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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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.Computability.DegreeRigidity.SetModels.InternalHartogs
import OAI.Computability.DegreeRigidity.SetModels.InternalChoice
import OAI.Computability.DegreeRigidity.Syntax.TermFunctionality
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel
universe u
namespace InternalWellOrder
def Remaining (a f i r : ZFSet.{u}) : Prop := r ⊆ a ∧
∀ x ∈ a, x ∈ r ↔ ¬ ∃ j ∈ i, ZFSet.pair j x ∈ f
structure Trace (α a q s f : ZFSet.{u}) : Prop where
shape : ∀ z ∈ f, ∃ i ∈ α, ∃ x ∈ a, z = ZFSet.pair i x
initial : ∀ i ∈ α, ∀ x ∈ a, ZFSet.pair i x ∈ f →
∀ j ∈ i, ∃ y ∈ a, ZFSet.pair j y ∈ f
choice : ∀ i ∈ α, ∀ x ∈ a, ZFSet.pair i x ∈ f →
∃ r ∈ q, Remaining a f i r ∧ ZFSet.pair r x ∈ s
theorem Trace.pair_mem {α a q s f i x : ZFSet.{u}} (hf : Trace α a q s f)
(hix : ZFSet.pair i x ∈ f) : i ∈ α ∧ x ∈ a := by
obtain ⟨j,hj,y,hy,he⟩ := hf.shape _ hix
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
exact ⟨hj,hy⟩
theorem choice_value_mem {q s r x : ZFSet.{u}} (hs : ChoiceGraph q s)
(hrx : ZFSet.pair r x ∈ s) : x ∈ r := by
obtain ⟨r',_,x',hx,he⟩ := hs.1 _ hrx
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
exact hx
theorem Remaining.eq {a f g i r t : ZFSet.{u}} (hr : Remaining a f i r)
(ht : Remaining a g i t)
(hfg : ∀ j ∈ i, ∀ x ∈ a, ZFSet.pair j x ∈ f ↔ ZFSet.pair j x ∈ g) : r = t := by
apply ZFSet.ext; intro x
constructor
· intro hx
have hxa := hr.1 hx
apply (ht.2 x hxa).mpr
rintro ⟨j,hji,hjx⟩
exact (hr.2 x hxa).mp hx ⟨j,hji,(hfg j hji x hxa).mpr hjx⟩
· intro hx
have hxa := ht.1 hx
apply (hr.2 x hxa).mpr
rintro ⟨j,hji,hjx⟩
exact (ht.2 x hxa).mp hx ⟨j,hji,(hfg j hji x hxa).mp hjx⟩
theorem Trace.compatible {α a q s f g : ZFSet.{u}} (_hα : Transitive α)
(hs : ChoiceGraph q s) (hf : Trace α a q s f) (hg : Trace α a q s g)
(i x y : ZFSet.{u}) (hix : ZFSet.pair i x ∈ f) (hiy : ZFSet.pair i y ∈ g) : x = y := by
induction i using ZFSet.inductionOn generalizing x y with
| h i ih =>
have hi := (hf.pair_mem hix).1
have hx := (hf.pair_mem hix).2
have hy := (hg.pair_mem hiy).2
obtain ⟨r,_,hr,hrx⟩ := hf.choice i hi x hx hix
obtain ⟨t,_,ht,hty⟩ := hg.choice i hi y hy hiy
have heq : r = t := hr.eq ht (by
intro j hji z _
constructor
· intro hjz
obtain ⟨w,_,hjw⟩ := hg.initial i hi y hy hiy j hji
have hzw := ih j hji z w hjz hjw
exact hzw.symm ▸ hjw
· intro hjz
obtain ⟨w,_,hjw⟩ := hf.initial i hi x hx hix j hji
have hwz := ih j hji w z hjw hjz
exact hwz ▸ hjw)
rw [heq] at hrx
exact hs.functional hrx hty
theorem Trace.functional {α a q s f : ZFSet.{u}} (hα : Transitive α)
(hs : ChoiceGraph q s) (hf : Trace α a q s f) : FiniteTerm.Functional f :=
fun i x y hx hy => hf.compatible hα hs hf i x y hx hy
theorem Trace.injective {α a q s f : ZFSet.{u}} (hα : α.IsOrdinal)
(hs : ChoiceGraph q s) (hf : Trace α a q s f)
(i j x : ZFSet.{u}) (hix : ZFSet.pair i x ∈ f) (hjx : ZFSet.pair j x ∈ f) : i = j := by
have hi := (hf.pair_mem hix).1
have hj := (hf.pair_mem hjx).1
have hx := (hf.pair_mem hix).2
rcases (hα.mem hi).mem_trichotomous (hα.mem hj) with hij|he|hji
· obtain ⟨r,_,hr,hrx⟩ := hf.choice j hj x hx hjx
exact False.elim ((hr.2 x hx).mp (choice_value_mem hs hrx) ⟨i,hij,hix⟩)
· exact he
· obtain ⟨r,_,hr,hrx⟩ := hf.choice i hi x hx hix
exact False.elim ((hr.2 x hx).mp (choice_value_mem hs hrx) ⟨j,hji,hjx⟩)
theorem Trace.sUnion {α a q s C : ZFSet.{u}} (hα : Transitive α)
(hs : ChoiceGraph q s) (hC : ∀ f ∈ C, Trace α a q s f) :
Trace α a q s (ZFSet.sUnion C) := by
have from_pair (i x : ZFSet.{u}) (hix : ZFSet.pair i x ∈ ZFSet.sUnion C) :
∃ f ∈ C, ZFSet.pair i x ∈ f := ZFSet.mem_sUnion.mp hix
refine ⟨?_,?_,?_⟩
· intro z hz
obtain ⟨f,hf,hzf⟩ := ZFSet.mem_sUnion.mp hz
exact (hC f hf).shape z hzf
· intro i hi x hx hix j hji
obtain ⟨f,hf,hif⟩ := from_pair i x hix
obtain ⟨y,hy,hjy⟩ := (hC f hf).initial i hi x hx hif j hji
exact ⟨y,hy,ZFSet.mem_sUnion.mpr ⟨f,hf,hjy⟩⟩
· intro i hi x hx hix
obtain ⟨f,hf,hif⟩ := from_pair i x hix
obtain ⟨r,hr,hrdef,hrx⟩ := (hC f hf).choice i hi x hx hif
refine ⟨r,hr,⟨hrdef.1,?_⟩,hrx⟩
intro y hy
rw [hrdef.2 y hy]
apply not_congr
constructor
· rintro ⟨j,hji,hjy⟩
exact ⟨j,hji,ZFSet.mem_sUnion.mpr ⟨f,hf,hjy⟩⟩
· rintro ⟨j,hji,hjy⟩
obtain ⟨g,hg,hjg⟩ := from_pair j y hjy
obtain ⟨z,_,hjz⟩ := (hC f hf).initial i hi x hx hif j hji
have hzy := (hC f hf).compatible hα hs (hC g hg) j z y hjz hjg
exact ⟨j,hji,hzy ▸ hjz⟩
end InternalWellOrder
end TuringRigidity.BoundedSetTheory
end OAI
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