/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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), 261 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), 103 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
        (18) BEST
            (19) proven lower bound
                (20) LowerBoundPropagationProof [FINISHED, 0 ms]
                (21) BOUNDS(n^2, INF)
            (22) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
   cond(true, x, y) -> s(0)
   cond(false, x, y) -> double(log(x, square(s(s(y)))))
   le(0, v) -> true
   le(s(u), 0) -> false
   le(s(u), s(v)) -> le(u, v)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   square(0) -> 0
   square(s(x)) -> s(plus(square(x), double(x)))
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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^2, INF).


The TRS R consists of the following rules:

   log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
   cond(true, x, y) -> s(0')
   cond(false, x, y) -> double(log(x, square(s(s(y)))))
   le(0', v) -> true
   le(s(u), 0') -> false
   le(s(u), s(v)) -> le(u, v)
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   square(0') -> 0'
   square(s(x)) -> s(plus(square(x), double(x)))
   plus(n, 0') -> n
   plus(n, s(m)) -> s(plus(n, m))

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

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

(4)
Obligation:
TRS:
Rules:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
log, le, double, square, plus

They will be analysed ascendingly in the following order:
le < log
double < log
square < log
double < square
plus < square

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

(6)
Obligation:
TRS:
Rules:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
le, log, double, square, plus

They will be analysed ascendingly in the following order:
le < log
double < log
square < log
double < square
plus < square

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
le(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_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:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
le, log, double, square, plus

They will be analysed ascendingly in the following order:
le < log
double < log
square < log
double < square
plus < square

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
double, log, square, plus

They will be analysed ascendingly in the following order:
double < log
square < log
double < square
plus < square

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_s:0'3_0(n264_0)) -> gen_s:0'3_0(*(2, n264_0)), rt in Omega(1 + n264_0)

Induction Base:
double(gen_s:0'3_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_s:0'3_0(+(n264_0, 1))) ->_R^Omega(1)
s(s(double(gen_s:0'3_0(n264_0)))) ->_IH
s(s(gen_s:0'3_0(*(2, c265_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:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
double(gen_s:0'3_0(n264_0)) -> gen_s:0'3_0(*(2, n264_0)), rt in Omega(1 + n264_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
plus, log, square

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n520_0)) -> gen_s:0'3_0(+(n520_0, a)), rt in Omega(1 + n520_0)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n520_0, 1))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n520_0))) ->_IH
s(gen_s:0'3_0(+(a, c521_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:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
double(gen_s:0'3_0(n264_0)) -> gen_s:0'3_0(*(2, n264_0)), rt in Omega(1 + n264_0)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n520_0)) -> gen_s:0'3_0(+(n520_0, a)), rt in Omega(1 + n520_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
square, log

They will be analysed ascendingly in the following order:
square < log

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
square(gen_s:0'3_0(n1099_0)) -> gen_s:0'3_0(*(n1099_0, n1099_0)), rt in Omega(1 + n1099_0 + n1099_0^2)

Induction Base:
square(gen_s:0'3_0(0)) ->_R^Omega(1)
0'

Induction Step:
square(gen_s:0'3_0(+(n1099_0, 1))) ->_R^Omega(1)
s(plus(square(gen_s:0'3_0(n1099_0)), double(gen_s:0'3_0(n1099_0)))) ->_IH
s(plus(gen_s:0'3_0(*(c1100_0, c1100_0)), double(gen_s:0'3_0(n1099_0)))) ->_L^Omega(1 + n1099_0)
s(plus(gen_s:0'3_0(*(n1099_0, n1099_0)), gen_s:0'3_0(*(2, n1099_0)))) ->_L^Omega(1 + 2*n1099_0)
s(gen_s:0'3_0(+(*(2, n1099_0), *(n1099_0, n1099_0))))

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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
double(gen_s:0'3_0(n264_0)) -> gen_s:0'3_0(*(2, n264_0)), rt in Omega(1 + n264_0)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n520_0)) -> gen_s:0'3_0(+(n520_0, a)), rt in Omega(1 + n520_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
square, log

They will be analysed ascendingly in the following order:
square < log

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

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

(21)
BOUNDS(n^2, INF)

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

(22)
Obligation:
TRS:
Rules:
log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
cond(true, x, y) -> s(0')
cond(false, x, y) -> double(log(x, square(s(s(y)))))
le(0', v) -> true
le(s(u), 0') -> false
le(s(u), s(v)) -> le(u, v)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
square(0') -> 0'
square(s(x)) -> s(plus(square(x), double(x)))
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
log :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
cond :: true:false -> s:0' -> s:0' -> s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
0' :: s:0'
false :: true:false
double :: s:0' -> s:0'
square :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
double(gen_s:0'3_0(n264_0)) -> gen_s:0'3_0(*(2, n264_0)), rt in Omega(1 + n264_0)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n520_0)) -> gen_s:0'3_0(+(n520_0, a)), rt in Omega(1 + n520_0)
square(gen_s:0'3_0(n1099_0)) -> gen_s:0'3_0(*(n1099_0, n1099_0)), rt in Omega(1 + n1099_0 + n1099_0^2)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
log