/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 396 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), 135 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 96 ms]
        (16) 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:

   ge(0, 0) -> true
   ge(s(x), 0) -> ge(x, 0)
   ge(0, s(0)) -> false
   ge(0, s(s(x))) -> ge(0, s(x))
   ge(s(x), s(y)) -> ge(x, y)
   minus(0, 0) -> 0
   minus(0, s(x)) -> minus(0, x)
   minus(s(x), 0) -> s(minus(x, 0))
   minus(s(x), s(y)) -> minus(x, y)
   plus(0, 0) -> 0
   plus(0, s(x)) -> s(plus(0, x))
   plus(s(x), y) -> s(plus(x, y))
   div(x, y) -> ify(ge(y, s(0)), x, y)
   ify(false, x, y) -> divByZeroError
   ify(true, x, y) -> if(ge(x, y), x, y)
   if(false, x, y) -> 0
   if(true, x, y) -> s(div(minus(x, y), 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:

   ge(0', 0') -> true
   ge(s(x), 0') -> ge(x, 0')
   ge(0', s(0')) -> false
   ge(0', s(s(x))) -> ge(0', s(x))
   ge(s(x), s(y)) -> ge(x, y)
   minus(0', 0') -> 0'
   minus(0', s(x)) -> minus(0', x)
   minus(s(x), 0') -> s(minus(x, 0'))
   minus(s(x), s(y)) -> minus(x, y)
   plus(0', 0') -> 0'
   plus(0', s(x)) -> s(plus(0', x))
   plus(s(x), y) -> s(plus(x, y))
   div(x, y) -> ify(ge(y, s(0')), x, y)
   ify(false, x, y) -> divByZeroError
   ify(true, x, y) -> if(ge(x, y), x, y)
   if(false, x, y) -> 0'
   if(true, x, y) -> s(div(minus(x, y), y))

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

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

(4)
Obligation:
TRS:
Rules:
ge(0', 0') -> true
ge(s(x), 0') -> ge(x, 0')
ge(0', s(0')) -> false
ge(0', s(s(x))) -> ge(0', s(x))
ge(s(x), s(y)) -> ge(x, y)
minus(0', 0') -> 0'
minus(0', s(x)) -> minus(0', x)
minus(s(x), 0') -> s(minus(x, 0'))
minus(s(x), s(y)) -> minus(x, y)
plus(0', 0') -> 0'
plus(0', s(x)) -> s(plus(0', x))
plus(s(x), y) -> s(plus(x, y))
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
plus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ge, minus, plus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

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

(6)
Obligation:
TRS:
Rules:
ge(0', 0') -> true
ge(s(x), 0') -> ge(x, 0')
ge(0', s(0')) -> false
ge(0', s(s(x))) -> ge(0', s(x))
ge(s(x), s(y)) -> ge(x, y)
minus(0', 0') -> 0'
minus(0', s(x)) -> minus(0', x)
minus(s(x), 0') -> s(minus(x, 0'))
minus(s(x), s(y)) -> minus(x, y)
plus(0', 0') -> 0'
plus(0', s(x)) -> s(plus(0', x))
plus(s(x), y) -> s(plus(x, y))
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
plus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
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:
ge, minus, plus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

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

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

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

Induction Step:
ge(gen_0':s:divByZeroError3_0(+(n5_0, 1)), gen_0':s:divByZeroError3_0(0)) ->_R^Omega(1)
ge(gen_0':s:divByZeroError3_0(n5_0), 0') ->_IH
true

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:
ge(0', 0') -> true
ge(s(x), 0') -> ge(x, 0')
ge(0', s(0')) -> false
ge(0', s(s(x))) -> ge(0', s(x))
ge(s(x), s(y)) -> ge(x, y)
minus(0', 0') -> 0'
minus(0', s(x)) -> minus(0', x)
minus(s(x), 0') -> s(minus(x, 0'))
minus(s(x), s(y)) -> minus(x, y)
plus(0', 0') -> 0'
plus(0', s(x)) -> s(plus(0', x))
plus(s(x), y) -> s(plus(x, y))
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
plus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
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:
ge, minus, plus, div

They will be analysed ascendingly in the following order:
ge < div
minus < div

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
ge(0', 0') -> true
ge(s(x), 0') -> ge(x, 0')
ge(0', s(0')) -> false
ge(0', s(s(x))) -> ge(0', s(x))
ge(s(x), s(y)) -> ge(x, y)
minus(0', 0') -> 0'
minus(0', s(x)) -> minus(0', x)
minus(s(x), 0') -> s(minus(x, 0'))
minus(s(x), s(y)) -> minus(x, y)
plus(0', 0') -> 0'
plus(0', s(x)) -> s(plus(0', x))
plus(s(x), y) -> s(plus(x, y))
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
plus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) -> true, 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:
minus, plus, div

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n6938_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n6938_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(0), gen_0':s:divByZeroError3_0(+(n6938_0, 1))) ->_R^Omega(1)
minus(0', gen_0':s:divByZeroError3_0(n6938_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).
----------------------------------------

(14)
Obligation:
TRS:
Rules:
ge(0', 0') -> true
ge(s(x), 0') -> ge(x, 0')
ge(0', s(0')) -> false
ge(0', s(s(x))) -> ge(0', s(x))
ge(s(x), s(y)) -> ge(x, y)
minus(0', 0') -> 0'
minus(0', s(x)) -> minus(0', x)
minus(s(x), 0') -> s(minus(x, 0'))
minus(s(x), s(y)) -> minus(x, y)
plus(0', 0') -> 0'
plus(0', s(x)) -> s(plus(0', x))
plus(s(x), y) -> s(plus(x, y))
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
plus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n6938_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n6938_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:
plus, div
----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n10209_0)) -> gen_0':s:divByZeroError3_0(n10209_0), rt in Omega(1 + n10209_0)

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

Induction Step:
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(+(n10209_0, 1))) ->_R^Omega(1)
s(plus(0', gen_0':s:divByZeroError3_0(n10209_0))) ->_IH
s(gen_0':s:divByZeroError3_0(c10210_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:
ge(0', 0') -> true
ge(s(x), 0') -> ge(x, 0')
ge(0', s(0')) -> false
ge(0', s(s(x))) -> ge(0', s(x))
ge(s(x), s(y)) -> ge(x, y)
minus(0', 0') -> 0'
minus(0', s(x)) -> minus(0', x)
minus(s(x), 0') -> s(minus(x, 0'))
minus(s(x), s(y)) -> minus(x, y)
plus(0', 0') -> 0'
plus(0', s(x)) -> s(plus(0', x))
plus(s(x), y) -> s(plus(x, y))
div(x, y) -> ify(ge(y, s(0')), x, y)
ify(false, x, y) -> divByZeroError
ify(true, x, y) -> if(ge(x, y), x, y)
if(false, x, y) -> 0'
if(true, x, y) -> s(div(minus(x, y), y))

Types:
ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
0' :: 0':s:divByZeroError
true :: true:false
s :: 0':s:divByZeroError -> 0':s:divByZeroError
false :: true:false
minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
plus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
divByZeroError :: 0':s:divByZeroError
if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
hole_true:false1_0 :: true:false
hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError


Lemmas:
ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n6938_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n6938_0)
plus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(n10209_0)) -> gen_0':s:divByZeroError3_0(n10209_0), rt in Omega(1 + n10209_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:
div