/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 45 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 448 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 112 ms]
        (18) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   d(a(x1)) -> b(d(x1))
   b(x1) -> a(a(a(x1)))
   c(d(c(x1))) -> a(d(x1))
   b(d(d(x1))) -> c(c(d(d(c(x1)))))

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

(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(a(x_1)) -> a(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

(2)
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:

   d(a(x1)) -> b(d(x1))
   b(x1) -> a(a(a(x1)))
   c(d(c(x1))) -> a(d(x1))
   b(d(d(x1))) -> c(c(d(d(c(x1)))))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
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:

   d(a(x1)) -> b(d(x1))
   b(x1) -> a(a(a(x1)))
   c(d(c(x1))) -> a(d(x1))
   b(d(d(x1))) -> c(c(d(d(c(x1)))))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST
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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
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:

   d(a(x1)) -> b(d(x1))
   b(x1) -> a(a(a(x1)))
   c(d(c(x1))) -> a(d(x1))
   b(d(d(x1))) -> c(c(d(d(c(x1)))))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

(8)
Obligation:
Innermost TRS:
Rules:
d(a(x1)) -> b(d(x1))
b(x1) -> a(a(a(x1)))
c(d(c(x1))) -> a(d(x1))
b(d(d(x1))) -> c(c(d(d(c(x1)))))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(cons_d(x_1)) -> d(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encArg :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
hole_a:cons_d:cons_b:cons_c1_0 :: a:cons_d:cons_b:cons_c
gen_a:cons_d:cons_b:cons_c2_0 :: Nat -> a:cons_d:cons_b:cons_c

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
d, b, c, encArg

They will be analysed ascendingly in the following order:
d = b
d = c
d < encArg
b = c
b < encArg
c < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
d(a(x1)) -> b(d(x1))
b(x1) -> a(a(a(x1)))
c(d(c(x1))) -> a(d(x1))
b(d(d(x1))) -> c(c(d(d(c(x1)))))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(cons_d(x_1)) -> d(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encArg :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
hole_a:cons_d:cons_b:cons_c1_0 :: a:cons_d:cons_b:cons_c
gen_a:cons_d:cons_b:cons_c2_0 :: Nat -> a:cons_d:cons_b:cons_c


Generator Equations:
gen_a:cons_d:cons_b:cons_c2_0(0) <=> hole_a:cons_d:cons_b:cons_c1_0
gen_a:cons_d:cons_b:cons_c2_0(+(x, 1)) <=> a(gen_a:cons_d:cons_b:cons_c2_0(x))


The following defined symbols remain to be analysed:
b, d, c, encArg

They will be analysed ascendingly in the following order:
d = b
d = c
d < encArg
b = c
b < encArg
c < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
d(gen_a:cons_d:cons_b:cons_c2_0(+(1, n14_0))) -> *3_0, rt in Omega(n14_0)

Induction Base:
d(gen_a:cons_d:cons_b:cons_c2_0(+(1, 0)))

Induction Step:
d(gen_a:cons_d:cons_b:cons_c2_0(+(1, +(n14_0, 1)))) ->_R^Omega(1)
b(d(gen_a:cons_d:cons_b:cons_c2_0(+(1, n14_0)))) ->_IH
b(*3_0)

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
d(a(x1)) -> b(d(x1))
b(x1) -> a(a(a(x1)))
c(d(c(x1))) -> a(d(x1))
b(d(d(x1))) -> c(c(d(d(c(x1)))))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(cons_d(x_1)) -> d(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encArg :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
hole_a:cons_d:cons_b:cons_c1_0 :: a:cons_d:cons_b:cons_c
gen_a:cons_d:cons_b:cons_c2_0 :: Nat -> a:cons_d:cons_b:cons_c


Generator Equations:
gen_a:cons_d:cons_b:cons_c2_0(0) <=> hole_a:cons_d:cons_b:cons_c1_0
gen_a:cons_d:cons_b:cons_c2_0(+(x, 1)) <=> a(gen_a:cons_d:cons_b:cons_c2_0(x))


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

They will be analysed ascendingly in the following order:
d = b
d = c
d < encArg
b = c
b < encArg
c < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
d(a(x1)) -> b(d(x1))
b(x1) -> a(a(a(x1)))
c(d(c(x1))) -> a(d(x1))
b(d(d(x1))) -> c(c(d(d(c(x1)))))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(cons_d(x_1)) -> d(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encArg :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
cons_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_d :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_a :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_b :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
encode_c :: a:cons_d:cons_b:cons_c -> a:cons_d:cons_b:cons_c
hole_a:cons_d:cons_b:cons_c1_0 :: a:cons_d:cons_b:cons_c
gen_a:cons_d:cons_b:cons_c2_0 :: Nat -> a:cons_d:cons_b:cons_c


Lemmas:
d(gen_a:cons_d:cons_b:cons_c2_0(+(1, n14_0))) -> *3_0, rt in Omega(n14_0)


Generator Equations:
gen_a:cons_d:cons_b:cons_c2_0(0) <=> hole_a:cons_d:cons_b:cons_c1_0
gen_a:cons_d:cons_b:cons_c2_0(+(x, 1)) <=> a(gen_a:cons_d:cons_b:cons_c2_0(x))


The following defined symbols remain to be analysed:
b, c, encArg

They will be analysed ascendingly in the following order:
d = b
d = c
d < encArg
b = c
b < encArg
c < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:cons_d:cons_b:cons_c2_0(+(1, n278_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_a:cons_d:cons_b:cons_c2_0(+(1, 0)))

Induction Step:
encArg(gen_a:cons_d:cons_b:cons_c2_0(+(1, +(n278_0, 1)))) ->_R^Omega(0)
a(encArg(gen_a:cons_d:cons_b:cons_c2_0(+(1, n278_0)))) ->_IH
a(*3_0)

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

(18)
BOUNDS(1, INF)