WORST_CASE(Omega(n^2),O(n^2))
proof of input_s6WwmpsyUB.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).

(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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 63 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 33 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 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), 6 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 293 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 63 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 733 ms]
            (34) proven lower bound
            (35) LowerBoundPropagationProof [FINISHED, 0 ms]
            (36) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).


The TRS R consists of the following rules:

   naiverev(Cons(x, xs)) -> app(naiverev(xs), Cons(x, Nil))
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   naiverev(Nil) -> Nil
   app(Nil, ys) -> ys
   goal(xs) -> naiverev(xs)

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:
   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> naiverev(z0)
Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   NAIVEREV(Nil) -> c1
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
   APP(Nil, z0) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   GOAL(z0) -> c6(NAIVEREV(z0))
S tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   NAIVEREV(Nil) -> c1
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
   APP(Nil, z0) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   GOAL(z0) -> c6(NAIVEREV(z0))
K tuples:none
Defined Rule Symbols:   naiverev_1, app_2, notEmpty_1, goal_1

Defined Pair Symbols:   NAIVEREV_1, APP_2, NOTEMPTY_1, GOAL_1

Compound Symbols:   c_2, c1, c2_1, c3, c4, c5, c6_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c6(NAIVEREV(z0))
Removed 4 trailing nodes:
   NAIVEREV(Nil) -> c1
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   APP(Nil, z0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> naiverev(z0)
Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
S tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
K tuples:none
Defined Rule Symbols:   naiverev_1, app_2, notEmpty_1, goal_1

Defined Pair Symbols:   NAIVEREV_1, APP_2

Compound Symbols:   c_2, c2_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> naiverev(z0)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
S tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
K tuples:none
Defined Rule Symbols:   naiverev_1, app_2

Defined Pair Symbols:   NAIVEREV_1, APP_2

Compound Symbols:   c_2, c2_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
We considered the (Usable) Rules:none
And the Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(APP(x_1, x_2)) = x_2
   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(NAIVEREV(x_1)) = [2]x_1
   POL(Nil) = 0
   POL(app(x_1, x_2)) = [3]
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(naiverev(x_1)) = [3]

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
S tuples:
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
K tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
Defined Rule Symbols:   naiverev_1, app_2

Defined Pair Symbols:   NAIVEREV_1, APP_2

Compound Symbols:   c_2, c2_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
We considered the (Usable) Rules:
   naiverev(Nil) -> Nil
   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   app(Nil, z0) -> z0
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
And the Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(APP(x_1, x_2)) = x_1
   POL(Cons(x_1, x_2)) = [2] + x_2
   POL(NAIVEREV(x_1)) = x_1^2
   POL(Nil) = 0
   POL(app(x_1, x_2)) = x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(naiverev(x_1)) = x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
S tuples:none
K tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
Defined Rule Symbols:   naiverev_1, app_2

Defined Pair Symbols:   NAIVEREV_1, APP_2

Compound Symbols:   c_2, c2_1


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

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

(12)
BOUNDS(1, 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:
   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> naiverev(z0)
Tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   NAIVEREV(Nil) -> c1
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
   APP(Nil, z0) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   GOAL(z0) -> c6(NAIVEREV(z0))
S tuples:
   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   NAIVEREV(Nil) -> c1
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
   APP(Nil, z0) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   GOAL(z0) -> c6(NAIVEREV(z0))
K tuples:none
Defined Rule Symbols:   naiverev_1, app_2, notEmpty_1, goal_1

Defined Pair Symbols:   NAIVEREV_1, APP_2, NOTEMPTY_1, GOAL_1

Compound Symbols:   c_2, c1, c2_1, c3, c4, c5, c6_1


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

(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^2, INF).


The TRS R consists of the following rules:

   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   NAIVEREV(Nil) -> c1
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
   APP(Nil, z0) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   GOAL(z0) -> c6(NAIVEREV(z0))

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

   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> naiverev(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^2, INF).


