/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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) 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), 233 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), 83 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 148 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 46 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:

   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
   ifPlus(true, x, y) -> p(y)
   ifPlus(false, x, y) -> plus(p(x), y)
   times(x, y) -> timesIter(0, x, y, 0)
   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
   ifTimes(true, i, x, y, z) -> z
   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))
   inc(x) -> s(x)
   p(0) -> 0
   p(s(x)) -> x
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   f0(0, y, x) -> f1(x, y, x)
   f1(x, y, z) -> f2(x, y, z)
   f2(x, 1, z) -> f0(x, z, z)
   f0(x, y, z) -> d
   f1(x, y, z) -> c

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:

   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
   ifPlus(true, x, y) -> p(y)
   ifPlus(false, x, y) -> plus(p(x), y)
   times(x, y) -> timesIter(0', x, y, 0')
   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
   ifTimes(true, i, x, y, z) -> z
   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
   isZero(0') -> true
   isZero(s(0')) -> false
   isZero(s(s(x))) -> isZero(s(x))
   inc(0') -> s(0')
   inc(s(x)) -> s(inc(x))
   inc(x) -> s(x)
   p(0') -> 0'
   p(s(x)) -> x
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   f0(0', y, x) -> f1(x, y, x)
   f1(x, y, z) -> f2(x, y, z)
   f2(x, 1', z) -> f0(x, z, z)
   f0(x, y, z) -> d
   f1(x, y, z) -> c

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

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

(4)
Obligation:
TRS:
Rules:
plus(x, y) -> ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) -> p(y)
ifPlus(false, x, y) -> plus(p(x), y)
times(x, y) -> timesIter(0', x, y, 0')
timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) -> z
ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))
inc(x) -> s(x)
p(0') -> 0'
p(s(x)) -> x
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
f0(0', y, x) -> f1(x, y, x)
f1(x, y, z) -> f2(x, y, z)
f2(x, 1', z) -> f0(x, z, z)
f0(x, y, z) -> d
f1(x, y, z) -> c

Types:
plus :: 0':s:1' -> 0':s:1' -> 0':s:1'
ifPlus :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1'
isZero :: 0':s:1' -> true:false
inc :: 0':s:1' -> 0':s:1'
true :: true:false
p :: 0':s:1' -> 0':s:1'
false :: true:false
times :: 0':s:1' -> 0':s:1' -> 0':s:1'
timesIter :: 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
ge :: 0':s:1' -> 0':s:1' -> true:false
s :: 0':s:1' -> 0':s:1'
f0 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f1 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f2 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat -> 0':s:1'

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, isZero, inc, p, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
isZero < plus
inc < plus
p < plus
plus < timesIter
inc < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

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

(6)
Obligation:
TRS:
Rules:
plus(x, y) -> ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) -> p(y)
ifPlus(false, x, y) -> plus(p(x), y)
times(x, y) -> timesIter(0', x, y, 0')
timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) -> z
ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))
inc(x) -> s(x)
p(0') -> 0'
p(s(x)) -> x
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
f0(0', y, x) -> f1(x, y, x)
f1(x, y, z) -> f2(x, y, z)
f2(x, 1', z) -> f0(x, z, z)
f0(x, y, z) -> d
f1(x, y, z) -> c

Types:
plus :: 0':s:1' -> 0':s:1' -> 0':s:1'
ifPlus :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1'
isZero :: 0':s:1' -> true:false
inc :: 0':s:1' -> 0':s:1'
true :: true:false
p :: 0':s:1' -> 0':s:1'
false :: true:false
times :: 0':s:1' -> 0':s:1' -> 0':s:1'
timesIter :: 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
ge :: 0':s:1' -> 0':s:1' -> true:false
s :: 0':s:1' -> 0':s:1'
f0 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f1 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f2 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat -> 0':s:1'


Generator Equations:
gen_0':s:1'4_3(0) <=> 0'
gen_0':s:1'4_3(+(x, 1)) <=> s(gen_0':s:1'4_3(x))


The following defined symbols remain to be analysed:
isZero, plus, inc, p, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
isZero < plus
inc < plus
p < plus
plus < timesIter
inc < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
isZero(gen_0':s:1'4_3(+(1, n6_3))) -> false, rt in Omega(1 + n6_3)

Induction Base:
isZero(gen_0':s:1'4_3(+(1, 0))) ->_R^Omega(1)
false

Induction Step:
isZero(gen_0':s:1'4_3(+(1, +(n6_3, 1)))) ->_R^Omega(1)
isZero(s(gen_0':s:1'4_3(n6_3))) ->_IH
false

