14.26/5.55	MAYBE
14.26/5.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
14.26/5.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
14.26/5.56	
14.26/5.56	
14.26/5.56	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, INF).
14.26/5.56	
14.26/5.56	(0) CpxIntTrs
14.26/5.56	(1) Loat Proof [FINISHED, 3935 ms]
14.26/5.56	(2) BOUNDS(1, INF)
14.26/5.56	
14.26/5.56	
14.26/5.56	----------------------------------------
14.26/5.56	
14.26/5.56	(0)
14.26/5.56	Obligation:
14.26/5.56	Complexity Int TRS consisting of the following rules:
14.26/5.56	f0(A, B, C, D, E, F, G) -> Com_1(f10(1, 1, H, 0, 2, 1, G)) :|: H >= 0
14.26/5.56	f10(A, B, C, D, E, F, G) -> Com_1(f21(A, B - 1, C, D, E, F, 0)) :|: F >= 1 && E >= F && B >= 1 && 0 >= C
14.26/5.56	f10(A, B, C, D, E, F, G) -> Com_1(f21(A + 1, A + 1, H, D, E, F, I)) :|: I >= 0 && 1 >= I && H >= 0 && F >= 1 && E >= F && 0 >= B && 0 >= C
14.26/5.56	f10(A, B, C, D, E, F, G) -> Com_1(f21(A, B, C - 1, D, E, F, H)) :|: H >= 0 && 1 >= H && F >= 1 && C >= 1 && E >= F
14.26/5.56	f21(A, B, C, D, E, F, G) -> Com_1(f10(A, B, C, D, E, F - 1, G)) :|: E + 1 >= A && 0 >= G
14.26/5.56	f21(A, B, C, D, E, F, G) -> Com_1(f10(A, B, C, D, E, F + 1, G)) :|: E + 1 >= A && G >= 1
14.26/5.56	f10(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, E, F, G)) :|: 0 >= F && E >= F
14.26/5.56	f10(A, B, C, D, E, F, G) -> Com_1(f32(A, B, C, D, E, F, G)) :|: F >= 1 + E
14.26/5.56	
14.26/5.56	The start-symbols are:[f0_7]
14.26/5.56	
14.26/5.56	
14.26/5.56	----------------------------------------
14.26/5.56	
14.26/5.56	(1) Loat Proof (FINISHED)
14.26/5.56	
14.26/5.56	
14.26/5.56	### Pre-processing the ITS problem ###
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Initial linear ITS problem
14.26/5.56	
14.26/5.56	   Start location: f0
14.26/5.56	
14.26/5.56	      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1
14.26/5.56	
14.26/5.56	      1: f10 -> f21 : B'=-1+B, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C ], cost: 1
14.26/5.56	
14.26/5.56	      2: f10 -> f21 : A'=1+A, B'=1+A, C'=free_2, G'=free_1, [ free_1>=0 && 1>=free_1 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C ], cost: 1
14.26/5.56	
14.26/5.56	      3: f10 -> f21 : C'=-1+C, G'=free_3, [ free_3>=0 && 1>=free_3 && F>=1 && C>=1 && E>=F ], cost: 1
14.26/5.56	
14.26/5.56	      6: f10 -> f32 : [ 0>=F && E>=F ], cost: 1
14.26/5.56	
14.26/5.56	      7: f10 -> f32 : [ F>=1+E ], cost: 1
14.26/5.56	
14.26/5.56	      4: f21 -> f10 : F'=-1+F, [ 1+E>=A && 0>=G ], cost: 1
14.26/5.56	
14.26/5.56	      5: f21 -> f10 : F'=1+F, [ 1+E>=A && G>=1 ], cost: 1
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Removed unreachable and leaf rules:
14.26/5.56	
14.26/5.56	   Start location: f0
14.26/5.56	
14.26/5.56	      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1
14.26/5.56	
14.26/5.56	      1: f10 -> f21 : B'=-1+B, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C ], cost: 1
14.26/5.56	
14.26/5.56	      2: f10 -> f21 : A'=1+A, B'=1+A, C'=free_2, G'=free_1, [ free_1>=0 && 1>=free_1 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C ], cost: 1
14.26/5.56	
14.26/5.56	      3: f10 -> f21 : C'=-1+C, G'=free_3, [ free_3>=0 && 1>=free_3 && F>=1 && C>=1 && E>=F ], cost: 1
14.26/5.56	
