/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 279 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), 30 ms]
        (14) 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:

   minus(x, x) -> 0
   minus(0, x) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   quot(x, y) -> if_quot(minus(x, y), y, le(y, 0), le(y, x))
   if_quot(x, y, true, z) -> divByZeroError
   if_quot(x, y, false, true) -> s(quot(x, y))
   if_quot(x, y, false, false) -> 0

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:

   minus(x, x) -> 0'
   minus(0', x) -> 0'
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   quot(x, y) -> if_quot(minus(x, y), y, le(y, 0'), le(y, x))
   if_quot(x, y, true, z) -> divByZeroError
   if_quot(x, y, false, true) -> s(quot(x, y))
   if_quot(x, y, false, false) -> 0'

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

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

(4)
Obligation:
TRS:
Rules:
minus(x, x) -> 0'
minus(0', x) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, y) -> if_quot(minus(x, y), y, le(y, 0'), le(y, x))
if_quot(x, y, true, z) -> divByZeroError
if_quot(x, y, false, true) -> s(quot(x, y))
if_quot(x, y, false, false) -> 0'

Types:
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
0' :: 0':s:divByZeroError
s :: 0':s:divByZeroError -> 0':s:divByZeroError
le :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
true :: true:false
false :: true:false
quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
if_quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false -> true:false -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
hole_0':s:divByZeroError1_0 :: 0':s:divByZeroError
hole_true:false2_0 :: true:false
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, le, quot

They will be analysed ascendingly in the following order:
minus < quot
le < quot

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

(6)
Obligation:
TRS:
Rules:
minus(x, x) -> 0'
minus(0', x) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, y) -> if_quot(minus(x, y), y, le(y, 0'), le(y, x))
if_quot(x, y, true, z) -> divByZeroError
if_quot(x, y, false, true) -> s(quot(x, y))
if_quot(x, y, false, false) -> 0'

Types:
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
0' :: 0':s:divByZeroError
s :: 0':s:divByZeroError -> 0':s:divByZeroError
le :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
true :: true:false
false :: true:false
quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
if_quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false -> true:false -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
hole_0':s:divByZeroError1_0 :: 0':s:divByZeroError
hole_true:false2_0 :: true:false
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


The following defined symbols remain to be analysed:
minus, le, quot

They will be analysed ascendingly in the following order:
minus < quot
le < quot

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n5_0)

Induction Base:
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) ->_R^Omega(1)
0'

Induction Step:
minus(gen_0':s:divByZeroError3_0(+(n5_0, 1)), gen_0':s:divByZeroError3_0(+(n5_0, 1))) ->_R^Omega(1)
minus(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) ->_IH
gen_0':s:divByZeroError3_0(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:
minus(x, x) -> 0'
minus(0', x) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, y) -> if_quot(minus(x, y), y, le(y, 0'), le(y, x))
if_quot(x, y, true, z) -> divByZeroError
if_quot(x, y, false, true) -> s(quot(x, y))
if_quot(x, y, false, false) -> 0'

Types:
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
0' :: 0':s:divByZeroError
s :: 0':s:divByZeroError -> 0':s:divByZeroError
le :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
true :: true:false
false :: true:false
quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
if_quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false -> true:false -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
hole_0':s:divByZeroError1_0 :: 0':s:divByZeroError
hole_true:false2_0 :: true:false
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


The following defined symbols remain to be analysed:
minus, le, quot

They will be analysed ascendingly in the following order:
minus < quot
le < quot

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

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

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

(12)
Obligation:
TRS:
Rules:
minus(x, x) -> 0'
minus(0', x) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, y) -> if_quot(minus(x, y), y, le(y, 0'), le(y, x))
if_quot(x, y, true, z) -> divByZeroError
if_quot(x, y, false, true) -> s(quot(x, y))
if_quot(x, y, false, false) -> 0'

Types:
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
0' :: 0':s:divByZeroError
s :: 0':s:divByZeroError -> 0':s:divByZeroError
le :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
true :: true:false
false :: true:false
quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
if_quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false -> true:false -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
hole_0':s:divByZeroError1_0 :: 0':s:divByZeroError
hole_true:false2_0 :: true:false
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Lemmas:
minus(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


The following defined symbols remain to be analysed:
le, quot

They will be analysed ascendingly in the following order:
le < quot

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s:divByZeroError3_0(n381_0), gen_0':s:divByZeroError3_0(n381_0)) -> true, rt in Omega(1 + n381_0)

Induction Base:
le(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s:divByZeroError3_0(+(n381_0, 1)), gen_0':s:divByZeroError3_0(+(n381_0, 1))) ->_R^Omega(1)
le(gen_0':s:divByZeroError3_0(n381_0), gen_0':s:divByZeroError3_0(n381_0)) ->_IH
true

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

(14)
Obligation:
TRS:
Rules:
minus(x, x) -> 0'
minus(0', x) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
quot(x, y) -> if_quot(minus(x, y), y, le(y, 0'), le(y, x))
if_quot(x, y, true, z) -> divByZeroError
if_quot(x, y, false, true) -> s(quot(x, y))
if_quot(x, y, false, false) -> 0'

Types:
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
0' :: 0':s:divByZeroError
s :: 0':s:divByZeroError -> 0':s:divByZeroError
le :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
true :: true:false
false :: true:false
quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
if_quot :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false -> true:false -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
hole_0':s:divByZeroError1_0 :: 0':s:divByZeroError
hole_true:false2_0 :: true:false
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Lemmas:
minus(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n5_0)
le(gen_0':s:divByZeroError3_0(n381_0), gen_0':s:divByZeroError3_0(n381_0)) -> true, rt in Omega(1 + n381_0)


Generator Equations:
gen_0':s:divByZeroError3_0(0) <=> 0'
gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))


The following defined symbols remain to be analysed:
quot