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


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 225 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(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), 268 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), 60 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 33 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 30 ms]
        (20) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   gt0(Cons(x, xs), Nil) -> True
   gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(x, xs)) -> Nil
   gcd(Cons(x, xs), Nil) -> Nil
   gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
   lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
   eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
   eqList(Cons(x, xs), Nil) -> False
   eqList(Nil, Cons(y, ys)) -> False
   eqList(Nil, Nil) -> True
   lgth(Nil) -> Nil
   gt0(Nil, y) -> False
   monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
   goal(x, y) -> gcd(x, y)

The (relative) TRS S consists of the following rules:

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
   monus[Ite](True, Cons(x, xs), y) -> xs
   gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
   gcd[Ite](True, x, y) -> x
   gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
   gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Rewrite Strategy: INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   gt0(Cons(x, xs), Nil) -> True
   gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(x, xs)) -> Nil
   gcd(Cons(x, xs), Nil) -> Nil
   gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
   lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
   eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
   eqList(Cons(x, xs), Nil) -> False
   eqList(Nil, Cons(y, ys)) -> False
   eqList(Nil, Nil) -> True
   lgth(Nil) -> Nil
   gt0(Nil, y) -> False
   monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
   goal(x, y) -> gcd(x, y)

The (relative) TRS S consists of the following rules:

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
   monus[Ite](True, Cons(x, xs), y) -> xs
   gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
   gcd[Ite](True, x, y) -> x
   gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
   gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Rewrite Strategy: INNERMOST
----------------------------------------

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   gt0(Cons(x, xs), Nil) -> True
   gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(x, xs)) -> Nil
   gcd(Cons(x, xs), Nil) -> Nil
   gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
   lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
   eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
   eqList(Cons(x, xs), Nil) -> False
   eqList(Nil, Cons(y, ys)) -> False
   eqList(Nil, Nil) -> True
   lgth(Nil) -> Nil
   gt0(Nil, y) -> False
   monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
   goal(x, y) -> gcd(x, y)

The (relative) TRS S consists of the following rules:

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
   monus[Ite](True, Cons(x, xs), y) -> xs
   gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
   gcd[Ite](True, x, y) -> x
   gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
   gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Rewrite Strategy: INNERMOST
----------------------------------------

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

(6)
Obligation:
Innermost TRS:
Rules:
@(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
@(Nil, ys) -> ys
gt0(Cons(x, xs), Nil) -> True
gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
gcd(Nil, Nil) -> Nil
gcd(Nil, Cons(x, xs)) -> Nil
gcd(Cons(x, xs), Nil) -> Nil
gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
eqList(Cons(x, xs), Nil) -> False
eqList(Nil, Cons(y, ys)) -> False
eqList(Nil, Nil) -> True
lgth(Nil) -> Nil
gt0(Nil, y) -> False
monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
goal(x, y) -> gcd(x, y)
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
monus[Ite](True, Cons(x, xs), y) -> xs
gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
gcd[Ite](True, x, y) -> x
gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Types:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
gt0 :: Cons:Nil -> Cons:Nil -> True:False
True :: True:False
gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
eqList :: Cons:Nil -> Cons:Nil -> True:False
lgth :: Cons:Nil -> Cons:Nil
and :: True:False -> True:False -> True:False
False :: True:False
monus :: Cons:Nil -> Cons:Nil -> Cons:Nil
monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat -> Cons:Nil

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
@, gt0, gcd, eqList, lgth, monus

They will be analysed ascendingly in the following order:
@ < lgth
gt0 < gcd
eqList < gcd
monus < gcd
eqList < monus
lgth < monus

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

(8)
Obligation:
Innermost TRS:
Rules:
@(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
@(Nil, ys) -> ys
gt0(Cons(x, xs), Nil) -> True
gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
gcd(Nil, Nil) -> Nil
gcd(Nil, Cons(x, xs)) -> Nil
gcd(Cons(x, xs), Nil) -> Nil
gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
eqList(Cons(x, xs), Nil) -> False
eqList(Nil, Cons(y, ys)) -> False
eqList(Nil, Nil) -> True
lgth(Nil) -> Nil
gt0(Nil, y) -> False
monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
goal(x, y) -> gcd(x, y)
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
monus[Ite](True, Cons(x, xs), y) -> xs
gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
gcd[Ite](True, x, y) -> x
gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Types:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
gt0 :: Cons:Nil -> Cons:Nil -> True:False
True :: True:False
gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
eqList :: Cons:Nil -> Cons:Nil -> True:False
lgth :: Cons:Nil -> Cons:Nil
and :: True:False -> True:False -> True:False
False :: True:False
monus :: Cons:Nil -> Cons:Nil -> Cons:Nil
monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_1(0) <=> Nil
gen_Cons:Nil3_1(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil3_1(x))


