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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 203 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), 54 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 23 ms]
        (16) 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:

   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> if_times(even(s(x)), s(x), y)
   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
   if_times(false, s(x), y) -> plus(y, times(x, y))

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:

   even(0') -> true
   even(s(0')) -> false
   even(s(s(x))) -> even(x)
   half(0') -> 0'
   half(s(s(x))) -> s(half(x))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(x), y) -> if_times(even(s(x)), s(x), y)
   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
   if_times(false, s(x), y) -> plus(y, times(x, y))

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

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

(4)
Obligation:
TRS:
Rules:
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
half(0') -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Types:
even :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
if_times :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
even, half, plus, times

They will be analysed ascendingly in the following order:
even < times
half < times
plus < times

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(6)
Obligation:
TRS:
Rules:
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
half(0') -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Types:
even :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
if_times :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
even, half, plus, times

They will be analysed ascendingly in the following order:
even < times
half < times
plus < times

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)

Induction Base:
even(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
true

Induction Step:
even(gen_0':s3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
even(gen_0':s3_0(*(2, n5_0))) ->_IH
true

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)

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

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

TRS:
Rules:
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
half(0') -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Types:
even :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
if_times :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
even, half, plus, times

They will be analysed ascendingly in the following order:
even < times
half < times
plus < times

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

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

(12)
Obligation:
TRS:
Rules:
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
half(0') -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Types:
even :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
if_times :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
half, plus, times

They will be analysed ascendingly in the following order:
half < times
plus < times

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n149_0))) -> gen_0':s3_0(n149_0), rt in Omega(1 + n149_0)

Induction Base:
half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n149_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n149_0)))) ->_IH
s(gen_0':s3_0(c150_0))

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

(14)
Obligation:
TRS:
Rules:
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
half(0') -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Types:
even :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
if_times :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)
half(gen_0':s3_0(*(2, n149_0))) -> gen_0':s3_0(n149_0), rt in Omega(1 + n149_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
plus, times

They will be analysed ascendingly in the following order:
plus < times

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n355_0, b)), rt in Omega(1 + n355_0)

Induction Base:
plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
gen_0':s3_0(b)

Induction Step:
plus(gen_0':s3_0(+(n355_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(plus(gen_0':s3_0(n355_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c356_0)))

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

(16)
Obligation:
TRS:
Rules:
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
half(0') -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Types:
even :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
if_times :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)
half(gen_0':s3_0(*(2, n149_0))) -> gen_0':s3_0(n149_0), rt in Omega(1 + n149_0)
plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n355_0, b)), rt in Omega(1 + n355_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
times