/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 78 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 467 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), 218 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:

   b(a(a(x1))) -> a(b(c(x1)))
   c(a(x1)) -> a(c(x1))
   c(b(x1)) -> b(a(x1))
   L(a(a(x1))) -> L(a(b(c(x1))))
   c(R(x1)) -> b(a(R(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(R(x_1)) -> R(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_L(x_1)) -> L(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_L(x_1) -> L(encArg(x_1))
   encode_R(x_1) -> R(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:

   b(a(a(x1))) -> a(b(c(x1)))
   c(a(x1)) -> a(c(x1))
   c(b(x1)) -> b(a(x1))
   L(a(a(x1))) -> L(a(b(c(x1))))
   c(R(x1)) -> b(a(R(x1)))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(R(x_1)) -> R(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_L(x_1)) -> L(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_L(x_1) -> L(encArg(x_1))
   encode_R(x_1) -> R(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:

   b(a(a(x1))) -> a(b(c(x1)))
   c(a(x1)) -> a(c(x1))
   c(b(x1)) -> b(a(x1))
   L(a(a(x1))) -> L(a(b(c(x1))))
   c(R(x1)) -> b(a(R(x1)))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(R(x_1)) -> R(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_L(x_1)) -> L(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_L(x_1) -> L(encArg(x_1))
   encode_R(x_1) -> R(encArg(x_1))

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

(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:

   b(a(a(x1))) -> a(b(c(x1)))
   c(a(x1)) -> a(c(x1))
   c(b(x1)) -> b(a(x1))
   L(a(a(x1))) -> L(a(b(c(x1))))
   c(R(x1)) -> b(a(R(x1)))

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(R(x_1)) -> R(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_L(x_1)) -> L(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_L(x_1) -> L(encArg(x_1))
   encode_R(x_1) -> R(encArg(x_1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
b(a(a(x1))) -> a(b(c(x1)))
c(a(x1)) -> a(c(x1))
c(b(x1)) -> b(a(x1))
L(a(a(x1))) -> L(a(b(c(x1))))
c(R(x1)) -> b(a(R(x1)))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(R(x_1)) -> R(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encArg(cons_L(x_1)) -> L(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_L(x_1) -> L(encArg(x_1))
encode_R(x_1) -> R(encArg(x_1))

Types:
b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encArg :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
hole_a:R:cons_b:cons_c:cons_L1_0 :: a:R:cons_b:cons_c:cons_L
gen_a:R:cons_b:cons_c:cons_L2_0 :: Nat -> a:R:cons_b:cons_c:cons_L

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
b(a(a(x1))) -> a(b(c(x1)))
c(a(x1)) -> a(c(x1))
c(b(x1)) -> b(a(x1))
L(a(a(x1))) -> L(a(b(c(x1))))
c(R(x1)) -> b(a(R(x1)))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(R(x_1)) -> R(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encArg(cons_L(x_1)) -> L(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_L(x_1) -> L(encArg(x_1))
encode_R(x_1) -> R(encArg(x_1))

Types:
b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encArg :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
hole_a:R:cons_b:cons_c:cons_L1_0 :: a:R:cons_b:cons_c:cons_L
gen_a:R:cons_b:cons_c:cons_L2_0 :: Nat -> a:R:cons_b:cons_c:cons_L


Generator Equations:
gen_a:R:cons_b:cons_c:cons_L2_0(0) <=> hole_a:R:cons_b:cons_c:cons_L1_0
gen_a:R:cons_b:cons_c:cons_L2_0(+(x, 1)) <=> a(gen_a:R:cons_b:cons_c:cons_L2_0(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
c(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
c(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, 0)))

Induction Step:
c(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
a(c(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, n4_0)))) ->_IH
a(*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:
b(a(a(x1))) -> a(b(c(x1)))
c(a(x1)) -> a(c(x1))
c(b(x1)) -> b(a(x1))
L(a(a(x1))) -> L(a(b(c(x1))))
c(R(x1)) -> b(a(R(x1)))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(R(x_1)) -> R(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encArg(cons_L(x_1)) -> L(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_L(x_1) -> L(encArg(x_1))
encode_R(x_1) -> R(encArg(x_1))

Types:
b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encArg :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
hole_a:R:cons_b:cons_c:cons_L1_0 :: a:R:cons_b:cons_c:cons_L
gen_a:R:cons_b:cons_c:cons_L2_0 :: Nat -> a:R:cons_b:cons_c:cons_L


Generator Equations:
gen_a:R:cons_b:cons_c:cons_L2_0(0) <=> hole_a:R:cons_b:cons_c:cons_L1_0
gen_a:R:cons_b:cons_c:cons_L2_0(+(x, 1)) <=> a(gen_a:R:cons_b:cons_c:cons_L2_0(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
b(a(a(x1))) -> a(b(c(x1)))
c(a(x1)) -> a(c(x1))
c(b(x1)) -> b(a(x1))
L(a(a(x1))) -> L(a(b(c(x1))))
c(R(x1)) -> b(a(R(x1)))
encArg(a(x_1)) -> a(encArg(x_1))
encArg(R(x_1)) -> R(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encArg(cons_c(x_1)) -> c(encArg(x_1))
encArg(cons_L(x_1)) -> L(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_L(x_1) -> L(encArg(x_1))
encode_R(x_1) -> R(encArg(x_1))

Types:
b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encArg :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
cons_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_b :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_a :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_c :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_L :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
encode_R :: a:R:cons_b:cons_c:cons_L -> a:R:cons_b:cons_c:cons_L
hole_a:R:cons_b:cons_c:cons_L1_0 :: a:R:cons_b:cons_c:cons_L
gen_a:R:cons_b:cons_c:cons_L2_0 :: Nat -> a:R:cons_b:cons_c:cons_L


Lemmas:
c(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_a:R:cons_b:cons_c:cons_L2_0(0) <=> hole_a:R:cons_b:cons_c:cons_L1_0
gen_a:R:cons_b:cons_c:cons_L2_0(+(x, 1)) <=> a(gen_a:R:cons_b:cons_c:cons_L2_0(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, n1951_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, 0)))

Induction Step:
encArg(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, +(n1951_0, 1)))) ->_R^Omega(0)
a(encArg(gen_a:R:cons_b:cons_c:cons_L2_0(+(1, n1951_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)