/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 246 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 45 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:

   table -> gen(s(0))
   gen(x) -> if1(le(x, 10), x)
   if1(false, x) -> nil
   if1(true, x) -> if2(x, x)
   if2(x, y) -> if3(le(y, 10), x, y)
   if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y)))
   if3(false, x, y) -> gen(s(x))
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   10 -> s(s(s(s(s(s(s(s(s(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^2, INF).


The TRS R consists of the following rules:

   table -> gen(s(0'))
   gen(x) -> if1(le(x, 10'), x)
   if1(false, x) -> nil
   if1(true, x) -> if2(x, x)
   if2(x, y) -> if3(le(y, 10'), x, y)
   if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y)))
   if3(false, x, y) -> gen(s(x))
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
entry/0
entry/1

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

(4)
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:

   table -> gen(s(0'))
   gen(x) -> if1(le(x, 10'), x)
   if1(false, x) -> nil
   if1(true, x) -> if2(x, x)
   if2(x, y) -> if3(le(y, 10'), x, y)
   if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
   if3(false, x, y) -> gen(s(x))
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

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

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

(6)
Obligation:
TRS:
Rules:
table -> gen(s(0'))
gen(x) -> if1(le(x, 10'), x)
if1(false, x) -> nil
if1(true, x) -> if2(x, x)
if2(x, y) -> if3(le(y, 10'), x, y)
if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) -> gen(s(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s -> nil:cons
s :: 0':s -> 0':s
0' :: 0':s
if1 :: false:true -> 0':s -> nil:cons
le :: 0':s -> 0':s -> false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s -> 0':s -> nil:cons
if3 :: false:true -> 0':s -> 0':s -> nil:cons
cons :: entry -> nil:cons -> nil:cons
entry :: 0':s -> entry
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat -> nil:cons
gen_0':s6_0 :: Nat -> 0':s

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
gen, le, if2, times, plus

They will be analysed ascendingly in the following order:
le < gen
gen = if2
le < if2
times < if2
plus < times

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

(8)
Obligation:
TRS:
Rules:
table -> gen(s(0'))
gen(x) -> if1(le(x, 10'), x)
if1(false, x) -> nil
if1(true, x) -> if2(x, x)
if2(x, y) -> if3(le(y, 10'), x, y)
if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) -> gen(s(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s -> nil:cons
s :: 0':s -> 0':s
0' :: 0':s
if1 :: false:true -> 0':s -> nil:cons
le :: 0':s -> 0':s -> false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s -> 0':s -> nil:cons
if3 :: false:true -> 0':s -> 0':s -> nil:cons
cons :: entry -> nil:cons -> nil:cons
entry :: 0':s -> entry
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat -> nil:cons
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
le, gen, if2, times, plus

They will be analysed ascendingly in the following order:
le < gen
gen = if2
le < if2
times < if2
plus < times

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) -> true, rt in Omega(1 + n8_0)

Induction Base:
le(gen_0':s6_0(0), gen_0':s6_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s6_0(+(n8_0, 1)), gen_0':s6_0(+(n8_0, 1))) ->_R^Omega(1)
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) ->_IH
true

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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
table -> gen(s(0'))
gen(x) -> if1(le(x, 10'), x)
if1(false, x) -> nil
if1(true, x) -> if2(x, x)
if2(x, y) -> if3(le(y, 10'), x, y)
if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) -> gen(s(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s -> nil:cons
s :: 0':s -> 0':s
0' :: 0':s
if1 :: false:true -> 0':s -> nil:cons
le :: 0':s -> 0':s -> false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s -> 0':s -> nil:cons
if3 :: false:true -> 0':s -> 0':s -> nil:cons
cons :: entry -> nil:cons -> nil:cons
entry :: 0':s -> entry
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat -> nil:cons
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
le, gen, if2, times, plus

They will be analysed ascendingly in the following order:
le < gen
gen = if2
le < if2
times < if2
plus < times

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
table -> gen(s(0'))
gen(x) -> if1(le(x, 10'), x)
if1(false, x) -> nil
if1(true, x) -> if2(x, x)
if2(x, y) -> if3(le(y, 10'), x, y)
if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) -> gen(s(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s -> nil:cons
s :: 0':s -> 0':s
0' :: 0':s
if1 :: false:true -> 0':s -> nil:cons
le :: 0':s -> 0':s -> false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s -> 0':s -> nil:cons
if3 :: false:true -> 0':s -> 0':s -> nil:cons
cons :: entry -> nil:cons -> nil:cons
entry :: 0':s -> entry
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat -> nil:cons
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) -> true, rt in Omega(1 + n8_0)


Generator Equations:
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
plus, gen, if2, times

They will be analysed ascendingly in the following order:
gen = if2
times < if2
plus < times

