/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^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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), 209 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), 75 ms]
        (14) proven lower bound
        (15) LowerBoundPropagationProof [FINISHED, 0 ms]
        (16) BOUNDS(n^2, INF)


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(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:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   U21(tt, M, N) -> U22(tt, activate(M), activate(N))
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)
   activate(X) -> X

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.
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(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:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   U21(tt, M, N) -> U22(tt, activate(M), activate(N))
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
   plus(N, 0') -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0') -> 0'
   x(N, s(M)) -> U21(tt, M, N)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
U21(tt, M, N) -> U22(tt, activate(M), activate(N))
U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
x(N, 0') -> 0'
x(N, s(M)) -> U21(tt, M, N)
activate(X) -> X

Types:
U11 :: tt -> s:0' -> s:0' -> s:0'
tt :: tt
U12 :: tt -> s:0' -> s:0' -> s:0'
activate :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
U21 :: tt -> s:0' -> s:0' -> s:0'
U22 :: tt -> s:0' -> s:0' -> s:0'
x :: s:0' -> s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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(6)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
U21(tt, M, N) -> U22(tt, activate(M), activate(N))
U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
x(N, 0') -> 0'
x(N, s(M)) -> U21(tt, M, N)
activate(X) -> X

Types:
U11 :: tt -> s:0' -> s:0' -> s:0'
tt :: tt
U12 :: tt -> s:0' -> s:0' -> s:0'
activate :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
U21 :: tt -> s:0' -> s:0' -> s:0'
U22 :: tt -> s:0' -> s:0' -> s:0'
x :: s:0' -> s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
U11(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) ->_R^Omega(1)
U12(tt, activate(gen_s:0'3_0(n5_0)), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U12(tt, gen_s:0'3_0(n5_0), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U12(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) ->_R^Omega(1)
s(plus(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n5_0)))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), activate(gen_s:0'3_0(n5_0)))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0))) ->_IH
s(gen_s:0'3_0(+(a, c6_0)))

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:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
U21(tt, M, N) -> U22(tt, activate(M), activate(N))
U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
x(N, 0') -> 0'
x(N, s(M)) -> U21(tt, M, N)
activate(X) -> X

Types:
U11 :: tt -> s:0' -> s:0' -> s:0'
tt :: tt
U12 :: tt -> s:0' -> s:0' -> s:0'
activate :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
U21 :: tt -> s:0' -> s:0' -> s:0'
U22 :: tt -> s:0' -> s:0' -> s:0'
x :: s:0' -> s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
plus, x

They will be analysed ascendingly in the following order:
plus < x

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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:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
U21(tt, M, N) -> U22(tt, activate(M), activate(N))
U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
x(N, 0') -> 0'
x(N, s(M)) -> U21(tt, M, N)
activate(X) -> X

Types:
U11 :: tt -> s:0' -> s:0' -> s:0'
tt :: tt
U12 :: tt -> s:0' -> s:0' -> s:0'
activate :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
U21 :: tt -> s:0' -> s:0' -> s:0'
U22 :: tt -> s:0' -> s:0' -> s:0'
x :: s:0' -> s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
x
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) -> gen_s:0'3_0(*(n580_0, a)), rt in Omega(1 + a*n580_0 + n580_0)

Induction Base:
x(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
0'

Induction Step:
x(gen_s:0'3_0(a), gen_s:0'3_0(+(n580_0, 1))) ->_R^Omega(1)
U21(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) ->_R^Omega(1)
U22(tt, activate(gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U22(tt, gen_s:0'3_0(n580_0), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
U22(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) ->_R^Omega(1)
plus(x(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
plus(x(gen_s:0'3_0(a), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
plus(x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) ->_IH
plus(gen_s:0'3_0(*(c581_0, a)), activate(gen_s:0'3_0(a))) ->_R^Omega(1)
plus(gen_s:0'3_0(*(n580_0, a)), gen_s:0'3_0(a)) ->_L^Omega(1 + a)
gen_s:0'3_0(+(a, *(n580_0, a)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(14)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
U21(tt, M, N) -> U22(tt, activate(M), activate(N))
U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
x(N, 0') -> 0'
x(N, s(M)) -> U21(tt, M, N)
activate(X) -> X

Types:
U11 :: tt -> s:0' -> s:0' -> s:0'
tt :: tt
U12 :: tt -> s:0' -> s:0' -> s:0'
activate :: s:0' -> s:0'
s :: s:0' -> s:0'
plus :: s:0' -> s:0' -> s:0'
U21 :: tt -> s:0' -> s:0' -> s:0'
U22 :: tt -> s:0' -> s:0' -> s:0'
x :: s:0' -> s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) -> gen_s:0'3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
x
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(15) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(16)
BOUNDS(n^2, INF)