WORST_CASE(Omega(n^1),O(n^1))
proof of input_lNHVLMrZgi.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTRS
    (9) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTRS
    (11) CpxTrsMatchBoundsTAProof [FINISHED, 36 ms]
    (12) BOUNDS(1, n^1)
(13) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
    (15) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 8 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 316 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs


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

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


The TRS R consists of the following rules:

   selects(x', revprefix, Cons(x, xs)) -> Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
   select(Cons(x, xs)) -> selects(x, Nil, xs)
   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
   selects(x, revprefix, Nil) -> Cons(Cons(x, revapp(revprefix, Nil)), Nil)
   select(Nil) -> Nil
   revapp(Nil, rest) -> rest

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
   selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
   select(Cons(z0, z1)) -> selects(z0, Nil, z1)
   select(Nil) -> Nil
   revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
   revapp(Nil, z0) -> z0
Tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
   SELECT(Nil) -> c4
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
   REVAPP(Nil, z0) -> c6
S tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
   SELECT(Nil) -> c4
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
   REVAPP(Nil, z0) -> c6
K tuples:none
Defined Rule Symbols:   selects_3, select_1, revapp_2

Defined Pair Symbols:   SELECTS_3, SELECT_1, REVAPP_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4, c5_1, c6


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
Removed 2 trailing nodes:
   SELECT(Nil) -> c4
   REVAPP(Nil, z0) -> c6

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
   selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
   select(Cons(z0, z1)) -> selects(z0, Nil, z1)
   select(Nil) -> Nil
   revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
   revapp(Nil, z0) -> z0
Tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
S tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
K tuples:none
Defined Rule Symbols:   selects_3, select_1, revapp_2

Defined Pair Symbols:   SELECTS_3, REVAPP_2

Compound Symbols:   c_1, c1_1, c2_1, c5_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
   selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
   select(Cons(z0, z1)) -> selects(z0, Nil, z1)
   select(Nil) -> Nil
   revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
   revapp(Nil, z0) -> z0

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
S tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   SELECTS_3, REVAPP_2

Compound Symbols:   c_1, c1_1, c2_1, c5_1


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(9) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(11) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
Cons0(0, 0) -> 0
c0(0) -> 0
c10(0) -> 0
Nil0() -> 0
c20(0) -> 0
c50(0) -> 0
SELECTS0(0, 0, 0) -> 1
REVAPP0(0, 0) -> 2
Cons1(0, 0) -> 4
REVAPP1(0, 4) -> 3
c1(3) -> 1
Cons1(0, 0) -> 6
SELECTS1(0, 6, 0) -> 5
c11(5) -> 1
Nil1() -> 8
REVAPP1(0, 8) -> 7
c21(7) -> 1
REVAPP1(0, 4) -> 9
c51(9) -> 2
REVAPP1(6, 4) -> 3
c1(3) -> 5
Cons1(0, 6) -> 6
c11(5) -> 5
REVAPP1(6, 8) -> 7
c21(7) -> 5
Cons1(0, 4) -> 4
c51(9) -> 3
Cons1(0, 8) -> 4
c51(9) -> 7
c51(9) -> 9
Cons2(0, 4) -> 11
REVAPP2(0, 11) -> 10
c52(10) -> 3
REVAPP2(6, 11) -> 10
Cons2(0, 8) -> 11
c52(10) -> 7
Cons1(0, 11) -> 4
c51(9) -> 10
Cons2(0, 11) -> 11
c52(10) -> 10

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

(12)
BOUNDS(1, n^1)

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

(13) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
   selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
   select(Cons(z0, z1)) -> selects(z0, Nil, z1)
   select(Nil) -> Nil
   revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
   revapp(Nil, z0) -> z0
Tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
   SELECT(Nil) -> c4
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
   REVAPP(Nil, z0) -> c6
S tuples:
   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
   SELECT(Nil) -> c4
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
   REVAPP(Nil, z0) -> c6
K tuples:none
Defined Rule Symbols:   selects_3, select_1, revapp_2

Defined Pair Symbols:   SELECTS_3, SELECT_1, REVAPP_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4, c5_1, c6


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

(15) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(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:

   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
   SELECT(Nil) -> c4
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
   REVAPP(Nil, z0) -> c6

