/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 577 ms]
(8) proven lower bound
(9) LowerBoundPropagationProof [FINISHED, 0 ms]
(10) BOUNDS(n^1, INF)


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

(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:

   ack(0, y) -> s(y)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))

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:

   ack(0', y) -> s(y)
   ack(s(x), 0') -> ack(x, s(0'))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))

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

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

(4)
Obligation:
TRS:
Rules:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Types:
ack :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

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

(6)
Obligation:
TRS:
Rules:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Types:
ack :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, 0)))

Induction Step:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
ack(gen_0':s2_0(0), ack(s(gen_0':s2_0(0)), gen_0':s2_0(+(1, n4_0)))) ->_IH
ack(gen_0':s2_0(0), *3_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:

TRS:
Rules:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Types:
ack :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

(10)
BOUNDS(n^1, INF)