WORST_CASE(Omega(n^1),?)
proof of input_eg5G5KJweg.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 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), 341 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), 154 ms]
        (18) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FIRST(0, z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))
S tuples:
   FIRST(0, z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   first_2, from_1

Defined Pair Symbols:   FIRST_2, FROM_1

Compound Symbols:   c, c1_1, c2_1


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(3) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to 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:

   FIRST(0, z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))

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

   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
   from(z0) -> cons(z0, from(s(z0)))

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

   FIRST(0', z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))

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

   first(0', z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
   from(z0) -> cons(z0, from(s(z0)))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(8)
Obligation:
Innermost TRS:
Rules:
FIRST(0', z0) -> c
FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
from(z0) -> cons(z0, from(s(z0)))

Types:
FIRST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
from :: 0':s -> cons:nil
hole_c:c11_3 :: c:c1
hole_0':s2_3 :: 0':s
hole_cons:nil3_3 :: cons:nil
hole_c24_3 :: c2
gen_c:c15_3 :: Nat -> c:c1
gen_0':s6_3 :: Nat -> 0':s
gen_cons:nil7_3 :: Nat -> cons:nil
gen_c28_3 :: Nat -> c2

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FIRST, FROM, first, from
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(10)
Obligation:
Innermost TRS:
Rules:
FIRST(0', z0) -> c
FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
from(z0) -> cons(z0, from(s(z0)))

Types:
FIRST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
from :: 0':s -> cons:nil
hole_c:c11_3 :: c:c1
hole_0':s2_3 :: 0':s
hole_cons:nil3_3 :: cons:nil
hole_c24_3 :: c2
gen_c:c15_3 :: Nat -> c:c1
gen_0':s6_3 :: Nat -> 0':s
gen_cons:nil7_3 :: Nat -> cons:nil
gen_c28_3 :: Nat -> c2


Generator Equations:
gen_c:c15_3(0) <=> c
gen_c:c15_3(+(x, 1)) <=> c1(gen_c:c15_3(x))
gen_0':s6_3(0) <=> 0'
gen_0':s6_3(+(x, 1)) <=> s(gen_0':s6_3(x))
gen_cons:nil7_3(0) <=> nil
gen_cons:nil7_3(+(x, 1)) <=> cons(0', gen_cons:nil7_3(x))
gen_c28_3(0) <=> hole_c24_3
gen_c28_3(+(x, 1)) <=> c2(gen_c28_3(x))


The following defined symbols remain to be analysed:
FIRST, FROM, first, from
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FIRST(gen_0':s6_3(n10_3), gen_cons:nil7_3(n10_3)) -> gen_c:c15_3(n10_3), rt in Omega(1 + n10_3)

Induction Base:
FIRST(gen_0':s6_3(0), gen_cons:nil7_3(0)) ->_R^Omega(1)
c

Induction Step:
FIRST(gen_0':s6_3(+(n10_3, 1)), gen_cons:nil7_3(+(n10_3, 1))) ->_R^Omega(1)
c1(FIRST(gen_0':s6_3(n10_3), gen_cons:nil7_3(n10_3))) ->_IH
c1(gen_c:c15_3(c11_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:
FIRST(0', z0) -> c
FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
from(z0) -> cons(z0, from(s(z0)))

Types:
FIRST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
from :: 0':s -> cons:nil
hole_c:c11_3 :: c:c1
hole_0':s2_3 :: 0':s
hole_cons:nil3_3 :: cons:nil
hole_c24_3 :: c2
gen_c:c15_3 :: Nat -> c:c1
gen_0':s6_3 :: Nat -> 0':s
gen_cons:nil7_3 :: Nat -> cons:nil
gen_c28_3 :: Nat -> c2


Generator Equations:
gen_c:c15_3(0) <=> c
gen_c:c15_3(+(x, 1)) <=> c1(gen_c:c15_3(x))
gen_0':s6_3(0) <=> 0'
gen_0':s6_3(+(x, 1)) <=> s(gen_0':s6_3(x))
gen_cons:nil7_3(0) <=> nil
gen_cons:nil7_3(+(x, 1)) <=> cons(0', gen_cons:nil7_3(x))
gen_c28_3(0) <=> hole_c24_3
gen_c28_3(+(x, 1)) <=> c2(gen_c28_3(x))


