WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) 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), 434 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs


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

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


The TRS R consists of the following rules:

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))
   f'(s(x), y, y) -> f'(y, x, s(x))

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

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

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


The TRS R consists of the following rules:

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))
   f'(s(x), y, y) -> f'(y, x, s(x))

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

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

(4)
Obligation:
TRS:
Rules:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Types:
f :: g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
f' :: s -> s -> s -> f'
s :: s -> s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat -> g:h
gen_s5_0 :: Nat -> s

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

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

(6)
Obligation:
TRS:
Rules:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Types:
f :: g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
f' :: s -> s -> s -> f'
s :: s -> s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat -> g:h
gen_s5_0 :: Nat -> s


Generator Equations:
gen_g:h4_0(0) <=> hole_g:h1_0
gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))
gen_s5_0(0) <=> hole_s3_0
gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))


The following defined symbols remain to be analysed:
f, f'
----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_g:h4_0(+(1, n7_0))) -> *6_0, rt in Omega(n7_0)

Induction Base:
f(gen_g:h4_0(+(1, 0)))

Induction Step:
f(gen_g:h4_0(+(1, +(n7_0, 1)))) ->_R^Omega(1)
g(f(f(gen_g:h4_0(+(1, n7_0))))) ->_IH
g(f(*6_0))

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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Types:
f :: g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
f' :: s -> s -> s -> f'
s :: s -> s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat -> g:h
gen_s5_0 :: Nat -> s


Generator Equations:
gen_g:h4_0(0) <=> hole_g:h1_0
gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))
gen_s5_0(0) <=> hole_s3_0
gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))


The following defined symbols remain to be analysed:
f, f'
----------------------------------------

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))

Types:
f :: g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
f' :: s -> s -> s -> f'
s :: s -> s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat -> g:h
gen_s5_0 :: Nat -> s


Lemmas:
f(gen_g:h4_0(+(1, n7_0))) -> *6_0, rt in Omega(n7_0)


Generator Equations:
gen_g:h4_0(0) <=> hole_g:h1_0
gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))
gen_s5_0(0) <=> hole_s3_0
gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))


The following defined symbols remain to be analysed:
f'