14.26/5.56	      4: f21 -> f10 : F'=-1+F, [ 1+E>=A && 0>=G ], cost: 1
14.26/5.56	
14.26/5.56	      5: f21 -> f10 : F'=1+F, [ 1+E>=A && G>=1 ], cost: 1
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	### Simplification by acceleration and chaining ###
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Eliminated locations (on tree-shaped paths):
14.26/5.56	
14.26/5.56	   Start location: f0
14.26/5.56	
14.26/5.56	      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1
14.26/5.56	
14.26/5.56	      8: f10 -> f10 : B'=-1+B, F'=-1+F, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A ], cost: 2
14.26/5.56	
14.26/5.56	      9: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=-1+F, G'=free_1, [ free_1>=0 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A && 0>=free_1 ], cost: 2
14.26/5.56	
14.26/5.56	     10: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=1+F, G'=free_1, [ 1>=free_1 && free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A && free_1>=1 ], cost: 2
14.26/5.56	
14.26/5.56	     11: f10 -> f10 : C'=-1+C, F'=-1+F, G'=free_3, [ free_3>=0 && F>=1 && C>=1 && E>=F && 1+E>=A && 0>=free_3 ], cost: 2
14.26/5.56	
14.26/5.56	     12: f10 -> f10 : C'=-1+C, F'=1+F, G'=free_3, [ 1>=free_3 && F>=1 && C>=1 && E>=F && 1+E>=A && free_3>=1 ], cost: 2
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Accelerating simple loops of location 1.
14.26/5.56	
14.26/5.56	   Simplified some of the simple loops (and removed duplicate rules).
14.26/5.56	
14.26/5.56	   Accelerating the following rules:
14.26/5.56	
14.26/5.56	      8: f10 -> f10 : B'=-1+B, F'=-1+F, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A ], cost: 2
14.26/5.56	
14.26/5.56	      9: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=-1+F, G'=0, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2
14.26/5.56	
14.26/5.56	     10: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=1+F, G'=1, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2
14.26/5.56	
14.26/5.56	     11: f10 -> f10 : C'=-1+C, F'=-1+F, G'=0, [ F>=1 && C>=1 && E>=F && 1+E>=A ], cost: 2
14.26/5.56	
14.26/5.56	     12: f10 -> f10 : C'=-1+C, F'=1+F, G'=1, [ F>=1 && C>=1 && E>=F && 1+E>=A ], cost: 2
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	   Accelerated rule 8 with metering function F (after adding B>=F), yielding the new rule 13.
14.26/5.56	
14.26/5.56	   Accelerated rule 8 with metering function B (after adding B<=F), yielding the new rule 14.
14.26/5.56	
14.26/5.56	   Found no metering function for rule 9.
14.26/5.56	
14.26/5.56	   Found no metering function for rule 10.
14.26/5.56	
14.26/5.56	   Accelerated rule 11 with metering function F (after adding C>=F), yielding the new rule 15.
14.26/5.56	
14.26/5.56	   Accelerated rule 11 with metering function C (after adding C<=F), yielding the new rule 16.
14.26/5.56	
14.26/5.56	   Accelerated rule 12 with backward acceleration, yielding the new rule 17.
14.26/5.56	
14.26/5.56	   Accelerated rule 12 with backward acceleration, yielding the new rule 18.
14.26/5.56	
14.26/5.56	      During metering: Instantiating temporary variables by {free_2==1}
14.26/5.56	
14.26/5.56	      During metering: Instantiating temporary variables by {free_2==1}
14.26/5.56	
14.26/5.56	   Removing the simple loops: 8 11 12.
