/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 194 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), 533 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), 40 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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0


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

(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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0', 0')) -> e(0')
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0') -> 0'
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0'

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

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

(8)
Obligation:
Innermost TRS:
Rules:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))
encArg(e(x_1)) -> e(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e(x_1) -> e(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
0' :: e:c:0':cons_h:cons_d:cons_g
encArg :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_0 :: e:c:0':cons_h:cons_d:cons_g
hole_e:c:0':cons_h:cons_d:cons_g1_3 :: e:c:0':cons_h:cons_d:cons_g
gen_e:c:0':cons_h:cons_d:cons_g2_3 :: Nat -> e:c:0':cons_h:cons_d:cons_g

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

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

They will be analysed ascendingly in the following order:
d < h
h < encArg
g < d
d < encArg
g < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))
encArg(e(x_1)) -> e(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e(x_1) -> e(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
0' :: e:c:0':cons_h:cons_d:cons_g
encArg :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_0 :: e:c:0':cons_h:cons_d:cons_g
hole_e:c:0':cons_h:cons_d:cons_g1_3 :: e:c:0':cons_h:cons_d:cons_g
gen_e:c:0':cons_h:cons_d:cons_g2_3 :: Nat -> e:c:0':cons_h:cons_d:cons_g


Generator Equations:
gen_e:c:0':cons_h:cons_d:cons_g2_3(0) <=> 0'
gen_e:c:0':cons_h:cons_d:cons_g2_3(+(x, 1)) <=> e(gen_e:c:0':cons_h:cons_d:cons_g2_3(x))


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

They will be analysed ascendingly in the following order:
d < h
h < encArg
g < d
d < encArg
g < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, n4_3)), gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)

Induction Base:
g(gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, 0)), gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, 0)))

Induction Step:
g(gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, +(n4_3, 1))), gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, +(n4_3, 1)))) ->_R^Omega(1)
e(g(gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, n4_3)), gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, n4_3)))) ->_IH
e(*3_3)

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:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))
encArg(e(x_1)) -> e(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e(x_1) -> e(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
0' :: e:c:0':cons_h:cons_d:cons_g
encArg :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_0 :: e:c:0':cons_h:cons_d:cons_g
hole_e:c:0':cons_h:cons_d:cons_g1_3 :: e:c:0':cons_h:cons_d:cons_g
gen_e:c:0':cons_h:cons_d:cons_g2_3 :: Nat -> e:c:0':cons_h:cons_d:cons_g


Generator Equations:
gen_e:c:0':cons_h:cons_d:cons_g2_3(0) <=> 0'
gen_e:c:0':cons_h:cons_d:cons_g2_3(+(x, 1)) <=> e(gen_e:c:0':cons_h:cons_d:cons_g2_3(x))


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

They will be analysed ascendingly in the following order:
d < h
h < encArg
g < d
d < encArg
g < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0', 0')) -> e(0')
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0')) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))
encArg(e(x_1)) -> e(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(0') -> 0'
encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e(x_1) -> e(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
0' :: e:c:0':cons_h:cons_d:cons_g
encArg :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
cons_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_h :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_e :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_c :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_d :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_g :: e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g -> e:c:0':cons_h:cons_d:cons_g
encode_0 :: e:c:0':cons_h:cons_d:cons_g
hole_e:c:0':cons_h:cons_d:cons_g1_3 :: e:c:0':cons_h:cons_d:cons_g
gen_e:c:0':cons_h:cons_d:cons_g2_3 :: Nat -> e:c:0':cons_h:cons_d:cons_g


Lemmas:
g(gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, n4_3)), gen_e:c:0':cons_h:cons_d:cons_g2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3)


Generator Equations:
gen_e:c:0':cons_h:cons_d:cons_g2_3(0) <=> 0'
gen_e:c:0':cons_h:cons_d:cons_g2_3(+(x, 1)) <=> e(gen_e:c:0':cons_h:cons_d:cons_g2_3(x))


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

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

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_e:c:0':cons_h:cons_d:cons_g2_3(n590_3)) -> gen_e:c:0':cons_h:cons_d:cons_g2_3(n590_3), rt in Omega(0)

Induction Base:
encArg(gen_e:c:0':cons_h:cons_d:cons_g2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_e:c:0':cons_h:cons_d:cons_g2_3(+(n590_3, 1))) ->_R^Omega(0)
e(encArg(gen_e:c:0':cons_h:cons_d:cons_g2_3(n590_3))) ->_IH
e(gen_e:c:0':cons_h:cons_d:cons_g2_3(c591_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
BOUNDS(1, INF)