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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 237 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), 6 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 16.5 s]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 8665 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 37 ms]
        (20) BOUNDS(1, INF)


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

(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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_sum1(x_1) -> sum1(encArg(x_1))


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

(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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_sum1(x_1) -> sum1(encArg(x_1))

Rewrite Strategy: INNERMOST
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(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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_sum1(x_1) -> sum1(encArg(x_1))

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:

   sum(0') -> 0'
   sum(s(x)) -> +'(sum(x), s(x))
   sum1(0') -> 0'
   sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
   encode_sum1(x_1) -> sum1(encArg(x_1))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_sum1(x_1) -> sum1(encArg(x_1))

Types:
sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
0' :: 0':s:+':cons_sum:cons_sum1
s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
+' :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encArg :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_0 :: 0':s:+':cons_sum:cons_sum1
encode_s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_+ :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
hole_0':s:+':cons_sum:cons_sum11_3 :: 0':s:+':cons_sum:cons_sum1
gen_0':s:+':cons_sum:cons_sum12_3 :: Nat -> 0':s:+':cons_sum:cons_sum1

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sum, sum1, encArg

They will be analysed ascendingly in the following order:
sum < encArg
sum1 < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_sum1(x_1) -> sum1(encArg(x_1))

Types:
sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
0' :: 0':s:+':cons_sum:cons_sum1
s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
+' :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encArg :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_0 :: 0':s:+':cons_sum:cons_sum1
encode_s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_+ :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
hole_0':s:+':cons_sum:cons_sum11_3 :: 0':s:+':cons_sum:cons_sum1
gen_0':s:+':cons_sum:cons_sum12_3 :: Nat -> 0':s:+':cons_sum:cons_sum1


Generator Equations:
gen_0':s:+':cons_sum:cons_sum12_3(0) <=> 0'
gen_0':s:+':cons_sum:cons_sum12_3(+(x, 1)) <=> s(gen_0':s:+':cons_sum:cons_sum12_3(x))


The following defined symbols remain to be analysed:
sum, sum1, encArg

They will be analysed ascendingly in the following order:
sum < encArg
sum1 < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)

Induction Base:
sum(gen_0':s:+':cons_sum:cons_sum12_3(+(1, 0)))

Induction Step:
sum(gen_0':s:+':cons_sum:cons_sum12_3(+(1, +(n4_3, 1)))) ->_R^Omega(1)
+'(sum(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n4_3))), s(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n4_3)))) ->_IH
+'(*3_3, s(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n4_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:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_sum1(x_1) -> sum1(encArg(x_1))

Types:
sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
0' :: 0':s:+':cons_sum:cons_sum1
s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
+' :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encArg :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_0 :: 0':s:+':cons_sum:cons_sum1
encode_s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_+ :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
hole_0':s:+':cons_sum:cons_sum11_3 :: 0':s:+':cons_sum:cons_sum1
gen_0':s:+':cons_sum:cons_sum12_3 :: Nat -> 0':s:+':cons_sum:cons_sum1


Generator Equations:
gen_0':s:+':cons_sum:cons_sum12_3(0) <=> 0'
gen_0':s:+':cons_sum:cons_sum12_3(+(x, 1)) <=> s(gen_0':s:+':cons_sum:cons_sum12_3(x))


The following defined symbols remain to be analysed:
sum, sum1, encArg

They will be analysed ascendingly in the following order:
sum < encArg
sum1 < encArg

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_sum1(x_1) -> sum1(encArg(x_1))

Types:
sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
0' :: 0':s:+':cons_sum:cons_sum1
s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
+' :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encArg :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_0 :: 0':s:+':cons_sum:cons_sum1
encode_s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_+ :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
hole_0':s:+':cons_sum:cons_sum11_3 :: 0':s:+':cons_sum:cons_sum1
gen_0':s:+':cons_sum:cons_sum12_3 :: Nat -> 0':s:+':cons_sum:cons_sum1


Lemmas:
sum(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)


Generator Equations:
gen_0':s:+':cons_sum:cons_sum12_3(0) <=> 0'
gen_0':s:+':cons_sum:cons_sum12_3(+(x, 1)) <=> s(gen_0':s:+':cons_sum:cons_sum12_3(x))


The following defined symbols remain to be analysed:
sum1, encArg

They will be analysed ascendingly in the following order:
sum1 < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum1(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n156190_3))) -> *3_3, rt in Omega(n156190_3)

Induction Base:
sum1(gen_0':s:+':cons_sum:cons_sum12_3(+(1, 0)))

Induction Step:
sum1(gen_0':s:+':cons_sum:cons_sum12_3(+(1, +(n156190_3, 1)))) ->_R^Omega(1)
s(+'(sum1(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n156190_3))), +'(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n156190_3)), gen_0':s:+':cons_sum:cons_sum12_3(+(1, n156190_3))))) ->_IH
s(+'(*3_3, +'(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n156190_3)), gen_0':s:+':cons_sum:cons_sum12_3(+(1, n156190_3)))))

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

(18)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_sum1(x_1)) -> sum1(encArg(x_1))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_sum1(x_1) -> sum1(encArg(x_1))

Types:
sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
0' :: 0':s:+':cons_sum:cons_sum1
s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
+' :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encArg :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
cons_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_0 :: 0':s:+':cons_sum:cons_sum1
encode_s :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_+ :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
encode_sum1 :: 0':s:+':cons_sum:cons_sum1 -> 0':s:+':cons_sum:cons_sum1
hole_0':s:+':cons_sum:cons_sum11_3 :: 0':s:+':cons_sum:cons_sum1
gen_0':s:+':cons_sum:cons_sum12_3 :: Nat -> 0':s:+':cons_sum:cons_sum1


Lemmas:
sum(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)
sum1(gen_0':s:+':cons_sum:cons_sum12_3(+(1, n156190_3))) -> *3_3, rt in Omega(n156190_3)


Generator Equations:
gen_0':s:+':cons_sum:cons_sum12_3(0) <=> 0'
gen_0':s:+':cons_sum:cons_sum12_3(+(x, 1)) <=> s(gen_0':s:+':cons_sum:cons_sum12_3(x))


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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:+':cons_sum:cons_sum12_3(n220417_3)) -> gen_0':s:+':cons_sum:cons_sum12_3(n220417_3), rt in Omega(0)

Induction Base:
encArg(gen_0':s:+':cons_sum:cons_sum12_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:+':cons_sum:cons_sum12_3(+(n220417_3, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:+':cons_sum:cons_sum12_3(n220417_3))) ->_IH
s(gen_0':s:+':cons_sum:cons_sum12_3(c220418_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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(20)
BOUNDS(1, INF)