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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 90 91 92 93 | import OAI.Computability.DegreeRigidity.Syntax.AtomicGraph
import OAI.Computability.DegreeRigidity.Syntax.AtomicEncoding
import OAI.Computability.DegreeRigidity.SetModels.NameUnionCode
namespace OAI
namespace TuringRigidity.AtomicForcing
open Set RecursiveNames TransitiveNameModel BoundedSetTheory
universe u
variable {c : ZFSet.{u}} [Preorder (Conditions c)]
theorem triple_pair {d f x y q : ZFSet.{u}} (hx : x ∈ d) (hy : y ∈ d) :
Triple (ZFSet.prod d d) f x y q ↔ ZFSet.pair (ZFSet.pair x y) q ∈ f := by
constructor
· rintro ⟨v,_,rfl,h⟩; exact h
· intro h
exact ⟨_,ZFSet.mem_prod.mpr ⟨x,hx,y,hy,rfl⟩,rfl,h⟩
omit [Preorder (Conditions c)] in
private theorem child_code_mem {d : ZFSet.{u}} (hd : Transitive d)
(a : Name (Conditions c)) (ha : a.encode (label c) ∈ d) (i : a.arity) :
(a.child i).encode (label c) ∈ d := by
apply names_childClosed d c hd a ha
cases a
exact ⟨i,rfl⟩
theorem match_encode (flip : Bool) {d k o f a : ZFSet.{u}} (hd : Transitive d)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(b : Name (Conditions c)) (hb : b.encode (label c) ∈ d) (q : Conditions c) :
Match flip d c k o f a (b.encode (label c)) (label c q) ↔
∃ r, r ≤ q ∧ ∃ j, r ≤ b.tag j ∧
Triple k f (if flip then (b.child j).encode (label c) else a)
(if flip then a else (b.child j).encode (label c)) (label c r) := by
constructor
· rintro ⟨r,hr,hrq,x,hx,t,ht,hxt,hrt,hf⟩
obtain ⟨s,rfl⟩ := label_surjective c hr
obtain ⟨j,hj,hjt⟩ := (pair_mem_encode (label c) b x t).mp hxt
subst x
subst t
exact ⟨s,(ho s q).mp hrq,j,(ho s (b.tag j)).mp hrt,hf⟩
· rintro ⟨r,hrq,j,hrj,hf⟩
exact ⟨label c r,label_mem c r,(ho r q).mpr hrq,
_,child_code_mem hd b hb j,_,label_mem c (b.tag j),
(pair_mem_encode _ _ _ _).mpr ⟨j,rfl,rfl⟩,(ho r (b.tag j)).mpr hrj,hf⟩
theorem side_encode (flip : Bool) {d k o f : ZFSet.{u}} (hd : Transitive d)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(a b : Name (Conditions c)) (ha : a.encode (label c) ∈ d)
(hb : b.encode (label c) ∈ d) (p : Conditions c) :
Side flip d c k o f (a.encode (label c)) (b.encode (label c)) (label c p) ↔
∀ i, Below p (a.tag i) (fun r => ∃ j, r ≤ b.tag j ∧
Triple k f (if flip then (b.child j).encode (label c) else (a.child i).encode (label c))
(if flip then (a.child i).encode (label c) else (b.child j).encode (label c)) (label c r)) := by
constructor
· intro h i q hqp hqi
have hm := h _ (child_code_mem hd a ha i) _ (label_mem c (a.tag i))
((pair_mem_encode _ _ _ _).mpr ⟨i,rfl,rfl⟩) _ (label_mem c q)
((ho q p).mpr hqp) ((ho q (a.tag i)).mpr hqi)
exact (match_encode flip hd ho b hb q).mp hm
· intro h x hx s hs hxs q hq hqp hqs
obtain ⟨i,hi,his⟩ := (pair_mem_encode (label c) a x s).mp hxs
subst x
subst s
obtain ⟨r,rfl⟩ := label_surjective c hq
exact (match_encode flip hd ho b hb r).mpr
(h i r ((ho r p).mp hqp) ((ho r (a.tag i)).mp hqs))
theorem Graph.correct {d o f : ZFSet.{u}} (hd : Transitive d)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(hf : Graph d c o f) (a b : Name (Conditions c))
(ha : a.encode (label c) ∈ d) (hb : b.encode (label c) ∈ d) (p : Conditions c) :
ZFSet.pair (ZFSet.pair (a.encode (label c)) (b.encode (label c))) (label c p) ∈ f ↔
EqForces a b p := by
induction a generalizing b p with
| mk ι a s ih =>
cases b with
| mk κ b t =>
have hai (i) : (a i).encode (label c) ∈ d := child_code_mem hd _ ha i
have hbj (j) : (b j).encode (label c) ∈ d := child_code_mem hd _ hb j
have hrel (i j) (q : Conditions c) :
Triple (ZFSet.prod d d) f ((a i).encode (label c)) ((b j).encode (label c)) (label c q) ↔
EqForces (a i) (b j) q :=
(triple_pair (hai i) (hbj j)).trans (ih i (b j) (hai i) (hbj j) q)
rw [hf.2 _ ha _ hb _ (label_mem c p)]
change (Side false _ _ _ _ _ _ _ _ ∧ Side true _ _ _ _ _ _ _ _) ↔ _
rw [side_encode false hd ho _ _ ha hb p, side_encode true hd ho _ _ hb ha p]
simp only [Bool.false_eq_true,ite_false,ite_true,Name.arity,Name.child,Name.tag]
simp only [Below,hrel,EqForces]
end TuringRigidity.AtomicForcing
end OAI
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