| import Mathlib |
| import OAI.SetTheory.PartitionConsistency.Algebra.Placements |
| import OAI.SetTheory.PartitionConsistency.Forcing.CohenForcing |
| import OAI.SetTheory.PartitionConsistency.Forcing.ForcingQuotient |
|
|
| namespace OAI |
|
|
| namespace PartitionConsistency |
| namespace SmallPlacements |
|
|
| /-- Only countably many members of a pairwise disjoint family meet a fixed |
| countable set. This will be applied to the nonroot supports of probe copies. -/ |
| theorem countable_obstructions {I A : Type*} (S : A → Set I) (C : Set I) |
| (hS : Pairwise (fun a b => Disjoint (S a) (S b))) (hC : C.Countable) : |
| {a | (S a ∩ C).Nonempty}.Countable := |
| Set.countable_ofPred_nonempty_of_disjoint |
| (fun _a _b hab => (hS hab).mono Set.inter_subset_left Set.inter_subset_left) |
| (fun _ => Set.inter_subset_right) hC |
|
|
| /-- An uncountable family of disjoint nonroot supports always has a member |
| avoiding any specified countable collection of coordinates. -/ |
| theorem exists_fresh_member {I A : Type*} [Uncountable A] (S : A → Set I) |
| (C : Set I) (hS : Pairwise (fun a b => Disjoint (S a) (S b))) (hC : C.Countable) : |
| ∃ a, Disjoint (S a) C := by |
| have hc := countable_obstructions S C hS hC |
| have hn : {a | (S a ∩ C).Nonempty} ≠ Set.univ := by |
| intro hu |
| exact Set.not_countable_univ (hu ▸ hc) |
| obtain ⟨a, ha⟩ := Set.ne_univ_iff_exists_notMem _ |>.1 hn |
| exact ⟨a, Set.disjoint_iff_inter_eq_empty.2 (Set.not_nonempty_iff_eq_empty.1 ha)⟩ |
|
|
| end SmallPlacements |
|
|
| namespace CohenForcing.Condition |
| universe u v w |
| variable {I : Type u} {κ : Type v} {A : Type w} |
|
|
| @[refl] theorem Compatible.refl (p : Condition I κ) : Compatible p p := p.functional |
|
|
| /-- Weakening either of two compatible partial functions preserves their |
| compatibility. Recall that the forcing order is reverse inclusion. -/ |
| theorem Compatible.weaken {p q p' q' : Condition I κ} (h : Compatible p q) |
| (hp : p ≤ p') (hq : q ≤ q') : Compatible p' q' := |
| fun a b c hb hc => h a b c (hp hb) (hq hc) |
|
|
| theorem Compatible.of_le {p q : Condition I κ} (h : p ≤ q) : Compatible p q := |
| (Compatible.refl p).weaken le_rfl h |
|
|
| /-- The explicit ground dense set in the probe argument for ordinal choice |
| (source equation `eq:probe-cohen-dense`). It contains conditions incompatible |
| with the common root and conditions extending one of the placed copies. -/ |
| def probeDenseSet (root : Condition I κ) (copies : A → Condition I κ) : |
| Set (Condition I κ) := |
| {c | ¬ Compatible c root ∨ ∃ a, c ≤ copies a} |
|
|
| /-- Pairwise independent copies with the same Cohen root give the probe dense |
| set. Uncountability is sufficient: a test condition has countable support. |
| No forcing name, generic extension, or choice in a symmetric model is assumed. -/ |
| theorem probeDenseSet_dense [Uncountable A] (J : Set I) (root : Condition I κ) |
| (copies : A → Condition I κ) (hroot : ∀ a, (copies a).restrict J = root) |
| (hdisj : Pairwise (fun a b => Disjoint ((copies a).support \ J) ((copies b).support \ J))) : |
| ForcingQuotient.Dense (probeDenseSet root copies) := by |
| intro c |
| by_cases hc : Compatible c root |
| · obtain ⟨a, ha⟩ := SmallPlacements.exists_fresh_member |
| (fun a => (copies a).support \ J) c.support hdisj c.countable_support |
| have hr : Compatible (c.restrict J) ((copies a).restrict J) := by |
| rw [hroot a] |
| exact hc.weaken (le_restrict c J) le_rfl |
| have hd : Disjoint (c.support \ J) ((copies a).support \ J) := |
| (ha.mono_right Set.sdiff_subset).symm |
| have h := compatible_of_disjoint_off_root c (copies a) J hr hd |
| exact ⟨union c (copies a) h, (union_le_left _ _ _), |
| Or.inr ⟨a, union_le_right _ _ _⟩⟩ |
| · exact ⟨c, le_rfl, Or.inl hc⟩ |
|
|
| /-- A filter containing the common root and meeting the displayed dense set |
| contains an extension of one of the copies. It is not necessary, and would be |
| incorrect, to assume that any preselected moved condition belongs to the filter. -/ |
| theorem meets_probe {root : Condition I κ} {copies : A → Condition I κ} |
| (G : ForcingQuotient.GenericFilter (Condition I κ)) (hroot : root ∈ G) |
| (hmeets : ∃ c : Condition I κ, c ∈ G ∧ c ∈ probeDenseSet root copies) : |
| ∃ a, ∃ c : Condition I κ, c ∈ G ∧ c ≤ copies a := by |
| obtain ⟨c, hcG, hc⟩ := hmeets |
| rcases hc with hn | ⟨a, ha⟩ |
| · obtain ⟨d, _, hdc, hdr⟩ := G.directed c hcG root hroot |
| exact (hn (compatible_iff.2 ⟨d, hdc, hdr⟩)).elim |
| · exact ⟨a, c, hcG, ha⟩ |
|
|
| end CohenForcing.Condition |
| end PartitionConsistency |
|
|
| end OAI |
|
|
