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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 8 ms]
    (8) CdtProblem
    (9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) BOUNDS(1, 1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) SlicingProof [LOWER BOUND(ID), 0 ms]
    (14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 265 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


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


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   duplicate(Cons(z0, z1)) -> Cons(z0, Cons(z0, duplicate(z1)))
   duplicate(Nil) -> Nil
   goal(z0) -> duplicate(z0)
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
   DUPLICATE(Nil) -> c1
   GOAL(z0) -> c2(DUPLICATE(z0))
S tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
   DUPLICATE(Nil) -> c1
   GOAL(z0) -> c2(DUPLICATE(z0))
K tuples:none
Defined Rule Symbols:   duplicate_1, goal_1

Defined Pair Symbols:   DUPLICATE_1, GOAL_1

Compound Symbols:   c_1, c1, c2_1


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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c2(DUPLICATE(z0))
Removed 1 trailing nodes:
   DUPLICATE(Nil) -> c1

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   duplicate(Cons(z0, z1)) -> Cons(z0, Cons(z0, duplicate(z1)))
   duplicate(Nil) -> Nil
   goal(z0) -> duplicate(z0)
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
S tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
K tuples:none
Defined Rule Symbols:   duplicate_1, goal_1

Defined Pair Symbols:   DUPLICATE_1

Compound Symbols:   c_1


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

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
S tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   DUPLICATE_1

Compound Symbols:   c_1


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

Polynomial interpretation :

   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(DUPLICATE(x_1)) = x_1
   POL(c(x_1)) = x_1

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
S tuples:none
K tuples:
   DUPLICATE(Cons(z0, z1)) -> c(DUPLICATE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   DUPLICATE_1

Compound Symbols:   c_1


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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(10)
BOUNDS(1, 1)

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(11) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   duplicate(Cons(x, xs)) -> Cons(x, Cons(x, duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

S is empty.
Rewrite Strategy: INNERMOST
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(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
   duplicate(Nil) -> Nil
   goal(x) -> duplicate(x)

S is empty.
Rewrite Strategy: INNERMOST
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(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(16)
Obligation:
Innermost TRS:
Rules:
duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
duplicate(Nil) -> Nil
goal(x) -> duplicate(x)

Types:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil

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(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
duplicate
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(18)
Obligation:
Innermost TRS:
Rules:
duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
duplicate(Nil) -> Nil
goal(x) -> duplicate(x)

Types:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
duplicate
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
duplicate(gen_Cons:Nil2_0(n4_0)) -> gen_Cons:Nil2_0(*(2, n4_0)), rt in Omega(1 + n4_0)

Induction Base:
duplicate(gen_Cons:Nil2_0(0)) ->_R^Omega(1)
Nil

Induction Step:
duplicate(gen_Cons:Nil2_0(+(n4_0, 1))) ->_R^Omega(1)
Cons(Cons(duplicate(gen_Cons:Nil2_0(n4_0)))) ->_IH
Cons(Cons(gen_Cons:Nil2_0(*(2, c5_0))))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
duplicate(Cons(xs)) -> Cons(Cons(duplicate(xs)))
duplicate(Nil) -> Nil
goal(x) -> duplicate(x)

Types:
duplicate :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
duplicate
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(21) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(22)
BOUNDS(n^1, INF)