/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^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 286 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), 82 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^2, INF)
            (18) typed CpxTrs
                (19) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
                (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 9 ms]
                (22) 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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(x, *(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0)))))

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:

   +'(0', y) -> y
   +'(s(x), y) -> s(+'(x, y))
   *'(x, 0') -> 0'
   *'(x, s(y)) -> +'(x, *'(x, y))
   twice(0') -> 0'
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

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

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

(4)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', *', twice, -, f

They will be analysed ascendingly in the following order:
+' < *'
+' < f
*' < f
- < f

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

(6)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
+', *', twice, -, f

They will be analysed ascendingly in the following order:
+' < *'
+' < f
*' < f
- < f

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)

Induction Base:
+'(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
gen_0':s3_0(b)

Induction Step:
+'(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(+'(gen_0':s3_0(n5_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c6_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:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
+', *', twice, -, f

They will be analysed ascendingly in the following order:
+' < *'
+' < f
*' < f
- < f

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
*', twice, -, f

They will be analysed ascendingly in the following order:
*' < f
- < f

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)

Induction Base:
*'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n494_0, 1))) ->_R^Omega(1)
+'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n494_0))) ->_IH
+'(gen_0':s3_0(a), gen_0':s3_0(*(c495_0, a))) ->_L^Omega(1 + a)
gen_0':s3_0(+(a, *(n494_0, a)))

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
*', twice, -, f

They will be analysed ascendingly in the following order:
*' < f
- < f

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
twice, -, f

They will be analysed ascendingly in the following order:
- < f

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
twice(gen_0':s3_0(n1110_0)) -> gen_0':s3_0(*(2, n1110_0)), rt in Omega(1 + n1110_0)

Induction Base:
twice(gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
twice(gen_0':s3_0(+(n1110_0, 1))) ->_R^Omega(1)
s(s(twice(gen_0':s3_0(n1110_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c1111_0))))

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

(20)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)
twice(gen_0':s3_0(n1110_0)) -> gen_0':s3_0(*(2, n1110_0)), rt in Omega(1 + n1110_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


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

They will be analysed ascendingly in the following order:
- < f

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n1358_0), gen_0':s3_0(n1358_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1358_0)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n1358_0, 1)), gen_0':s3_0(+(n1358_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n1358_0), gen_0':s3_0(n1358_0)) ->_IH
gen_0':s3_0(0)

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

(22)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(x, *'(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
f(s(x)) -> f(-(*'(s(s(x)), s(s(x))), +'(*'(s(x), s(s(x))), s(s(0')))))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
f :: 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
*'(gen_0':s3_0(a), gen_0':s3_0(n494_0)) -> gen_0':s3_0(*(n494_0, a)), rt in Omega(1 + a*n494_0 + n494_0)
twice(gen_0':s3_0(n1110_0)) -> gen_0':s3_0(*(2, n1110_0)), rt in Omega(1 + n1110_0)
-(gen_0':s3_0(n1358_0), gen_0':s3_0(n1358_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1358_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
f