| import Mathlib |
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| namespace OAI |
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| section |
| section |
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| noncomputable section |
| namespace HyperbolicObstruction |
| namespace SparsePattern |
| open scoped BigOperators |
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| /-- Independent table entry types, with their coordinate supports and critical costs. |
| The table model of Sections 2 and 4 is a concrete instance. -/ |
| structure Pattern (F R : Type*) [Fintype F] [Fintype R] where |
| support : R → Finset F |
| cost : R → ℤ |
| nonneg : ∀ r, 0 ≤ cost r |
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| variable {F R : Type*} [Fintype F] [Fintype R] [DecidableEq F] |
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| namespace Pattern |
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| def exponent (P : Pattern F R) (W : Finset F) : ℤ := |
| W.card - ∑ r, if P.support r ⊆ W then P.cost r else 0 |
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| lemma exponent_submodular (P : Pattern F R) (S T : Finset F) : |
| P.exponent (S ∪ T) + P.exponent (S ∩ T) ≤ P.exponent S + P.exponent T := by |
| classical |
| have hcost : (∑ r, if P.support r ⊆ S then P.cost r else 0) + |
| (∑ r, if P.support r ⊆ T then P.cost r else 0) ≤ |
| (∑ r, if P.support r ⊆ S ∪ T then P.cost r else 0) + |
| (∑ r, if P.support r ⊆ S ∩ T then P.cost r else 0) := by |
| rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib] |
| apply Finset.sum_le_sum |
| intro r _ |
| by_cases hS : P.support r ⊆ S <;> by_cases hT : P.support r ⊆ T |
| · simp [hS, hT, Finset.subset_union_left.trans' hS, Finset.subset_inter hS hT] |
| · have hu : P.support r ⊆ S ∪ T := hS.trans Finset.subset_union_left |
| have hi : ¬P.support r ⊆ S ∩ T := fun h => hT (h.trans Finset.inter_subset_right) |
| simp [hS, hT, hu, hi] |
| · have hu : P.support r ⊆ S ∪ T := hT.trans Finset.subset_union_right |
| have hi : ¬P.support r ⊆ S ∩ T := fun h => hS (h.trans Finset.inter_subset_left) |
| simp [hS, hT, hu, hi] |
| · have hi : ¬P.support r ⊆ S ∩ T := fun h => hS (h.trans Finset.inter_subset_left) |
| simp only [hS, hT, hi, ite_false, add_zero] |
| split_ifs <;> simp [P.nonneg] |
| have hc : ((S ∪ T).card : ℤ) + (S ∩ T).card = S.card + T.card := by |
| exact_mod_cast Finset.card_union_add_card_inter S T |
| unfold exponent |
| linarith |
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| lemma exponent_eq_card (P : Pattern F R) (I : Finset F) |
| (hI : ∀ r, ¬P.support r ⊆ I) : P.exponent I = I.card := by |
| simp [exponent, hI] |
|
|
| /-- The maximal critical closure in Section 4. The conclusions retain a full |
| unit integer margin, not merely strictness. -/ |
| theorem critical_closure (P : Pattern F R) (I : Finset F) (e : F) |
| (he : e ∉ I) (hI : P.exponent I = I.card) |
| (hbudget : ∀ W, I ⊆ W → (I.card : ℤ) ≤ P.exponent W) |
| (htest : ∀ W, insert e I ⊆ W → ((insert e I).card : ℤ) ≤ P.exponent W) : |
| ∃ U : Finset F, I ⊆ U ∧ P.exponent U = I.card ∧ e ∉ U ∧ |
| ∀ V : Finset F, V.Nonempty → Disjoint V U → |
| 1 ≤ P.exponent (U ∪ V) - P.exponent U := by |
| classical |
| let B : Finset (Finset F) := Finset.univ.filter (fun U => I ⊆ U ∧ P.exponent U = I.card) |
| have hmem : I ∈ B := by simp [B, hI] |
| obtain ⟨U, hU, hmax⟩ := B.exists_max_image Finset.card ⟨I, hmem⟩ |
| have hU' : I ⊆ U ∧ P.exponent U = I.card := (Finset.mem_filter.mp hU).2 |
| refine ⟨U, hU'.1, hU'.2, ?_, ?_⟩ |
| · intro heU |
| have hh := htest U (Finset.insert_subset heU hU'.1) |
| rw [Finset.card_insert_of_notMem he, Nat.cast_add, Nat.cast_one, hU'.2] at hh |
| omega |
| · intro V hV hd |
| have hsub : I ⊆ U ∪ V := hU'.1.trans Finset.subset_union_left |
| have hb := hbudget (U ∪ V) hsub |
| have hne : P.exponent (U ∪ V) ≠ (I.card : ℤ) := by |
| intro heq |
| have hm : U ∪ V ∈ B := by simp [B, hsub, heq] |
| have hm' := hmax (U ∪ V) hm |
| have hc : (U ∪ V).card = U.card + V.card := |
| Finset.card_union_of_disjoint hd.symm |
| have hv := Finset.card_pos.mpr hV |
| omega |
| rw [hU'.2] |
| omega |
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| end Pattern |
| end SparsePattern |
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| end HyperbolicObstruction |
| end |
| end |
| end |
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| end OAI |
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