The following defined symbols remain to be analysed:
@, gt0, gcd, eqList, lgth, monus

They will be analysed ascendingly in the following order:
@ < lgth
gt0 < gcd
eqList < gcd
monus < gcd
eqList < monus
lgth < monus

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
@(gen_Cons:Nil3_1(n5_1), gen_Cons:Nil3_1(b)) -> gen_Cons:Nil3_1(+(n5_1, b)), rt in Omega(1 + n5_1)

Induction Base:
@(gen_Cons:Nil3_1(0), gen_Cons:Nil3_1(b)) ->_R^Omega(1)
gen_Cons:Nil3_1(b)

Induction Step:
@(gen_Cons:Nil3_1(+(n5_1, 1)), gen_Cons:Nil3_1(b)) ->_R^Omega(1)
Cons(Nil, @(gen_Cons:Nil3_1(n5_1), gen_Cons:Nil3_1(b))) ->_IH
Cons(Nil, gen_Cons:Nil3_1(+(b, c6_1)))

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:

Innermost TRS:
Rules:
@(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
@(Nil, ys) -> ys
gt0(Cons(x, xs), Nil) -> True
gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
gcd(Nil, Nil) -> Nil
gcd(Nil, Cons(x, xs)) -> Nil
gcd(Cons(x, xs), Nil) -> Nil
gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
eqList(Cons(x, xs), Nil) -> False
eqList(Nil, Cons(y, ys)) -> False
eqList(Nil, Nil) -> True
lgth(Nil) -> Nil
gt0(Nil, y) -> False
monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
goal(x, y) -> gcd(x, y)
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
monus[Ite](True, Cons(x, xs), y) -> xs
gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
gcd[Ite](True, x, y) -> x
gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Types:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
gt0 :: Cons:Nil -> Cons:Nil -> True:False
True :: True:False
gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
eqList :: Cons:Nil -> Cons:Nil -> True:False
lgth :: Cons:Nil -> Cons:Nil
and :: True:False -> True:False -> True:False
False :: True:False
monus :: Cons:Nil -> Cons:Nil -> Cons:Nil
monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_1(0) <=> Nil
gen_Cons:Nil3_1(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil3_1(x))


The following defined symbols remain to be analysed:
@, gt0, gcd, eqList, lgth, monus

They will be analysed ascendingly in the following order:
@ < lgth
gt0 < gcd
eqList < gcd
monus < gcd
eqList < monus
lgth < monus

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
Innermost TRS:
Rules:
@(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
@(Nil, ys) -> ys
gt0(Cons(x, xs), Nil) -> True
gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
gcd(Nil, Nil) -> Nil
gcd(Nil, Cons(x, xs)) -> Nil
gcd(Cons(x, xs), Nil) -> Nil
gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
eqList(Cons(x, xs), Nil) -> False
eqList(Nil, Cons(y, ys)) -> False
eqList(Nil, Nil) -> True
lgth(Nil) -> Nil
gt0(Nil, y) -> False
monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
goal(x, y) -> gcd(x, y)
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
monus[Ite](True, Cons(x, xs), y) -> xs
gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
gcd[Ite](True, x, y) -> x
gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Types:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
gt0 :: Cons:Nil -> Cons:Nil -> True:False
True :: True:False
gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
eqList :: Cons:Nil -> Cons:Nil -> True:False
lgth :: Cons:Nil -> Cons:Nil
and :: True:False -> True:False -> True:False
False :: True:False
monus :: Cons:Nil -> Cons:Nil -> Cons:Nil
monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat -> Cons:Nil


Lemmas:
@(gen_Cons:Nil3_1(n5_1), gen_Cons:Nil3_1(b)) -> gen_Cons:Nil3_1(+(n5_1, b)), rt in Omega(1 + n5_1)


Generator Equations:
gen_Cons:Nil3_1(0) <=> Nil
gen_Cons:Nil3_1(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil3_1(x))


The following defined symbols remain to be analysed:
gt0, gcd, eqList, lgth, monus

They will be analysed ascendingly in the following order:
gt0 < gcd
eqList < gcd
monus < gcd
eqList < monus
lgth < monus

