| import Mathlib | |
| namespace OAI | |
| namespace LargeIndependentSets | |
| theorem extend_partialIso {X Y : Type*} [LinearOrder X] [LinearOrder Y] | |
| [Countable X] [DenselyOrdered X] [NoMinOrder X] [NoMaxOrder X] [Nonempty X] | |
| [Countable Y] [DenselyOrdered Y] [NoMinOrder Y] [NoMaxOrder Y] [Nonempty Y] | |
| (seed : Order.PartialIso X Y) : | |
| ∃ e : X ≃o Y, ∀ x y, (x, y) ∈ seed.val → e x = y := by | |
| classical | |
| cases nonempty_encodable X | |
| cases nonempty_encodable Y | |
| let cof : X ⊕ Y → Order.Cofinal (Order.PartialIso X Y) := fun p => | |
| Sum.recOn p (Order.PartialIso.definedAtLeft Y) (Order.PartialIso.definedAtRight X) | |
| let ideal : Order.Ideal (Order.PartialIso X Y) := Order.idealOfCofinals seed cof | |
| let F x := Order.PartialIso.funOfIdeal x ideal | |
| (Order.cofinal_meets_idealOfCofinals seed cof (Sum.inl x)) | |
| let G y := Order.PartialIso.invOfIdeal y ideal | |
| (Order.cofinal_meets_idealOfCofinals seed cof (Sum.inr y)) | |
| let e := OrderIso.ofCmpEqCmp (fun x => (F x).val) (fun y => (G y).val) (by | |
| intro x y | |
| rcases (F x).property with ⟨f, hf, hx⟩ | |
| rcases (G y).property with ⟨g, hg, hy⟩ | |
| rcases ideal.directed _ hf _ hg with ⟨m, _, fm, gm⟩ | |
| exact m.property (x, _) (fm hx) (_, y) (gm hy)) | |
| refine ⟨e, ?_⟩ | |
| intro x y hxy | |
| rcases (F x).property with ⟨f, hf, hx⟩ | |
| rcases ideal.directed _ (Order.mem_idealOfCofinals seed cof) _ hf with ⟨m, _, sm, fm⟩ | |
| have hh := m.property (x, y) (sm hxy) (x, (F x).val) (fm hx) | |
| have hresult : y = (F x).val := (eq_iff_eq_of_cmp_eq_cmp hh).mp rfl | |
| exact hresult.symm | |
| theorem rational_automorphism_fixing (I : Finset ℚ) (a b : ℚ) | |
| (ha : a ∉ I) (hb : b ∉ I) | |
| (hcut : ∀ i ∈ I, i < a ↔ i < b) : | |
| ∃ e : ℚ ≃o ℚ, e a = b ∧ ∀ i ∈ I, e i = i := by | |
| classical | |
| have hcmp (i : ℚ) (hi : i ∈ I) : cmp i a = cmp i b := by | |
| by_cases hia : i < a | |
| · rw [(cmp_eq_lt_iff i a).mpr hia, (cmp_eq_lt_iff i b).mpr ((hcut i hi).mp hia)] | |
| · have hai : a < i := lt_of_le_of_ne (le_of_not_gt hia) (fun h => ha (h ▸ hi)) | |
| have hbi : b < i := lt_of_le_of_ne (le_of_not_gt (fun h => hia ((hcut i hi).mpr h))) | |
| (fun h => hb (h ▸ hi)) | |
| rw [(cmp_eq_gt_iff i a).mpr hai, (cmp_eq_gt_iff i b).mpr hbi] | |
| let seed : Order.PartialIso ℚ ℚ := | |
| ⟨insert (a, b) (I.image (fun i => (i, i))), by | |
| intro p hp q hq | |
| rcases Finset.mem_insert.mp hp with rfl | hp | |
| · rcases Finset.mem_insert.mp hq with rfl | hq | |
| · simp | |
| · obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hq | |
| simpa only [Prod.fst, Prod.snd, cmp_swap] using congrArg Ordering.swap (hcmp i hi) | |
| · obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hp | |
| rcases Finset.mem_insert.mp hq with rfl | hq | |
| · exact hcmp i hi | |
| · obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hq | |
| rfl⟩ | |
| obtain ⟨e, he⟩ := extend_partialIso seed | |
| refine ⟨e, he _ _ (Finset.mem_insert_self _ _), ?_⟩ | |
| intro i hi | |
| exact he i i (Finset.mem_insert_of_mem (Finset.mem_image_of_mem _ hi)) | |
| end LargeIndependentSets | |
| end OAI | |