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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 455 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs


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(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(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs)))
   select(cons(ap, xs)) -> ap
   select(cons(ap, xs)) -> select(xs)
   addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys))

S is empty.
Rewrite Strategy: FULL
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(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(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs)))
   select(cons(ap, xs)) -> ap
   select(cons(ap, xs)) -> select(xs)
   addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys))

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) -> ap
select(cons(ap, xs)) -> select(xs)
addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys))

Types:
f :: node -> f
node :: s -> cons -> node
s :: s -> s
addchild :: node -> node -> node
select :: cons -> node
cons :: node -> cons -> cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat -> s
gen_cons6_0 :: Nat -> cons

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

They will be analysed ascendingly in the following order:
select < f

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(6)
Obligation:
TRS:
Rules:
f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) -> ap
select(cons(ap, xs)) -> select(xs)
addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys))

Types:
f :: node -> f
node :: s -> cons -> node
s :: s -> s
addchild :: node -> node -> node
select :: cons -> node
cons :: node -> cons -> cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat -> s
gen_cons6_0 :: Nat -> cons


Generator Equations:
gen_s5_0(0) <=> hole_s3_0
gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))
gen_cons6_0(0) <=> hole_cons4_0
gen_cons6_0(+(x, 1)) <=> cons(node(hole_s3_0, hole_cons4_0), gen_cons6_0(x))


The following defined symbols remain to be analysed:
select, f

They will be analysed ascendingly in the following order:
select < f

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
select(gen_cons6_0(+(1, n8_0))) -> *7_0, rt in Omega(n8_0)

Induction Base:
select(gen_cons6_0(+(1, 0)))

Induction Step:
select(gen_cons6_0(+(1, +(n8_0, 1)))) ->_R^Omega(1)
select(gen_cons6_0(+(1, n8_0))) ->_IH
*7_0

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

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

TRS:
Rules:
f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) -> ap
select(cons(ap, xs)) -> select(xs)
addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys))

Types:
f :: node -> f
node :: s -> cons -> node
s :: s -> s
addchild :: node -> node -> node
select :: cons -> node
cons :: node -> cons -> cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat -> s
gen_cons6_0 :: Nat -> cons


Generator Equations:
gen_s5_0(0) <=> hole_s3_0
gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))
gen_cons6_0(0) <=> hole_cons4_0
gen_cons6_0(+(x, 1)) <=> cons(node(hole_s3_0, hole_cons4_0), gen_cons6_0(x))


The following defined symbols remain to be analysed:
select, f

They will be analysed ascendingly in the following order:
select < f

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

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
TRS:
Rules:
f(node(s(n), xs)) -> f(addchild(select(xs), node(n, xs)))
select(cons(ap, xs)) -> ap
select(cons(ap, xs)) -> select(xs)
addchild(node(y, ys), node(n, xs)) -> node(y, cons(node(n, xs), ys))

Types:
f :: node -> f
node :: s -> cons -> node
s :: s -> s
addchild :: node -> node -> node
select :: cons -> node
cons :: node -> cons -> cons
hole_f1_0 :: f
hole_node2_0 :: node
hole_s3_0 :: s
hole_cons4_0 :: cons
gen_s5_0 :: Nat -> s
gen_cons6_0 :: Nat -> cons


Lemmas:
select(gen_cons6_0(+(1, n8_0))) -> *7_0, rt in Omega(n8_0)


Generator Equations:
gen_s5_0(0) <=> hole_s3_0
gen_s5_0(+(x, 1)) <=> s(gen_s5_0(x))
gen_cons6_0(0) <=> hole_cons4_0
gen_cons6_0(+(x, 1)) <=> cons(node(hole_s3_0, hole_cons4_0), gen_cons6_0(x))


The following defined symbols remain to be analysed:
f