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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 278 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), 1199 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 2061 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 2370 ms]
        (18) BOUNDS(1, INF)


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

(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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   f(0) -> s(0)
   f(s(x)) -> minus(s(x), g(f(x)))
   g(0) -> 0
   g(s(x)) -> minus(s(x), f(g(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:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   f(0') -> s(0')
   f(s(x)) -> minus(s(x), g(f(x)))
   g(0') -> 0'
   g(s(x)) -> minus(s(x), f(g(x)))

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

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

(4)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
f(0') -> s(0')
f(s(x)) -> minus(s(x), g(f(x)))
g(0') -> 0'
g(s(x)) -> minus(s(x), f(g(x)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s
g :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, f, g

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

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

(6)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
f(0') -> s(0')
f(s(x)) -> minus(s(x), g(f(x)))
g(0') -> 0'
g(s(x)) -> minus(s(x), f(g(x)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s
g :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)

Induction Base:
minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(0)

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
gen_0':s2_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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
f(0') -> s(0')
f(s(x)) -> minus(s(x), g(f(x)))
g(0') -> 0'
g(s(x)) -> minus(s(x), f(g(x)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s
g :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
f(0') -> s(0')
f(s(x)) -> minus(s(x), g(f(x)))
g(0') -> 0'
g(s(x)) -> minus(s(x), f(g(x)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s
g :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':s2_0(+(1, n216_0))) -> *3_0, rt in Omega(n216_0)

Induction Base:
g(gen_0':s2_0(+(1, 0)))

Induction Step:
g(gen_0':s2_0(+(1, +(n216_0, 1)))) ->_R^Omega(1)
minus(s(gen_0':s2_0(+(1, n216_0))), f(g(gen_0':s2_0(+(1, n216_0))))) ->_IH
minus(s(gen_0':s2_0(+(1, n216_0))), f(*3_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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
f(0') -> s(0')
f(s(x)) -> minus(s(x), g(f(x)))
g(0') -> 0'
g(s(x)) -> minus(s(x), f(g(x)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s
g :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
g(gen_0':s2_0(+(1, n216_0))) -> *3_0, rt in Omega(n216_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
f

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':s2_0(+(1, n3745_0))) -> *3_0, rt in Omega(n3745_0)

Induction Base:
f(gen_0':s2_0(+(1, 0)))

Induction Step:
f(gen_0':s2_0(+(1, +(n3745_0, 1)))) ->_R^Omega(1)
minus(s(gen_0':s2_0(+(1, n3745_0))), g(f(gen_0':s2_0(+(1, n3745_0))))) ->_IH
minus(s(gen_0':s2_0(+(1, n3745_0))), g(*3_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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
f(0') -> s(0')
f(s(x)) -> minus(s(x), g(f(x)))
g(0') -> 0'
g(s(x)) -> minus(s(x), f(g(x)))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s
g :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
g(gen_0':s2_0(+(1, n216_0))) -> *3_0, rt in Omega(n216_0)
f(gen_0':s2_0(+(1, n3745_0))) -> *3_0, rt in Omega(n3745_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
g

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':s2_0(+(1, n63581_0))) -> *3_0, rt in Omega(n63581_0)

Induction Base:
g(gen_0':s2_0(+(1, 0)))

Induction Step:
g(gen_0':s2_0(+(1, +(n63581_0, 1)))) ->_R^Omega(1)
minus(s(gen_0':s2_0(+(1, n63581_0))), f(g(gen_0':s2_0(+(1, n63581_0))))) ->_IH
minus(s(gen_0':s2_0(+(1, n63581_0))), f(*3_0))

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

(18)
BOUNDS(1, INF)