/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 280 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), 41 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), 37 ms]
                (20) BEST
                    (21) proven lower bound
                        (22) LowerBoundPropagationProof [FINISHED, 0 ms]
                        (23) BOUNDS(n^3, INF)
                    (24) typed CpxTrs
                        (25) RewriteLemmaProof [LOWER BOUND(ID), 534 ms]
                        (26) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) 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(p(s(x)), y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(p(s(x)), y))
   exp(x, 0) -> s(0)
   exp(x, s(y)) -> times(x, exp(x, y))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   tower(x, y) -> towerIter(x, y, s(0))
   towerIter(0, y, z) -> z
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

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

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) 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(p(s(x)), y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(p(s(x)), y))
   exp(x, 0') -> s(0')
   exp(x, s(y)) -> times(x, exp(x, y))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   tower(x, y) -> towerIter(x, y, s(0'))
   towerIter(0', y, z) -> z
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

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

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

(4)
Obligation:
Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, p, times, exp, towerIter

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

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

(6)
Obligation:
Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
p(gen_0':s2_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_0':s2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
s(p(s(gen_0':s2_0(n4_0)))) ->_IH
s(gen_0':s2_0(c5_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:

Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2)

Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n245_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
s(plus(p(s(gen_0':s2_0(n245_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n245_0)
s(plus(gen_0':s2_0(n245_0), gen_0':s2_0(b))) ->_IH
s(gen_0':s2_0(+(b, c246_0)))

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:

Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
times, exp, towerIter

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s2_0(n767_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n767_0, b)), rt in Omega(1 + b*n767_0 + b^2*n767_0 + n767_0 + n767_0^2)

Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s2_0(+(n767_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
plus(gen_0':s2_0(b), times(p(s(gen_0':s2_0(n767_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n767_0)
plus(gen_0':s2_0(b), times(gen_0':s2_0(n767_0), gen_0':s2_0(b))) ->_IH
plus(gen_0':s2_0(b), gen_0':s2_0(*(c768_0, b))) ->_L^Omega(1 + b + b^2)
gen_0':s2_0(+(b, *(n767_0, b)))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
times, exp, towerIter

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

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

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

(23)
BOUNDS(n^3, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2)
times(gen_0':s2_0(n767_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n767_0, b)), rt in Omega(1 + b*n767_0 + b^2*n767_0 + n767_0 + n767_0^2)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
exp, towerIter

They will be analysed ascendingly in the following order:
exp < towerIter

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1497_0))) -> *3_0, rt in Omega(n1497_0)

Induction Base:
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, 0)))

Induction Step:
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, +(n1497_0, 1)))) ->_R^Omega(1)
times(gen_0':s2_0(a), exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1497_0)))) ->_IH
times(gen_0':s2_0(a), *3_0)

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

(26)
Obligation:
Innermost TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
exp(x, 0') -> s(0')
exp(x, s(y)) -> times(x, exp(x, y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
tower(x, y) -> towerIter(x, y, s(0'))
towerIter(0', y, z) -> z
towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
exp :: 0':s -> 0':s -> 0':s
tower :: 0':s -> 0':s -> 0':s
towerIter :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n245_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n245_0, b)), rt in Omega(1 + n245_0 + n245_0^2)
times(gen_0':s2_0(n767_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n767_0, b)), rt in Omega(1 + b*n767_0 + b^2*n767_0 + n767_0 + n767_0^2)
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1497_0))) -> *3_0, rt in Omega(n1497_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
towerIter