| import OAI.Combinatorics.DiscreteConvexity.Model |
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| namespace OAI |
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| namespace TalagrandDiscreteConvexity |
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| open scoped BigOperators |
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| variable {α : Type*} [Fintype α] [DecidableEq α] |
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| noncomputable section |
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| theorem bernoulliMass_nonneg {p : ℝ} (hp0 : 0 ≤ p) (hp1 : p ≤ 1) |
| (s : Finset α) : 0 ≤ bernoulliMass p s := |
| mul_nonneg (pow_nonneg hp0 _) (pow_nonneg (sub_nonneg.mpr hp1) _) |
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| theorem familyMeasure_nonneg {p : ℝ} (hp0 : 0 ≤ p) (hp1 : p ≤ 1) |
| (D : Family α) : 0 ≤ familyMeasure p D := |
| Finset.sum_nonneg (fun s _ => bernoulliMass_nonneg hp0 hp1 s) |
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| @[simp] theorem familyMeasure_empty (p : ℝ) : |
| familyMeasure p (∅ : Family α) = 0 := by simp [familyMeasure] |
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| theorem familyMeasure_mono {p : ℝ} (hp0 : 0 ≤ p) (hp1 : p ≤ 1) |
| {D E : Family α} (h : D ⊆ E) : familyMeasure p D ≤ familyMeasure p E := by |
| exact Finset.sum_le_sum_of_subset_of_nonneg h |
| (fun s _ _ => bernoulliMass_nonneg hp0 hp1 s) |
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| omit [Fintype α] [DecidableEq α] in |
| @[simp] theorem cost_empty (p : ℝ) : cost p (∅ : Family α) = 0 := by |
| simp [cost] |
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| omit [Fintype α] [DecidableEq α] in |
| @[simp] theorem cost_singleton (p : ℝ) (I : Finset α) : |
| cost p {I} = p ^ I.card := by simp [cost] |
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| omit [Fintype α] [DecidableEq α] in |
| theorem cost_nonneg {p : ℝ} (hp : 0 ≤ p) (G : Family α) : 0 ≤ cost p G := |
| Finset.sum_nonneg (fun generator _ => pow_nonneg hp generator.card) |
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| omit [Fintype α] [DecidableEq α] in |
| theorem cost_mono {p : ℝ} (hp : 0 ≤ p) {G H : Family α} (h : G ⊆ H) : |
| cost p G ≤ cost p H := |
| Finset.sum_le_sum_of_subset_of_nonneg h (fun generator _ _ => pow_nonneg hp generator.card) |
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| omit [Fintype α] in |
| theorem cost_union_le {p : ℝ} (hp : 0 ≤ p) (G H : Family α) : |
| cost p (G ∪ H) ≤ cost p G + cost p H := by |
| calc |
| cost p (G ∪ H) ≤ cost p (G ∪ H) + cost p (G ∩ H) := |
| le_add_of_nonneg_right (cost_nonneg hp _) |
| _ = cost p G + cost p H := by |
| exact Finset.sum_union_inter |
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| omit [Fintype α] in |
| theorem cost_biUnion_le {ι : Type*} [DecidableEq ι] {p : ℝ} (hp : 0 ≤ p) |
| (T : Finset ι) (G : ι → Family α) : |
| cost p (T.biUnion G) ≤ ∑ i ∈ T, cost p (G i) := by |
| induction T using Finset.induction_on with |
| | empty => simp |
| | @insert a T ha ih => |
| rw [Finset.biUnion_insert, Finset.sum_insert ha] |
| exact (cost_union_le hp _ _).trans (add_le_add le_rfl ih) |
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| omit [Fintype α] in |
| theorem cost_image_le {ι : Type*} [DecidableEq ι] {p : ℝ} (hp : 0 ≤ p) |
| (T : Finset ι) (G : ι → Finset α) : |
| cost p (T.image G) ≤ ∑ i ∈ T, p ^ (G i).card := by |
| exact Finset.sum_image_le_of_nonneg (fun I _ => pow_nonneg hp I.card) |
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| omit [Fintype α] [DecidableEq α] in |
| theorem covers_self (A : Family α) : Covers A A := |
| fun S hS => ⟨S, hS, Finset.Subset.refl S⟩ |
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| omit [Fintype α] [DecidableEq α] in |
| @[simp] theorem covers_empty (G : Family α) : Covers G ∅ := by |
| intro S hS |
| exact False.elim (Finset.notMem_empty S hS) |
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| omit [Fintype α] [DecidableEq α] in |
| theorem Covers.mono_target {G A B : Family α} (h : Covers G A) (hBA : B ⊆ A) : |
| Covers G B := fun S hS => h S (hBA hS) |
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| omit [Fintype α] [DecidableEq α] in |
| theorem Covers.mono_cover {G H A : Family α} (h : Covers G A) (hGH : G ⊆ H) : |
| Covers H A := by |
| intro S hS |
| obtain ⟨I, hI, hIS⟩ := h S hS |
| exact ⟨I, hGH hI, hIS⟩ |
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| omit [Fintype α] in |
