WORST_CASE(NON_POLY,?)
proof of input_qMFpYVgt1B.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, 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), 17 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 267 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), 38 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [FINISHED, 986 ms]
        (20) BOUNDS(EXP, INF)


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

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


The TRS R consists of the following rules:

   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   rev1(0, nil) -> 0
   rev1(s(x), nil) -> s(x)
   rev1(x, cons(y, l)) -> rev1(y, l)
   rev2(x, nil) -> nil
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
Tuples:
   REV(nil) -> c
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(0, nil) -> c3
   REV1(s(z0), nil) -> c4
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
S tuples:
   REV(nil) -> c
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(0, nil) -> c3
   REV1(s(z0), nil) -> c4
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   rev_1, rev1_2, rev2_2

Defined Pair Symbols:   REV_1, REV1_2, REV2_2

Compound Symbols:   c, c1_1, c2_1, c3, c4, c5_1, c6, c7_2


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

(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(EXP, INF).


The TRS R consists of the following rules:

   REV(nil) -> c
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(0, nil) -> c3
   REV1(s(z0), nil) -> c4
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))

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

   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))

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

(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(EXP, INF).


The TRS R consists of the following rules:

   REV(nil) -> c
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(0', nil) -> c3
   REV1(s(z0), nil) -> c4
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))

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

   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0', nil) -> 0'
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
REV(nil) -> c
REV(cons(z0, z1)) -> c1(REV1(z0, z1))
REV(cons(z0, z1)) -> c2(REV2(z0, z1))
REV1(0', nil) -> c3
REV1(s(z0), nil) -> c4
REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))

Types:
REV :: nil:cons -> c:c1:c2
nil :: nil:cons
c :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c3:c4:c5 -> c:c1:c2
REV1 :: 0':s -> nil:cons -> c3:c4:c5
c2 :: c6:c7 -> c:c1:c2
REV2 :: 0':s -> nil:cons -> c6:c7
0' :: 0':s
c3 :: c3:c4:c5
s :: a -> 0':s
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c:c1:c2 -> c6:c7 -> c6:c7
rev2 :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:cons2_8 :: nil:cons
hole_0':s3_8 :: 0':s
hole_c3:c4:c54_8 :: c3:c4:c5
hole_c6:c75_8 :: c6:c7
hole_a6_8 :: a
gen_nil:cons7_8 :: Nat -> nil:cons
gen_c3:c4:c58_8 :: Nat -> c3:c4:c5
gen_c6:c79_8 :: Nat -> c6:c7

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
REV, REV1, REV2, rev2, rev, rev1

They will be analysed ascendingly in the following order:
REV1 < REV
REV = REV2
rev2 < REV2
rev2 = rev
rev1 < rev

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

(10)
Obligation:
Innermost TRS:
Rules:
REV(nil) -> c
REV(cons(z0, z1)) -> c1(REV1(z0, z1))
REV(cons(z0, z1)) -> c2(REV2(z0, z1))
REV1(0', nil) -> c3
REV1(s(z0), nil) -> c4
REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))

Types:
REV :: nil:cons -> c:c1:c2
nil :: nil:cons
c :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c3:c4:c5 -> c:c1:c2
REV1 :: 0':s -> nil:cons -> c3:c4:c5
c2 :: c6:c7 -> c:c1:c2
REV2 :: 0':s -> nil:cons -> c6:c7
0' :: 0':s
c3 :: c3:c4:c5
s :: a -> 0':s
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c:c1:c2 -> c6:c7 -> c6:c7
rev2 :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:cons2_8 :: nil:cons
hole_0':s3_8 :: 0':s
hole_c3:c4:c54_8 :: c3:c4:c5
hole_c6:c75_8 :: c6:c7
hole_a6_8 :: a
gen_nil:cons7_8 :: Nat -> nil:cons
gen_c3:c4:c58_8 :: Nat -> c3:c4:c5
gen_c6:c79_8 :: Nat -> c6:c7


