/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 252 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), 85 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 34 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 588 ms]
        (20) BEST
            (21) proven lower bound
                (22) LowerBoundPropagationProof [FINISHED, 0 ms]
                (23) BOUNDS(n^2, INF)
            (24) typed CpxTrs
                (25) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
                (26) 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:

   g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0))), k(n(s(x), s(y)), s(s(0))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
   n(0, y) -> 0
   n(x, 0) -> 0
   n(s(x), s(y)) -> s(n(x, y))
   m(0, y) -> y
   m(x, 0) -> x
   m(s(x), s(y)) -> s(m(x, y))
   k(0, s(y)) -> 0
   k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
   t(x) -> p(x, x)
   p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
   p(0, y) -> y
   p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   and(x, false) -> false
   and(true, true) -> true
   f(0) -> true
   f(s(x)) -> h(x)
   h(0) -> false
   h(s(x)) -> f(x)
   gt(s(x), 0) -> true
   gt(0, y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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:

   g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
   n(0', y) -> 0'
   n(x, 0') -> 0'
   n(s(x), s(y)) -> s(n(x, y))
   m(0', y) -> y
   m(x, 0') -> x
   m(s(x), s(y)) -> s(m(x, y))
   k(0', s(y)) -> 0'
   k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
   t(x) -> p(x, x)
   p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
   p(0', y) -> y
   p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   and(x, false) -> false
   and(true, true) -> true
   f(0') -> true
   f(s(x)) -> h(x)
   h(0') -> false
   h(s(x)) -> f(x)
   gt(s(x), 0') -> true
   gt(0', y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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

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

(4)
Obligation:
TRS:
Rules:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g, f, k, minus, m, n, p, gt, h

They will be analysed ascendingly in the following order:
f < g
k < g
minus < g
m < g
n < g
f = h
minus < k
gt < p

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

(6)
Obligation:
TRS:
Rules:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
minus, g, f, k, m, n, p, gt, h

They will be analysed ascendingly in the following order:
f < g
k < g
minus < g
m < g
n < g
f = h
minus < k
gt < p

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)

Induction Base:
minus(gen_s:0':true:false2_0(0), gen_s:0':true:false2_0(0)) ->_R^Omega(1)
gen_s:0':true:false2_0(0)

Induction Step:
minus(gen_s:0':true:false2_0(+(n4_0, 1)), gen_s:0':true:false2_0(+(n4_0, 1))) ->_R^Omega(1)
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) ->_IH
gen_s:0':true:false2_0(0)

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:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
minus, g, f, k, m, n, p, gt, h

They will be analysed ascendingly in the following order:
f < g
k < g
minus < g
m < g
n < g
f = h
minus < k
gt < p

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Lemmas:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
k, g, f, m, n, p, gt, h

They will be analysed ascendingly in the following order:
f < g
k < g
m < g
n < g
f = h
gt < p

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0)) -> gen_s:0':true:false2_0(n472_0), rt in Omega(1 + n472_0)

Induction Base:
m(gen_s:0':true:false2_0(0), gen_s:0':true:false2_0(0)) ->_R^Omega(1)
gen_s:0':true:false2_0(0)

Induction Step:
m(gen_s:0':true:false2_0(+(n472_0, 1)), gen_s:0':true:false2_0(+(n472_0, 1))) ->_R^Omega(1)
s(m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0))) ->_IH
s(gen_s:0':true:false2_0(c473_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:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Lemmas:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)
m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0)) -> gen_s:0':true:false2_0(n472_0), rt in Omega(1 + n472_0)


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
n, g, f, p, gt, h

They will be analysed ascendingly in the following order:
f < g
n < g
f = h
gt < p

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
n(gen_s:0':true:false2_0(n1010_0), gen_s:0':true:false2_0(n1010_0)) -> gen_s:0':true:false2_0(n1010_0), rt in Omega(1 + n1010_0)

Induction Base:
n(gen_s:0':true:false2_0(0), gen_s:0':true:false2_0(0)) ->_R^Omega(1)
0'

Induction Step:
n(gen_s:0':true:false2_0(+(n1010_0, 1)), gen_s:0':true:false2_0(+(n1010_0, 1))) ->_R^Omega(1)
s(n(gen_s:0':true:false2_0(n1010_0), gen_s:0':true:false2_0(n1010_0))) ->_IH
s(gen_s:0':true:false2_0(c1011_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:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Lemmas:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)
m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0)) -> gen_s:0':true:false2_0(n472_0), rt in Omega(1 + n472_0)
n(gen_s:0':true:false2_0(n1010_0), gen_s:0':true:false2_0(n1010_0)) -> gen_s:0':true:false2_0(n1010_0), rt in Omega(1 + n1010_0)


