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9dbc64e | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import OAI.Computability.DegreeRigidity.OrdinalCodes.LevelOrdinalHeight
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel
universe u
noncomputable def largestOrdinalSentence : SentenceForm :=
.conj ordinalSentence (.all (.imp ordinalSentence (.disj (.member 0 1) (.equal 0 1))))
theorem largestOrdinalSentence_bound : largestOrdinalSentence.bound = 1 := by rfl
theorem largestOrdinalSentence_spec (A : ZFSet.{u}) (hA : Transitive A)
(e : ℕ → ZFSet.{u}) (he : e 0 ∈ A) :
largestOrdinalSentence.Sat (A : Set ZFSet) e ↔
(e 0).IsOrdinal ∧ ∀ x ∈ A, x.IsOrdinal → x ∈ e 0 ∨ x = e 0 := by
simp only [largestOrdinalSentence,SentenceForm.Sat,SentenceForm.sat_all,
SentenceForm.sat_imp,SentenceForm.sat_disj,cons_zero,cons_succ]
rw [ordinalSentence_spec A hA e he]
apply and_congr_right; intro _
apply forall_congr'; intro x
apply forall_congr'; intro hx
rw [ordinalSentence_spec A hA (cons x e) hx]
rfl
theorem largestOrdinalSentence_unique (A : ZFSet.{u}) (hA : Transitive A)
(a : Ordinal.{u}) (ha : a.toZFSet ∈ A)
(hmax : ∀ b : Ordinal.{u}, b.toZFSet ∈ A → b ≤ a)
(e : ℕ → ZFSet.{u}) (he : e 0 ∈ A) :
largestOrdinalSentence.Sat (A : Set ZFSet) e ↔ e 0 = a.toZFSet := by
rw [largestOrdinalSentence_spec A hA e he]
constructor
· rintro ⟨ho,hm⟩
have heq := ho.toZFSet_rank_eq
have hle := hmax (e 0).rank (by rwa [heq])
have hge : a ≤ (e 0).rank := by
rcases hm a.toZFSet ha (ZFSet.isOrdinal_toZFSet a) with h|h
· rw [← heq,Ordinal.toZFSet_mem_toZFSet_iff] at h
exact h.le
· have := congrArg ZFSet.rank h
simpa only [Ordinal.rank_toZFSet] using this.le
exact heq.symm.trans (congrArg Ordinal.toZFSet (le_antisymm hle hge))
· intro heq
refine ⟨heq ▸ ZFSet.isOrdinal_toZFSet a,?_⟩
intro x hx ho
have hle : x.rank ≤ a := hmax x.rank (by rwa [ho.toZFSet_rank_eq])
rcases lt_or_eq_of_le hle with h|h
· left
rw [heq,← ho.toZFSet_rank_eq]
exact Ordinal.toZFSet_mem_toZFSet_iff.mpr h
· right
rw [← ho.toZFSet_rank_eq,h,heq]
theorem largestOrdinalSentence_level (R : ZFSet.{u}) (o : Ordinal.{u})
(e : ℕ → ZFSet.{u}) (he : e 0 ∈ level R (o+1)) :
largestOrdinalSentence.Sat (level R (o+1) : Set ZFSet) e ↔
e 0 = (ordinalHeight (seed R) + o).toZFSet := by
have hm := largest_ordinal_level_succ R o
exact largestOrdinalSentence_unique _ (level_transitive R _) _ hm.1
(fun b hb => (hm.2 b).mp hb) e he
noncomputable def bindLargestOrdinal (p : SentenceForm) : SentenceForm :=
.ex (.conj largestOrdinalSentence p)
theorem bindLargestOrdinal_bound (p : SentenceForm) :
(bindLargestOrdinal p).bound = p.bound - 1 := by
change max largestOrdinalSentence.bound p.bound - 1 = p.bound - 1
rw [largestOrdinalSentence_bound]
omega
theorem bindLargestOrdinal_level (p : SentenceForm) (R : ZFSet.{u}) (o : Ordinal.{u})
(e : ℕ → ZFSet.{u}) :
(bindLargestOrdinal p).Sat (level R (o+1) : Set ZFSet) e ↔
p.Sat (level R (o+1) : Set ZFSet) (cons (ordinalHeight (seed R) + o).toZFSet e) := by
change (∃ a ∈ level R (o+1), largestOrdinalSentence.Sat _ (cons a e) ∧ p.Sat _ (cons a e)) ↔ _
constructor
· rintro ⟨a,ha,hm,hp⟩
have heq := (largestOrdinalSentence_level R o (cons a e) ha).mp hm
change a = _ at heq
simpa only [heq] using hp
· intro hp
have hm := largest_ordinal_level_succ R o
exact ⟨_,hm.1,(largestOrdinalSentence_level R o _ hm.1).mpr rfl,hp⟩
end TuringRigidity.RelativeConstructible
end OAI
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