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


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


WORST_CASE(Omega(n^3), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, 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), 278 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), 75 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^2, INF)
            (18) typed CpxTrs
                (19) RewriteLemmaProof [LOWER BOUND(ID), 437 ms]
                (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 297 ms]
                (22) proven lower bound
                (23) LowerBoundPropagationProof [FINISHED, 0 ms]
                (24) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   p(s(x)) -> x
   p(0) -> 0
   minus(x, 0) -> x
   minus(0, x) -> 0
   minus(x, s(y)) -> p(minus(x, y))
   isZero(0) -> true
   isZero(s(x)) -> false
   facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> facIter(x, z)
   factorial(x) -> facIter(x, s(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^3, INF).


The TRS R consists of the following rules:

   plus(0', x) -> x
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   p(s(x)) -> x
   p(0') -> 0'
   minus(x, 0') -> x
   minus(0', x) -> 0'
   minus(x, s(y)) -> p(minus(x, y))
   isZero(0') -> true
   isZero(s(x)) -> false
   facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> facIter(x, z)
   factorial(x) -> facIter(x, s(0'))

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

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

(4)
Obligation:
TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

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

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

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

(6)
Obligation:
TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
plus, times, minus, facIter

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_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(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(plus(gen_0':s3_0(n5_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c6_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:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
plus, times, minus, facIter

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), 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:
times, minus, facIter

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_0)

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

Induction Step:
times(gen_0':s3_0(+(n602_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
plus(gen_0':s3_0(b), times(gen_0':s3_0(n602_0), gen_0':s3_0(b))) ->_IH
plus(gen_0':s3_0(b), gen_0':s3_0(*(c603_0, b))) ->_L^Omega(1 + b)
gen_0':s3_0(+(b, *(n602_0, b)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), 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:
times, minus, facIter

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

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_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:
minus, facIter

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0))) -> *4_0, rt in Omega(n1362_0)

Induction Base:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, 0)))

Induction Step:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n1362_0, 1)))) ->_R^Omega(1)
p(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0)))) ->_IH
p(*4_0)

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

(20)
Obligation:
TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_0)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0))) -> *4_0, rt in Omega(n1362_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:
facIter
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
facIter(gen_0':s3_0(n4187_0), gen_0':s3_0(b)) -> *4_0, rt in Omega(n4187_0 + n4187_0^2 + n4187_0^3)

Induction Base:
facIter(gen_0':s3_0(0), gen_0':s3_0(b))

Induction Step:
facIter(gen_0':s3_0(+(n4187_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
if(isZero(gen_0':s3_0(+(n4187_0, 1))), minus(gen_0':s3_0(+(n4187_0, 1)), s(0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(n4187_0, 1)))) ->_R^Omega(1)
if(false, minus(gen_0':s3_0(+(1, n4187_0)), s(0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_R^Omega(1)
if(false, p(minus(gen_0':s3_0(+(1, n4187_0)), 0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_R^Omega(1)
if(false, p(gen_0':s3_0(+(1, n4187_0))), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_R^Omega(1)
if(false, gen_0':s3_0(n4187_0), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_L^Omega(3 + 3*n4187_0 + n4187_0^2)
if(false, gen_0':s3_0(n4187_0), gen_0':s3_0(+(1, n4187_0)), gen_0':s3_0(*(b, +(1, n4187_0)))) ->_R^Omega(1)
facIter(gen_0':s3_0(n4187_0), gen_0':s3_0(+(b, *(b, n4187_0)))) ->_IH
*4_0

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(22)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
p(s(x)) -> x
p(0') -> 0'
minus(x, 0') -> x
minus(0', x) -> 0'
minus(x, s(y)) -> p(minus(x, y))
isZero(0') -> true
isZero(s(x)) -> false
facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) -> y
if(false, x, y, z) -> facIter(x, z)
factorial(x) -> facIter(x, s(0'))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
isZero :: 0':s -> true:false
true :: true:false
false :: true:false
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_0)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0))) -> *4_0, rt in Omega(n1362_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:
facIter
----------------------------------------

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

(24)
BOUNDS(n^3, INF)