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
gt, g, f, p, h

They will be analysed ascendingly in the following order:
f < g
f = h
gt < p

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0':true:false2_0(+(1, n1446_0)), gen_s:0':true:false2_0(n1446_0)) -> true, rt in Omega(1 + n1446_0)

Induction Base:
gt(gen_s:0':true:false2_0(+(1, 0)), gen_s:0':true:false2_0(0)) ->_R^Omega(1)
true

Induction Step:
gt(gen_s:0':true:false2_0(+(1, +(n1446_0, 1))), gen_s:0':true:false2_0(+(n1446_0, 1))) ->_R^Omega(1)
gt(gen_s:0':true:false2_0(+(1, n1446_0)), gen_s:0':true:false2_0(n1446_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:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Lemmas:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)
m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0)) -> gen_s:0':true:false2_0(n472_0), rt in Omega(1 + n472_0)
n(gen_s:0':true:false2_0(n1010_0), gen_s:0':true:false2_0(n1010_0)) -> gen_s:0':true:false2_0(n1010_0), rt in Omega(1 + n1010_0)
gt(gen_s:0':true:false2_0(+(1, n1446_0)), gen_s:0':true:false2_0(n1446_0)) -> true, rt in Omega(1 + n1446_0)


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
p, g, f, h

They will be analysed ascendingly in the following order:
f < g
f = h

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0))) -> *3_0, rt in Omega(n1825_0 + n1825_0^2)

Induction Base:
p(gen_s:0':true:false2_0(+(2, 0)), gen_s:0':true:false2_0(+(1, 0)))

Induction Step:
p(gen_s:0':true:false2_0(+(2, +(n1825_0, 1))), gen_s:0':true:false2_0(+(1, +(n1825_0, 1)))) ->_R^Omega(1)
s(s(p(if(gt(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0))), gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0))), if(not(gt(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0)))), id(gen_s:0':true:false2_0(+(2, n1825_0))), id(gen_s:0':true:false2_0(+(1, n1825_0))))))) ->_L^Omega(2 + n1825_0)
s(s(p(if(true, gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0))), if(not(gt(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0)))), id(gen_s:0':true:false2_0(+(2, n1825_0))), id(gen_s:0':true:false2_0(+(1, n1825_0))))))) ->_R^Omega(1)
s(s(p(gen_s:0':true:false2_0(+(2, n1825_0)), if(not(gt(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0)))), id(gen_s:0':true:false2_0(+(2, n1825_0))), id(gen_s:0':true:false2_0(+(1, n1825_0))))))) ->_L^Omega(2 + n1825_0)
s(s(p(gen_s:0':true:false2_0(+(2, n1825_0)), if(not(true), id(gen_s:0':true:false2_0(+(2, n1825_0))), id(gen_s:0':true:false2_0(+(1, n1825_0))))))) ->_R^Omega(1)
s(s(p(gen_s:0':true:false2_0(+(2, n1825_0)), if(if(true, false, true), id(gen_s:0':true:false2_0(+(2, n1825_0))), id(gen_s:0':true:false2_0(+(1, n1825_0))))))) ->_R^Omega(1)
s(s(p(gen_s:0':true:false2_0(+(2, n1825_0)), if(false, id(gen_s:0':true:false2_0(+(2, n1825_0))), id(gen_s:0':true:false2_0(+(1, n1825_0))))))) ->_R^Omega(1)
s(s(p(gen_s:0':true:false2_0(+(2, n1825_0)), if(false, gen_s:0':true:false2_0(+(2, n1825_0)), id(gen_s:0':true:false2_0(+(1, n1825_0))))))) ->_R^Omega(1)
s(s(p(gen_s:0':true:false2_0(+(2, n1825_0)), if(false, gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0)))))) ->_R^Omega(1)
s(s(p(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0))))) ->_IH
s(s(*3_0))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Lemmas:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)
m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0)) -> gen_s:0':true:false2_0(n472_0), rt in Omega(1 + n472_0)
n(gen_s:0':true:false2_0(n1010_0), gen_s:0':true:false2_0(n1010_0)) -> gen_s:0':true:false2_0(n1010_0), rt in Omega(1 + n1010_0)
gt(gen_s:0':true:false2_0(+(1, n1446_0)), gen_s:0':true:false2_0(n1446_0)) -> true, rt in Omega(1 + n1446_0)


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
p, g, f, h

They will be analysed ascendingly in the following order:
f < g
f = h

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

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

(23)
BOUNDS(n^2, INF)

