File size: 3,070 Bytes
c270adf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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
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