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

   selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
   selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
   select(Cons(z0, z1)) -> selects(z0, Nil, z1)
   select(Nil) -> Nil
   revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
   revapp(Nil, z0) -> z0

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

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

(18)
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:

   SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
   SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
   SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
   SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
   SELECT(Nil) -> c4
   REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
   REVAPP(Nil, z0) -> c6

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

   selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
   selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
   select(Cons(z0, z1)) -> selects(z0, Nil, z1)
   select(Nil) -> Nil
   revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
   revapp(Nil, z0) -> z0

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

(19) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
SELECT(Nil) -> c4
REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
REVAPP(Nil, z0) -> c6
selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
select(Cons(z0, z1)) -> selects(z0, Nil, z1)
select(Nil) -> Nil
revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
revapp(Nil, z0) -> z0

Types:
SELECTS :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c:c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c5:c6 -> c:c1:c2
REVAPP :: Cons:Nil -> Cons:Nil -> c5:c6
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c5:c6 -> c:c1:c2
SELECT :: Cons:Nil -> c3:c4
c3 :: c:c1:c2 -> c3:c4
c4 :: c3:c4
c5 :: c5:c6 -> c5:c6
c6 :: c5:c6
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
hole_c:c1:c21_7 :: c:c1:c2
hole_Cons:Nil2_7 :: Cons:Nil
hole_c5:c63_7 :: c5:c6
hole_c3:c44_7 :: c3:c4
gen_c:c1:c25_7 :: Nat -> c:c1:c2
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c5:c67_7 :: Nat -> c5:c6

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SELECTS, REVAPP, selects, revapp

They will be analysed ascendingly in the following order:
REVAPP < SELECTS
revapp < selects

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

(22)
Obligation:
Innermost TRS:
Rules:
SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
SELECT(Nil) -> c4
REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
REVAPP(Nil, z0) -> c6
selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
select(Cons(z0, z1)) -> selects(z0, Nil, z1)
select(Nil) -> Nil
revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
revapp(Nil, z0) -> z0

Types:
SELECTS :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c:c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c5:c6 -> c:c1:c2
REVAPP :: Cons:Nil -> Cons:Nil -> c5:c6
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c5:c6 -> c:c1:c2
SELECT :: Cons:Nil -> c3:c4
c3 :: c:c1:c2 -> c3:c4
c4 :: c3:c4
c5 :: c5:c6 -> c5:c6
c6 :: c5:c6
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
hole_c:c1:c21_7 :: c:c1:c2
hole_Cons:Nil2_7 :: Cons:Nil
hole_c5:c63_7 :: c5:c6
hole_c3:c44_7 :: c3:c4
gen_c:c1:c25_7 :: Nat -> c:c1:c2
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c5:c67_7 :: Nat -> c5:c6


Generator Equations:
gen_c:c1:c25_7(0) <=> c(c6)
gen_c:c1:c25_7(+(x, 1)) <=> c1(gen_c:c1:c25_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c5:c67_7(0) <=> c6
gen_c5:c67_7(+(x, 1)) <=> c5(gen_c5:c67_7(x))


The following defined symbols remain to be analysed:
REVAPP, SELECTS, selects, revapp

They will be analysed ascendingly in the following order:
REVAPP < SELECTS
revapp < selects

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
REVAPP(gen_Cons:Nil6_7(n9_7), gen_Cons:Nil6_7(b)) -> gen_c5:c67_7(n9_7), rt in Omega(1 + n9_7)

Induction Base:
REVAPP(gen_Cons:Nil6_7(0), gen_Cons:Nil6_7(b)) ->_R^Omega(1)
c6

Induction Step:
REVAPP(gen_Cons:Nil6_7(+(n9_7, 1)), gen_Cons:Nil6_7(b)) ->_R^Omega(1)
c5(REVAPP(gen_Cons:Nil6_7(n9_7), Cons(Nil, gen_Cons:Nil6_7(b)))) ->_IH
c5(gen_c5:c67_7(c10_7))

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
SELECT(Nil) -> c4
REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
REVAPP(Nil, z0) -> c6
selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
select(Cons(z0, z1)) -> selects(z0, Nil, z1)
select(Nil) -> Nil
revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
revapp(Nil, z0) -> z0

