/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 165 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (12) CdtProblem
    (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 115 ms]
    (16) CdtProblem
    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) BOUNDS(1, 1)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 171 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 63 ms]
            (32) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(0, y) -> 0
   f(s(x), y) -> f(f(x, y), 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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))


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

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


The TRS R consists of the following rules:

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

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, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(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, n^2).


The TRS R consists of the following rules:

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

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, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   ENCARG(0) -> c
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1) -> c3(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_0 -> c4
   ENCODE_S(z0) -> c5(ENCARG(z0))
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
S tuples:
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_0, encode_s_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_0, ENCODE_S_1, F_2

Compound Symbols:   c, c1_1, c2_3, c3_3, c4, c5_1, c6, c7_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_S(z0) -> c5(ENCARG(z0))
Removed 2 trailing nodes:
   ENCARG(0) -> c
   ENCODE_0 -> c4

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1) -> c3(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
S tuples:
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_0, encode_s_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, F_2

Compound Symbols:   c1_1, c2_3, c3_3, c6, c7_2


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

(9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1) -> c(ENCARG(z0))
   ENCODE_F(z0, z1) -> c(ENCARG(z1))
S tuples:
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_0, encode_s_1

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c1_1, c2_3, c6, c7_2, c_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0, z1) -> c(ENCARG(z0))
   ENCODE_F(z0, z1) -> c(ENCARG(z1))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
S tuples:
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2, encode_0, encode_s_1

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c1_1, c2_3, c6, c7_2, c_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_0 -> 0
   encode_s(z0) -> s(encArg(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
S tuples:
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c1_1, c2_3, c6, c7_2, c_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
We considered the (Usable) Rules:
   f(0, z0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_F(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + x_2^2 + x_1*x_2 + x_1^2
   POL(F(x_1, x_2)) = [1] + [2]x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c6) = 0
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(cons_f(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(s(x_1)) = [2] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(0, z0) -> 0
   f(s(z0), z1) -> f(f(z0, z1), z1)
Tuples:
   ENCARG(s(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
   ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   F(0, z0) -> c6
   F(s(z0), z1) -> c7(F(f(z0, z1), z1), F(z0, z1))
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c1_1, c2_3, c6, c7_2, c_1


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

(17) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(18)
BOUNDS(1, 1)

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

(19) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(20)
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:

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

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, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))

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

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

(22)
Obligation:
Innermost TRS:
Rules:
f(0', y) -> 0'
f(s(x), y) -> f(f(x, y), y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_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 -> 0':s:cons_f
encode_f :: 0':s:cons_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_3 :: 0':s:cons_f
gen_0':s:cons_f2_3 :: Nat -> 0':s:cons_f

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

(23) 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

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

(24)
Obligation:
Innermost TRS:
Rules:
f(0', y) -> 0'
f(s(x), y) -> f(f(x, y), y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_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 -> 0':s:cons_f
encode_f :: 0':s:cons_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_3 :: 0':s:cons_f
gen_0':s:cons_f2_3 :: Nat -> 0':s:cons_f


Generator Equations:
gen_0':s:cons_f2_3(0) <=> 0'
gen_0':s:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_f2_3(x))


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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':s:cons_f2_3(n4_3), gen_0':s:cons_f2_3(b)) -> gen_0':s:cons_f2_3(0), rt in Omega(1 + n4_3)

Induction Base:
f(gen_0':s:cons_f2_3(0), gen_0':s:cons_f2_3(b)) ->_R^Omega(1)
0'

Induction Step:
f(gen_0':s:cons_f2_3(+(n4_3, 1)), gen_0':s:cons_f2_3(b)) ->_R^Omega(1)
f(f(gen_0':s:cons_f2_3(n4_3), gen_0':s:cons_f2_3(b)), gen_0':s:cons_f2_3(b)) ->_IH
f(gen_0':s:cons_f2_3(0), gen_0':s:cons_f2_3(b)) ->_R^Omega(1)
0'

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

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(0', y) -> 0'
f(s(x), y) -> f(f(x, y), y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_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 -> 0':s:cons_f
encode_f :: 0':s:cons_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_3 :: 0':s:cons_f
gen_0':s:cons_f2_3 :: Nat -> 0':s:cons_f


Generator Equations:
gen_0':s:cons_f2_3(0) <=> 0'
gen_0':s:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_f2_3(x))


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

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

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

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
Innermost TRS:
Rules:
f(0', y) -> 0'
f(s(x), y) -> f(f(x, y), y)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
f :: 0':s:cons_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 -> 0':s:cons_f
encode_f :: 0':s:cons_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_3 :: 0':s:cons_f
gen_0':s:cons_f2_3 :: Nat -> 0':s:cons_f


Lemmas:
f(gen_0':s:cons_f2_3(n4_3), gen_0':s:cons_f2_3(b)) -> gen_0':s:cons_f2_3(0), rt in Omega(1 + n4_3)


Generator Equations:
gen_0':s:cons_f2_3(0) <=> 0'
gen_0':s:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_f2_3(x))


The following defined symbols remain to be analysed:
encArg
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(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_f2_3(n395_3)) -> gen_0':s:cons_f2_3(n395_3), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_f2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_f2_3(+(n395_3, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_f2_3(n395_3))) ->_IH
s(gen_0':s:cons_f2_3(c396_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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(32)
BOUNDS(1, INF)