
938.59/291.52	  with precondition: [V1=0,V=0,Out=0] 
938.59/291.52	
938.59/291.52	* Chain [24]: 1
938.59/291.52	  with precondition: [V1=0,Out=1,V>=1] 
938.59/291.52	
938.59/291.52	* Chain [23]: 1
938.59/291.52	  with precondition: [V=0,Out=1,V1>=1] 
938.59/291.52	
938.59/291.52	
938.59/291.52	#### Cost of chains of start(V1,V,V5):
938.59/291.52	* Chain [30]: 12*s(14)+2*s(17)+4*s(21)+1*s(22)+5
938.59/291.52	  Such that:s(18) =< V
938.59/291.52	aux(6) =< V5
938.59/291.52	s(14) =< aux(6)
938.59/291.52	s(21) =< s(18)
938.59/291.52	s(22) =< s(21)*s(18)
938.59/291.52	s(17) =< s(14)*aux(6)
938.59/291.52	
938.59/291.52	  with precondition: [V1=0] 
938.59/291.52	
938.59/291.52	* Chain [29]: 10*s(26)+2*s(28)+8*s(32)+6*s(33)+1*s(34)+1*s(35)+4
938.59/291.52	  Such that:aux(7) =< V5
938.59/291.52	aux(8) =< V1
938.59/291.52	aux(9) =< V
938.59/291.52	s(33) =< aux(8)
938.59/291.52	s(32) =< aux(9)
938.59/291.52	s(34) =< s(33)*aux(8)
938.59/291.52	s(35) =< s(32)*aux(9)
938.59/291.52	s(26) =< aux(7)
938.59/291.52	s(28) =< s(26)*aux(7)
938.59/291.52	
938.59/291.52	  with precondition: [V1>=0,V>=0] 
938.59/291.52	
938.59/291.52	* Chain [28]: 1
938.59/291.52	  with precondition: [V1>=1] 
938.59/291.52	
938.59/291.52	* Chain [27]: 1*s(40)+1
938.59/291.52	  Such that:s(40) =< V
938.59/291.52	
938.59/291.52	  with precondition: [V1=V,V1>=1] 
938.59/291.52	
938.59/291.52	
938.59/291.52	Closed-form bounds of start(V1,V,V5): 
938.59/291.52	-------------------------------------
938.59/291.52	* Chain [30] with precondition: [V1=0] 
938.59/291.52	    - Upper bound: nat(V)*4+5+nat(V)*nat(V)+nat(V5)*12+nat(V5)*2*nat(V5) 
938.59/291.52	    - Complexity: n^2 
938.59/291.52	* Chain [29] with precondition: [V1>=0,V>=0] 
938.59/291.52	    - Upper bound: 6*V1+4+V1*V1+8*V+V*V+nat(V5)*10+nat(V5)*2*nat(V5) 
938.59/291.52	    - Complexity: n^2 
938.59/291.52	* Chain [28] with precondition: [V1>=1] 
938.59/291.52	    - Upper bound: 1 
938.59/291.52	    - Complexity: constant 
938.59/291.52	* Chain [27] with precondition: [V1=V,V1>=1] 
938.59/291.52	    - Upper bound: V+1 
938.59/291.52	    - Complexity: n 
938.59/291.52	
938.59/291.52	### Maximum cost of start(V1,V,V5): nat(V)*3+3+nat(V)*nat(V)+nat(V5)*10+nat(V5)*2*nat(V5)+max([nat(V5)*2+1,6*V1+V1*V1+nat(V)*4])+nat(V)+1 
938.59/291.52	Asymptotic class: n^2 
938.59/291.52	* Total analysis performed in 247 ms.
938.59/291.52	
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(10)
938.59/291.52	BOUNDS(1, n^2)
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(11) RenamingProof (BOTH BOUNDS(ID, ID))
938.59/291.52	Renamed function symbols to avoid clashes with predefined symbol.
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(12)
938.59/291.52	Obligation:
938.59/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
938.59/291.52	
938.59/291.52	
938.59/291.52	The TRS R consists of the following rules:
938.59/291.52	
938.59/291.52	   cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
938.59/291.52	   cond2(true, x, y) -> cond1(y, y)
938.59/291.52	   cond2(false, x, y) -> cond1(p(x), y)
938.59/291.52	   gr(0', x) -> false
938.59/291.52	   gr(s(x), 0') -> true
938.59/291.52	   gr(s(x), s(y)) -> gr(x, y)
938.59/291.52	   neq(0', 0') -> false
938.59/291.52	   neq(0', s(x)) -> true
938.59/291.52	   neq(s(x), 0') -> true
938.59/291.52	   neq(s(x), s(y)) -> neq(x, y)
938.59/291.52	   p(0') -> 0'
938.59/291.52	   p(s(x)) -> x
938.59/291.52	
938.59/291.52	S is empty.
938.59/291.52	Rewrite Strategy: INNERMOST
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
938.59/291.52	Infered types.
