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Finite list tail statistics

Lax342547.ListTails · concepts/Lax342547/ListTails.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Nonnegative decreasing finite lists extend by zero to decreasing sequences; their finite sums and dyadic tail sums agree with the actual list tails.

    Concept map
    6 concepts
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 7 statements. Each proof establishes one of them relative to its assumptions.

    1 sum_values proven

    2 tail_sum_list proven

    3 value_antitone proven

    4 value_drop proven

    5 value_in_range proven

    6 value_nonneg proven

    7 value_out_of_range proven

    Lean source view on GitHub

    1import Lax342547.GreedySpans
    2import Lax342547.DyadicTails
    3import Mathlib.Data.List.Sort
    4
    5/-!
    6---
    7title: Finite list tail statistics
    8type: lemma
    9---
    10Nonnegative decreasing finite lists extend by zero to decreasing sequences; their finite sums and dyadic tail sums agree with the actual list tails.
    11-/
    12
    13namespace Lax342547.ListTails
    14
    15open scoped BigOperators
    16
    17noncomputable def value (l : List ℝ) (t : ℕ) : ℝ := l[t]?.getD 0
    18
    19axiom value_in_range (l : List ℝ) (t : ℕ) (ht : t < l.length) : value l t = l[t]
    20
    21axiom value_out_of_range (l : List ℝ) (t : ℕ) (ht : l.length ≤ t) : value l t = 0
    22
    23axiom value_nonneg (l : List ℝ) (h : ∀ x ∈ l, 0 ≤ x) (t : ℕ) : 0 ≤ value l t
    24
    25axiom value_antitone (l : List ℝ) (h : l.Pairwise (fun a b => b ≤ a))
    26 (h0 : ∀ x ∈ l, 0 ≤ x) : Antitone (value l)
    27
    28axiom value_drop (l : List ℝ) (n t : ℕ) : value (l.drop n) t = value l (n+t)
    29
    30axiom sum_values (l : List ℝ) : (∑ t ∈ Finset.range l.length, value l t) = l.sum
    31
    32axiom tail_sum_list (l : List ℝ) (b : ℕ) (hb : b ≤ l.length) :
    33 Lax342547.DyadicTails.tailSum (value l) l.length b = (l.drop (l.length-b)).sum
    34
    35end Lax342547.ListTails
    36
    Show ProofShow ProofShow ProofShow ProofShow ProofShow ProofShow Proof
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