/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 239 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), 57 ms]
        (14) 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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(neq(x, 0), y, y)
   cond2(false, x, y) -> cond1(neq(x, 0), p(x), y)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   neq(0, 0) -> false
   neq(0, s(x)) -> true
   neq(s(x), 0) -> true
   neq(s(x), s(y)) -> neq(x, y)
   p(0) -> 0
   p(s(x)) -> x

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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
   cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   neq(0', 0') -> false
   neq(0', s(x)) -> true
   neq(s(x), 0') -> true
   neq(s(x), s(y)) -> neq(x, y)
   p(0') -> 0'
   p(s(x)) -> x

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

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

(4)
Obligation:
TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
neq :: 0':s -> 0':s -> true:false
0' :: 0':s
false :: true:false
p :: 0':s -> 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr, neq

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
neq < cond2

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

(6)
Obligation:
TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
neq :: 0':s -> 0':s -> true:false
0' :: 0':s
false :: true:false
p :: 0':s -> 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2, neq

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
neq < cond2

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
false

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_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:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
neq :: 0':s -> 0':s -> true:false
0' :: 0':s
false :: true:false
p :: 0':s -> 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2, neq

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
neq < cond2

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
neq :: 0':s -> 0':s -> true:false
0' :: 0':s
false :: true:false
p :: 0':s -> 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
neq, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
neq < cond2

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
neq(gen_0':s4_0(n265_0), gen_0':s4_0(n265_0)) -> false, rt in Omega(1 + n265_0)

Induction Base:
neq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
false

Induction Step:
neq(gen_0':s4_0(+(n265_0, 1)), gen_0':s4_0(+(n265_0, 1))) ->_R^Omega(1)
neq(gen_0':s4_0(n265_0), gen_0':s4_0(n265_0)) ->_IH
false

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

(14)
Obligation:
TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
neq :: 0':s -> 0':s -> true:false
0' :: 0':s
false :: true:false
p :: 0':s -> 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
neq(gen_0':s4_0(n265_0), gen_0':s4_0(n265_0)) -> false, rt in Omega(1 + n265_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2