WORST_CASE(Omega(n^1),?)
proof of input_DFx30qa8vV.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 522 ms]
(12) typed CpxTrs
(13) RewriteLemmaProof [LOWER BOUND(ID), 723 ms]
(14) BEST
    (15) proven lower bound
        (16) LowerBoundPropagationProof [FINISHED, 0 ms]
        (17) BOUNDS(n^1, INF)
    (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 464 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 530 ms]
        (22) 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)))
   length(nil) -> 0
   length(cons(X, Y)) -> s(length1(Y))
   length1(X) -> length(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:
   from(z0) -> cons(z0, from(s(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length1(z1))
   length1(z0) -> length(z0)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   LENGTH(nil) -> c1
   LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
   LENGTH1(z0) -> c3(LENGTH(z0))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   LENGTH(nil) -> c1
   LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
   LENGTH1(z0) -> c3(LENGTH(z0))
K tuples:none
Defined Rule Symbols:   from_1, length_1, length1_1

Defined Pair Symbols:   FROM_1, LENGTH_1, LENGTH1_1

Compound Symbols:   c_1, c1, c2_1, c3_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.
----------------------------------------

(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)))
   LENGTH(nil) -> c1
   LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
   LENGTH1(z0) -> c3(LENGTH(z0))

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

   from(z0) -> cons(z0, from(s(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length1(z1))
   length1(z0) -> length(z0)

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:

   FROM(z0) -> c(FROM(s(z0)))
   LENGTH(nil) -> c1
   LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
   LENGTH1(z0) -> c3(LENGTH(z0))

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

   from(z0) -> cons(z0, from(s(z0)))
   length(nil) -> 0'
   length(cons(z0, z1)) -> s(length1(z1))
   length1(z0) -> length(z0)

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

(8)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
LENGTH(nil) -> c1
LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
LENGTH1(z0) -> c3(LENGTH(z0))
from(z0) -> cons(z0, from(s(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length1(z1))
length1(z0) -> length(z0)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
LENGTH :: nil:cons -> c1:c2
nil :: nil:cons
c1 :: c1:c2
cons :: s:0' -> nil:cons -> nil:cons
c2 :: c3 -> c1:c2
LENGTH1 :: nil:cons -> c3
c3 :: c1:c2 -> c3
from :: s:0' -> nil:cons
length :: nil:cons -> s:0'
0' :: s:0'
length1 :: nil:cons -> s:0'
hole_c1_4 :: c
hole_s:0'2_4 :: s:0'
hole_c1:c23_4 :: c1:c2
hole_nil:cons4_4 :: nil:cons
hole_c35_4 :: c3
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_nil:cons8_4 :: Nat -> nil:cons

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FROM, LENGTH, LENGTH1, from, length, length1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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

(10)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
LENGTH(nil) -> c1
LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
LENGTH1(z0) -> c3(LENGTH(z0))
from(z0) -> cons(z0, from(s(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length1(z1))
length1(z0) -> length(z0)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
LENGTH :: nil:cons -> c1:c2
nil :: nil:cons
c1 :: c1:c2
cons :: s:0' -> nil:cons -> nil:cons
c2 :: c3 -> c1:c2
LENGTH1 :: nil:cons -> c3
c3 :: c1:c2 -> c3
from :: s:0' -> nil:cons
length :: nil:cons -> s:0'
0' :: s:0'
length1 :: nil:cons -> s:0'
hole_c1_4 :: c
hole_s:0'2_4 :: s:0'
hole_c1:c23_4 :: c1:c2
hole_nil:cons4_4 :: nil:cons
hole_c35_4 :: c3
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_nil:cons8_4 :: Nat -> nil:cons


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x))
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_nil:cons8_4(0) <=> nil
gen_nil:cons8_4(+(x, 1)) <=> cons(0', gen_nil:cons8_4(x))


The following defined symbols remain to be analysed:
FROM, LENGTH, LENGTH1, from, length, length1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length1(gen_nil:cons8_4(n230_4)) -> gen_s:0'7_4(n230_4), rt in Omega(0)

