WORST_CASE(Omega(n^1),?)
proof of input_YjRsTvlTbG.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 14 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 358 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), 296 ms]
        (18) BOUNDS(1, INF)


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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:

   from(X) -> cons(X, from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, XS)

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:
   from(z0) -> cons(z0, from(s(z0)))
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, z2)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   AFTER(0, z0) -> c1
   AFTER(s(z0), cons(z1, z2)) -> c2(AFTER(z0, z2))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   AFTER(0, z0) -> c1
   AFTER(s(z0), cons(z1, z2)) -> c2(AFTER(z0, z2))
K tuples:none
Defined Rule Symbols:   from_1, after_2

Defined Pair Symbols:   FROM_1, AFTER_2

Compound Symbols:   c_1, c1, 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:

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

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

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

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:

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

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

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

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

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
AFTER :: s:0' -> cons -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons -> cons
c2 :: c1:c2 -> c1:c2
from :: s:0' -> cons
after :: s:0' -> cons -> cons
hole_c1_3 :: c
hole_s:0'2_3 :: s:0'
hole_c1:c23_3 :: c1:c2
hole_cons4_3 :: cons
gen_c5_3 :: Nat -> c
gen_s:0'6_3 :: Nat -> s:0'
gen_c1:c27_3 :: Nat -> c1:c2
gen_cons8_3 :: Nat -> cons

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

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
AFTER :: s:0' -> cons -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons -> cons
c2 :: c1:c2 -> c1:c2
from :: s:0' -> cons
after :: s:0' -> cons -> cons
hole_c1_3 :: c
hole_s:0'2_3 :: s:0'
hole_c1:c23_3 :: c1:c2
hole_cons4_3 :: cons
gen_c5_3 :: Nat -> c
gen_s:0'6_3 :: Nat -> s:0'
gen_c1:c27_3 :: Nat -> c1:c2
gen_cons8_3 :: Nat -> cons


Generator Equations:
gen_c5_3(0) <=> hole_c1_3
gen_c5_3(+(x, 1)) <=> c(gen_c5_3(x))
gen_s:0'6_3(0) <=> 0'
gen_s:0'6_3(+(x, 1)) <=> s(gen_s:0'6_3(x))
gen_c1:c27_3(0) <=> c1
gen_c1:c27_3(+(x, 1)) <=> c2(gen_c1:c27_3(x))
gen_cons8_3(0) <=> hole_cons4_3
gen_cons8_3(+(x, 1)) <=> cons(0', gen_cons8_3(x))


The following defined symbols remain to be analysed:
FROM, AFTER, from, after
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
AFTER(gen_s:0'6_3(n123_3), gen_cons8_3(n123_3)) -> gen_c1:c27_3(n123_3), rt in Omega(1 + n123_3)

Induction Base:
AFTER(gen_s:0'6_3(0), gen_cons8_3(0)) ->_R^Omega(1)
c1

Induction Step:
AFTER(gen_s:0'6_3(+(n123_3, 1)), gen_cons8_3(+(n123_3, 1))) ->_R^Omega(1)
c2(AFTER(gen_s:0'6_3(n123_3), gen_cons8_3(n123_3))) ->_IH
c2(gen_c1:c27_3(c124_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:
FROM(z0) -> c(FROM(s(z0)))
AFTER(0', z0) -> c1
AFTER(s(z0), cons(z1, z2)) -> c2(AFTER(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
after(0', z0) -> z0
after(s(z0), cons(z1, z2)) -> after(z0, z2)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
AFTER :: s:0' -> cons -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons -> cons
c2 :: c1:c2 -> c1:c2
from :: s:0' -> cons
after :: s:0' -> cons -> cons
hole_c1_3 :: c
hole_s:0'2_3 :: s:0'
hole_c1:c23_3 :: c1:c2
hole_cons4_3 :: cons
gen_c5_3 :: Nat -> c
gen_s:0'6_3 :: Nat -> s:0'
gen_c1:c27_3 :: Nat -> c1:c2
gen_cons8_3 :: Nat -> cons


Generator Equations:
gen_c5_3(0) <=> hole_c1_3
gen_c5_3(+(x, 1)) <=> c(gen_c5_3(x))
gen_s:0'6_3(0) <=> 0'
gen_s:0'6_3(+(x, 1)) <=> s(gen_s:0'6_3(x))
gen_c1:c27_3(0) <=> c1
gen_c1:c27_3(+(x, 1)) <=> c2(gen_c1:c27_3(x))
gen_cons8_3(0) <=> hole_cons4_3
gen_cons8_3(+(x, 1)) <=> cons(0', gen_cons8_3(x))


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

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
AFTER :: s:0' -> cons -> c1:c2
0' :: s:0'
c1 :: c1:c2
cons :: s:0' -> cons -> cons
c2 :: c1:c2 -> c1:c2
from :: s:0' -> cons
after :: s:0' -> cons -> cons
hole_c1_3 :: c
hole_s:0'2_3 :: s:0'
hole_c1:c23_3 :: c1:c2
hole_cons4_3 :: cons
gen_c5_3 :: Nat -> c
gen_s:0'6_3 :: Nat -> s:0'
gen_c1:c27_3 :: Nat -> c1:c2
gen_cons8_3 :: Nat -> cons


Lemmas:
AFTER(gen_s:0'6_3(n123_3), gen_cons8_3(n123_3)) -> gen_c1:c27_3(n123_3), rt in Omega(1 + n123_3)


Generator Equations:
gen_c5_3(0) <=> hole_c1_3
gen_c5_3(+(x, 1)) <=> c(gen_c5_3(x))
gen_s:0'6_3(0) <=> 0'
gen_s:0'6_3(+(x, 1)) <=> s(gen_s:0'6_3(x))
gen_c1:c27_3(0) <=> c1
gen_c1:c27_3(+(x, 1)) <=> c2(gen_c1:c27_3(x))
gen_cons8_3(0) <=> hole_cons4_3
gen_cons8_3(+(x, 1)) <=> cons(0', gen_cons8_3(x))


The following defined symbols remain to be analysed:
from, after
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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
after(gen_s:0'6_3(n618_3), gen_cons8_3(n618_3)) -> gen_cons8_3(0), rt in Omega(0)

Induction Base:
after(gen_s:0'6_3(0), gen_cons8_3(0)) ->_R^Omega(0)
gen_cons8_3(0)

Induction Step:
after(gen_s:0'6_3(+(n618_3, 1)), gen_cons8_3(+(n618_3, 1))) ->_R^Omega(0)
after(gen_s:0'6_3(n618_3), gen_cons8_3(n618_3)) ->_IH
gen_cons8_3(0)

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)