openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Pivots /MarkedPivotSpecification.lean
| 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 | |