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt0(gen_Cons:Nil3_1(+(1, n946_1)), gen_Cons:Nil3_1(n946_1)) -> True, rt in Omega(1 + n946_1)

Induction Base:
gt0(gen_Cons:Nil3_1(+(1, 0)), gen_Cons:Nil3_1(0)) ->_R^Omega(1)
True

Induction Step:
gt0(gen_Cons:Nil3_1(+(1, +(n946_1, 1))), gen_Cons:Nil3_1(+(n946_1, 1))) ->_R^Omega(1)
gt0(gen_Cons:Nil3_1(+(1, n946_1)), gen_Cons:Nil3_1(n946_1)) ->_IH
True

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

(16)
Obligation:
Innermost TRS:
Rules:
@(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
@(Nil, ys) -> ys
gt0(Cons(x, xs), Nil) -> True
gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
gcd(Nil, Nil) -> Nil
gcd(Nil, Cons(x, xs)) -> Nil
gcd(Cons(x, xs), Nil) -> Nil
gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
eqList(Cons(x, xs), Nil) -> False
eqList(Nil, Cons(y, ys)) -> False
eqList(Nil, Nil) -> True
lgth(Nil) -> Nil
gt0(Nil, y) -> False
monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
goal(x, y) -> gcd(x, y)
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
monus[Ite](True, Cons(x, xs), y) -> xs
gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
gcd[Ite](True, x, y) -> x
gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Types:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
gt0 :: Cons:Nil -> Cons:Nil -> True:False
True :: True:False
gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
eqList :: Cons:Nil -> Cons:Nil -> True:False
lgth :: Cons:Nil -> Cons:Nil
and :: True:False -> True:False -> True:False
False :: True:False
monus :: Cons:Nil -> Cons:Nil -> Cons:Nil
monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat -> Cons:Nil


Lemmas:
@(gen_Cons:Nil3_1(n5_1), gen_Cons:Nil3_1(b)) -> gen_Cons:Nil3_1(+(n5_1, b)), rt in Omega(1 + n5_1)
gt0(gen_Cons:Nil3_1(+(1, n946_1)), gen_Cons:Nil3_1(n946_1)) -> True, rt in Omega(1 + n946_1)


Generator Equations:
gen_Cons:Nil3_1(0) <=> Nil
gen_Cons:Nil3_1(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil3_1(x))


The following defined symbols remain to be analysed:
eqList, gcd, lgth, monus

They will be analysed ascendingly in the following order:
eqList < gcd
monus < gcd
eqList < monus
lgth < monus

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eqList(gen_Cons:Nil3_1(+(1, n1415_1)), gen_Cons:Nil3_1(n1415_1)) -> False, rt in Omega(1 + n1415_1)

Induction Base:
eqList(gen_Cons:Nil3_1(+(1, 0)), gen_Cons:Nil3_1(0)) ->_R^Omega(1)
False

Induction Step:
eqList(gen_Cons:Nil3_1(+(1, +(n1415_1, 1))), gen_Cons:Nil3_1(+(n1415_1, 1))) ->_R^Omega(1)
and(eqList(Nil, Nil), eqList(gen_Cons:Nil3_1(+(1, n1415_1)), gen_Cons:Nil3_1(n1415_1))) ->_R^Omega(1)
and(True, eqList(gen_Cons:Nil3_1(+(1, n1415_1)), gen_Cons:Nil3_1(n1415_1))) ->_IH
and(True, False) ->_R^Omega(0)
False

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

(18)
Obligation:
Innermost TRS:
Rules:
@(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
@(Nil, ys) -> ys
gt0(Cons(x, xs), Nil) -> True
gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
gcd(Nil, Nil) -> Nil
gcd(Nil, Cons(x, xs)) -> Nil
gcd(Cons(x, xs), Nil) -> Nil
gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
eqList(Cons(x, xs), Nil) -> False
eqList(Nil, Cons(y, ys)) -> False
eqList(Nil, Nil) -> True
lgth(Nil) -> Nil
gt0(Nil, y) -> False
monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
goal(x, y) -> gcd(x, y)
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
monus[Ite](True, Cons(x, xs), y) -> xs
gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
gcd[Ite](True, x, y) -> x
gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Types:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
gt0 :: Cons:Nil -> Cons:Nil -> True:False
True :: True:False
gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
eqList :: Cons:Nil -> Cons:Nil -> True:False
lgth :: Cons:Nil -> Cons:Nil
and :: True:False -> True:False -> True:False
False :: True:False
monus :: Cons:Nil -> Cons:Nil -> Cons:Nil
monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat -> Cons:Nil


