WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 1289 ms]
(8) proven lower bound
(9) LowerBoundPropagationProof [FINISHED, 0 ms]
(10) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

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

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

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

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Innermost TRS:
Rules:
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Types:
dfib :: s -> dfib -> dfib
s :: s -> s
hole_dfib1_0 :: dfib
hole_s2_0 :: s
gen_s3_0 :: Nat -> s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
dfib
----------------------------------------

(6)
Obligation:
Innermost TRS:
Rules:
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Types:
dfib :: s -> dfib -> dfib
s :: s -> s
hole_dfib1_0 :: dfib
hole_s2_0 :: s
gen_s3_0 :: Nat -> s


Generator Equations:
gen_s3_0(0) <=> hole_s2_0
gen_s3_0(+(x, 1)) <=> s(gen_s3_0(x))


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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0) -> *4_0, rt in Omega(n5_0)

Induction Base:
dfib(gen_s3_0(+(2, *(2, 0))), hole_dfib1_0)

Induction Step:
dfib(gen_s3_0(+(2, *(2, +(n5_0, 1)))), hole_dfib1_0) ->_R^Omega(1)
dfib(s(gen_s3_0(+(2, *(2, n5_0)))), dfib(gen_s3_0(+(2, *(2, n5_0))), hole_dfib1_0)) ->_IH
dfib(s(gen_s3_0(+(2, *(2, n5_0)))), *4_0)

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

(8)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
dfib(s(s(x)), y) -> dfib(s(x), dfib(x, y))

Types:
dfib :: s -> dfib -> dfib
s :: s -> s
hole_dfib1_0 :: dfib
hole_s2_0 :: s
gen_s3_0 :: Nat -> s


Generator Equations:
gen_s3_0(0) <=> hole_s2_0
gen_s3_0(+(x, 1)) <=> s(gen_s3_0(x))


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

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

(10)
BOUNDS(n^1, INF)