/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 183 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

The (relative) TRS S consists of the following rules:

   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

The (relative) TRS S consists of the following rules:

   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

The (relative) TRS S consists of the following rules:

   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

and(x, or(y, z)) ->^+ or(and(x, y), and(x, z))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [y / or(y, z)].

The result substitution is [ ].




----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

The (relative) TRS S consists of the following rules:

   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

The (relative) TRS S consists of the following rules:

   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST