WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 163 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 3 ms]
(4) CdtProblem
    (5) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 3 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 284 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 27 ms]
    (24) proven lower bound
    (25) LowerBoundPropagationProof [FINISHED, 0 ms]
    (26) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   member(x, Nil) -> False
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x, xs) -> member(x, xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
   member[Ite][True][Ite](True, x, xs) -> True

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   member(x, Nil) -> False
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x, xs) -> member(x, xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
   member[Ite][True][Ite](True, x, xs) -> True

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
   member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2)
   member[Ite][True][Ite](True, z0, z1) -> True
   member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   member(z0, Nil) -> False
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0, z1) -> member(z0, z1)
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   !EQ'(0, S(z0)) -> c1
   !EQ'(S(z0), 0) -> c2
   !EQ'(0, 0) -> c3
   MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c4(MEMBER(z0, z2))
   MEMBER[ITE][TRUE][ITE](True, z0, z1) -> c5
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
   NOTEMPTY(Cons(z0, z1)) -> c8
   NOTEMPTY(Nil) -> c9
   GOAL(z0, z1) -> c10(MEMBER(z0, z1))
S tuples:
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
   NOTEMPTY(Cons(z0, z1)) -> c8
   NOTEMPTY(Nil) -> c9
   GOAL(z0, z1) -> c10(MEMBER(z0, z1))
K tuples:none
Defined Rule Symbols:   member_2, notEmpty_1, goal_2, !EQ_2, member[Ite][True][Ite]_3

Defined Pair Symbols:   !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2, NOTEMPTY_1, GOAL_2

Compound Symbols:   c_1, c1, c2, c3, c4_1, c5, c6_2, c7, c8, c9, c10_1


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

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c10(MEMBER(z0, z1))
Removed 6 trailing nodes:
   NOTEMPTY(Cons(z0, z1)) -> c8
   !EQ'(S(z0), 0) -> c2
   NOTEMPTY(Nil) -> c9
   !EQ'(0, 0) -> c3
   !EQ'(0, S(z0)) -> c1
   MEMBER[ITE][TRUE][ITE](True, z0, z1) -> c5

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
   member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2)
   member[Ite][True][Ite](True, z0, z1) -> True
   member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   member(z0, Nil) -> False
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0, z1) -> member(z0, z1)
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c4(MEMBER(z0, z2))
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
S tuples:
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
K tuples:none
Defined Rule Symbols:   member_2, notEmpty_1, goal_2, !EQ_2, member[Ite][True][Ite]_3

Defined Pair Symbols:   !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2

Compound Symbols:   c_1, c4_1, c6_2, c7


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   member[Ite][True][Ite](False, z0, Cons(z1, z2)) -> member(z0, z2)
   member[Ite][True][Ite](True, z0, z1) -> True
   member(z0, Cons(z1, z2)) -> member[Ite][True][Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   member(z0, Nil) -> False
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0, z1) -> member(z0, z1)

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c4(MEMBER(z0, z2))
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
S tuples:
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
K tuples:none
Defined Rule Symbols:   !EQ_2

Defined Pair Symbols:   !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2

Compound Symbols:   c_1, c4_1, c6_2, c7


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MEMBER(z0, Nil) -> c7
We considered the (Usable) Rules:none
And the Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c4(MEMBER(z0, z2))
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(!EQ(x_1, x_2)) = [3]
   POL(!EQ'(x_1, x_2)) = 0
   POL(0) = [3]
   POL(Cons(x_1, x_2)) = x_2
   POL(False) = [3]
   POL(MEMBER(x_1, x_2)) = [3]x_1 + x_2
   POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = [3]x_2 + x_3
   POL(Nil) = [3]
   POL(S(x_1)) = [3] + x_1
   POL(True) = [3]
   POL(c(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7) = 0

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c4(MEMBER(z0, z2))
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
S tuples:
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
K tuples:
   MEMBER(z0, Nil) -> c7
Defined Rule Symbols:   !EQ_2

Defined Pair Symbols:   !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2

Compound Symbols:   c_1, c4_1, c6_2, c7


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c4(MEMBER(z0, z2))
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(!EQ(x_1, x_2)) = [1] + x_1 + x_2
   POL(!EQ'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(False) = [1]
   POL(MEMBER(x_1, x_2)) = [1] + x_1 + x_2
   POL(MEMBER[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_2 + x_3
   POL(Nil) = 0
   POL(S(x_1)) = [1] + x_1
   POL(True) = [1]
   POL(c(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7) = 0