Types:
SELECTS :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c:c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c5:c6 -> c:c1:c2
REVAPP :: Cons:Nil -> Cons:Nil -> c5:c6
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c5:c6 -> c:c1:c2
SELECT :: Cons:Nil -> c3:c4
c3 :: c:c1:c2 -> c3:c4
c4 :: c3:c4
c5 :: c5:c6 -> c5:c6
c6 :: c5:c6
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
hole_c:c1:c21_7 :: c:c1:c2
hole_Cons:Nil2_7 :: Cons:Nil
hole_c5:c63_7 :: c5:c6
hole_c3:c44_7 :: c3:c4
gen_c:c1:c25_7 :: Nat -> c:c1:c2
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c5:c67_7 :: Nat -> c5:c6


Generator Equations:
gen_c:c1:c25_7(0) <=> c(c6)
gen_c:c1:c25_7(+(x, 1)) <=> c1(gen_c:c1:c25_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c5:c67_7(0) <=> c6
gen_c5:c67_7(+(x, 1)) <=> c5(gen_c5:c67_7(x))


The following defined symbols remain to be analysed:
REVAPP, SELECTS, selects, revapp

They will be analysed ascendingly in the following order:
REVAPP < SELECTS
revapp < selects

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
SELECTS(z0, z1, Cons(z2, z3)) -> c(REVAPP(z1, Cons(z2, z3)))
SELECTS(z0, z1, Cons(z2, z3)) -> c1(SELECTS(z2, Cons(z0, z1), z3))
SELECTS(z0, z1, Nil) -> c2(REVAPP(z1, Nil))
SELECT(Cons(z0, z1)) -> c3(SELECTS(z0, Nil, z1))
SELECT(Nil) -> c4
REVAPP(Cons(z0, z1), z2) -> c5(REVAPP(z1, Cons(z0, z2)))
REVAPP(Nil, z0) -> c6
selects(z0, z1, Cons(z2, z3)) -> Cons(Cons(z0, revapp(z1, Cons(z2, z3))), selects(z2, Cons(z0, z1), z3))
selects(z0, z1, Nil) -> Cons(Cons(z0, revapp(z1, Nil)), Nil)
select(Cons(z0, z1)) -> selects(z0, Nil, z1)
select(Nil) -> Nil
revapp(Cons(z0, z1), z2) -> revapp(z1, Cons(z0, z2))
revapp(Nil, z0) -> z0

Types:
SELECTS :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c:c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c5:c6 -> c:c1:c2
REVAPP :: Cons:Nil -> Cons:Nil -> c5:c6
c1 :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c2 :: c5:c6 -> c:c1:c2
SELECT :: Cons:Nil -> c3:c4
c3 :: c:c1:c2 -> c3:c4
c4 :: c3:c4
c5 :: c5:c6 -> c5:c6
c6 :: c5:c6
selects :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
revapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
select :: Cons:Nil -> Cons:Nil
hole_c:c1:c21_7 :: c:c1:c2
hole_Cons:Nil2_7 :: Cons:Nil
hole_c5:c63_7 :: c5:c6
hole_c3:c44_7 :: c3:c4
gen_c:c1:c25_7 :: Nat -> c:c1:c2
gen_Cons:Nil6_7 :: Nat -> Cons:Nil
gen_c5:c67_7 :: Nat -> c5:c6


Lemmas:
REVAPP(gen_Cons:Nil6_7(n9_7), gen_Cons:Nil6_7(b)) -> gen_c5:c67_7(n9_7), rt in Omega(1 + n9_7)


Generator Equations:
gen_c:c1:c25_7(0) <=> c(c6)
gen_c:c1:c25_7(+(x, 1)) <=> c1(gen_c:c1:c25_7(x))
gen_Cons:Nil6_7(0) <=> Nil
gen_Cons:Nil6_7(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_7(x))
gen_c5:c67_7(0) <=> c6
gen_c5:c67_7(+(x, 1)) <=> c5(gen_c5:c67_7(x))


The following defined symbols remain to be analysed:
SELECTS, selects, revapp

They will be analysed ascendingly in the following order:
revapp < selects