The following defined symbols remain to be analysed:
FIRST, FROM, first, from
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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:
FIRST(0', z0) -> c
FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
from(z0) -> cons(z0, from(s(z0)))

Types:
FIRST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
from :: 0':s -> cons:nil
hole_c:c11_3 :: c:c1
hole_0':s2_3 :: 0':s
hole_cons:nil3_3 :: cons:nil
hole_c24_3 :: c2
gen_c:c15_3 :: Nat -> c:c1
gen_0':s6_3 :: Nat -> 0':s
gen_cons:nil7_3 :: Nat -> cons:nil
gen_c28_3 :: Nat -> c2


Lemmas:
FIRST(gen_0':s6_3(n10_3), gen_cons:nil7_3(n10_3)) -> gen_c:c15_3(n10_3), rt in Omega(1 + n10_3)


Generator Equations:
gen_c:c15_3(0) <=> c
gen_c:c15_3(+(x, 1)) <=> c1(gen_c:c15_3(x))
gen_0':s6_3(0) <=> 0'
gen_0':s6_3(+(x, 1)) <=> s(gen_0':s6_3(x))
gen_cons:nil7_3(0) <=> nil
gen_cons:nil7_3(+(x, 1)) <=> cons(0', gen_cons:nil7_3(x))
gen_c28_3(0) <=> hole_c24_3
gen_c28_3(+(x, 1)) <=> c2(gen_c28_3(x))


The following defined symbols remain to be analysed:
FROM, first, from
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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
first(gen_0':s6_3(n413_3), gen_cons:nil7_3(n413_3)) -> gen_cons:nil7_3(n413_3), rt in Omega(0)

Induction Base:
first(gen_0':s6_3(0), gen_cons:nil7_3(0)) ->_R^Omega(0)
nil

Induction Step:
first(gen_0':s6_3(+(n413_3, 1)), gen_cons:nil7_3(+(n413_3, 1))) ->_R^Omega(0)
cons(0', first(gen_0':s6_3(n413_3), gen_cons:nil7_3(n413_3))) ->_IH
cons(0', gen_cons:nil7_3(c414_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)
Obligation:
Innermost TRS:
Rules:
FIRST(0', z0) -> c
FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
from(z0) -> cons(z0, from(s(z0)))

Types:
FIRST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
first :: 0':s -> cons:nil -> cons:nil
nil :: cons:nil
from :: 0':s -> cons:nil
hole_c:c11_3 :: c:c1
hole_0':s2_3 :: 0':s
hole_cons:nil3_3 :: cons:nil
hole_c24_3 :: c2
gen_c:c15_3 :: Nat -> c:c1
gen_0':s6_3 :: Nat -> 0':s
gen_cons:nil7_3 :: Nat -> cons:nil
gen_c28_3 :: Nat -> c2


Lemmas:
FIRST(gen_0':s6_3(n10_3), gen_cons:nil7_3(n10_3)) -> gen_c:c15_3(n10_3), rt in Omega(1 + n10_3)
first(gen_0':s6_3(n413_3), gen_cons:nil7_3(n413_3)) -> gen_cons:nil7_3(n413_3), rt in Omega(0)


Generator Equations:
gen_c:c15_3(0) <=> c
gen_c:c15_3(+(x, 1)) <=> c1(gen_c:c15_3(x))
gen_0':s6_3(0) <=> 0'
gen_0':s6_3(+(x, 1)) <=> s(gen_0':s6_3(x))
gen_cons:nil7_3(0) <=> nil
gen_cons:nil7_3(+(x, 1)) <=> cons(0', gen_cons:nil7_3(x))
gen_c28_3(0) <=> hole_c24_3
gen_c28_3(+(x, 1)) <=> c2(gen_c28_3(x))


The following defined symbols remain to be analysed:
from