The TRS R consists of the following rules:

   NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
   NAIVEREV(Nil) -> c1
   APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
   APP(Nil, z0) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   GOAL(z0) -> c6(NAIVEREV(z0))

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

   naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
   naiverev(Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> naiverev(z0)

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

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

(20)
Obligation:
Innermost TRS:
Rules:
NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
NAIVEREV(Nil) -> c1
APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
APP(Nil, z0) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
GOAL(z0) -> c6(NAIVEREV(z0))
naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
naiverev(Nil) -> Nil
app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
app(Nil, z0) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> naiverev(z0)

Types:
NAIVEREV :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
APP :: Cons:Nil -> Cons:Nil -> c2:c3
naiverev :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
GOAL :: Cons:Nil -> c6
c6 :: c:c1 -> c6
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_True:False7_7 :: True:False
gen_c:c18_7 :: Nat -> c:c1
gen_Cons:Nil9_7 :: Nat -> Cons:Nil
gen_c2:c310_7 :: Nat -> c2:c3

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
NAIVEREV, APP, naiverev, app

They will be analysed ascendingly in the following order:
APP < NAIVEREV
naiverev < NAIVEREV
app < naiverev

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

(22)
Obligation:
Innermost TRS:
Rules:
NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
NAIVEREV(Nil) -> c1
APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
APP(Nil, z0) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
GOAL(z0) -> c6(NAIVEREV(z0))
naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
naiverev(Nil) -> Nil
app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
app(Nil, z0) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> naiverev(z0)

Types:
NAIVEREV :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
APP :: Cons:Nil -> Cons:Nil -> c2:c3
naiverev :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
GOAL :: Cons:Nil -> c6
c6 :: c:c1 -> c6
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_True:False7_7 :: True:False
gen_c:c18_7 :: Nat -> c:c1
gen_Cons:Nil9_7 :: Nat -> Cons:Nil
gen_c2:c310_7 :: Nat -> c2:c3


Generator Equations:
gen_c:c18_7(0) <=> c1
gen_c:c18_7(+(x, 1)) <=> c(c3, gen_c:c18_7(x))
gen_Cons:Nil9_7(0) <=> Nil
gen_Cons:Nil9_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil9_7(x))
gen_c2:c310_7(0) <=> c3
gen_c2:c310_7(+(x, 1)) <=> c2(gen_c2:c310_7(x))


The following defined symbols remain to be analysed:
APP, NAIVEREV, naiverev, app

They will be analysed ascendingly in the following order:
APP < NAIVEREV
naiverev < NAIVEREV
app < naiverev

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APP(gen_Cons:Nil9_7(n12_7), gen_Cons:Nil9_7(b)) -> gen_c2:c310_7(n12_7), rt in Omega(1 + n12_7)

Induction Base:
APP(gen_Cons:Nil9_7(0), gen_Cons:Nil9_7(b)) ->_R^Omega(1)
c3

Induction Step:
APP(gen_Cons:Nil9_7(+(n12_7, 1)), gen_Cons:Nil9_7(b)) ->_R^Omega(1)
c2(APP(gen_Cons:Nil9_7(n12_7), gen_Cons:Nil9_7(b))) ->_IH
c2(gen_c2:c310_7(c13_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:
NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
NAIVEREV(Nil) -> c1
APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
APP(Nil, z0) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
GOAL(z0) -> c6(NAIVEREV(z0))
naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
naiverev(Nil) -> Nil
app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
app(Nil, z0) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> naiverev(z0)

Types:
NAIVEREV :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
APP :: Cons:Nil -> Cons:Nil -> c2:c3
naiverev :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
GOAL :: Cons:Nil -> c6
c6 :: c:c1 -> c6
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_True:False7_7 :: True:False
gen_c:c18_7 :: Nat -> c:c1
gen_Cons:Nil9_7 :: Nat -> Cons:Nil
gen_c2:c310_7 :: Nat -> c2:c3


Generator Equations:
gen_c:c18_7(0) <=> c1
gen_c:c18_7(+(x, 1)) <=> c(c3, gen_c:c18_7(x))
gen_Cons:Nil9_7(0) <=> Nil
gen_Cons:Nil9_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil9_7(x))
gen_c2:c310_7(0) <=> c3
gen_c2:c310_7(+(x, 1)) <=> c2(gen_c2:c310_7(x))


The following defined symbols remain to be analysed:
APP, NAIVEREV, naiverev, app

They will be analysed ascendingly in the following order:
APP < NAIVEREV
naiverev < NAIVEREV
app < naiverev

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
NAIVEREV(Nil) -> c1
APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
APP(Nil, z0) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
GOAL(z0) -> c6(NAIVEREV(z0))
naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
naiverev(Nil) -> Nil
app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
app(Nil, z0) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> naiverev(z0)

Types:
NAIVEREV :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
APP :: Cons:Nil -> Cons:Nil -> c2:c3
naiverev :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
GOAL :: Cons:Nil -> c6
c6 :: c:c1 -> c6
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_True:False7_7 :: True:False
gen_c:c18_7 :: Nat -> c:c1
gen_Cons:Nil9_7 :: Nat -> Cons:Nil
gen_c2:c310_7 :: Nat -> c2:c3