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Accelerated all simple loops using metering functions (where possible):
14.26/5.56	
14.26/5.56	   Start location: f0
14.26/5.56	
14.26/5.56	      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1
14.26/5.56	
14.26/5.56	      9: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=-1+F, G'=0, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2
14.26/5.56	
14.26/5.56	     10: f10 -> f10 : A'=1+A, B'=1+A, C'=free_2, F'=1+F, G'=1, [ free_2>=0 && F>=1 && E>=F && 0>=B && 0>=C && 1+E>=1+A ], cost: 2
14.26/5.56	
14.26/5.56	     13: f10 -> f10 : B'=-F+B, F'=0, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A && B>=F ], cost: 2*F
14.26/5.56	
14.26/5.56	     14: f10 -> f10 : B'=0, F'=F-B, G'=0, [ F>=1 && E>=F && B>=1 && 0>=C && 1+E>=A && B<=F ], cost: 2*B
14.26/5.56	
14.26/5.56	     15: f10 -> f10 : C'=-F+C, F'=0, G'=0, [ F>=1 && C>=1 && E>=F && 1+E>=A && C>=F ], cost: 2*F
14.26/5.56	
14.26/5.56	     16: f10 -> f10 : C'=0, F'=F-C, G'=0, [ F>=1 && C>=1 && E>=F && 1+E>=A && C<=F ], cost: 2*C
14.26/5.56	
14.26/5.56	     17: f10 -> f10 : C'=0, F'=F+C, G'=1, [ F>=1 && C>=1 && E>=F && 1+E>=A && -1+F+C>=1 && E>=-1+F+C ], cost: 2*C
14.26/5.56	
14.26/5.56	     18: f10 -> f10 : C'=-1+F+C-E, F'=1+E, G'=1, [ F>=1 && C>=1 && E>=F && 1+E>=A && E>=1 && F+C-E>=1 ], cost: 2-2*F+2*E
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Chained accelerated rules (with incoming rules):
14.26/5.56	
14.26/5.56	   Start location: f0
14.26/5.56	
14.26/5.56	      0: f0 -> f10 : A'=1, B'=1, C'=free, D'=0, E'=2, F'=1, [ free>=0 ], cost: 1
14.26/5.56	
14.26/5.56	     19: f0 -> f10 : A'=1, B'=0, C'=0, D'=0, E'=2, F'=0, G'=0, [], cost: 3
14.26/5.56	
14.26/5.56	     20: f0 -> f10 : A'=1, B'=0, C'=0, D'=0, E'=2, F'=0, G'=0, [], cost: 3
14.26/5.56	
14.26/5.56	     21: f0 -> f10 : A'=1, B'=1, C'=-1+free, D'=0, E'=2, F'=0, G'=0, [ free>=1 ], cost: 3
14.26/5.56	
14.26/5.56	     22: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=0, G'=0, [], cost: 3
14.26/5.56	
14.26/5.56	     23: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=1+free, G'=1, [ free>=1 && 2>=free ], cost: 1+2*free
14.26/5.56	
14.26/5.56	     24: f0 -> f10 : A'=1, B'=1, C'=-2+free, D'=0, E'=2, F'=3, G'=1, [ -1+free>=1 ], cost: 5
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Removed unreachable locations (and leaf rules with constant cost):
14.26/5.56	
14.26/5.56	   Start location: f0
14.26/5.56	
14.26/5.56	     23: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=1+free, G'=1, [ free>=1 && 2>=free ], cost: 1+2*free
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	### Computing asymptotic complexity ###
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Fully simplified ITS problem
14.26/5.56	
14.26/5.56	   Start location: f0
14.26/5.56	
14.26/5.56	     23: f0 -> f10 : A'=1, B'=1, C'=0, D'=0, E'=2, F'=1+free, G'=1, [ free>=1 && 2>=free ], cost: 1+2*free
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Computing asymptotic complexity for rule 23
14.26/5.56	
14.26/5.56	   Could not solve the limit problem.
14.26/5.56	
14.26/5.56	   Resulting cost 0 has complexity: Unknown
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	Obtained the following overall complexity (w.r.t. the length of the input n):
14.26/5.56	
14.26/5.56	   Complexity:  Unknown
14.26/5.56	
14.26/5.56	   Cpx degree:  ?
14.26/5.56	
14.26/5.56	   Solved cost: 0
14.26/5.56	
14.26/5.56	   Rule cost:   0
14.26/5.56	
14.26/5.56	   Rule guard:  []
14.26/5.56	
14.26/5.56	
14.26/5.56	
14.26/5.56	WORST_CASE(Omega(0),?)
14.26/5.56	
14.26/5.56	
14.26/5.56	----------------------------------------
14.26/5.56	
14.26/5.56	(2)
14.26/5.56	BOUNDS(1, INF)
14.26/5.58	EOF