WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) 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), 46 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsProof [FINISHED, 9 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), 217 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), 41 ms]
            (22) BOUNDS(1, INF)


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


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

S is empty.
Rewrite Strategy: FULL
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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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))


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


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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


The TRS R consists of the following rules:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

(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 4. 
The certificate found is represented by the following graph.

"[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
{(5,6,[f_1|0, encArg_1|0, encode_f_1|0, encode_0|0, encode_s_1|0, 0|1]), (5,7,[s_1|1, f_1|1]), (5,8,[f_1|1]), (5,10,[s_1|2]), (5,12,[s_1|2]), (5,13,[f_1|2]), (5,17,[s_1|3]), (6,6,[0|0, s_1|0, cons_f_1|0]), (7,6,[0|1, encArg_1|1]), (7,7,[s_1|1, f_1|1]), (7,10,[s_1|2]), (7,13,[f_1|2]), (7,12,[s_1|2]), (7,17,[s_1|3]), (8,9,[f_1|1]), (8,11,[s_1|1]), (8,8,[f_1|1]), (8,12,[s_1|2]), (9,6,[s_1|1]), (10,6,[0|2]), (11,6,[0|1]), (12,6,[0|2]), (13,14,[f_1|2]), (13,12,[s_1|2]), (13,13,[f_1|2]), (13,15,[f_1|3]), (13,17,[s_1|3]), (13,18,[s_1|4]), (14,7,[s_1|2]), (14,10,[s_1|2]), (14,12,[s_1|2]), (14,17,[s_1|2]), (15,16,[f_1|3]), (15,17,[s_1|3]), (15,18,[s_1|4]), (16,10,[s_1|3]), (16,12,[s_1|3]), (16,17,[s_1|3]), (17,6,[0|3]), (18,6,[0|4])}"
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(0') -> s(0')
   f(s(0')) -> s(0')
   f(s(s(x))) -> f(f(s(x)))

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: FULL
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(12)
Obligation:
TRS:
Rules:
f(0') -> s(0')
f(s(0')) -> s(0')
f(s(s(x))) -> f(f(s(x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_f -> 0':s:cons_f
0' :: 0':s:cons_f
s :: 0':s:cons_f -> 0':s:cons_f
encArg :: 0':s:cons_f -> 0':s:cons_f
cons_f :: 0':s:cons_f -> 0':s:cons_f
encode_f :: 0':s:cons_f -> 0':s:cons_f
encode_0 :: 0':s:cons_f
encode_s :: 0':s:cons_f -> 0':s:cons_f
hole_0':s:cons_f1_2 :: 0':s:cons_f
gen_0':s:cons_f2_2 :: Nat -> 0':s: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:
TRS:
Rules:
f(0') -> s(0')
f(s(0')) -> s(0')
f(s(s(x))) -> f(f(s(x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_f -> 0':s:cons_f
0' :: 0':s:cons_f
s :: 0':s:cons_f -> 0':s:cons_f
encArg :: 0':s:cons_f -> 0':s:cons_f
cons_f :: 0':s:cons_f -> 0':s:cons_f
encode_f :: 0':s:cons_f -> 0':s:cons_f
encode_0 :: 0':s:cons_f
encode_s :: 0':s:cons_f -> 0':s:cons_f
hole_0':s:cons_f1_2 :: 0':s:cons_f
gen_0':s:cons_f2_2 :: Nat -> 0':s:cons_f


Generator Equations:
gen_0':s:cons_f2_2(0) <=> 0'
gen_0':s:cons_f2_2(+(x, 1)) <=> s(gen_0':s:cons_f2_2(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_0':s:cons_f2_2(+(1, n4_2))) -> gen_0':s:cons_f2_2(1), rt in Omega(1 + n4_2)

Induction Base:
f(gen_0':s:cons_f2_2(+(1, 0))) ->_R^Omega(1)
s(0')

Induction Step:
f(gen_0':s:cons_f2_2(+(1, +(n4_2, 1)))) ->_R^Omega(1)
f(f(s(gen_0':s:cons_f2_2(n4_2)))) ->_IH
f(gen_0':s:cons_f2_2(1)) ->_R^Omega(1)
s(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:

TRS:
Rules:
f(0') -> s(0')
f(s(0')) -> s(0')
f(s(s(x))) -> f(f(s(x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_f -> 0':s:cons_f
0' :: 0':s:cons_f
s :: 0':s:cons_f -> 0':s:cons_f
encArg :: 0':s:cons_f -> 0':s:cons_f
cons_f :: 0':s:cons_f -> 0':s:cons_f
encode_f :: 0':s:cons_f -> 0':s:cons_f
encode_0 :: 0':s:cons_f
encode_s :: 0':s:cons_f -> 0':s:cons_f
hole_0':s:cons_f1_2 :: 0':s:cons_f
gen_0':s:cons_f2_2 :: Nat -> 0':s:cons_f


Generator Equations:
gen_0':s:cons_f2_2(0) <=> 0'
gen_0':s:cons_f2_2(+(x, 1)) <=> s(gen_0':s:cons_f2_2(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:
TRS:
Rules:
f(0') -> s(0')
f(s(0')) -> s(0')
f(s(s(x))) -> f(f(s(x)))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_f -> 0':s:cons_f
0' :: 0':s:cons_f
s :: 0':s:cons_f -> 0':s:cons_f
encArg :: 0':s:cons_f -> 0':s:cons_f
cons_f :: 0':s:cons_f -> 0':s:cons_f
encode_f :: 0':s:cons_f -> 0':s:cons_f
encode_0 :: 0':s:cons_f
encode_s :: 0':s:cons_f -> 0':s:cons_f
hole_0':s:cons_f1_2 :: 0':s:cons_f
gen_0':s:cons_f2_2 :: Nat -> 0':s:cons_f


Lemmas:
f(gen_0':s:cons_f2_2(+(1, n4_2))) -> gen_0':s:cons_f2_2(1), rt in Omega(1 + n4_2)


Generator Equations:
gen_0':s:cons_f2_2(0) <=> 0'
gen_0':s:cons_f2_2(+(x, 1)) <=> s(gen_0':s:cons_f2_2(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_0':s:cons_f2_2(n244_2)) -> gen_0':s:cons_f2_2(n244_2), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_f2_2(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_f2_2(+(n244_2, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_f2_2(n244_2))) ->_IH
s(gen_0':s:cons_f2_2(c245_2))

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)