Lemmas:
APP(gen_Cons:Nil9_7(n12_7), gen_Cons:Nil9_7(b)) -> gen_c2:c310_7(n12_7), rt in Omega(1 + n12_7)


Generator Equations:
gen_c:c18_7(0) <=> c1
gen_c:c18_7(+(x, 1)) <=> c(c3, gen_c:c18_7(x))
gen_Cons:Nil9_7(0) <=> Nil
gen_Cons:Nil9_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil9_7(x))
gen_c2:c310_7(0) <=> c3
gen_c2:c310_7(+(x, 1)) <=> c2(gen_c2:c310_7(x))


The following defined symbols remain to be analysed:
app, NAIVEREV, naiverev

They will be analysed ascendingly in the following order:
naiverev < NAIVEREV
app < naiverev

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_Cons:Nil9_7(n513_7), gen_Cons:Nil9_7(b)) -> gen_Cons:Nil9_7(+(n513_7, b)), rt in Omega(0)

Induction Base:
app(gen_Cons:Nil9_7(0), gen_Cons:Nil9_7(b)) ->_R^Omega(0)
gen_Cons:Nil9_7(b)

Induction Step:
app(gen_Cons:Nil9_7(+(n513_7, 1)), gen_Cons:Nil9_7(b)) ->_R^Omega(0)
Cons(hole_a3_7, app(gen_Cons:Nil9_7(n513_7), gen_Cons:Nil9_7(b))) ->_IH
Cons(hole_a3_7, gen_Cons:Nil9_7(+(b, c514_7)))

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

(30)
Obligation:
Innermost TRS:
Rules:
NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
NAIVEREV(Nil) -> c1
APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
APP(Nil, z0) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
GOAL(z0) -> c6(NAIVEREV(z0))
naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
naiverev(Nil) -> Nil
app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
app(Nil, z0) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> naiverev(z0)

Types:
NAIVEREV :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
APP :: Cons:Nil -> Cons:Nil -> c2:c3
naiverev :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
GOAL :: Cons:Nil -> c6
c6 :: c:c1 -> c6
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_True:False7_7 :: True:False
gen_c:c18_7 :: Nat -> c:c1
gen_Cons:Nil9_7 :: Nat -> Cons:Nil
gen_c2:c310_7 :: Nat -> c2:c3


Lemmas:
APP(gen_Cons:Nil9_7(n12_7), gen_Cons:Nil9_7(b)) -> gen_c2:c310_7(n12_7), rt in Omega(1 + n12_7)
app(gen_Cons:Nil9_7(n513_7), gen_Cons:Nil9_7(b)) -> gen_Cons:Nil9_7(+(n513_7, b)), rt in Omega(0)


Generator Equations:
gen_c:c18_7(0) <=> c1
gen_c:c18_7(+(x, 1)) <=> c(c3, gen_c:c18_7(x))
gen_Cons:Nil9_7(0) <=> Nil
gen_Cons:Nil9_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil9_7(x))
gen_c2:c310_7(0) <=> c3
gen_c2:c310_7(+(x, 1)) <=> c2(gen_c2:c310_7(x))


The following defined symbols remain to be analysed:
naiverev, NAIVEREV

They will be analysed ascendingly in the following order:
naiverev < NAIVEREV

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
naiverev(gen_Cons:Nil9_7(n1454_7)) -> gen_Cons:Nil9_7(n1454_7), rt in Omega(0)

Induction Base:
naiverev(gen_Cons:Nil9_7(0)) ->_R^Omega(0)
Nil

Induction Step:
naiverev(gen_Cons:Nil9_7(+(n1454_7, 1))) ->_R^Omega(0)
app(naiverev(gen_Cons:Nil9_7(n1454_7)), Cons(hole_a3_7, Nil)) ->_IH
app(gen_Cons:Nil9_7(c1455_7), Cons(hole_a3_7, Nil)) ->_L^Omega(0)
gen_Cons:Nil9_7(+(n1454_7, +(0, 1)))

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

(32)
Obligation:
Innermost TRS:
Rules:
NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
NAIVEREV(Nil) -> c1
APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
APP(Nil, z0) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
GOAL(z0) -> c6(NAIVEREV(z0))
naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
naiverev(Nil) -> Nil
app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
app(Nil, z0) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> naiverev(z0)