Generator Equations:
gen_nil:cons7_8(0) <=> nil
gen_nil:cons7_8(+(x, 1)) <=> cons(0', gen_nil:cons7_8(x))
gen_c3:c4:c58_8(0) <=> c3
gen_c3:c4:c58_8(+(x, 1)) <=> c5(gen_c3:c4:c58_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
REV1, REV, REV2, rev2, rev, rev1

They will be analysed ascendingly in the following order:
REV1 < REV
REV = REV2
rev2 < REV2
rev2 = rev
rev1 < rev

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
REV1(0', gen_nil:cons7_8(n11_8)) -> gen_c3:c4:c58_8(n11_8), rt in Omega(1 + n11_8)

Induction Base:
REV1(0', gen_nil:cons7_8(0)) ->_R^Omega(1)
c3

Induction Step:
REV1(0', gen_nil:cons7_8(+(n11_8, 1))) ->_R^Omega(1)
c5(REV1(0', gen_nil:cons7_8(n11_8))) ->_IH
c5(gen_c3:c4:c58_8(c12_8))

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

(12)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
REV(nil) -> c
REV(cons(z0, z1)) -> c1(REV1(z0, z1))
REV(cons(z0, z1)) -> c2(REV2(z0, z1))
REV1(0', nil) -> c3
REV1(s(z0), nil) -> c4
REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))

Types:
REV :: nil:cons -> c:c1:c2
nil :: nil:cons
c :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c3:c4:c5 -> c:c1:c2
REV1 :: 0':s -> nil:cons -> c3:c4:c5
c2 :: c6:c7 -> c:c1:c2
REV2 :: 0':s -> nil:cons -> c6:c7
0' :: 0':s
c3 :: c3:c4:c5
s :: a -> 0':s
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c:c1:c2 -> c6:c7 -> c6:c7
rev2 :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:cons2_8 :: nil:cons
hole_0':s3_8 :: 0':s
hole_c3:c4:c54_8 :: c3:c4:c5
hole_c6:c75_8 :: c6:c7
hole_a6_8 :: a
gen_nil:cons7_8 :: Nat -> nil:cons
gen_c3:c4:c58_8 :: Nat -> c3:c4:c5
gen_c6:c79_8 :: Nat -> c6:c7


Generator Equations:
gen_nil:cons7_8(0) <=> nil
gen_nil:cons7_8(+(x, 1)) <=> cons(0', gen_nil:cons7_8(x))
gen_c3:c4:c58_8(0) <=> c3
gen_c3:c4:c58_8(+(x, 1)) <=> c5(gen_c3:c4:c58_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
REV1, REV, REV2, rev2, rev, rev1

They will be analysed ascendingly in the following order:
REV1 < REV
REV = REV2
rev2 < REV2
rev2 = rev
rev1 < rev

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
REV(nil) -> c
REV(cons(z0, z1)) -> c1(REV1(z0, z1))
REV(cons(z0, z1)) -> c2(REV2(z0, z1))
REV1(0', nil) -> c3
REV1(s(z0), nil) -> c4
REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))

Types:
REV :: nil:cons -> c:c1:c2
nil :: nil:cons
c :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c3:c4:c5 -> c:c1:c2
REV1 :: 0':s -> nil:cons -> c3:c4:c5
c2 :: c6:c7 -> c:c1:c2
REV2 :: 0':s -> nil:cons -> c6:c7
0' :: 0':s
c3 :: c3:c4:c5
s :: a -> 0':s
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c:c1:c2 -> c6:c7 -> c6:c7
rev2 :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:cons2_8 :: nil:cons
hole_0':s3_8 :: 0':s
hole_c3:c4:c54_8 :: c3:c4:c5
hole_c6:c75_8 :: c6:c7
hole_a6_8 :: a
gen_nil:cons7_8 :: Nat -> nil:cons
gen_c3:c4:c58_8 :: Nat -> c3:c4:c5
gen_c6:c79_8 :: Nat -> c6:c7


