WORST_CASE(Omega(n^1),O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) SlicingProof [LOWER BOUND(ID), 0 ms]
    (8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 4 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 314 ms]
    (14) proven lower bound
    (15) LowerBoundPropagationProof [FINISHED, 0 ms]
    (16) 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:

   anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y))
   anchored(Nil, y) -> y
   goal(x, y) -> anchored(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
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:

   anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y))
   anchored(Nil, y) -> y
   goal(x, y) -> anchored(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(3) 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 1. 

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
Nil0() -> 0
anchored0(0, 0) -> 1
goal0(0, 0) -> 2
Nil1() -> 5
Nil1() -> 6
Cons1(5, 6) -> 4
Cons1(4, 0) -> 3
anchored1(0, 3) -> 1
anchored1(0, 0) -> 2
Cons1(4, 3) -> 3
anchored1(0, 3) -> 2
0 -> 1
0 -> 2
3 -> 1
3 -> 2

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(4)
BOUNDS(1, n^1)

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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(6)
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:

   anchored(Cons(x, xs), y) -> anchored(xs, Cons(Cons(Nil, Nil), y))
   anchored(Nil, y) -> y
   goal(x, y) -> anchored(x, y)

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

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(8)
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:

   anchored(Cons(xs), y) -> anchored(xs, Cons(y))
   anchored(Nil, y) -> y
   goal(x, y) -> anchored(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(10)
Obligation:
Innermost TRS:
Rules:
anchored(Cons(xs), y) -> anchored(xs, Cons(y))
anchored(Nil, y) -> y
goal(x, y) -> anchored(x, y)

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

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(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
anchored
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(12)
Obligation:
Innermost TRS:
Rules:
anchored(Cons(xs), y) -> anchored(xs, Cons(y))
anchored(Nil, y) -> y
goal(x, y) -> anchored(x, y)

Types:
anchored :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> 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:
anchored
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
anchored(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
anchored(gen_Cons:Nil2_0(0), gen_Cons:Nil2_0(b)) ->_R^Omega(1)
gen_Cons:Nil2_0(b)

Induction Step:
anchored(gen_Cons:Nil2_0(+(n4_0, 1)), gen_Cons:Nil2_0(b)) ->_R^Omega(1)
anchored(gen_Cons:Nil2_0(n4_0), Cons(gen_Cons:Nil2_0(b))) ->_IH
gen_Cons:Nil2_0(+(+(b, 1), 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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(14)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
anchored(Cons(xs), y) -> anchored(xs, Cons(y))
anchored(Nil, y) -> y
goal(x, y) -> anchored(x, y)

Types:
anchored :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> 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:
anchored
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(15) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(16)
BOUNDS(n^1, INF)