Lemmas:
@(gen_Cons:Nil3_1(n5_1), gen_Cons:Nil3_1(b)) -> gen_Cons:Nil3_1(+(n5_1, b)), rt in Omega(1 + n5_1)
gt0(gen_Cons:Nil3_1(+(1, n946_1)), gen_Cons:Nil3_1(n946_1)) -> True, rt in Omega(1 + n946_1)
eqList(gen_Cons:Nil3_1(+(1, n1415_1)), gen_Cons:Nil3_1(n1415_1)) -> False, rt in Omega(1 + n1415_1)


Generator Equations:
gen_Cons:Nil3_1(0) <=> Nil
gen_Cons:Nil3_1(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil3_1(x))


The following defined symbols remain to be analysed:
lgth, gcd, monus

They will be analysed ascendingly in the following order:
monus < gcd
lgth < monus

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lgth(gen_Cons:Nil3_1(n2029_1)) -> gen_Cons:Nil3_1(n2029_1), rt in Omega(1 + n2029_1)

Induction Base:
lgth(gen_Cons:Nil3_1(0)) ->_R^Omega(1)
Nil

Induction Step:
lgth(gen_Cons:Nil3_1(+(n2029_1, 1))) ->_R^Omega(1)
@(Cons(Nil, Nil), lgth(gen_Cons:Nil3_1(n2029_1))) ->_IH
@(Cons(Nil, Nil), gen_Cons:Nil3_1(c2030_1)) ->_L^Omega(2)
gen_Cons:Nil3_1(+(+(0, 1), n2029_1))

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

(20)
Obligation:
Innermost TRS:
Rules:
@(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
@(Nil, ys) -> ys
gt0(Cons(x, xs), Nil) -> True
gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs)
gcd(Nil, Nil) -> Nil
gcd(Nil, Cons(x, xs)) -> Nil
gcd(Cons(x, xs), Nil) -> Nil
gcd(Cons(x', xs'), Cons(x, xs)) -> gcd[Ite](eqList(Cons(x', xs'), Cons(x, xs)), Cons(x', xs'), Cons(x, xs))
lgth(Cons(x, xs)) -> @(Cons(Nil, Nil), lgth(xs))
eqList(Cons(x, xs), Cons(y, ys)) -> and(eqList(x, y), eqList(xs, ys))
eqList(Cons(x, xs), Nil) -> False
eqList(Nil, Cons(y, ys)) -> False
eqList(Nil, Nil) -> True
lgth(Nil) -> Nil
gt0(Nil, y) -> False
monus(x, y) -> monus[Ite](eqList(lgth(y), Cons(Nil, Nil)), x, y)
goal(x, y) -> gcd(x, y)
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
monus[Ite](False, Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs)
monus[Ite](True, Cons(x, xs), y) -> xs
gcd[Ite](False, x, y) -> gcd[False][Ite](gt0(x, y), x, y)
gcd[Ite](True, x, y) -> x
gcd[False][Ite](False, x, y) -> gcd(x, monus(y, x))
gcd[False][Ite](True, x, y) -> gcd(monus(x, y), y)

Types:
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
gt0 :: Cons:Nil -> Cons:Nil -> True:False
True :: True:False
gcd :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
eqList :: Cons:Nil -> Cons:Nil -> True:False
lgth :: Cons:Nil -> Cons:Nil
and :: True:False -> True:False -> True:False
False :: True:False
monus :: Cons:Nil -> Cons:Nil -> Cons:Nil
monus[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
gcd[False][Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat -> Cons:Nil


Lemmas:
@(gen_Cons:Nil3_1(n5_1), gen_Cons:Nil3_1(b)) -> gen_Cons:Nil3_1(+(n5_1, b)), rt in Omega(1 + n5_1)
gt0(gen_Cons:Nil3_1(+(1, n946_1)), gen_Cons:Nil3_1(n946_1)) -> True, rt in Omega(1 + n946_1)
eqList(gen_Cons:Nil3_1(+(1, n1415_1)), gen_Cons:Nil3_1(n1415_1)) -> False, rt in Omega(1 + n1415_1)
lgth(gen_Cons:Nil3_1(n2029_1)) -> gen_Cons:Nil3_1(n2029_1), rt in Omega(1 + n2029_1)


Generator Equations:
gen_Cons:Nil3_1(0) <=> Nil
gen_Cons:Nil3_1(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil3_1(x))


The following defined symbols remain to be analysed:
monus, gcd

They will be analysed ascendingly in the following order:
monus < gcd