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


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 387 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 278 ms]
        (14) BOUNDS(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:

   sort(nil) -> nil
   sort(cons(x, y)) -> insert(x, sort(y))
   insert(x, nil) -> cons(x, nil)
   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
   choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
   choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

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:

   sort(nil) -> nil
   sort(cons(x, y)) -> insert(x, sort(y))
   insert(x, nil) -> cons(x, nil)
   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
   choose(x, cons(v, w), y, 0') -> cons(x, cons(v, w))
   choose(x, cons(v, w), 0', s(z)) -> cons(v, insert(x, w))
   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

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

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

(4)
Obligation:
TRS:
Rules:
sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') -> cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
0' :: 0':s
s :: 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_0':s4_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sort, insert, choose

They will be analysed ascendingly in the following order:
insert < sort
insert = choose

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

(6)
Obligation:
TRS:
Rules:
sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') -> cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
0' :: 0':s
s :: 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
choose, sort, insert

They will be analysed ascendingly in the following order:
insert < sort
insert = choose

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt in Omega(1 + n6_0)

Induction Base:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
cons(gen_0':s4_0(a), cons(0', gen_nil:cons3_0(0)))

Induction Step:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
choose(gen_0':s4_0(a), cons(0', gen_nil:cons3_0(0)), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
cons(gen_0':s4_0(a), gen_nil:cons3_0(1))

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:
sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') -> cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
0' :: 0':s
s :: 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
choose, sort, insert

They will be analysed ascendingly in the following order:
insert < sort
insert = choose

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0') -> cons(x, cons(v, w))
choose(x, cons(v, w), 0', s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

Types:
sort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
0' :: 0':s
s :: 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
choose(gen_0':s4_0(a), gen_nil:cons3_0(1), gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> cons(gen_0':s4_0(a), gen_nil:cons3_0(1)), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
insert, sort

They will be analysed ascendingly in the following order:
insert < sort
insert = choose

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sort(gen_nil:cons3_0(n3102_0)) -> *5_0, rt in Omega(n3102_0)

Induction Base:
sort(gen_nil:cons3_0(0))

Induction Step:
sort(gen_nil:cons3_0(+(n3102_0, 1))) ->_R^Omega(1)
insert(0', sort(gen_nil:cons3_0(n3102_0))) ->_IH
insert(0', *5_0)

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

(14)
BOUNDS(1, INF)