WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 170 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), 3451 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), 0 ms]
        (18) BOUNDS(1, INF)


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(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:

   p(s(x)) -> x
   fac(0) -> s(0)
   fac(s(x)) -> times(s(x), fac(p(s(x))))

S is empty.
Rewrite Strategy: INNERMOST
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(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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_0 -> 0
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))


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(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:

   p(s(x)) -> x
   fac(0) -> s(0)
   fac(s(x)) -> times(s(x), fac(p(s(x))))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_0 -> 0
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

   p(s(x)) -> x
   fac(0) -> s(0)
   fac(s(x)) -> times(s(x), fac(p(s(x))))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_0 -> 0
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))

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

(6)
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:

   p(s(x)) -> x
   fac(0') -> s(0')
   fac(s(x)) -> times(s(x), fac(p(s(x))))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0') -> 0'
   encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_fac(x_1)) -> fac(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fac(x_1) -> fac(encArg(x_1))
   encode_0 -> 0'
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
fac(0') -> s(0')
fac(s(x)) -> times(s(x), fac(p(s(x))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_0 -> 0'
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))

Types:
p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
0' :: s:0':times:cons_p:cons_fac
times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encArg :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_0 :: s:0':times:cons_p:cons_fac
encode_times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
hole_s:0':times:cons_p:cons_fac1_3 :: s:0':times:cons_p:cons_fac
gen_s:0':times:cons_p:cons_fac2_3 :: Nat -> s:0':times:cons_p:cons_fac

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
fac, encArg

They will be analysed ascendingly in the following order:
fac < encArg

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(10)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
fac(0') -> s(0')
fac(s(x)) -> times(s(x), fac(p(s(x))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_0 -> 0'
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))

Types:
p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
0' :: s:0':times:cons_p:cons_fac
times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encArg :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_0 :: s:0':times:cons_p:cons_fac
encode_times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
hole_s:0':times:cons_p:cons_fac1_3 :: s:0':times:cons_p:cons_fac
gen_s:0':times:cons_p:cons_fac2_3 :: Nat -> s:0':times:cons_p:cons_fac


Generator Equations:
gen_s:0':times:cons_p:cons_fac2_3(0) <=> 0'
gen_s:0':times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_s:0':times:cons_p:cons_fac2_3(x))


The following defined symbols remain to be analysed:
fac, encArg

They will be analysed ascendingly in the following order:
fac < encArg

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fac(gen_s:0':times:cons_p:cons_fac2_3(n4_3)) -> *3_3, rt in Omega(n4_3)

Induction Base:
fac(gen_s:0':times:cons_p:cons_fac2_3(0))

Induction Step:
fac(gen_s:0':times:cons_p:cons_fac2_3(+(n4_3, 1))) ->_R^Omega(1)
times(s(gen_s:0':times:cons_p:cons_fac2_3(n4_3)), fac(p(s(gen_s:0':times:cons_p:cons_fac2_3(n4_3))))) ->_R^Omega(1)
times(s(gen_s:0':times:cons_p:cons_fac2_3(n4_3)), fac(gen_s:0':times:cons_p:cons_fac2_3(n4_3))) ->_IH
times(s(gen_s:0':times:cons_p:cons_fac2_3(n4_3)), *3_3)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(12)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
p(s(x)) -> x
fac(0') -> s(0')
fac(s(x)) -> times(s(x), fac(p(s(x))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_0 -> 0'
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))

Types:
p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
0' :: s:0':times:cons_p:cons_fac
times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encArg :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_0 :: s:0':times:cons_p:cons_fac
encode_times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
hole_s:0':times:cons_p:cons_fac1_3 :: s:0':times:cons_p:cons_fac
gen_s:0':times:cons_p:cons_fac2_3 :: Nat -> s:0':times:cons_p:cons_fac


Generator Equations:
gen_s:0':times:cons_p:cons_fac2_3(0) <=> 0'
gen_s:0':times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_s:0':times:cons_p:cons_fac2_3(x))


The following defined symbols remain to be analysed:
fac, encArg

They will be analysed ascendingly in the following order:
fac < encArg

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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
fac(0') -> s(0')
fac(s(x)) -> times(s(x), fac(p(s(x))))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encArg(cons_fac(x_1)) -> fac(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_fac(x_1) -> fac(encArg(x_1))
encode_0 -> 0'
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))

Types:
p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
0' :: s:0':times:cons_p:cons_fac
times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encArg :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
cons_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_p :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_s :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_fac :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
encode_0 :: s:0':times:cons_p:cons_fac
encode_times :: s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac -> s:0':times:cons_p:cons_fac
hole_s:0':times:cons_p:cons_fac1_3 :: s:0':times:cons_p:cons_fac
gen_s:0':times:cons_p:cons_fac2_3 :: Nat -> s:0':times:cons_p:cons_fac


Lemmas:
fac(gen_s:0':times:cons_p:cons_fac2_3(n4_3)) -> *3_3, rt in Omega(n4_3)


Generator Equations:
gen_s:0':times:cons_p:cons_fac2_3(0) <=> 0'
gen_s:0':times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_s:0':times:cons_p:cons_fac2_3(x))


The following defined symbols remain to be analysed:
encArg
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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:0':times:cons_p:cons_fac2_3(n21996_3)) -> gen_s:0':times:cons_p:cons_fac2_3(n21996_3), rt in Omega(0)

Induction Base:
encArg(gen_s:0':times:cons_p:cons_fac2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_s:0':times:cons_p:cons_fac2_3(+(n21996_3, 1))) ->_R^Omega(0)
s(encArg(gen_s:0':times:cons_p:cons_fac2_3(n21996_3))) ->_IH
s(gen_s:0':times:cons_p:cons_fac2_3(c21997_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
BOUNDS(1, INF)