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

(24)
Obligation:
TRS:
Rules:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Lemmas:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)
m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0)) -> gen_s:0':true:false2_0(n472_0), rt in Omega(1 + n472_0)
n(gen_s:0':true:false2_0(n1010_0), gen_s:0':true:false2_0(n1010_0)) -> gen_s:0':true:false2_0(n1010_0), rt in Omega(1 + n1010_0)
gt(gen_s:0':true:false2_0(+(1, n1446_0)), gen_s:0':true:false2_0(n1446_0)) -> true, rt in Omega(1 + n1446_0)
p(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0))) -> *3_0, rt in Omega(n1825_0 + n1825_0^2)


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
h, g, f

They will be analysed ascendingly in the following order:
f < g
f = h

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_s:0':true:false2_0(*(2, n11616_0))) -> false, rt in Omega(1 + n11616_0)

Induction Base:
h(gen_s:0':true:false2_0(*(2, 0))) ->_R^Omega(1)
false

Induction Step:
h(gen_s:0':true:false2_0(*(2, +(n11616_0, 1)))) ->_R^Omega(1)
f(gen_s:0':true:false2_0(+(1, *(2, n11616_0)))) ->_R^Omega(1)
h(gen_s:0':true:false2_0(*(2, n11616_0))) ->_IH
false

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

(26)
Obligation:
TRS:
Rules:
g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0'))), k(n(s(x), s(y)), s(s(0'))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))
n(0', y) -> 0'
n(x, 0') -> 0'
n(s(x), s(y)) -> s(n(x, y))
m(0', y) -> y
m(x, 0') -> x
m(s(x), s(y)) -> s(m(x, y))
k(0', s(y)) -> 0'
k(s(x), s(y)) -> s(k(minus(x, y), s(y)))
t(x) -> p(x, x)
p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x))
p(0', y) -> y
p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y))))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
id(x) -> x
if(true, x, y) -> x
if(false, x, y) -> y
not(x) -> if(x, false, true)
and(x, false) -> false
and(true, true) -> true
f(0') -> true
f(s(x)) -> h(x)
h(0') -> false
h(s(x)) -> f(x)
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)

Types:
g :: s:0':true:false -> s:0':true:false -> s:0':true:false
s :: s:0':true:false -> s:0':true:false
if :: s:0':true:false -> s:0':true:false -> s:0':true:false -> s:0':true:false
and :: s:0':true:false -> s:0':true:false -> s:0':true:false
f :: s:0':true:false -> s:0':true:false
t :: s:0':true:false -> s:0':true:false
k :: s:0':true:false -> s:0':true:false -> s:0':true:false
minus :: s:0':true:false -> s:0':true:false -> s:0':true:false
m :: s:0':true:false -> s:0':true:false -> s:0':true:false
n :: s:0':true:false -> s:0':true:false -> s:0':true:false
0' :: s:0':true:false
p :: s:0':true:false -> s:0':true:false -> s:0':true:false
gt :: s:0':true:false -> s:0':true:false -> s:0':true:false
not :: s:0':true:false -> s:0':true:false
id :: s:0':true:false -> s:0':true:false
true :: s:0':true:false
false :: s:0':true:false
h :: s:0':true:false -> s:0':true:false
hole_s:0':true:false1_0 :: s:0':true:false
gen_s:0':true:false2_0 :: Nat -> s:0':true:false


Lemmas:
minus(gen_s:0':true:false2_0(n4_0), gen_s:0':true:false2_0(n4_0)) -> gen_s:0':true:false2_0(0), rt in Omega(1 + n4_0)
m(gen_s:0':true:false2_0(n472_0), gen_s:0':true:false2_0(n472_0)) -> gen_s:0':true:false2_0(n472_0), rt in Omega(1 + n472_0)
n(gen_s:0':true:false2_0(n1010_0), gen_s:0':true:false2_0(n1010_0)) -> gen_s:0':true:false2_0(n1010_0), rt in Omega(1 + n1010_0)
gt(gen_s:0':true:false2_0(+(1, n1446_0)), gen_s:0':true:false2_0(n1446_0)) -> true, rt in Omega(1 + n1446_0)
p(gen_s:0':true:false2_0(+(2, n1825_0)), gen_s:0':true:false2_0(+(1, n1825_0))) -> *3_0, rt in Omega(n1825_0 + n1825_0^2)
h(gen_s:0':true:false2_0(*(2, n11616_0))) -> false, rt in Omega(1 + n11616_0)


Generator Equations:
gen_s:0':true:false2_0(0) <=> 0'
gen_s:0':true:false2_0(+(x, 1)) <=> s(gen_s:0':true:false2_0(x))


The following defined symbols remain to be analysed:
f, g

They will be analysed ascendingly in the following order:
f < g
f = h