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:
plus(x, y) -> ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) -> p(y)
ifPlus(false, x, y) -> plus(p(x), y)
times(x, y) -> timesIter(0', x, y, 0')
timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) -> z
ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))
inc(x) -> s(x)
p(0') -> 0'
p(s(x)) -> x
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
f0(0', y, x) -> f1(x, y, x)
f1(x, y, z) -> f2(x, y, z)
f2(x, 1', z) -> f0(x, z, z)
f0(x, y, z) -> d
f1(x, y, z) -> c

Types:
plus :: 0':s:1' -> 0':s:1' -> 0':s:1'
ifPlus :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1'
isZero :: 0':s:1' -> true:false
inc :: 0':s:1' -> 0':s:1'
true :: true:false
p :: 0':s:1' -> 0':s:1'
false :: true:false
times :: 0':s:1' -> 0':s:1' -> 0':s:1'
timesIter :: 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
ge :: 0':s:1' -> 0':s:1' -> true:false
s :: 0':s:1' -> 0':s:1'
f0 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f1 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f2 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat -> 0':s:1'


Generator Equations:
gen_0':s:1'4_3(0) <=> 0'
gen_0':s:1'4_3(+(x, 1)) <=> s(gen_0':s:1'4_3(x))


The following defined symbols remain to be analysed:
isZero, plus, inc, p, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
isZero < plus
inc < plus
p < plus
plus < timesIter
inc < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
plus(x, y) -> ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) -> p(y)
ifPlus(false, x, y) -> plus(p(x), y)
times(x, y) -> timesIter(0', x, y, 0')
timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) -> z
ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))
inc(x) -> s(x)
p(0') -> 0'
p(s(x)) -> x
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
f0(0', y, x) -> f1(x, y, x)
f1(x, y, z) -> f2(x, y, z)
f2(x, 1', z) -> f0(x, z, z)
f0(x, y, z) -> d
f1(x, y, z) -> c

Types:
plus :: 0':s:1' -> 0':s:1' -> 0':s:1'
ifPlus :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1'
isZero :: 0':s:1' -> true:false
inc :: 0':s:1' -> 0':s:1'
true :: true:false
p :: 0':s:1' -> 0':s:1'
false :: true:false
times :: 0':s:1' -> 0':s:1' -> 0':s:1'
timesIter :: 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
ge :: 0':s:1' -> 0':s:1' -> true:false
s :: 0':s:1' -> 0':s:1'
f0 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f1 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f2 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat -> 0':s:1'


Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) -> false, rt in Omega(1 + n6_3)


Generator Equations:
gen_0':s:1'4_3(0) <=> 0'
gen_0':s:1'4_3(+(x, 1)) <=> s(gen_0':s:1'4_3(x))


The following defined symbols remain to be analysed:
inc, plus, p, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
inc < plus
p < plus
plus < timesIter
inc < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s:1'4_3(n163_3)) -> gen_0':s:1'4_3(+(1, n163_3)), rt in Omega(1 + n163_3)

Induction Base:
inc(gen_0':s:1'4_3(0)) ->_R^Omega(1)
s(0')

Induction Step:
inc(gen_0':s:1'4_3(+(n163_3, 1))) ->_R^Omega(1)
s(inc(gen_0':s:1'4_3(n163_3))) ->_IH
s(gen_0':s:1'4_3(+(1, c164_3)))

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

(14)
Obligation:
TRS:
Rules:
plus(x, y) -> ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) -> p(y)
ifPlus(false, x, y) -> plus(p(x), y)
times(x, y) -> timesIter(0', x, y, 0')
timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) -> z
ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))
inc(x) -> s(x)
p(0') -> 0'
p(s(x)) -> x
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
f0(0', y, x) -> f1(x, y, x)
f1(x, y, z) -> f2(x, y, z)
f2(x, 1', z) -> f0(x, z, z)
f0(x, y, z) -> d
f1(x, y, z) -> c

Types:
plus :: 0':s:1' -> 0':s:1' -> 0':s:1'
ifPlus :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1'
isZero :: 0':s:1' -> true:false
inc :: 0':s:1' -> 0':s:1'
true :: true:false
p :: 0':s:1' -> 0':s:1'
false :: true:false
times :: 0':s:1' -> 0':s:1' -> 0':s:1'
timesIter :: 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
ge :: 0':s:1' -> 0':s:1' -> true:false
s :: 0':s:1' -> 0':s:1'
f0 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f1 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f2 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat -> 0':s:1'


Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) -> false, rt in Omega(1 + n6_3)
inc(gen_0':s:1'4_3(n163_3)) -> gen_0':s:1'4_3(+(1, n163_3)), rt in Omega(1 + n163_3)


Generator Equations:
gen_0':s:1'4_3(0) <=> 0'
gen_0':s:1'4_3(+(x, 1)) <=> s(gen_0':s:1'4_3(x))


