File size: 3,086 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import Mathlib.Combinatorics.Matroid.Closure
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Order.Interval.Finset.Nat
namespace OAI
namespace MatroidProphet.MarkedPivots
open scoped BigOperators
abbrev Occurrences (r q : ℕ) := Fin r ⊕ Fin q
def label {α : Type*} {r q : ℕ} (old : Fin r → α) (movable : Fin q → α) :
Occurrences r q → α := Sum.elim old movable
def mark {r q : ℕ} (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) :
Occurrences r q → Bool := Sum.elim oldMark movableMark
def OldOrdered {r q : ℕ} (time : Occurrences r q → ℕ) : Prop :=
∀ i j : Fin r, i < j → time (Sum.inl i) < time (Sum.inl j)
/-- The label is first spanned at this occurrence, and not before it. -/
def FirstSpans {α : Type*} {r q : ℕ} (M : Matroid α)
(labels : Occurrences r q → α) (time : Occurrences r q → ℕ)
(e : α) (o : Occurrences r q) : Prop :=
e ∈ M.closure (labels '' {p | time p ≤ time o}) ∧
e ∉ M.closure (labels '' {p | time p < time o})
/-- The finite set encodes precisely those test coordinates whose pivot has mark one. -/
def Records {α : Type*} {r q n : ℕ} (M : Matroid α)
(old : Fin r → α) (movable : Fin q → α) (test : Fin n → α)
(time : Occurrences r q → ℕ) (oldMark : Fin r → Bool)
(movableMark : Fin q → Bool) (pattern : Finset (Fin n)) : Prop :=
∀ f, ∃ o, FirstSpans M (label old movable) time (test f) o ∧
(mark oldMark movableMark o = true ↔ f ∈ pattern)
/-- All actual pivot-mark vectors over all interleavings and all movable marks. -/
noncomputable def patterns {α : Type*} {r q n : ℕ} (M : Matroid α)
(old : Fin r → α) (movable : Fin q → α) (test : Fin n → α)
(oldMark : Fin r → Bool) : Finset (Finset (Fin n)) := by
classical
exact Finset.univ.filter fun pattern =>
∃ time : Occurrences r q → ℕ, OldOrdered time ∧ Function.Injective time ∧
∃ movableMark : Fin q → Bool, Records M old movable test time oldMark movableMark pattern
/-- Full named source obligation, including well-definedness for every interleaving.
The test function is injective because the manuscript tests a set of `n` nonloops.
The conclusion actually remains true without that injectivity assumption. -/
def SourceContract.{u} : Prop :=
∀ {α : Type u} [Fintype α] (M : Matroid α) (r q n s : ℕ)
(old : Fin r → α) (movable : Fin q → α) (test : Fin n → α)
(oldMark : Fin r → Bool),
q ≤ s → (∀ i, old i ∈ M.E) → (∀ i, movable i ∈ M.E) →
M.closure (Set.range old) = M.E → Function.Injective test →
(∀ f, test f ∈ M.E ∧ test f ∉ M.closure ∅) →
(patterns M old movable test oldMark).card ≤
4 ^ s * ∑ k ∈ Finset.Iic s, n.choose k ∧
∀ (time : Occurrences r q → ℕ), OldOrdered time → Function.Injective time →
∀ movableMark : Fin q → Bool,
∃! pattern, Records M old movable test time oldMark movableMark pattern
end MatroidProphet.MarkedPivots
end OAI
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