Induction Base:
length1(gen_nil:cons8_4(0)) ->_R^Omega(0)
length(gen_nil:cons8_4(0)) ->_R^Omega(0)
0'

Induction Step:
length1(gen_nil:cons8_4(+(n230_4, 1))) ->_R^Omega(0)
length(gen_nil:cons8_4(+(n230_4, 1))) ->_R^Omega(0)
s(length1(gen_nil:cons8_4(n230_4))) ->_IH
s(gen_s:0'7_4(c231_4))

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

(12)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
LENGTH(nil) -> c1
LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
LENGTH1(z0) -> c3(LENGTH(z0))
from(z0) -> cons(z0, from(s(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length1(z1))
length1(z0) -> length(z0)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
LENGTH :: nil:cons -> c1:c2
nil :: nil:cons
c1 :: c1:c2
cons :: s:0' -> nil:cons -> nil:cons
c2 :: c3 -> c1:c2
LENGTH1 :: nil:cons -> c3
c3 :: c1:c2 -> c3
from :: s:0' -> nil:cons
length :: nil:cons -> s:0'
0' :: s:0'
length1 :: nil:cons -> s:0'
hole_c1_4 :: c
hole_s:0'2_4 :: s:0'
hole_c1:c23_4 :: c1:c2
hole_nil:cons4_4 :: nil:cons
hole_c35_4 :: c3
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_nil:cons8_4 :: Nat -> nil:cons


Lemmas:
length1(gen_nil:cons8_4(n230_4)) -> gen_s:0'7_4(n230_4), rt in Omega(0)


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x))
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_nil:cons8_4(0) <=> nil
gen_nil:cons8_4(+(x, 1)) <=> cons(0', gen_nil:cons8_4(x))


The following defined symbols remain to be analysed:
length, LENGTH, LENGTH1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH1(gen_nil:cons8_4(n518_4)) -> *9_4, rt in Omega(n518_4)

Induction Base:
LENGTH1(gen_nil:cons8_4(0))

Induction Step:
LENGTH1(gen_nil:cons8_4(+(n518_4, 1))) ->_R^Omega(1)
c3(LENGTH(gen_nil:cons8_4(+(n518_4, 1)))) ->_R^Omega(1)
c3(c2(LENGTH1(gen_nil:cons8_4(n518_4)))) ->_IH
c3(c2(*9_4))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
LENGTH(nil) -> c1
LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
LENGTH1(z0) -> c3(LENGTH(z0))
from(z0) -> cons(z0, from(s(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length1(z1))
length1(z0) -> length(z0)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
LENGTH :: nil:cons -> c1:c2
nil :: nil:cons
c1 :: c1:c2
cons :: s:0' -> nil:cons -> nil:cons
c2 :: c3 -> c1:c2
LENGTH1 :: nil:cons -> c3
c3 :: c1:c2 -> c3
from :: s:0' -> nil:cons
length :: nil:cons -> s:0'
0' :: s:0'
length1 :: nil:cons -> s:0'
hole_c1_4 :: c
hole_s:0'2_4 :: s:0'
hole_c1:c23_4 :: c1:c2
hole_nil:cons4_4 :: nil:cons
hole_c35_4 :: c3
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_nil:cons8_4 :: Nat -> nil:cons


Lemmas:
length1(gen_nil:cons8_4(n230_4)) -> gen_s:0'7_4(n230_4), rt in Omega(0)


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x))
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_nil:cons8_4(0) <=> nil
gen_nil:cons8_4(+(x, 1)) <=> cons(0', gen_nil:cons8_4(x))


The following defined symbols remain to be analysed:
LENGTH1, LENGTH

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1

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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
LENGTH(nil) -> c1
LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
LENGTH1(z0) -> c3(LENGTH(z0))
from(z0) -> cons(z0, from(s(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length1(z1))
length1(z0) -> length(z0)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
LENGTH :: nil:cons -> c1:c2
nil :: nil:cons
c1 :: c1:c2
cons :: s:0' -> nil:cons -> nil:cons
c2 :: c3 -> c1:c2
LENGTH1 :: nil:cons -> c3
c3 :: c1:c2 -> c3
from :: s:0' -> nil:cons
length :: nil:cons -> s:0'
0' :: s:0'
length1 :: nil:cons -> s:0'
hole_c1_4 :: c
hole_s:0'2_4 :: s:0'
hole_c1:c23_4 :: c1:c2
hole_nil:cons4_4 :: nil:cons
hole_c35_4 :: c3
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_nil:cons8_4 :: Nat -> nil:cons