The following defined symbols remain to be analysed:
p, plus, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
p < plus
plus < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':s:1'4_3(+(2, n652_3))) -> *5_3, rt in Omega(n652_3)

Induction Base:
p(gen_0':s:1'4_3(+(2, 0)))

Induction Step:
p(gen_0':s:1'4_3(+(2, +(n652_3, 1)))) ->_R^Omega(1)
s(p(s(gen_0':s:1'4_3(+(1, n652_3))))) ->_IH
s(*5_3)

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

(16)
Obligation:
TRS:
Rules:
plus(x, y) -> ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) -> p(y)
ifPlus(false, x, y) -> plus(p(x), y)
times(x, y) -> timesIter(0', x, y, 0')
timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) -> z
ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))
inc(x) -> s(x)
p(0') -> 0'
p(s(x)) -> x
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
f0(0', y, x) -> f1(x, y, x)
f1(x, y, z) -> f2(x, y, z)
f2(x, 1', z) -> f0(x, z, z)
f0(x, y, z) -> d
f1(x, y, z) -> c

Types:
plus :: 0':s:1' -> 0':s:1' -> 0':s:1'
ifPlus :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1'
isZero :: 0':s:1' -> true:false
inc :: 0':s:1' -> 0':s:1'
true :: true:false
p :: 0':s:1' -> 0':s:1'
false :: true:false
times :: 0':s:1' -> 0':s:1' -> 0':s:1'
timesIter :: 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
ge :: 0':s:1' -> 0':s:1' -> true:false
s :: 0':s:1' -> 0':s:1'
f0 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f1 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f2 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat -> 0':s:1'


Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) -> false, rt in Omega(1 + n6_3)
inc(gen_0':s:1'4_3(n163_3)) -> gen_0':s:1'4_3(+(1, n163_3)), rt in Omega(1 + n163_3)
p(gen_0':s:1'4_3(+(2, n652_3))) -> *5_3, rt in Omega(n652_3)


Generator Equations:
gen_0':s:1'4_3(0) <=> 0'
gen_0':s:1'4_3(+(x, 1)) <=> s(gen_0':s:1'4_3(x))


The following defined symbols remain to be analysed:
plus, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s:1'4_3(n1702_3), gen_0':s:1'4_3(n1702_3)) -> true, rt in Omega(1 + n1702_3)

Induction Base:
ge(gen_0':s:1'4_3(0), gen_0':s:1'4_3(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s:1'4_3(+(n1702_3, 1)), gen_0':s:1'4_3(+(n1702_3, 1))) ->_R^Omega(1)
ge(gen_0':s:1'4_3(n1702_3), gen_0':s:1'4_3(n1702_3)) ->_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:
plus(x, y) -> ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) -> p(y)
ifPlus(false, x, y) -> plus(p(x), y)
times(x, y) -> timesIter(0', x, y, 0')
timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) -> z
ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
isZero(0') -> true
isZero(s(0')) -> false
isZero(s(s(x))) -> isZero(s(x))
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))
inc(x) -> s(x)
p(0') -> 0'
p(s(x)) -> x
p(s(s(x))) -> s(p(s(x)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
f0(0', y, x) -> f1(x, y, x)
f1(x, y, z) -> f2(x, y, z)
f2(x, 1', z) -> f0(x, z, z)
f0(x, y, z) -> d
f1(x, y, z) -> c

Types:
plus :: 0':s:1' -> 0':s:1' -> 0':s:1'
ifPlus :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1'
isZero :: 0':s:1' -> true:false
inc :: 0':s:1' -> 0':s:1'
true :: true:false
p :: 0':s:1' -> 0':s:1'
false :: true:false
times :: 0':s:1' -> 0':s:1' -> 0':s:1'
timesIter :: 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1' -> 0':s:1'
ge :: 0':s:1' -> 0':s:1' -> true:false
s :: 0':s:1' -> 0':s:1'
f0 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f1 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
f2 :: 0':s:1' -> 0':s:1' -> 0':s:1' -> d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat -> 0':s:1'


Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) -> false, rt in Omega(1 + n6_3)
inc(gen_0':s:1'4_3(n163_3)) -> gen_0':s:1'4_3(+(1, n163_3)), rt in Omega(1 + n163_3)
p(gen_0':s:1'4_3(+(2, n652_3))) -> *5_3, rt in Omega(n652_3)
ge(gen_0':s:1'4_3(n1702_3), gen_0':s:1'4_3(n1702_3)) -> true, rt in Omega(1 + n1702_3)


Generator Equations:
gen_0':s:1'4_3(0) <=> 0'
gen_0':s:1'4_3(+(x, 1)) <=> s(gen_0':s:1'4_3(x))


The following defined symbols remain to be analysed:
timesIter, f0, f1, f2

They will be analysed ascendingly in the following order:
f0 = f1
f0 = f2
f1 = f2