| import OAI.Computability.DegreeRigidity.SetModels.InternalHartogs |
| import OAI.Computability.DegreeRigidity.SetModels.InternalChoice |
| import OAI.Computability.DegreeRigidity.Syntax.TermFunctionality |
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| namespace OAI |
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| namespace TuringRigidity.BoundedSetTheory |
| open TransitiveNameModel |
| universe u |
| namespace InternalWellOrder |
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|
| def Remaining (a f i r : ZFSet.{u}) : Prop := r ⊆ a ∧ |
| ∀ x ∈ a, x ∈ r ↔ ¬ ∃ j ∈ i, ZFSet.pair j x ∈ f |
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| 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 |
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| 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 |
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| 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⟩ |
|
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| 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 |
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| 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 |
|
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| end OAI |
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