Types:
NAIVEREV :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
APP :: Cons:Nil -> Cons:Nil -> c2:c3
naiverev :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
GOAL :: Cons:Nil -> c6
c6 :: c:c1 -> c6
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_True:False7_7 :: True:False
gen_c:c18_7 :: Nat -> c:c1
gen_Cons:Nil9_7 :: Nat -> Cons:Nil
gen_c2:c310_7 :: Nat -> c2:c3


Lemmas:
APP(gen_Cons:Nil9_7(n12_7), gen_Cons:Nil9_7(b)) -> gen_c2:c310_7(n12_7), rt in Omega(1 + n12_7)
app(gen_Cons:Nil9_7(n513_7), gen_Cons:Nil9_7(b)) -> gen_Cons:Nil9_7(+(n513_7, b)), rt in Omega(0)
naiverev(gen_Cons:Nil9_7(n1454_7)) -> gen_Cons:Nil9_7(n1454_7), rt in Omega(0)


Generator Equations:
gen_c:c18_7(0) <=> c1
gen_c:c18_7(+(x, 1)) <=> c(c3, gen_c:c18_7(x))
gen_Cons:Nil9_7(0) <=> Nil
gen_Cons:Nil9_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil9_7(x))
gen_c2:c310_7(0) <=> c3
gen_c2:c310_7(+(x, 1)) <=> c2(gen_c2:c310_7(x))


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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
NAIVEREV(gen_Cons:Nil9_7(+(1, n1742_7))) -> *11_7, rt in Omega(n1742_7 + n1742_7^2)

Induction Base:
NAIVEREV(gen_Cons:Nil9_7(+(1, 0)))

Induction Step:
NAIVEREV(gen_Cons:Nil9_7(+(1, +(n1742_7, 1)))) ->_R^Omega(1)
c(APP(naiverev(gen_Cons:Nil9_7(+(1, n1742_7))), Cons(hole_a3_7, Nil)), NAIVEREV(gen_Cons:Nil9_7(+(1, n1742_7)))) ->_L^Omega(0)
c(APP(gen_Cons:Nil9_7(+(1, n1742_7)), Cons(hole_a3_7, Nil)), NAIVEREV(gen_Cons:Nil9_7(+(1, n1742_7)))) ->_L^Omega(2 + n1742_7)
c(gen_c2:c310_7(+(1, n1742_7)), NAIVEREV(gen_Cons:Nil9_7(+(1, n1742_7)))) ->_IH
c(gen_c2:c310_7(+(1, n1742_7)), *11_7)

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(34)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
NAIVEREV(Cons(z0, z1)) -> c(APP(naiverev(z1), Cons(z0, Nil)), NAIVEREV(z1))
NAIVEREV(Nil) -> c1
APP(Cons(z0, z1), z2) -> c2(APP(z1, z2))
APP(Nil, z0) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
GOAL(z0) -> c6(NAIVEREV(z0))
naiverev(Cons(z0, z1)) -> app(naiverev(z1), Cons(z0, Nil))
naiverev(Nil) -> Nil
app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
app(Nil, z0) -> z0
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> naiverev(z0)

Types:
NAIVEREV :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
APP :: Cons:Nil -> Cons:Nil -> c2:c3
naiverev :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
GOAL :: Cons:Nil -> c6
c6 :: c:c1 -> c6
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_True:False7_7 :: True:False
gen_c:c18_7 :: Nat -> c:c1
gen_Cons:Nil9_7 :: Nat -> Cons:Nil
gen_c2:c310_7 :: Nat -> c2:c3


Lemmas:
APP(gen_Cons:Nil9_7(n12_7), gen_Cons:Nil9_7(b)) -> gen_c2:c310_7(n12_7), rt in Omega(1 + n12_7)
app(gen_Cons:Nil9_7(n513_7), gen_Cons:Nil9_7(b)) -> gen_Cons:Nil9_7(+(n513_7, b)), rt in Omega(0)
naiverev(gen_Cons:Nil9_7(n1454_7)) -> gen_Cons:Nil9_7(n1454_7), rt in Omega(0)


Generator Equations:
gen_c:c18_7(0) <=> c1
gen_c:c18_7(+(x, 1)) <=> c(c3, gen_c:c18_7(x))
gen_Cons:Nil9_7(0) <=> Nil
gen_Cons:Nil9_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil9_7(x))
gen_c2:c310_7(0) <=> c3
gen_c2:c310_7(+(x, 1)) <=> c2(gen_c2:c310_7(x))


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

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

(36)
BOUNDS(n^2, INF)