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(14)
938.59/291.52	Obligation:
938.59/291.52	Innermost TRS:
938.59/291.52	Rules:
938.59/291.52	cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
938.59/291.52	cond2(true, x, y) -> cond1(y, y)
938.59/291.52	cond2(false, x, y) -> cond1(p(x), y)
938.59/291.52	gr(0', x) -> false
938.59/291.52	gr(s(x), 0') -> true
938.59/291.52	gr(s(x), s(y)) -> gr(x, y)
938.59/291.52	neq(0', 0') -> false
938.59/291.52	neq(0', s(x)) -> true
938.59/291.52	neq(s(x), 0') -> true
938.59/291.52	neq(s(x), s(y)) -> neq(x, y)
938.59/291.52	p(0') -> 0'
938.59/291.52	p(s(x)) -> x
938.59/291.52	
938.59/291.52	Types:
938.59/291.52	cond1 :: s:0' -> s:0' -> cond1:cond2
938.59/291.52	s :: s:0' -> s:0'
938.59/291.52	cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
938.59/291.52	gr :: s:0' -> s:0' -> true:false
938.59/291.52	true :: true:false
938.59/291.52	false :: true:false
938.59/291.52	p :: s:0' -> s:0'
938.59/291.52	0' :: s:0'
938.59/291.52	neq :: s:0' -> s:0' -> true:false
938.59/291.52	hole_cond1:cond21_0 :: cond1:cond2
938.59/291.52	hole_s:0'2_0 :: s:0'
938.59/291.52	hole_true:false3_0 :: true:false
938.59/291.52	gen_s:0'4_0 :: Nat -> s:0'
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(15) OrderProof (LOWER BOUND(ID))
938.59/291.52	Heuristically decided to analyse the following defined symbols:
938.59/291.52	cond1, gr, neq
938.59/291.52	
938.59/291.52	They will be analysed ascendingly in the following order:
938.59/291.52	gr < cond1
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(16)
938.59/291.52	Obligation:
938.59/291.52	Innermost TRS:
938.59/291.52	Rules:
938.59/291.52	cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
938.59/291.52	cond2(true, x, y) -> cond1(y, y)
938.59/291.52	cond2(false, x, y) -> cond1(p(x), y)
938.59/291.52	gr(0', x) -> false
938.59/291.52	gr(s(x), 0') -> true
938.59/291.52	gr(s(x), s(y)) -> gr(x, y)
938.59/291.52	neq(0', 0') -> false
938.59/291.52	neq(0', s(x)) -> true
938.59/291.52	neq(s(x), 0') -> true
938.59/291.52	neq(s(x), s(y)) -> neq(x, y)
938.59/291.52	p(0') -> 0'
938.59/291.52	p(s(x)) -> x
938.59/291.52	
938.59/291.52	Types:
938.59/291.52	cond1 :: s:0' -> s:0' -> cond1:cond2
938.59/291.52	s :: s:0' -> s:0'
938.59/291.52	cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
938.59/291.52	gr :: s:0' -> s:0' -> true:false
938.59/291.52	true :: true:false
938.59/291.52	false :: true:false
938.59/291.52	p :: s:0' -> s:0'
938.59/291.52	0' :: s:0'
938.59/291.52	neq :: s:0' -> s:0' -> true:false
938.59/291.52	hole_cond1:cond21_0 :: cond1:cond2
938.59/291.52	hole_s:0'2_0 :: s:0'
938.59/291.52	hole_true:false3_0 :: true:false
938.59/291.52	gen_s:0'4_0 :: Nat -> s:0'
938.59/291.52	
938.59/291.52	
938.59/291.52	Generator Equations:
938.59/291.52	gen_s:0'4_0(0) <=> 0'
938.59/291.52	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
938.59/291.52	
938.59/291.52	
938.59/291.52	The following defined symbols remain to be analysed:
938.59/291.52	gr, cond1, neq
938.59/291.52	
938.59/291.52	They will be analysed ascendingly in the following order:
938.59/291.52	gr < cond1
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(17) RewriteLemmaProof (LOWER BOUND(ID))
938.59/291.52	Proved the following rewrite lemma:
938.59/291.52	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
938.59/291.52	
938.59/291.52	Induction Base:
938.59/291.52	gr(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
938.59/291.52	false
938.59/291.52	
938.59/291.52	Induction Step:
938.59/291.52	gr(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
938.59/291.52	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
938.59/291.52	false
938.59/291.52	
938.59/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(18)
938.59/291.52	Complex Obligation (BEST)
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(19)
938.59/291.52	Obligation:
938.59/291.52	Proved the lower bound n^1 for the following obligation:
938.59/291.52	
938.59/291.52	Innermost TRS:
938.59/291.52	Rules:
938.59/291.52	cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
938.59/291.52	cond2(true, x, y) -> cond1(y, y)
