/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF).

(0) CpxIntTrs
(1) Loat Proof [FINISHED, 639 ms]
(2) BOUNDS(INF, INF)


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(0)
Obligation:
Complexity Int TRS consisting of the following rules:
f0(A, B) -> Com_1(f3(0, B)) :|: TRUE
f3(A, B) -> Com_1(f3(C + 1, B)) :|: A >= 5 && A <= 5
f3(A, B) -> Com_1(f3(A + 1, A)) :|: 9 >= A && 4 >= A
f3(A, B) -> Com_1(f3(A + 1, A)) :|: 9 >= A && A >= 6
f3(A, B) -> Com_1(f12(A, B)) :|: A >= 10

The start-symbols are:[f0_2]


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(1) Loat Proof (FINISHED)


### Pre-processing the ITS problem ###



Initial linear ITS problem

   Start location: f0

      0: f0 -> f3 : A'=0, [], cost: 1

      1: f3 -> f3 : A'=1+free, [ A==5 ], cost: 1

      2: f3 -> f3 : A'=1+A, B'=A, [ 9>=A && 4>=A ], cost: 1

      3: f3 -> f3 : A'=1+A, B'=A, [ 9>=A && A>=6 ], cost: 1

      4: f3 -> f12 : [ A>=10 ], cost: 1



Checking for constant complexity:

   The following rule is satisfiable with cost >= 1, yielding constant complexity:

      0: f0 -> f3 : A'=0, [], cost: 1



Removed unreachable and leaf rules:

   Start location: f0

      0: f0 -> f3 : A'=0, [], cost: 1

      1: f3 -> f3 : A'=1+free, [ A==5 ], cost: 1

      2: f3 -> f3 : A'=1+A, B'=A, [ 9>=A && 4>=A ], cost: 1

      3: f3 -> f3 : A'=1+A, B'=A, [ 9>=A && A>=6 ], cost: 1



Simplified all rules, resulting in:

   Start location: f0

      0: f0 -> f3 : A'=0, [], cost: 1

      1: f3 -> f3 : A'=1+free, [ A==5 ], cost: 1

      2: f3 -> f3 : A'=1+A, B'=A, [ 4>=A ], cost: 1

      3: f3 -> f3 : A'=1+A, B'=A, [ 9>=A && A>=6 ], cost: 1



### Simplification by acceleration and chaining ###



Accelerating simple loops of location 1.

   Accelerating the following rules:

      1: f3 -> f3 : A'=1+free, [ A==5 ], cost: 1

      2: f3 -> f3 : A'=1+A, B'=A, [ 4>=A ], cost: 1

      3: f3 -> f3 : A'=1+A, B'=A, [ 9>=A && A>=6 ], cost: 1



   Accelerated rule 1 with NONTERM (after strengthening guard), yielding the new rule 5.

   Accelerated rule 2 with metering function 5-A, yielding the new rule 6.

   Accelerated rule 3 with metering function 10-A, yielding the new rule 7.

   Nested simple loops 1 (outer loop) and 6 (inner loop) with NONTERM, resulting in the new rules: 8, 9.

   Nested simple loops 1 (outer loop) and 7 (inner loop) with metering function meter (where 5*meter==5-A), resulting in the new rules: 10.

   Removing the simple loops: 1 2 3.



Accelerated all simple loops using metering functions (where possible):

   Start location: f0

      0: f0 -> f3 : A'=0, [], cost: 1

      5: f3 -> [3] : [ A==5 && 1+free==5 ], cost: NONTERM

      6: f3 -> f3 : A'=5, B'=4, [ 4>=A ], cost: 5-A

      7: f3 -> f3 : A'=10, B'=9, [ 9>=A && A>=6 ], cost: 10-A

      8: f3 -> [3] : [ A==5 && 4>=1+free ], cost: NONTERM

      9: f3 -> [3] : A'=5, B'=4, [ 4>=A && 4>=1+free ], cost: NONTERM

     10: f3 -> f3 : A'=10, B'=9, [ A==5 && 9>=1+free && 1+free>=6 && 5*meter==5-A && meter>=1 ], cost: 10*meter-meter*free



Chained accelerated rules (with incoming rules):

   Start location: f0

      0: f0 -> f3 : A'=0, [], cost: 1

     11: f0 -> f3 : A'=5, B'=4, [], cost: 6

     12: f0 -> [3] : A'=5, B'=4, [], cost: NONTERM



Removed unreachable locations (and leaf rules with constant cost):

   Start location: f0

     12: f0 -> [3] : A'=5, B'=4, [], cost: NONTERM



### Computing asymptotic complexity ###



Fully simplified ITS problem

   Start location: f0

     12: f0 -> [3] : A'=5, B'=4, [], cost: NONTERM



Computing asymptotic complexity for rule 12

   Guard is satisfiable, yielding nontermination

   Resulting cost NONTERM has complexity: Nonterm



Found new complexity Nonterm.



Obtained the following overall complexity (w.r.t. the length of the input n):

   Complexity:  Nonterm

   Cpx degree:  Nonterm

   Solved cost: NONTERM

   Rule cost:   NONTERM

   Rule guard:  []



NO


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(2)
BOUNDS(INF, INF)