| import OAI.Probability.MatroidProphet.Pivots.Arbitrary |
| import OAI.Probability.MatroidProphet.Pivots.Encoding |
| import OAI.Probability.MatroidProphet.Pivots.OccurrenceSupport |
| import OAI.Probability.MatroidProphet.Pivots.Scores |
| import OAI.Probability.MatroidProphet.Pivots.Sign |
| import Mathlib.LinearAlgebra.Dimension.Constructions |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| namespace Pivots |
|
|
| open Set Finset |
| open scoped BigOperators |
|
|
| variable {α : Type*} [Fintype α] {r q n : ℕ} |
|
|
| def occurrenceMark (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) : |
| Occurrence r q → Bool := Sum.elim oldMark movableMark |
|
|
| noncomputable def markedScore (s : ℕ) (time : Occurrence r q → ℕ) |
| (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) (o : Occurrence r q) : ℝ := |
| if occurrenceMark oldMark movableMark o then (2 : ℝ) ^ scoreExponent s time o |
| else -(2 : ℝ) ^ scoreExponent s time o |
|
|
| lemma abs_markedScore (s : ℕ) (time : Occurrence r q → ℕ) |
| (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) (o : Occurrence r q) : |
| |markedScore s time oldMark movableMark o| = (2 : ℝ) ^ scoreExponent s time o := by |
| unfold markedScore |
| split <;> simp [abs_of_nonneg (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) _)] |
|
|
| noncomputable def fixedOldScore (s : ℕ) (oldMark : Fin r → Bool) (i : Fin r) : ℝ := |
| if oldMark i then (2 : ℝ)^((s+1)*(i.val+1)) else -(2 : ℝ)^((s+1)*(i.val+1)) |
|
|
| noncomputable def constantScore (s : ℕ) (oldMark : Fin r → Bool) : Occurrence r q → ℝ := |
| Sum.elim (fixedOldScore s oldMark) (fun _ => 0) |
|
|
| noncomputable def coordinateScore : Occurrence r q → ((Fin q → ℝ) →ₗ[ℝ] ℝ) := |
| Sum.elim (fun _ => 0) (fun i => LinearMap.proj i) |
|
|
| noncomputable def patternLinear (M : Matroid α) (label : Occurrence r q → α) |
| (J : Finset (Occurrence r q)) (e : α) : (Fin q → ℝ) →ₗ[ℝ] ℝ := |
| ∑ o ∈ occurrenceSupport M label J e, coordinateScore o |
|
|
| noncomputable def patternConstant (M : Matroid α) (label : Occurrence r q → α) |
| (J : Finset (Occurrence r q)) (e : α) (s : ℕ) (oldMark : Fin r → Bool) : ℝ := |
| ∑ o ∈ occurrenceSupport M label J e, constantScore s oldMark o |
|
|
| lemma affineScore_eq_sum |
| {α : Type u_1} [Fintype α] {r : ℕ} {q : ℕ} (M : Matroid α) (label : Occurrence r q → α) |
| (J : Finset (Occurrence r q)) (e : α) (s : ℕ) (time : Occurrence r q → ℕ) |
| (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) : |
| patternLinear M label J e (fun i => markedScore s time oldMark movableMark (Sum.inr i)) + |
| patternConstant M label J e s oldMark = |
| ∑ o ∈ occurrenceSupport M label J e, markedScore s time oldMark movableMark o := by |
| simp only [patternLinear, patternConstant, LinearMap.sum_apply, ← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro o _ |
| cases o <;> simp [coordinateScore, constantScore, fixedOldScore, markedScore, |
| occurrenceMark, scoreExponent] |
| all_goals rfl |
|
|
| def RecordsPivots (M : Matroid α) (label : Occurrence r q → α) |
| (test : Fin n → α) (time : Occurrence r q → ℕ) |
| (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) (t : Finset (Fin n)) : Prop := |
| ∀ f, ∃ o, IsPivot M label time (test f) o ∧ |
| (occurrenceMark oldMark movableMark o = true ↔ f ∈ t) |
|
|
| lemma recordsPivots_affine (M : Matroid α) (b : ℕ → α) (a : Fin q → α) |
| (test : Fin n → α) (s : ℕ) (hqs : q ≤ s) |
| (oldMark : Fin r → Bool) (movableMark : Fin q → Bool) |
| (time : Occurrence r q → ℕ) (horder : OldOrdered time) (htime : Function.Injective time) |
| (hground : ∀ o : Occurrence r q, occurrenceLabel b a o ∈ M.E) (t : Finset (Fin n)) |
| (hrecords : RecordsPivots M (occurrenceLabel b a) test time oldMark movableMark t) : |
| ∃ x, RealizesSigns |
| (fun f => patternLinear M (occurrenceLabel b a) (retained M b a time) (test f)) |
| (fun f => patternConstant M (occurrenceLabel b a) (retained M b a time) (test f) s oldMark) |
| t x := by |
| classical |
| let label : Occurrence r q → α := occurrenceLabel b a |
| let J := retained M b a time |
| let w := markedScore s time oldMark movableMark |
| let exponent := scoreExponent s time |
| refine ⟨fun i => w (Sum.inr i), ?_⟩ |
| intro f |
| rw [affineScore_eq_sum] |
| obtain ⟨o, hpivot, hmark⟩ := hrecords f |
| have hp := pivot_mem_support_and_max M label time hground htime hpivot |
| change o ∈ occurrenceSupport M label J (test f) ∧ |
| (∀ p ∈ occurrenceSupport M label J (test f), time p ≤ time o) at hp |
| have hmax : ∀ p ∈ occurrenceSupport M label J (test f), p ≠ o → exponent p < exponent o := by |
| intro p hp' hne |
| apply scoreExponent_preserves_order s hqs time horder |
| exact lt_of_le_of_ne (hp.2 p hp') (fun heq => hne (htime heq)) |
| have hexp := scoreExponent_injective s hqs time horder htime |
| have habs : ∀ p ∈ occurrenceSupport M label J (test f), |w p| = 2 ^ exponent p := |
| fun p _ => abs_markedScore s time oldMark movableMark p |
| have hpos := sum_positive_iff_max_exponent _ w exponent hexp hp.1 hmax habs |
| have hneg := sum_positive_iff_max_exponent _ (fun p => -w p) exponent hexp hp.1 hmax |
| (by intro p hp'; simpa using habs p hp') |
| simp only [Finset.sum_neg_distrib, neg_pos] at hneg |
| have hpow : (0 : ℝ) < 2 ^ exponent o := pow_pos (by norm_num) _ |
| by_cases hf : f ∈ t |
| · simp only [hf, ↓reduceIte] |
| apply hpos.mpr |
| have hm := hmark.mpr hf |
| simp [w, markedScore, hm] |
| · simp only [hf, ↓reduceIte] |
| apply hneg.mpr |
| have hm : occurrenceMark oldMark movableMark o = false := by |
| cases h : occurrenceMark oldMark movableMark o |
| · rfl |
| · exact False.elim (hf (hmark.mp h)) |
| simp [w, markedScore, hm] |
|
|
| end Pivots |
| end MatroidProphet |
|
|
| end OAI |
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|