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   MEMBER[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c4(MEMBER(z0, z2))
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   MEMBER(z0, Nil) -> c7
S tuples:none
K tuples:
   MEMBER(z0, Nil) -> c7
   MEMBER(z0, Cons(z1, z2)) -> c6(MEMBER[ITE][TRUE][ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
Defined Rule Symbols:   !EQ_2

Defined Pair Symbols:   !EQ'_2, MEMBER[ITE][TRUE][ITE]_3, MEMBER_2

Compound Symbols:   c_1, c4_1, c6_2, c7


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

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

(14)
BOUNDS(1, 1)

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

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

(16)
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:

   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
   member(x, Nil) -> False
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x, xs) -> member(x, xs)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0', S(y)) -> False
   !EQ(S(x), 0') -> False
   !EQ(0', 0') -> True
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
   member[Ite][True][Ite](True, x, xs) -> True

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

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

(18)
Obligation:
Innermost TRS:
Rules:
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(x, xs) -> member(x, xs)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
member[Ite][True][Ite](True, x, xs) -> True

Types:
member :: S:0' -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: S:0' -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat -> S:0'
gen_Cons:Nil5_0 :: Nat -> Cons:Nil

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
member, !EQ

They will be analysed ascendingly in the following order:
!EQ < member

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

(20)
Obligation:
Innermost TRS:
Rules:
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(x, xs) -> member(x, xs)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
member[Ite][True][Ite](True, x, xs) -> True

Types:
member :: S:0' -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: S:0' -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat -> S:0'
gen_Cons:Nil5_0 :: Nat -> Cons:Nil


Generator Equations:
gen_S:0'4_0(0) <=> 0'
gen_S:0'4_0(+(x, 1)) <=> S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) <=> Nil
gen_Cons:Nil5_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil5_0(x))


The following defined symbols remain to be analysed:
!EQ, member

They will be analysed ascendingly in the following order:
!EQ < member

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) -> False, rt in Omega(0)

Induction Base:
!EQ(gen_S:0'4_0(0), gen_S:0'4_0(+(1, 0))) ->_R^Omega(0)
False

Induction Step:
!EQ(gen_S:0'4_0(+(n7_0, 1)), gen_S:0'4_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) ->_IH
False

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

(22)
Obligation:
Innermost TRS:
Rules:
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(x, xs) -> member(x, xs)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
member[Ite][True][Ite](True, x, xs) -> True

Types:
member :: S:0' -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: S:0' -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat -> S:0'
gen_Cons:Nil5_0 :: Nat -> Cons:Nil


Lemmas:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) -> False, rt in Omega(0)


Generator Equations:
gen_S:0'4_0(0) <=> 0'
gen_S:0'4_0(+(x, 1)) <=> S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) <=> Nil
gen_Cons:Nil5_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil5_0(x))


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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n260_0)) -> False, rt in Omega(1 + n260_0)

Induction Base:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(0)) ->_R^Omega(1)
False

Induction Step:
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(+(n260_0, 1))) ->_R^Omega(1)
member[Ite][True][Ite](!EQ(gen_S:0'4_0(1), 0'), gen_S:0'4_0(1), Cons(0', gen_Cons:Nil5_0(n260_0))) ->_R^Omega(0)
member[Ite][True][Ite](False, gen_S:0'4_0(1), Cons(0', gen_Cons:Nil5_0(n260_0))) ->_R^Omega(0)
member(gen_S:0'4_0(1), gen_Cons:Nil5_0(n260_0)) ->_IH
False

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

(24)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x', x), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(x, xs) -> member(x, xs)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
member[Ite][True][Ite](True, x, xs) -> True

Types:
member :: S:0' -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: S:0' -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_S:0'2_0 :: S:0'
hole_Cons:Nil3_0 :: Cons:Nil
gen_S:0'4_0 :: Nat -> S:0'
gen_Cons:Nil5_0 :: Nat -> Cons:Nil


Lemmas:
!EQ(gen_S:0'4_0(n7_0), gen_S:0'4_0(+(1, n7_0))) -> False, rt in Omega(0)


Generator Equations:
gen_S:0'4_0(0) <=> 0'
gen_S:0'4_0(+(x, 1)) <=> S(gen_S:0'4_0(x))
gen_Cons:Nil5_0(0) <=> Nil
gen_Cons:Nil5_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil5_0(x))


The following defined symbols remain to be analysed:
member
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(25) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(26)
BOUNDS(n^1, INF)