938.59/291.52	cond2(false, x, y) -> cond1(p(x), y)
938.59/291.52	gr(0', x) -> false
938.59/291.52	gr(s(x), 0') -> true
938.59/291.52	gr(s(x), s(y)) -> gr(x, y)
938.59/291.52	neq(0', 0') -> false
938.59/291.52	neq(0', s(x)) -> true
938.59/291.52	neq(s(x), 0') -> true
938.59/291.52	neq(s(x), s(y)) -> neq(x, y)
938.59/291.52	p(0') -> 0'
938.59/291.52	p(s(x)) -> x
938.59/291.52	
938.59/291.52	Types:
938.59/291.52	cond1 :: s:0' -> s:0' -> cond1:cond2
938.59/291.52	s :: s:0' -> s:0'
938.59/291.52	cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
938.59/291.52	gr :: s:0' -> s:0' -> true:false
938.59/291.52	true :: true:false
938.59/291.52	false :: true:false
938.59/291.52	p :: s:0' -> s:0'
938.59/291.52	0' :: s:0'
938.59/291.52	neq :: s:0' -> s:0' -> true:false
938.59/291.52	hole_cond1:cond21_0 :: cond1:cond2
938.59/291.52	hole_s:0'2_0 :: s:0'
938.59/291.52	hole_true:false3_0 :: true:false
938.59/291.52	gen_s:0'4_0 :: Nat -> s:0'
938.59/291.52	
938.59/291.52	
938.59/291.52	Generator Equations:
938.59/291.52	gen_s:0'4_0(0) <=> 0'
938.59/291.52	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
938.59/291.52	
938.59/291.52	
938.59/291.52	The following defined symbols remain to be analysed:
938.59/291.52	gr, cond1, neq
938.59/291.52	
938.59/291.52	They will be analysed ascendingly in the following order:
938.59/291.52	gr < cond1
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(20) LowerBoundPropagationProof (FINISHED)
938.59/291.52	Propagated lower bound.
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(21)
938.59/291.52	BOUNDS(n^1, INF)
938.59/291.52	
938.59/291.52	----------------------------------------
938.59/291.52	
938.59/291.52	(22)
938.59/291.52	Obligation:
938.59/291.52	Innermost TRS:
938.59/291.52	Rules:
938.59/291.52	cond1(s(x), y) -> cond2(gr(s(x), y), s(x), y)
938.59/291.52	cond2(true, x, y) -> cond1(y, y)
938.59/291.52	cond2(false, x, y) -> cond1(p(x), y)
938.59/291.52	gr(0', x) -> false
938.59/291.52	gr(s(x), 0') -> true
938.59/291.52	gr(s(x), s(y)) -> gr(x, y)
938.59/291.52	neq(0', 0') -> false
938.59/291.52	neq(0', s(x)) -> true
938.59/291.53	neq(s(x), 0') -> true
938.59/291.53	neq(s(x), s(y)) -> neq(x, y)
938.59/291.53	p(0') -> 0'
938.59/291.53	p(s(x)) -> x
938.59/291.53	
938.59/291.53	Types:
938.59/291.53	cond1 :: s:0' -> s:0' -> cond1:cond2
938.59/291.53	s :: s:0' -> s:0'
938.59/291.53	cond2 :: true:false -> s:0' -> s:0' -> cond1:cond2
938.59/291.53	gr :: s:0' -> s:0' -> true:false
938.59/291.53	true :: true:false
938.59/291.53	false :: true:false
938.59/291.53	p :: s:0' -> s:0'
938.59/291.53	0' :: s:0'
938.59/291.53	neq :: s:0' -> s:0' -> true:false
938.59/291.53	hole_cond1:cond21_0 :: cond1:cond2
938.59/291.53	hole_s:0'2_0 :: s:0'
938.59/291.53	hole_true:false3_0 :: true:false
938.59/291.53	gen_s:0'4_0 :: Nat -> s:0'
938.59/291.53	
938.59/291.53	
938.59/291.53	Lemmas:
938.59/291.53	gr(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
938.59/291.53	
938.59/291.53	
938.59/291.53	Generator Equations:
938.59/291.53	gen_s:0'4_0(0) <=> 0'
938.59/291.53	gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))
938.59/291.53	
938.59/291.53	
938.59/291.53	The following defined symbols remain to be analysed:
938.59/291.53	cond1, neq
938.59/291.53	----------------------------------------
938.59/291.53	
938.59/291.53	(23) RewriteLemmaProof (LOWER BOUND(ID))
938.59/291.53	Proved the following rewrite lemma:
938.59/291.53	neq(gen_s:0'4_0(n16104_0), gen_s:0'4_0(n16104_0)) -> false, rt in Omega(1 + n16104_0)
938.59/291.53	
938.59/291.53	Induction Base:
938.59/291.53	neq(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
938.59/291.53	false
938.59/291.53	
938.59/291.53	Induction Step:
938.59/291.53	neq(gen_s:0'4_0(+(n16104_0, 1)), gen_s:0'4_0(+(n16104_0, 1))) ->_R^Omega(1)
938.59/291.53	neq(gen_s:0'4_0(n16104_0), gen_s:0'4_0(n16104_0)) ->_IH
938.59/291.53	false
938.59/291.53	
938.59/291.53	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
938.59/291.53	----------------------------------------
938.59/291.53	
938.59/291.53	(24)
938.59/291.53	BOUNDS(1, INF)
938.66/291.57	EOF