WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 36 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (8) BOUNDS(1, n^1)
(9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 370 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 105 ms]
            (22) BOUNDS(1, INF)


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(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(g(X)) -> f(X)

S is empty.
Rewrite Strategy: INNERMOST
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(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))


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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(g(X)) -> f(X)

The (relative) TRS S consists of the following rules:

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(g(X)) -> f(X)

The (relative) TRS S consists of the following rules:

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(g(X)) -> f(X)
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

S is empty.
Rewrite Strategy: INNERMOST
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(7) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
The certificate found is represented by the following graph.

"[5, 6, 11]
{(5,6,[f_1|0, encArg_1|0, encode_f_1|0, encode_g_1|0, f_1|1]), (5,11,[g_1|1, f_1|1, f_1|2]), (6,6,[g_1|0, cons_f_1|0]), (11,6,[encArg_1|1]), (11,11,[g_1|1, f_1|1, f_1|2])}"
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(8)
BOUNDS(1, n^1)

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(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(g(X)) -> f(X)

The (relative) TRS S consists of the following rules:

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(12)
Obligation:
Innermost TRS:
Rules:
f(g(X)) -> f(X)
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: g:cons_f -> g:cons_f
g :: g:cons_f -> g:cons_f
encArg :: g:cons_f -> g:cons_f
cons_f :: g:cons_f -> g:cons_f
encode_f :: g:cons_f -> g:cons_f
encode_g :: g:cons_f -> g:cons_f
hole_g:cons_f1_0 :: g:cons_f
gen_g:cons_f2_0 :: Nat -> g:cons_f

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(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, encArg

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

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(14)
Obligation:
Innermost TRS:
Rules:
f(g(X)) -> f(X)
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: g:cons_f -> g:cons_f
g :: g:cons_f -> g:cons_f
encArg :: g:cons_f -> g:cons_f
cons_f :: g:cons_f -> g:cons_f
encode_f :: g:cons_f -> g:cons_f
encode_g :: g:cons_f -> g:cons_f
hole_g:cons_f1_0 :: g:cons_f
gen_g:cons_f2_0 :: Nat -> g:cons_f


Generator Equations:
gen_g:cons_f2_0(0) <=> hole_g:cons_f1_0
gen_g:cons_f2_0(+(x, 1)) <=> g(gen_g:cons_f2_0(x))


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

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

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_g:cons_f2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
f(gen_g:cons_f2_0(+(1, 0)))

Induction Step:
f(gen_g:cons_f2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
f(gen_g:cons_f2_0(+(1, n4_0))) ->_IH
*3_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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(16)
Complex Obligation (BEST)

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(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(g(X)) -> f(X)
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: g:cons_f -> g:cons_f
g :: g:cons_f -> g:cons_f
encArg :: g:cons_f -> g:cons_f
cons_f :: g:cons_f -> g:cons_f
encode_f :: g:cons_f -> g:cons_f
encode_g :: g:cons_f -> g:cons_f
hole_g:cons_f1_0 :: g:cons_f
gen_g:cons_f2_0 :: Nat -> g:cons_f


Generator Equations:
gen_g:cons_f2_0(0) <=> hole_g:cons_f1_0
gen_g:cons_f2_0(+(x, 1)) <=> g(gen_g:cons_f2_0(x))


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

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

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(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(19)
BOUNDS(n^1, INF)

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(20)
Obligation:
Innermost TRS:
Rules:
f(g(X)) -> f(X)
encArg(g(x_1)) -> g(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: g:cons_f -> g:cons_f
g :: g:cons_f -> g:cons_f
encArg :: g:cons_f -> g:cons_f
cons_f :: g:cons_f -> g:cons_f
encode_f :: g:cons_f -> g:cons_f
encode_g :: g:cons_f -> g:cons_f
hole_g:cons_f1_0 :: g:cons_f
gen_g:cons_f2_0 :: Nat -> g:cons_f


Lemmas:
f(gen_g:cons_f2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_g:cons_f2_0(0) <=> hole_g:cons_f1_0
gen_g:cons_f2_0(+(x, 1)) <=> g(gen_g:cons_f2_0(x))


The following defined symbols remain to be analysed:
encArg
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_g:cons_f2_0(+(1, n149_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_g:cons_f2_0(+(1, 0)))

Induction Step:
encArg(gen_g:cons_f2_0(+(1, +(n149_0, 1)))) ->_R^Omega(0)
g(encArg(gen_g:cons_f2_0(+(1, n149_0)))) ->_IH
g(*3_0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(22)
BOUNDS(1, INF)