4.37/2.10	WORST_CASE(NON_POLY, ?)
4.37/2.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
4.37/2.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.37/2.11	
4.37/2.11	
4.37/2.11	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).
4.37/2.11	
4.37/2.11	(0) CpxIntTrs
4.37/2.11	(1) Loat Proof [FINISHED, 429 ms]
4.37/2.11	(2) BOUNDS(INF, INF)
4.37/2.11	
4.37/2.11	
4.37/2.11	----------------------------------------
4.37/2.11	
4.37/2.11	(0)
4.37/2.11	Obligation:
4.37/2.11	Complexity Int TRS consisting of the following rules:
4.37/2.11	f2(A, B, C, D) -> Com_1(f300(A, B, C, D)) :|: TRUE
4.37/2.11	f300(A, B, C, D) -> Com_1(f300(1 + A, B, E, D)) :|: E >= 1 && B >= 1 + A
4.37/2.11	f300(A, B, C, D) -> Com_1(f300(1 + A, B, E, D)) :|: 0 >= E + 1 && B >= 1 + A
4.37/2.11	f300(A, B, C, D) -> Com_1(f300(A, B, 0, D)) :|: B >= 1 + A
4.37/2.11	f300(A, B, C, D) -> Com_1(f1(A, B, C, E)) :|: A >= B
4.37/2.11	
4.37/2.11	The start-symbols are:[f2_4]
4.37/2.11	
4.37/2.11	
4.37/2.11	----------------------------------------
4.37/2.11	
4.37/2.11	(1) Loat Proof (FINISHED)
4.37/2.11	
4.37/2.11	
4.37/2.11	### Pre-processing the ITS problem ###
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Initial linear ITS problem
4.37/2.11	
4.37/2.11	   Start location: f2
4.37/2.11	
4.37/2.11	      0: f2 -> f300 : [], cost: 1
4.37/2.11	
4.37/2.11	      1: f300 -> f300 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	      2: f300 -> f300 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	      3: f300 -> f300 : C'=0, [ A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	      4: f300 -> f1 : D'=free_2, [ B>=A ], cost: 1
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Removed unreachable and leaf rules:
4.37/2.11	
4.37/2.11	   Start location: f2
4.37/2.11	
4.37/2.11	      0: f2 -> f300 : [], cost: 1
4.37/2.11	
4.37/2.11	      1: f300 -> f300 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	      2: f300 -> f300 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	      3: f300 -> f300 : C'=0, [ A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	### Simplification by acceleration and chaining ###
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Accelerating simple loops of location 1.
4.37/2.11	
4.37/2.11	   Accelerating the following rules:
4.37/2.11	
4.37/2.11	      1: f300 -> f300 : B'=1+B, C'=free, [ free>=1 && A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	      2: f300 -> f300 : B'=1+B, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	      3: f300 -> f300 : C'=0, [ A>=1+B ], cost: 1
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	   Accelerated rule 1 with metering function A-B, yielding the new rule 5.
4.37/2.11	
4.37/2.11	   Accelerated rule 2 with metering function A-B, yielding the new rule 6.
4.37/2.11	
4.37/2.11	   Accelerated rule 3 with NONTERM, yielding the new rule 7.
4.37/2.11	
4.37/2.11	   Removing the simple loops: 1 2 3.
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Accelerated all simple loops using metering functions (where possible):
4.37/2.11	
4.37/2.11	   Start location: f2
4.37/2.11	
4.37/2.11	      0: f2 -> f300 : [], cost: 1
4.37/2.11	
4.37/2.11	      5: f300 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: A-B
4.37/2.11	
4.37/2.11	      6: f300 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: A-B
4.37/2.11	
4.37/2.11	      7: f300 -> [3] : [ A>=1+B ], cost: INF
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Chained accelerated rules (with incoming rules):
4.37/2.11	
4.37/2.11	   Start location: f2
4.37/2.11	
4.37/2.11	      0: f2 -> f300 : [], cost: 1
4.37/2.11	
4.37/2.11	      8: f2 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
4.37/2.11	
4.37/2.11	      9: f2 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
4.37/2.11	
4.37/2.11	     10: f2 -> [3] : [ A>=1+B ], cost: INF
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Removed unreachable locations (and leaf rules with constant cost):
4.37/2.11	
4.37/2.11	   Start location: f2
4.37/2.11	
4.37/2.11	      8: f2 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
4.37/2.11	
4.37/2.11	      9: f2 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
4.37/2.11	
4.37/2.11	     10: f2 -> [3] : [ A>=1+B ], cost: INF
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	### Computing asymptotic complexity ###
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Fully simplified ITS problem
4.37/2.11	
4.37/2.11	   Start location: f2
4.37/2.11	
4.37/2.11	      8: f2 -> f300 : B'=A, C'=free, [ free>=1 && A>=1+B ], cost: 1+A-B
4.37/2.11	
4.37/2.11	      9: f2 -> f300 : B'=A, C'=free_1, [ 0>=1+free_1 && A>=1+B ], cost: 1+A-B
4.37/2.11	
4.37/2.11	     10: f2 -> [3] : [ A>=1+B ], cost: INF
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Computing asymptotic complexity for rule 8
4.37/2.11	
4.37/2.11	   Solved the limit problem by the following transformations:
4.37/2.11	
4.37/2.11	      Created initial limit problem:
4.37/2.11	
4.37/2.11	      free (+/+!), 1+A-B (+), A-B (+/+!) [not solved]
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	      removing all constraints (solved by SMT)
4.37/2.11	
4.37/2.11	      resulting limit problem: [solved]
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	      applying transformation rule (C) using substitution {free==1,A==0,B==-n}
4.37/2.11	
4.37/2.11	      resulting limit problem:
4.37/2.11	
4.37/2.11	      [solved]
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	   Solution:
4.37/2.11	
4.37/2.11	      free / 1
4.37/2.11	
4.37/2.11	      A / 0
4.37/2.11	
4.37/2.11	      B / -n
4.37/2.11	
4.37/2.11	   Resulting cost 1+n has complexity: Poly(n^1)
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Found new complexity Poly(n^1).
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Computing asymptotic complexity for rule 10
4.37/2.11	
4.37/2.11	   Resulting cost INF has complexity: Nonterm
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Found new complexity Nonterm.
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	Obtained the following overall complexity (w.r.t. the length of the input n):
4.37/2.11	
4.37/2.11	   Complexity:  Nonterm
4.37/2.11	
4.37/2.11	   Cpx degree:  Nonterm
4.37/2.11	
4.37/2.11	   Solved cost: INF
4.37/2.11	
4.37/2.11	   Rule cost:   INF
4.37/2.11	
4.37/2.11	   Rule guard:  [ A>=1+B ]
4.37/2.11	
4.37/2.11	
4.37/2.11	
4.37/2.11	NO
4.37/2.11	
4.37/2.11	
4.37/2.11	----------------------------------------
4.37/2.11	
4.37/2.11	(2)
4.37/2.11	BOUNDS(INF, INF)
4.44/2.15	EOF