Lemmas:
REV1(0', gen_nil:cons7_8(n11_8)) -> gen_c3:c4:c58_8(n11_8), rt in Omega(1 + n11_8)


Generator Equations:
gen_nil:cons7_8(0) <=> nil
gen_nil:cons7_8(+(x, 1)) <=> cons(0', gen_nil:cons7_8(x))
gen_c3:c4:c58_8(0) <=> c3
gen_c3:c4:c58_8(+(x, 1)) <=> c5(gen_c3:c4:c58_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
rev1, REV, REV2, rev2, rev

They will be analysed ascendingly in the following order:
REV = REV2
rev2 < REV2
rev2 = rev
rev1 < rev

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev1(0', gen_nil:cons7_8(n417_8)) -> 0', rt in Omega(0)

Induction Base:
rev1(0', gen_nil:cons7_8(0)) ->_R^Omega(0)
0'

Induction Step:
rev1(0', gen_nil:cons7_8(+(n417_8, 1))) ->_R^Omega(0)
rev1(0', gen_nil:cons7_8(n417_8)) ->_IH
0'

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

(18)
Obligation:
Innermost TRS:
Rules:
REV(nil) -> c
REV(cons(z0, z1)) -> c1(REV1(z0, z1))
REV(cons(z0, z1)) -> c2(REV2(z0, z1))
REV1(0', nil) -> c3
REV1(s(z0), nil) -> c4
REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))

Types:
REV :: nil:cons -> c:c1:c2
nil :: nil:cons
c :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c3:c4:c5 -> c:c1:c2
REV1 :: 0':s -> nil:cons -> c3:c4:c5
c2 :: c6:c7 -> c:c1:c2
REV2 :: 0':s -> nil:cons -> c6:c7
0' :: 0':s
c3 :: c3:c4:c5
s :: a -> 0':s
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c:c1:c2 -> c6:c7 -> c6:c7
rev2 :: 0':s -> nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:cons2_8 :: nil:cons
hole_0':s3_8 :: 0':s
hole_c3:c4:c54_8 :: c3:c4:c5
hole_c6:c75_8 :: c6:c7
hole_a6_8 :: a
gen_nil:cons7_8 :: Nat -> nil:cons
gen_c3:c4:c58_8 :: Nat -> c3:c4:c5
gen_c6:c79_8 :: Nat -> c6:c7


Lemmas:
REV1(0', gen_nil:cons7_8(n11_8)) -> gen_c3:c4:c58_8(n11_8), rt in Omega(1 + n11_8)
rev1(0', gen_nil:cons7_8(n417_8)) -> 0', rt in Omega(0)


Generator Equations:
gen_nil:cons7_8(0) <=> nil
gen_nil:cons7_8(+(x, 1)) <=> cons(0', gen_nil:cons7_8(x))
gen_c3:c4:c58_8(0) <=> c3
gen_c3:c4:c58_8(+(x, 1)) <=> c5(gen_c3:c4:c58_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
rev, REV, REV2, rev2

They will be analysed ascendingly in the following order:
REV = REV2
rev2 < REV2
rev2 = rev

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

(19) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
rev2(0', gen_nil:cons7_8(n7712_8)) -> gen_nil:cons7_8(n7712_8), rt in Omega(EXP)

Induction Base:
rev2(0', gen_nil:cons7_8(0)) ->_R^Omega(0)
nil

Induction Step:
rev2(0', gen_nil:cons7_8(+(n7712_8, 1))) ->_R^Omega(0)
rev(cons(0', rev2(0', gen_nil:cons7_8(n7712_8)))) ->_IH
rev(cons(0', gen_nil:cons7_8(c7713_8))) ->_R^Omega(0)
cons(rev1(0', gen_nil:cons7_8(n7712_8)), rev2(0', gen_nil:cons7_8(n7712_8))) ->_L^Omega(0)
cons(0', rev2(0', gen_nil:cons7_8(n7712_8))) ->_IH
cons(0', gen_nil:cons7_8(c7713_8))

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

(20)
BOUNDS(EXP, INF)