WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 191 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), 763 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), 125 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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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(g(x_1)) -> g(encArg(x_1))
   encArg(b) -> b
   encArg(c) -> c
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_b -> b
   encode_c -> c
   encode_a__b -> a__b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))


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

(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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(b) -> b
   encArg(c) -> c
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_b -> b
   encode_c -> c
   encode_a__b -> a__b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(b) -> b
   encArg(c) -> c
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_b -> b
   encode_c -> c
   encode_a__b -> a__b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(b) -> b
   encArg(c) -> c
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a__g(x_1) -> a__g(encArg(x_1))
   encode_b -> b
   encode_c -> c
   encode_a__b -> a__b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
a__f(X, g(X), Y) -> a__f(Y, Y, Y)
a__g(b) -> c
a__b -> c
mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
mark(g(X)) -> a__g(mark(X))
mark(b) -> a__b
mark(c) -> c
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__g(X) -> g(X)
a__b -> b
encArg(g(x_1)) -> g(encArg(x_1))
encArg(b) -> b
encArg(c) -> c
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_a__g(x_1) -> a__g(encArg(x_1))
encode_b -> b
encode_c -> c
encode_a__b -> a__b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encArg :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
hole_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark1_0 :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0 :: Nat -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__f, mark, encArg

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

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

(10)
Obligation:
Innermost TRS:
Rules:
a__f(X, g(X), Y) -> a__f(Y, Y, Y)
a__g(b) -> c
a__b -> c
mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
mark(g(X)) -> a__g(mark(X))
mark(b) -> a__b
mark(c) -> c
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__g(X) -> g(X)
a__b -> b
encArg(g(x_1)) -> g(encArg(x_1))
encArg(b) -> b
encArg(c) -> c
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_a__g(x_1) -> a__g(encArg(x_1))
encode_b -> b
encode_c -> c
encode_a__b -> a__b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encArg :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
hole_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark1_0 :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0 :: Nat -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark


Generator Equations:
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(0) <=> b
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(x, 1)) <=> g(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(x))


The following defined symbols remain to be analysed:
a__f, mark, encArg

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(1, n101_0))) -> *3_0, rt in Omega(n101_0)

Induction Base:
mark(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(1, 0)))

Induction Step:
mark(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(1, +(n101_0, 1)))) ->_R^Omega(1)
a__g(mark(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(1, n101_0)))) ->_IH
a__g(*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:
a__f(X, g(X), Y) -> a__f(Y, Y, Y)
a__g(b) -> c
a__b -> c
mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
mark(g(X)) -> a__g(mark(X))
mark(b) -> a__b
mark(c) -> c
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__g(X) -> g(X)
a__b -> b
encArg(g(x_1)) -> g(encArg(x_1))
encArg(b) -> b
encArg(c) -> c
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_a__g(x_1) -> a__g(encArg(x_1))
encode_b -> b
encode_c -> c
encode_a__b -> a__b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encArg :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
hole_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark1_0 :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0 :: Nat -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark


Generator Equations:
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(0) <=> b
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(x, 1)) <=> g(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
a__f(X, g(X), Y) -> a__f(Y, Y, Y)
a__g(b) -> c
a__b -> c
mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
mark(g(X)) -> a__g(mark(X))
mark(b) -> a__b
mark(c) -> c
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__g(X) -> g(X)
a__b -> b
encArg(g(x_1)) -> g(encArg(x_1))
encArg(b) -> b
encArg(c) -> c
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__g(x_1)) -> a__g(encArg(x_1))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1) -> g(encArg(x_1))
encode_a__g(x_1) -> a__g(encArg(x_1))
encode_b -> b
encode_c -> c
encode_a__b -> a__b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encArg :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
cons_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__g :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_c :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_a__b :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_mark :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
encode_f :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
hole_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark1_0 :: g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0 :: Nat -> g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark


Lemmas:
mark(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(1, n101_0))) -> *3_0, rt in Omega(n101_0)


Generator Equations:
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(0) <=> b
gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(x, 1)) <=> g(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(n3569_0)) -> gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(n3569_0), rt in Omega(0)

Induction Base:
encArg(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(0)) ->_R^Omega(0)
b

Induction Step:
encArg(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(+(n3569_0, 1))) ->_R^Omega(0)
g(encArg(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(n3569_0))) ->_IH
g(gen_g:b:c:f:cons_a__f:cons_a__g:cons_a__b:cons_mark2_0(c3570_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)