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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 287 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), 32 ms]
        (14) typed CpxTrs


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(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:

   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0, x) -> x
   add(s(x), y) -> s(add(x, y))

S is empty.
Rewrite Strategy: FULL
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(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:

   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0', x) -> x
   add(s(x), y) -> s(add(x, y))

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

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

(4)
Obligation:
TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond, gr, add

They will be analysed ascendingly in the following order:
gr < cond
add < cond

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(6)
Obligation:
TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
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, cond, add

They will be analysed ascendingly in the following order:
gr < cond
add < cond

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(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).
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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
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, cond, add

They will be analysed ascendingly in the following order:
gr < cond
add < cond

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
TRS:
Rules:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
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:
add, cond

They will be analysed ascendingly in the following order:
add < cond

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s4_0(n229_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n229_0, b)), rt in Omega(1 + n229_0)

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

Induction Step:
add(gen_0':s4_0(+(n229_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
s(add(gen_0':s4_0(n229_0), gen_0':s4_0(b))) ->_IH
s(gen_0':s4_0(+(b, c230_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:
cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))

Types:
cond :: true:false -> 0':s -> 0':s -> cond
true :: true:false
gr :: 0':s -> 0':s -> true:false
add :: 0':s -> 0':s -> 0':s
0' :: 0':s
false :: true:false
s :: 0':s -> 0':s
hole_cond1_0 :: cond
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)
add(gen_0':s4_0(n229_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n229_0, b)), rt in Omega(1 + n229_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:
cond