/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 55 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)), 75 ms]
    (16) CdtProblem
    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) BOUNDS(1, 1)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 1 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), 437 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), 102 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:

   a(x1) -> x1
   a(a(x1)) -> a(b(c(a(b(x1)))))
   c(b(x1)) -> a(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(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(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, n^2).


The TRS R consists of the following rules:

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

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(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, n^2).


The TRS R consists of the following rules:

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

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(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
----------------------------------------

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> z0
   a(a(z0)) -> a(b(c(a(b(z0)))))
   c(b(z0)) -> a(c(z0))
Tuples:
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0))
   ENCODE_B(z0) -> c5(ENCARG(z0))
   ENCODE_C(z0) -> c6(C(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   A(a(z0)) -> c8(A(b(c(a(b(z0))))), C(a(b(z0))), A(b(z0)))
   C(b(z0)) -> c9(A(c(z0)), C(z0))
S tuples:
   A(z0) -> c7
   A(a(z0)) -> c8(A(b(c(a(b(z0))))), C(a(b(z0))), A(b(z0)))
   C(b(z0)) -> c9(A(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:   a_1, c_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_B_1, ENCODE_C_1, A_1, C_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5_1, c6_2, c7, c8_3, c9_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_B(z0) -> c5(ENCARG(z0))
   A(a(z0)) -> c8(A(b(c(a(b(z0))))), C(a(b(z0))), A(b(z0)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> z0
   a(a(z0)) -> a(b(c(a(b(z0)))))
   c(b(z0)) -> a(c(z0))
Tuples:
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0) -> c4(A(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c6(C(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
S tuples:
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:   a_1, c_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A_1, ENCODE_C_1, A_1, C_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c6_2, c7, c9_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> z0
   a(a(z0)) -> a(b(c(a(b(z0)))))
   c(b(z0)) -> a(c(z0))
Tuples:
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_A(z0) -> c5(ENCARG(z0))
   ENCODE_C(z0) -> c5(C(encArg(z0)))
   ENCODE_C(z0) -> c5(ENCARG(z0))
S tuples:
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:   a_1, c_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c1_1, c2_2, c3_2, c7, c9_2, c5_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A(z0) -> c5(ENCARG(z0))
   ENCODE_C(z0) -> c5(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(z0) -> z0
   a(a(z0)) -> a(b(c(a(b(z0)))))
   c(b(z0)) -> a(c(z0))
Tuples:
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_C(z0) -> c5(C(encArg(z0)))
S tuples:
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:   a_1, c_1, encArg_1, encode_a_1, encode_b_1, encode_c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c1_1, c2_2, c3_2, c7, c9_2, c5_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_a(z0) -> a(encArg(z0))
   encode_b(z0) -> b(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   a(a(z0)) -> a(b(c(a(b(z0)))))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(z0) -> z0
   c(b(z0)) -> a(c(z0))
Tuples:
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_C(z0) -> c5(C(encArg(z0)))
S tuples:
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c1_1, c2_2, c3_2, c7, c9_2, c5_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.
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
We considered the (Usable) Rules:
   a(z0) -> z0
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   c(b(z0)) -> a(c(z0))
And the Tuples:
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_C(z0) -> c5(C(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A(x_1)) = [1]
   POL(C(x_1)) = [2]x_1
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A(x_1)) = [1] + [2]x_1 + x_1^2
   POL(ENCODE_C(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(a(x_1)) = x_1
   POL(b(x_1)) = [2] + x_1
   POL(c(x_1)) = [2] + x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7) = 0
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons_a(x_1)) = [2] + x_1
   POL(cons_c(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b(z0)) -> b(encArg(z0))
   encArg(cons_a(z0)) -> a(encArg(z0))
   encArg(cons_c(z0)) -> c(encArg(z0))
   a(z0) -> z0
   c(b(z0)) -> a(c(z0))
Tuples:
   ENCARG(b(z0)) -> c1(ENCARG(z0))
   ENCARG(cons_a(z0)) -> c2(A(encArg(z0)), ENCARG(z0))
   ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0))
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
   ENCODE_A(z0) -> c5(A(encArg(z0)))
   ENCODE_C(z0) -> c5(C(encArg(z0)))
S tuples:none
K tuples:
   A(z0) -> c7
   C(b(z0)) -> c9(A(c(z0)), C(z0))
Defined Rule Symbols:   encArg_1, a_1, c_1

Defined Pair Symbols:   ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1

Compound Symbols:   c1_1, c2_2, c3_2, c7, c9_2, c5_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:

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

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(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
----------------------------------------