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s6_0(n289_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n289_0, b)), rt in Omega(1 + n289_0)

Induction Base:
plus(gen_0':s6_0(0), gen_0':s6_0(b)) ->_R^Omega(1)
gen_0':s6_0(b)

Induction Step:
plus(gen_0':s6_0(+(n289_0, 1)), gen_0':s6_0(b)) ->_R^Omega(1)
s(plus(gen_0':s6_0(n289_0), gen_0':s6_0(b))) ->_IH
s(gen_0':s6_0(+(b, c290_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:
table -> gen(s(0'))
gen(x) -> if1(le(x, 10'), x)
if1(false, x) -> nil
if1(true, x) -> if2(x, x)
if2(x, y) -> if3(le(y, 10'), x, y)
if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) -> gen(s(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s -> nil:cons
s :: 0':s -> 0':s
0' :: 0':s
if1 :: false:true -> 0':s -> nil:cons
le :: 0':s -> 0':s -> false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s -> 0':s -> nil:cons
if3 :: false:true -> 0':s -> 0':s -> nil:cons
cons :: entry -> nil:cons -> nil:cons
entry :: 0':s -> entry
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat -> nil:cons
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) -> true, rt in Omega(1 + n8_0)
plus(gen_0':s6_0(n289_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n289_0, b)), rt in Omega(1 + n289_0)


Generator Equations:
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
times, gen, if2

They will be analysed ascendingly in the following order:
gen = if2
times < if2

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s6_0(n1106_0), gen_0':s6_0(b)) -> gen_0':s6_0(*(n1106_0, b)), rt in Omega(1 + b*n1106_0 + n1106_0)

Induction Base:
times(gen_0':s6_0(0), gen_0':s6_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s6_0(+(n1106_0, 1)), gen_0':s6_0(b)) ->_R^Omega(1)
plus(gen_0':s6_0(b), times(gen_0':s6_0(n1106_0), gen_0':s6_0(b))) ->_IH
plus(gen_0':s6_0(b), gen_0':s6_0(*(c1107_0, b))) ->_L^Omega(1 + b)
gen_0':s6_0(+(b, *(n1106_0, b)))

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:
table -> gen(s(0'))
gen(x) -> if1(le(x, 10'), x)
if1(false, x) -> nil
if1(true, x) -> if2(x, x)
if2(x, y) -> if3(le(y, 10'), x, y)
if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) -> gen(s(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s -> nil:cons
s :: 0':s -> 0':s
0' :: 0':s
if1 :: false:true -> 0':s -> nil:cons
le :: 0':s -> 0':s -> false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s -> 0':s -> nil:cons
if3 :: false:true -> 0':s -> 0':s -> nil:cons
cons :: entry -> nil:cons -> nil:cons
entry :: 0':s -> entry
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat -> nil:cons
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) -> true, rt in Omega(1 + n8_0)
plus(gen_0':s6_0(n289_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n289_0, b)), rt in Omega(1 + n289_0)


Generator Equations:
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
times, gen, if2

They will be analysed ascendingly in the following order:
gen = if2
times < if2

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

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

(21)
BOUNDS(n^2, INF)

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

(22)
Obligation:
TRS:
Rules:
table -> gen(s(0'))
gen(x) -> if1(le(x, 10'), x)
if1(false, x) -> nil
if1(true, x) -> if2(x, x)
if2(x, y) -> if3(le(y, 10'), x, y)
if3(true, x, y) -> cons(entry(times(x, y)), if2(x, s(y)))
if3(false, x, y) -> gen(s(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
10' -> s(s(s(s(s(s(s(s(s(s(0'))))))))))

Types:
table :: nil:cons
gen :: 0':s -> nil:cons
s :: 0':s -> 0':s
0' :: 0':s
if1 :: false:true -> 0':s -> nil:cons
le :: 0':s -> 0':s -> false:true
10' :: 0':s
false :: false:true
nil :: nil:cons
true :: false:true
if2 :: 0':s -> 0':s -> nil:cons
if3 :: false:true -> 0':s -> 0':s -> nil:cons
cons :: entry -> nil:cons -> nil:cons
entry :: 0':s -> entry
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_false:true3_0 :: false:true
hole_entry4_0 :: entry
gen_nil:cons5_0 :: Nat -> nil:cons
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s6_0(n8_0), gen_0':s6_0(n8_0)) -> true, rt in Omega(1 + n8_0)
plus(gen_0':s6_0(n289_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n289_0, b)), rt in Omega(1 + n289_0)
times(gen_0':s6_0(n1106_0), gen_0':s6_0(b)) -> gen_0':s6_0(*(n1106_0, b)), rt in Omega(1 + b*n1106_0 + n1106_0)


Generator Equations:
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(entry(0'), gen_nil:cons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
if2, gen

They will be analysed ascendingly in the following order:
gen = if2