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A fixed RAM program constructing the complete moment LP

Lax263307.NativeMomentLP · concepts/Lax263307/NativeMomentLP.lean · lax-263307

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    Natural Language Statement

    Theorem

    One finite RAM program constructs the exact dense rational moment LP from its graph, target k and level r. It generates the coordinate catalog, computes cardinality and independence metadata, writes all six matrix row families, and appends the bound and objective vectors. If N is the catalog size, both execution time and sufficient word width are bounded by c(n+k+r+N+2)^4 for one fixed constant c, uniformly at every larger width.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax263307.MomentLP
    2import Lax808846.Ram
    3import Lax109476.RationalEncoding
    4
    5/-!
    6---
    7title: A fixed RAM program constructing the complete moment LP
    8type: theorem
    9---
    10One finite RAM program constructs the exact dense rational moment LP from
    11its graph, target k and level r. It generates the coordinate catalog,
    12computes cardinality and independence metadata, writes all six matrix row
    13families, and appends the bound and objective vectors. If N is the catalog
    14size, both execution time and sufficient word width are bounded by
    15c(n+k+r+N+2)^4 for one fixed constant c, uniformly at every larger width.
    16
    17# Formalization notes
    18
    19Physical input is n, k, r, n squared, followed by the row-major adjacency
    20bits. Output is the registered RationalEncoding.encodeProgram format,
    21including its dimension header and canonical rational triples. The program
    22uses only the N catalog masks, without any array indexed by all 2^n masks.
    23Solver invocation and the recursive QPTAS driver are separate obligations.
    24-/
    25
    26namespace Lax263307.NativeMomentLP
    27
    28def encodeInput {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] (k r : ℕ) : List ℕ :=
    29 n::k::r::(n*n)::(List.ofFn fun u => List.ofFn fun v => if G.Adj u v then 1 else 0).flatten
    30
    31axiom exists_program : ∃ p : Lax808846.Ram.Program, ∃ c : ℕ,0 < c ∧
    32 ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] (k r width : ℕ),
    33 c*(n+k+r+MomentLP.dimension n r+2)^4 ≤ width →
    34 ∃ t ≤ c*(n+k+r+MomentLP.dimension n r+2)^4,Lax808846.Ram.RunsTo width p (encodeInput G k r)
    35 (Lax109476.RationalEncoding.encodeProgram (MomentLP.program G k r)) t
    36
    37end Lax263307.NativeMomentLP
    38
    Show Proof
    Formalization notes

    Physical input is n, k, r, n squared, followed by the row-major adjacency bits. Output is the registered RationalEncoding.encodeProgram format, including its dimension header and canonical rational triples. The program uses only the N catalog masks, without any array indexed by all 2^n masks. Solver invocation and the recursive QPTAS driver are separate obligations.

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