6.41/2.98	WORST_CASE(NON_POLY, ?)
6.41/2.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
6.41/2.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.41/2.99	
6.41/2.99	
6.41/2.99	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).
6.41/2.99	
6.41/2.99	(0) CpxIntTrs
6.41/2.99	(1) Loat Proof [FINISHED, 1187 ms]
6.41/2.99	(2) BOUNDS(INF, INF)
6.41/2.99	
6.41/2.99	
6.41/2.99	----------------------------------------
6.41/2.99	
6.41/2.99	(0)
6.41/2.99	Obligation:
6.41/2.99	Complexity Int TRS consisting of the following rules:
6.41/2.99	f2(A, B, C, D) -> Com_1(f1(A, B, C, D)) :|: TRUE
6.41/2.99	f1(A, B, C, D) -> Com_1(f1(1 + A, B, E, D)) :|: E >= 1 && B >= 1 + A
6.41/2.99	f1(A, B, C, D) -> Com_1(f1(1 + A, B, E, D)) :|: 0 >= E + 1 && B >= 1 + A
6.41/2.99	f1(A, B, C, D) -> Com_1(f1(A, B, 0, D)) :|: B >= 1 + A
6.41/2.99	f1(A, B, C, D) -> Com_1(f1(1 + A, A, E, D)) :|: E >= 1 && B >= F && A >= B && A <= B
6.41/2.99	f1(A, B, C, D) -> Com_1(f1(1 + A, A, E, D)) :|: 0 >= E + 1 && B >= F && A >= B && A <= B
6.41/2.99	f1(A, B, C, D) -> Com_1(f1(A, A, 0, D)) :|: B >= E && A >= B && A <= B
6.41/2.99	f1(A, B, C, D) -> Com_1(f300(A, B, C, E)) :|: A >= B && A >= B + 1
6.41/2.99	f1(A, B, C, D) -> Com_1(f300(A, B, C, E)) :|: A >= B && B >= A + 1
6.41/2.99	
6.41/2.99	The start-symbols are:[f2_4]
6.41/2.99	
6.41/2.99	
6.41/2.99	----------------------------------------
6.41/2.99	
6.41/2.99	(1) Loat Proof (FINISHED)
6.41/2.99	
6.41/2.99	
6.41/2.99	### Pre-processing the ITS problem ###
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Initial linear ITS problem
6.41/2.99	
6.41/2.99	   Start location: f2
6.41/2.99	
6.41/2.99	      0: f2 -> f1 : [], cost: 1
6.41/2.99	
6.41/2.99	      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && A>=free_2 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && A>=free_4 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      6: f1 -> f1 : A'=B, C'=0, [ A>=free_6 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      7: f1 -> f300 : D'=free_7, [ B>=A && B>=1+A ], cost: 1
6.41/2.99	
6.41/2.99	      8: f1 -> f300 : D'=free_8, [ B>=A && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Removed unreachable and leaf rules:
6.41/2.99	
6.41/2.99	   Start location: f2
6.41/2.99	
6.41/2.99	      0: f2 -> f1 : [], cost: 1
6.41/2.99	
6.41/2.99	      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && A>=free_2 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && A>=free_4 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      6: f1 -> f1 : A'=B, C'=0, [ A>=free_6 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Simplified all rules, resulting in:
6.41/2.99	
6.41/2.99	   Start location: f2
6.41/2.99	
6.41/2.99	      0: f2 -> f1 : [], cost: 1
6.41/2.99	
6.41/2.99	      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      6: f1 -> f1 : A'=B, C'=0, [ B==A ], cost: 1
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	### Simplification by acceleration and chaining ###
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Accelerating simple loops of location 1.
6.41/2.99	
6.41/2.99	   Accelerating the following rules:
6.41/2.99	
6.41/2.99	      1: f1 -> f1 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      2: f1 -> f1 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      3: f1 -> f1 : C'=0, [ A>=1+B ], cost: 1
6.41/2.99	
6.41/2.99	      4: f1 -> f1 : A'=B, B'=1+B, C'=free_3, [ free_3>=1 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      5: f1 -> f1 : A'=B, B'=1+B, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 1
6.41/2.99	
6.41/2.99	      6: f1 -> f1 : A'=B, C'=0, [ B==A ], cost: 1
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	   Accelerated rule 1 with metering function A-B, yielding the new rule 9.
6.41/2.99	
6.41/2.99	   Accelerated rule 2 with metering function A-B, yielding the new rule 10.
6.41/2.99	
6.41/2.99	   Accelerated rule 3 with NONTERM, yielding the new rule 11.
6.41/2.99	
6.41/2.99	   Accelerated rule 4 with metering function 1+A-B, yielding the new rule 12.
6.41/2.99	
6.41/2.99	   Accelerated rule 5 with metering function 1+A-B, yielding the new rule 13.
