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


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


WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, 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), 295 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), 23 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
        (18) 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:

   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   if(true, c, l, ys, zs) -> nil
   if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
   greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
   smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
   helpc(true, ys, zs) -> ys
   helpc(false, ys, zs) -> zs
   helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))

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:

   app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))
   length(nil) -> 0'
   length(cons(x, y)) -> s(length(y))
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   if(true, c, l, ys, zs) -> nil
   if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
   greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
   smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
   helpc(true, ys, zs) -> ys
   helpc(false, ys, zs) -> zs
   helpb(c, l, cons(y, ys), zs) -> cons(y, helpa(s(c), l, ys, zs))

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

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

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

(4)
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:

   app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))
   length(nil) -> 0'
   length(cons(y)) -> s(length(y))
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   if(true, c, l, ys, zs) -> nil
   if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
   greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
   smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
   helpc(true, ys, zs) -> ys
   helpc(false, ys, zs) -> zs
   helpb(c, l, cons(ys), zs) -> cons(helpa(s(c), l, ys, zs))

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

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

(6)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
helpc(true, ys, zs) -> ys
helpc(false, ys, zs) -> zs
helpb(c, l, cons(ys), zs) -> cons(helpa(s(c), l, ys, zs))

Types:
app :: nil:cons -> nil:cons -> nil:cons
helpa :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons
cons :: nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
greater :: nil:cons -> nil:cons -> nil:cons
smaller :: nil:cons -> nil:cons -> nil:cons
helpc :: true:false -> nil:cons -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
helpa, plus, length, ge, helpb

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb

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

(8)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
helpc(true, ys, zs) -> ys
helpc(false, ys, zs) -> zs
helpb(c, l, cons(ys), zs) -> cons(helpa(s(c), l, ys, zs))

Types:
app :: nil:cons -> nil:cons -> nil:cons
helpa :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons
cons :: nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
greater :: nil:cons -> nil:cons -> nil:cons
smaller :: nil:cons -> nil:cons -> nil:cons
helpc :: true:false -> nil:cons -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
plus, helpa, length, ge, helpb

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s5_0(a), gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(n7_0, a)), rt in Omega(1 + n7_0)

Induction Base:
plus(gen_0':s5_0(a), gen_0':s5_0(0)) ->_R^Omega(1)
gen_0':s5_0(a)

Induction Step:
plus(gen_0':s5_0(a), gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s5_0(a), gen_0':s5_0(n7_0))) ->_IH
s(gen_0':s5_0(+(a, c8_0)))

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:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
helpc(true, ys, zs) -> ys
helpc(false, ys, zs) -> zs
helpb(c, l, cons(ys), zs) -> cons(helpa(s(c), l, ys, zs))

Types:
app :: nil:cons -> nil:cons -> nil:cons
helpa :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons
cons :: nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
greater :: nil:cons -> nil:cons -> nil:cons
smaller :: nil:cons -> nil:cons -> nil:cons
helpc :: true:false -> nil:cons -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
plus, helpa, length, ge, helpb

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
helpc(true, ys, zs) -> ys
helpc(false, ys, zs) -> zs
helpb(c, l, cons(ys), zs) -> cons(helpa(s(c), l, ys, zs))

Types:
app :: nil:cons -> nil:cons -> nil:cons
helpa :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons
cons :: nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
greater :: nil:cons -> nil:cons -> nil:cons
smaller :: nil:cons -> nil:cons -> nil:cons
helpc :: true:false -> nil:cons -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s5_0(a), gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(n7_0, a)), rt in Omega(1 + n7_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
length, helpa, ge, helpb

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons4_0(n824_0)) -> gen_0':s5_0(n824_0), rt in Omega(1 + n824_0)

Induction Base:
length(gen_nil:cons4_0(0)) ->_R^Omega(1)
0'

Induction Step:
length(gen_nil:cons4_0(+(n824_0, 1))) ->_R^Omega(1)
s(length(gen_nil:cons4_0(n824_0))) ->_IH
s(gen_0':s5_0(c825_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:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
helpc(true, ys, zs) -> ys
helpc(false, ys, zs) -> zs
helpb(c, l, cons(ys), zs) -> cons(helpa(s(c), l, ys, zs))

Types:
app :: nil:cons -> nil:cons -> nil:cons
helpa :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons
cons :: nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
greater :: nil:cons -> nil:cons -> nil:cons
smaller :: nil:cons -> nil:cons -> nil:cons
helpc :: true:false -> nil:cons -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s5_0(a), gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(n7_0, a)), rt in Omega(1 + n7_0)
length(gen_nil:cons4_0(n824_0)) -> gen_0':s5_0(n824_0), rt in Omega(1 + n824_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
ge, helpa, helpb

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s5_0(n1080_0), gen_0':s5_0(n1080_0)) -> true, rt in Omega(1 + n1080_0)

Induction Base:
ge(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s5_0(+(n1080_0, 1)), gen_0':s5_0(+(n1080_0, 1))) ->_R^Omega(1)
ge(gen_0':s5_0(n1080_0), gen_0':s5_0(n1080_0)) ->_IH
true

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

(18)
Obligation:
TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, greater(ys, zs), smaller(ys, zs))
greater(ys, zs) -> helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs) -> helpc(ge(length(ys), length(zs)), zs, ys)
helpc(true, ys, zs) -> ys
helpc(false, ys, zs) -> zs
helpb(c, l, cons(ys), zs) -> cons(helpa(s(c), l, ys, zs))

Types:
app :: nil:cons -> nil:cons -> nil:cons
helpa :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons
cons :: nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
greater :: nil:cons -> nil:cons -> nil:cons
smaller :: nil:cons -> nil:cons -> nil:cons
helpc :: true:false -> nil:cons -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s5_0(a), gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(n7_0, a)), rt in Omega(1 + n7_0)
length(gen_nil:cons4_0(n824_0)) -> gen_0':s5_0(n824_0), rt in Omega(1 + n824_0)
ge(gen_0':s5_0(n1080_0), gen_0':s5_0(n1080_0)) -> true, rt in Omega(1 + n1080_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
helpb, helpa

They will be analysed ascendingly in the following order:
helpa = helpb