Lemmas:
length1(gen_nil:cons8_4(n230_4)) -> gen_s:0'7_4(n230_4), rt in Omega(0)
LENGTH1(gen_nil:cons8_4(n518_4)) -> *9_4, rt in Omega(n518_4)


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x))
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_nil:cons8_4(0) <=> nil
gen_nil:cons8_4(+(x, 1)) <=> cons(0', gen_nil:cons8_4(x))


The following defined symbols remain to be analysed:
LENGTH

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1

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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_nil:cons8_4(n811_4)) -> *9_4, rt in Omega(n811_4)

Induction Base:
LENGTH(gen_nil:cons8_4(0))

Induction Step:
LENGTH(gen_nil:cons8_4(+(n811_4, 1))) ->_R^Omega(1)
c2(LENGTH1(gen_nil:cons8_4(n811_4))) ->_R^Omega(1)
c2(c3(LENGTH(gen_nil:cons8_4(n811_4)))) ->_IH
c2(c3(*9_4))

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

(20)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(FROM(s(z0)))
LENGTH(nil) -> c1
LENGTH(cons(z0, z1)) -> c2(LENGTH1(z1))
LENGTH1(z0) -> c3(LENGTH(z0))
from(z0) -> cons(z0, from(s(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length1(z1))
length1(z0) -> length(z0)

Types:
FROM :: s:0' -> c
c :: c -> c
s :: s:0' -> s:0'
LENGTH :: nil:cons -> c1:c2
nil :: nil:cons
c1 :: c1:c2
cons :: s:0' -> nil:cons -> nil:cons
c2 :: c3 -> c1:c2
LENGTH1 :: nil:cons -> c3
c3 :: c1:c2 -> c3
from :: s:0' -> nil:cons
length :: nil:cons -> s:0'
0' :: s:0'
length1 :: nil:cons -> s:0'
hole_c1_4 :: c
hole_s:0'2_4 :: s:0'
hole_c1:c23_4 :: c1:c2
hole_nil:cons4_4 :: nil:cons
hole_c35_4 :: c3
gen_c6_4 :: Nat -> c
gen_s:0'7_4 :: Nat -> s:0'
gen_nil:cons8_4 :: Nat -> nil:cons


Lemmas:
length1(gen_nil:cons8_4(n230_4)) -> gen_s:0'7_4(n230_4), rt in Omega(0)
LENGTH1(gen_nil:cons8_4(n518_4)) -> *9_4, rt in Omega(n518_4)
LENGTH(gen_nil:cons8_4(n811_4)) -> *9_4, rt in Omega(n811_4)


Generator Equations:
gen_c6_4(0) <=> hole_c1_4
gen_c6_4(+(x, 1)) <=> c(gen_c6_4(x))
gen_s:0'7_4(0) <=> 0'
gen_s:0'7_4(+(x, 1)) <=> s(gen_s:0'7_4(x))
gen_nil:cons8_4(0) <=> nil
gen_nil:cons8_4(+(x, 1)) <=> cons(0', gen_nil:cons8_4(x))


The following defined symbols remain to be analysed:
LENGTH1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH1(gen_nil:cons8_4(n1773_4)) -> *9_4, rt in Omega(n1773_4)

Induction Base:
LENGTH1(gen_nil:cons8_4(0))

Induction Step:
LENGTH1(gen_nil:cons8_4(+(n1773_4, 1))) ->_R^Omega(1)
c3(LENGTH(gen_nil:cons8_4(+(n1773_4, 1)))) ->_R^Omega(1)
c3(c2(LENGTH1(gen_nil:cons8_4(n1773_4)))) ->_IH
c3(c2(*9_4))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(22)
BOUNDS(1, INF)