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

(22)
Obligation:
Innermost TRS:
Rules:
a(x1) -> x1
a(a(x1)) -> a(b(c(a(b(x1)))))
c(b(x1)) -> a(c(x1))
encArg(b(x_1)) -> b(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_c(x_1)) -> c(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:
a :: b:cons_a:cons_c -> b:cons_a:cons_c
b :: b:cons_a:cons_c -> b:cons_a:cons_c
c :: b:cons_a:cons_c -> b:cons_a:cons_c
encArg :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_a :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_c :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_a :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_b :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_c :: b:cons_a:cons_c -> b:cons_a:cons_c
hole_b:cons_a:cons_c1_0 :: b:cons_a:cons_c
gen_b:cons_a:cons_c2_0 :: Nat -> b:cons_a:cons_c

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a, c, encArg

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

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

(24)
Obligation:
Innermost TRS:
Rules:
a(x1) -> x1
a(a(x1)) -> a(b(c(a(b(x1)))))
c(b(x1)) -> a(c(x1))
encArg(b(x_1)) -> b(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_c(x_1)) -> c(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:
a :: b:cons_a:cons_c -> b:cons_a:cons_c
b :: b:cons_a:cons_c -> b:cons_a:cons_c
c :: b:cons_a:cons_c -> b:cons_a:cons_c
encArg :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_a :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_c :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_a :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_b :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_c :: b:cons_a:cons_c -> b:cons_a:cons_c
hole_b:cons_a:cons_c1_0 :: b:cons_a:cons_c
gen_b:cons_a:cons_c2_0 :: Nat -> b:cons_a:cons_c


Generator Equations:
gen_b:cons_a:cons_c2_0(0) <=> hole_b:cons_a:cons_c1_0
gen_b:cons_a:cons_c2_0(+(x, 1)) <=> b(gen_b:cons_a:cons_c2_0(x))


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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
c(gen_b:cons_a:cons_c2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
c(gen_b:cons_a:cons_c2_0(+(1, 0)))

Induction Step:
c(gen_b:cons_a:cons_c2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
a(c(gen_b:cons_a:cons_c2_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).
----------------------------------------

(26)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
a(x1) -> x1
a(a(x1)) -> a(b(c(a(b(x1)))))
c(b(x1)) -> a(c(x1))
encArg(b(x_1)) -> b(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_c(x_1)) -> c(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:
a :: b:cons_a:cons_c -> b:cons_a:cons_c
b :: b:cons_a:cons_c -> b:cons_a:cons_c
c :: b:cons_a:cons_c -> b:cons_a:cons_c
encArg :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_a :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_c :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_a :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_b :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_c :: b:cons_a:cons_c -> b:cons_a:cons_c
hole_b:cons_a:cons_c1_0 :: b:cons_a:cons_c
gen_b:cons_a:cons_c2_0 :: Nat -> b:cons_a:cons_c


Generator Equations:
gen_b:cons_a:cons_c2_0(0) <=> hole_b:cons_a:cons_c1_0
gen_b:cons_a:cons_c2_0(+(x, 1)) <=> b(gen_b:cons_a:cons_c2_0(x))


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

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

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(28) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(29)
BOUNDS(n^1, INF)

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(30)
Obligation:
Innermost TRS:
Rules:
a(x1) -> x1
a(a(x1)) -> a(b(c(a(b(x1)))))
c(b(x1)) -> a(c(x1))
encArg(b(x_1)) -> b(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_c(x_1)) -> c(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:
a :: b:cons_a:cons_c -> b:cons_a:cons_c
b :: b:cons_a:cons_c -> b:cons_a:cons_c
c :: b:cons_a:cons_c -> b:cons_a:cons_c
encArg :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_a :: b:cons_a:cons_c -> b:cons_a:cons_c
cons_c :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_a :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_b :: b:cons_a:cons_c -> b:cons_a:cons_c
encode_c :: b:cons_a:cons_c -> b:cons_a:cons_c
hole_b:cons_a:cons_c1_0 :: b:cons_a:cons_c
gen_b:cons_a:cons_c2_0 :: Nat -> b:cons_a:cons_c


Lemmas:
c(gen_b:cons_a:cons_c2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_b:cons_a:cons_c2_0(0) <=> hole_b:cons_a:cons_c1_0
gen_b:cons_a:cons_c2_0(+(x, 1)) <=> b(gen_b:cons_a:cons_c2_0(x))


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

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

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(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_b:cons_a:cons_c2_0(+(1, n227_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_b:cons_a:cons_c2_0(+(1, 0)))

Induction Step:
encArg(gen_b:cons_a:cons_c2_0(+(1, +(n227_0, 1)))) ->_R^Omega(0)
b(encArg(gen_b:cons_a:cons_c2_0(+(1, n227_0)))) ->_IH
b(*3_0)

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)