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


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


WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (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)), 78 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 65 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (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), 282 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 13 ms]
            (28) proven lower bound
            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
            (30) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (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: INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS 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:
   APP(Nil, z0) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   NAIVEREV(Nil) -> c1

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

(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)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(NAIVEREV(x_1)) = x_1
   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)) = [1] + x_1

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

(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)) = [1] + x_2
   POL(NAIVEREV(x_1)) = 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) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


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: INNERMOST
----------------------------------------

(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

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

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

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

(18)
Obligation:
Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil

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

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

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

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

(20)
Obligation:
Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) -> gen_Cons:Nil3_0(+(n5_0, b)), rt in Omega(1 + n5_0)

Induction Base:
app(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(b)) ->_R^Omega(1)
gen_Cons:Nil3_0(b)

Induction Step:
app(gen_Cons:Nil3_0(+(n5_0, 1)), gen_Cons:Nil3_0(b)) ->_R^Omega(1)
Cons(app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b))) ->_IH
Cons(gen_Cons:Nil3_0(+(b, c6_0)))

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

(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


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

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

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) -> gen_Cons:Nil3_0(+(n5_0, b)), rt in Omega(1 + n5_0)


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
naiverev(gen_Cons:Nil3_0(n489_0)) -> gen_Cons:Nil3_0(n489_0), rt in Omega(1 + n489_0 + n489_0^2)

Induction Base:
naiverev(gen_Cons:Nil3_0(0)) ->_R^Omega(1)
Nil

Induction Step:
naiverev(gen_Cons:Nil3_0(+(n489_0, 1))) ->_R^Omega(1)
app(naiverev(gen_Cons:Nil3_0(n489_0)), Cons(Nil)) ->_IH
app(gen_Cons:Nil3_0(c490_0), Cons(Nil)) ->_L^Omega(1 + n489_0)
gen_Cons:Nil3_0(+(n489_0, +(0, 1)))

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

(28)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) -> gen_Cons:Nil3_0(+(n5_0, b)), rt in Omega(1 + n5_0)


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


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

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

(30)
BOUNDS(n^2, INF)