Lax Lean Archive

Computing the far tree

Lax185574.ToFarTree · concepts/Lax185574/ToFarTree.lean · lax-185574

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    Natural Language Statement

    Lemma

    Fix k ∈ ℕ and q ≥ 1, let G be a partial graph, and let ā ∈ V(G)^{≤k} be nonempty. Then ftree_{k,q}(ā) is computable from tree_{k,q}(ā) and stp_{k,q} in time

    ‖stp_{k,q}‖ · poly_{k,q}(‖tree_{k,q}(ā)‖).

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    Evidence

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    In the paper

    • page 46 of this submission's paper

    Lean source view on GitHub

    1import Lax185574.ScatterTypes
    2import Lax185574.FarTrees
    3import Lax185574.Algorithms
    4
    5/-!
    6---
    7title: Computing the far tree
    8type: lemma
    9---
    10Fix k ∈ ℕ and q ≥ 1, let G be a partial graph, and let ā ∈ V(G)^{≤k} be
    11nonempty. Then ftree_{k,q}(ā) is computable from tree_{k,q}(ā) and
    12stp_{k,q} in time
    13
    14 ‖stp_{k,q}‖ · poly_{k,q}(‖tree_{k,q}(ā)‖).
    15
    16# Formalization notes
    17
    18The algorithm is an algorithm in the sense of `Lax185574.Algorithms`,
    19one for all k and q. q ≥ 1 is written as q + 1. The input is the pair of
    20the type tree and the scatter type. Sizes are lengths of encodings. The
    21time is `O((‖stp‖ + 1) · (‖τ‖ + 1)^a)`, where the degree `a` and the
    22constant `c` may depend on k and q. The factors are increased by one, so
    23that the bound is positive.
    24-/
    25
    26namespace Lax185574.ToFarTree
    27
    28open Lax185574.Encoding Lax185574.Algorithms
    29open Lax185574.PartialGraphs Lax185574.TypeTrees Lax185574.ScatterTypes Lax185574.FarTrees
    30
    31/-- **Computing the far tree.** One algorithm computes `ftree_{k,q}(u)`
    32from `tree_{k,q}(u)` and `stp_{k,q}`, in time linear in the size of the
    33scatter type and polynomial in the size of the tree. -/
    34axiom exists_program_ftree :
    35 ∃ A : Algorithm, ∀ k q : ℕ, ∃ c a : ℕ,
    36 ∀ (n : ℕ) (G : PartialGraph (Fin n)) (ℓ : ℕ) (u : Fin ℓ → Fin n), 1 ≤ ℓ → ℓ ≤ k →
    37 A.Computes c (tree G k (q + 1) u, stp G k (q + 1)) (ftree G k (q + 1) u)
    38 (((encode (stp G k (q + 1))).length + 1) *
    39 ((encode (tree G k (q + 1) u)).length + 1) ^ a)
    40
    41end Lax185574.ToFarTree
    42
    Show Proof
    Formalization notes

    The algorithm is an algorithm in the sense of Lax185574.AlgorithmsLax185574.Algorithms, one for all k and q. q ≥ 1 is written as q + 1. The input is the pair of the type tree and the scatter type. Sizes are lengths of encodings. The time is O((‖stp‖+1)⋅(‖τ‖+1)a)O((‖stp‖ + 1) · (‖τ‖ + 1)^a), where the degree aa and the constant cc may depend on k and q. The factors are increased by one, so that the bound is positive.

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