| theorem Covers.union {G H A B : Family α} (hGA : Covers G A) (hHB : Covers H B) : |
| Covers (G ∪ H) (A ∪ B) := by |
| intro S hS |
| rcases Finset.mem_union.mp hS with hS | hS |
| · obtain ⟨I, hI, hIS⟩ := hGA S hS |
| exact ⟨I, Finset.mem_union_left H hI, hIS⟩ |
| · obtain ⟨I, hI, hIS⟩ := hHB S hS |
| exact ⟨I, Finset.mem_union_right G hI, hIS⟩ |
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| omit [Fintype α] [DecidableEq α] in |
| theorem small_empty (p : ℝ) : Small p (∅ : Family α) := by |
| refine ⟨∅, covers_empty _, ?_⟩ |
| simp |
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| omit [Fintype α] in |
| @[simp] theorem mem_unionTuple {k : ℕ} {f : Fin k → Finset α} {a : α} : |
| a ∈ unionTuple f ↔ ∃ j, a ∈ f j := by |
| simp [unionTuple] |
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| omit [Fintype α] in |
| theorem subset_unionTuple {k : ℕ} (f : Fin k → Finset α) (j : Fin k) : |
| f j ⊆ unionTuple f := fun _ ha => mem_unionTuple.mpr ⟨j, ha⟩ |
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| omit [Fintype α] in |
| theorem unionTuple_subset_iff {k : ℕ} (f : Fin k → Finset α) (S : Finset α) : |
| unionTuple f ⊆ S ↔ ∀ j, f j ⊆ S := by |
| constructor |
| · intro h j |
| exact (subset_unionTuple f j).trans h |
| · intro h a ha |
| obtain ⟨j, hj⟩ := mem_unionTuple.mp ha |
| exact h j hj |
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| @[simp] theorem mem_unionFamily {k : ℕ} {D : Family α} {S : Finset α} : |
| S ∈ unionFamily k D ↔ |
| ∃ f : Fin k → Finset α, (∀ j, f j ∈ D) ∧ unionTuple f = S := by |
| classical |
| simp [unionFamily] |
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| @[simp] theorem mem_exceptional {k : ℕ} {D : Family α} {S : Finset α} : |
| S ∈ exceptional k D ↔ |
| ∀ f : Fin k → Finset α, (∀ j, f j ∈ D) → ¬ S ⊆ unionTuple f := by |
| classical |
| simp [exceptional] |
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| theorem mem_exceptional_iff_unionFamily {k : ℕ} {D : Family α} {S : Finset α} : |
| S ∈ exceptional k D ↔ ∀ T ∈ unionFamily k D, ¬ S ⊆ T := by |
| constructor |
| · intro h T hT |
| obtain ⟨f, hf, rfl⟩ := mem_unionFamily.mp hT |
| exact mem_exceptional.mp h f hf |
| · intro h |
| apply mem_exceptional.mpr |
| intro f hf |
| exact h (unionTuple f) (mem_unionFamily.mpr ⟨f, hf, rfl⟩) |
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| theorem exceptional_tuple_misses {k : ℕ} {D : Family α} {S : Finset α} |
| (hS : S ∈ exceptional k D) (f : Fin k → Finset α) (hf : ∀ j, f j ∈ D) : |
| ∃ a ∈ S, ∀ j, a ∉ f j := by |
| obtain ⟨a, ha, hmiss⟩ := Finset.not_subset.mp (mem_exceptional.mp hS f hf) |
| exact ⟨a, ha, fun j hj => hmiss (mem_unionTuple.mpr ⟨j, hj⟩)⟩ |
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| theorem exceptional_antitone {k : ℕ} {D E : Family α} (hDE : D ⊆ E) : |
| exceptional k E ⊆ exceptional k D := by |
| intro S hS |
| apply mem_exceptional.mpr |
| intro f hf |
| exact mem_exceptional.mp hS f (fun j => hDE (hf j)) |
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| theorem exceptional_eq_empty_of_full_union {k : ℕ} {D : Family α} |
| (f : Fin k → Finset α) (hf : ∀ j, f j ∈ D) (hfull : unionTuple f = Finset.univ) : |
| exceptional k D = ∅ := by |
| apply Finset.eq_empty_iff_forall_notMem.mpr |
| intro S hS |
| exact mem_exceptional.mp hS f hf (hfull.symm ▸ Finset.subset_univ S) |
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| omit [DecidableEq α] in |
| @[simp] theorem mem_cubeSet {x : α → Bool} {a : α} : |
| a ∈ cubeSet x ↔ x a = true := by simp [cubeSet] |
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| @[simp] theorem cubeSet_decide (S : Finset α) : |
| cubeSet (fun a => decide (a ∈ S)) = S := by |
| ext a |
| simp |
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| theorem familyIndicator_nonneg (D : Family α) (x : α → Bool) : |
| 0 ≤ familyIndicator D x := by |
| unfold familyIndicator |
| split_ifs <;> simp |
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| theorem familyIndicator_le_one (D : Family α) (x : α → Bool) : |
| familyIndicator D x ≤ 1 := by |
| unfold familyIndicator |
| split_ifs <;> simp |
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| theorem familyIndicator_sq (D : Family α) (x : α → Bool) : |
| familyIndicator D x ^ 2 = familyIndicator D x := by |
| unfold familyIndicator |
| split_ifs <;> simp |
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| end |
| end TalagrandDiscreteConvexity |
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| end OAI |
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