/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) 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), 175 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 492 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 429 ms]
        (18) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

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

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

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


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

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


The TRS R consists of the following rules:

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

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

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

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

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

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

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

Rewrite Strategy: FULL
----------------------------------------

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

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

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

Rewrite Strategy: FULL
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
TRS:
Rules:
not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))
encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_not(x_1)) -> not(encArg(x_1))
encode_not(x_1) -> not(encArg(x_1))
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))

Types:
not :: or:and:cons_not -> or:and:cons_not
or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encArg :: or:and:cons_not -> or:and:cons_not
cons_not :: or:and:cons_not -> or:and:cons_not
encode_not :: or:and:cons_not -> or:and:cons_not
encode_or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encode_and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
hole_or:and:cons_not1_0 :: or:and:cons_not
gen_or:and:cons_not2_0 :: Nat -> or:and:cons_not

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
not, encArg

They will be analysed ascendingly in the following order:
not < encArg

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

(10)
Obligation:
TRS:
Rules:
not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))
encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_not(x_1)) -> not(encArg(x_1))
encode_not(x_1) -> not(encArg(x_1))
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))

Types:
not :: or:and:cons_not -> or:and:cons_not
or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encArg :: or:and:cons_not -> or:and:cons_not
cons_not :: or:and:cons_not -> or:and:cons_not
encode_not :: or:and:cons_not -> or:and:cons_not
encode_or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encode_and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
hole_or:and:cons_not1_0 :: or:and:cons_not
gen_or:and:cons_not2_0 :: Nat -> or:and:cons_not


Generator Equations:
gen_or:and:cons_not2_0(0) <=> hole_or:and:cons_not1_0
gen_or:and:cons_not2_0(+(x, 1)) <=> or(hole_or:and:cons_not1_0, gen_or:and:cons_not2_0(x))


The following defined symbols remain to be analysed:
not, encArg

They will be analysed ascendingly in the following order:
not < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
not(gen_or:and:cons_not2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
not(gen_or:and:cons_not2_0(0))

Induction Step:
not(gen_or:and:cons_not2_0(+(n4_0, 1))) ->_R^Omega(1)
and(not(not(not(hole_or:and:cons_not1_0))), not(not(not(gen_or:and:cons_not2_0(n4_0))))) ->_R^Omega(1)
and(not(hole_or:and:cons_not1_0), not(not(not(gen_or:and:cons_not2_0(n4_0))))) ->_IH
and(not(hole_or:and:cons_not1_0), not(not(*3_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))
encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_not(x_1)) -> not(encArg(x_1))
encode_not(x_1) -> not(encArg(x_1))
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))

Types:
not :: or:and:cons_not -> or:and:cons_not
or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encArg :: or:and:cons_not -> or:and:cons_not
cons_not :: or:and:cons_not -> or:and:cons_not
encode_not :: or:and:cons_not -> or:and:cons_not
encode_or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encode_and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
hole_or:and:cons_not1_0 :: or:and:cons_not
gen_or:and:cons_not2_0 :: Nat -> or:and:cons_not


Generator Equations:
gen_or:and:cons_not2_0(0) <=> hole_or:and:cons_not1_0
gen_or:and:cons_not2_0(+(x, 1)) <=> or(hole_or:and:cons_not1_0, gen_or:and:cons_not2_0(x))


The following defined symbols remain to be analysed:
not, encArg

They will be analysed ascendingly in the following order:
not < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
TRS:
Rules:
not(not(x)) -> x
not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))
encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_not(x_1)) -> not(encArg(x_1))
encode_not(x_1) -> not(encArg(x_1))
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))

Types:
not :: or:and:cons_not -> or:and:cons_not
or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encArg :: or:and:cons_not -> or:and:cons_not
cons_not :: or:and:cons_not -> or:and:cons_not
encode_not :: or:and:cons_not -> or:and:cons_not
encode_or :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
encode_and :: or:and:cons_not -> or:and:cons_not -> or:and:cons_not
hole_or:and:cons_not1_0 :: or:and:cons_not
gen_or:and:cons_not2_0 :: Nat -> or:and:cons_not


Lemmas:
not(gen_or:and:cons_not2_0(n4_0)) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_or:and:cons_not2_0(0) <=> hole_or:and:cons_not1_0
gen_or:and:cons_not2_0(+(x, 1)) <=> or(hole_or:and:cons_not1_0, gen_or:and:cons_not2_0(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_or:and:cons_not2_0(+(1, n9906_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_or:and:cons_not2_0(+(1, 0)))

Induction Step:
encArg(gen_or:and:cons_not2_0(+(1, +(n9906_0, 1)))) ->_R^Omega(0)
or(encArg(hole_or:and:cons_not1_0), encArg(gen_or:and:cons_not2_0(+(1, n9906_0)))) ->_IH
or(encArg(hole_or:and:cons_not1_0), *3_0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(18)
BOUNDS(1, INF)