6.41/2.99	
6.41/2.99	   Accelerated rule 6 with NONTERM, yielding the new rule 14.
6.41/2.99	
6.41/2.99	   Removing the simple loops: 1 2 3 4 5 6.
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Accelerated all simple loops using metering functions (where possible):
6.41/2.99	
6.41/2.99	   Start location: f2
6.41/2.99	
6.41/2.99	      0: f2 -> f1 : [], cost: 1
6.41/2.99	
6.41/2.99	      9: f1 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: A-B
6.41/2.99	
6.41/2.99	     10: f1 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: A-B
6.41/2.99	
6.41/2.99	     11: f1 -> [3] : [ A>=1+B ], cost: INF
6.41/2.99	
6.41/2.99	     12: f1 -> f1 : A'=A, B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     13: f1 -> f1 : A'=A, B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     14: f1 -> [3] : [ B==A ], cost: INF
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Chained accelerated rules (with incoming rules):
6.41/2.99	
6.41/2.99	   Start location: f2
6.41/2.99	
6.41/2.99	      0: f2 -> f1 : [], cost: 1
6.41/2.99	
6.41/2.99	     15: f2 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     16: f2 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     17: f2 -> [3] : [ A>=1+B ], cost: INF
6.41/2.99	
6.41/2.99	     18: f2 -> f1 : B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 2+A-B
6.41/2.99	
6.41/2.99	     19: f2 -> f1 : B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 2+A-B
6.41/2.99	
6.41/2.99	     20: f2 -> [3] : [ B==A ], cost: INF
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Removed unreachable locations (and leaf rules with constant cost):
6.41/2.99	
6.41/2.99	   Start location: f2
6.41/2.99	
6.41/2.99	     15: f2 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     16: f2 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     17: f2 -> [3] : [ A>=1+B ], cost: INF
6.41/2.99	
6.41/2.99	     18: f2 -> f1 : B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 2+A-B
6.41/2.99	
6.41/2.99	     19: f2 -> f1 : B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 2+A-B
6.41/2.99	
6.41/2.99	     20: f2 -> [3] : [ B==A ], cost: INF
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	### Computing asymptotic complexity ###
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Fully simplified ITS problem
6.41/2.99	
6.41/2.99	   Start location: f2
6.41/2.99	
6.41/2.99	     15: f2 -> f1 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     16: f2 -> f1 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
6.41/2.99	
6.41/2.99	     17: f2 -> [3] : [ A>=1+B ], cost: INF
6.41/2.99	
6.41/2.99	     18: f2 -> f1 : B'=1+A, C'=free_3, [ free_3>=1 && B==A ], cost: 2+A-B
6.41/2.99	
6.41/2.99	     19: f2 -> f1 : B'=1+A, C'=free_5, [ 0>=1+free_5 && B==A ], cost: 2+A-B
6.41/2.99	
6.41/2.99	     20: f2 -> [3] : [ B==A ], cost: INF
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Computing asymptotic complexity for rule 15
6.41/2.99	
6.41/2.99	   Solved the limit problem by the following transformations:
6.41/2.99	
6.41/2.99	      Created initial limit problem:
6.41/2.99	
6.41/2.99	      free (+/+!), 1+A-B (+), A-B (+/+!) [not solved]
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	      removing all constraints (solved by SMT)
6.41/2.99	
6.41/2.99	      resulting limit problem: [solved]
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	      applying transformation rule (C) using substitution {free==1,A==0,B==-n}
6.41/2.99	
6.41/2.99	      resulting limit problem:
6.41/2.99	
6.41/2.99	      [solved]
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	   Solution:
6.41/2.99	
6.41/2.99	      free / 1
6.41/2.99	
6.41/2.99	      A / 0
6.41/2.99	
6.41/2.99	      B / -n
6.41/2.99	
6.41/2.99	   Resulting cost 1+n has complexity: Poly(n^1)
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Found new complexity Poly(n^1).
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Computing asymptotic complexity for rule 17
6.41/2.99	
6.41/2.99	   Resulting cost INF has complexity: Nonterm
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Found new complexity Nonterm.
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	Obtained the following overall complexity (w.r.t. the length of the input n):
6.41/2.99	
6.41/2.99	   Complexity:  Nonterm
6.41/2.99	
6.41/2.99	   Cpx degree:  Nonterm
6.41/2.99	
6.41/2.99	   Solved cost: INF
6.41/2.99	
6.41/2.99	   Rule cost:   INF
6.41/2.99	
6.41/2.99	   Rule guard:  [ A>=1+B ]
6.41/2.99	
6.41/2.99	
6.41/2.99	
6.41/2.99	NO
6.41/2.99	
6.41/2.99	
6.41/2.99	----------------------------------------
6.41/2.99	
6.41/2.99	(2)
6.41/2.99	BOUNDS(INF, INF)
6.54/3.01	EOF