| import OAI.Computability.DegreeRigidity.SetModels.NameEncoding |
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| namespace OAI |
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| namespace TuringRigidity.AtomicForcing |
| open Set RecursiveNames CountableForcing |
| universe u |
| variable {P : Type u} [Preorder P] |
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| def Below (p t : P) (A : P → Prop) : Prop := |
| ∀ q, q ≤ p → q ≤ t → ∃ r, r ≤ q ∧ A r |
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| def EqForces : Name P → Name P → P → Prop |
| | .mk _ a s, .mk _ b t, p => |
| (∀ i, Below p (s i) (fun r => ∃ j, r ≤ t j ∧ EqForces (a i) (b j) r)) ∧ |
| (∀ j, Below p (t j) (fun r => ∃ i, r ≤ s i ∧ EqForces (a i) (b j) r)) |
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| def MemForces (a : Name P) : Name P → P → Prop |
| | .mk _ b t, p => ∀ q, q ≤ p → ∃ r, r ≤ q ∧ ∃ j, r ≤ t j ∧ EqForces a (b j) r |
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| theorem eq_mono (a b : Name P) {p q : P} (hqp : q ≤ p) |
| (h : EqForces a b p) : EqForces a b q := by |
| cases a with |
| | mk ι a s => |
| cases b with |
| | mk κ b t => |
| exact ⟨fun i r hr hs => h.1 i r (hr.trans hqp) hs, |
| fun j r hr ht => h.2 j r (hr.trans hqp) ht⟩ |
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| theorem mem_mono (a b : Name P) {p q : P} (hqp : q ≤ p) |
| (h : MemForces a b p) : MemForces a b q := by |
| cases b with |
| | mk κ b t => exact fun r hr => h r (hr.trans hqp) |
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| theorem eq_refl (a : Name P) (p : P) : EqForces a a p := by |
| induction a generalizing p with |
| | mk ι a s ih => |
| exact ⟨fun i q _ hq => ⟨q,le_rfl,i,hq,ih i q⟩, |
| fun i q _ hq => ⟨q,le_rfl,i,hq,ih i q⟩⟩ |
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| theorem eq_symm (a b : Name P) (p : P) : EqForces a b p → EqForces b a p := by |
| induction a generalizing b p with |
| | mk ι a s ih => |
| cases b with |
| | mk κ b t => |
| intro h |
| constructor |
| · intro j q hq ht |
| obtain ⟨r,hr,i,hi,he⟩ := h.2 j q hq ht |
| exact ⟨r,hr,i,hi,ih i (b j) r he⟩ |
| · intro i q hq hs |
| obtain ⟨r,hr,j,hj,he⟩ := h.1 i q hq hs |
| exact ⟨r,hr,j,hj,ih i (b j) r he⟩ |
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| theorem eq_trans (a b c : Name P) (p : P) |
| (hab : EqForces a b p) (hbc : EqForces b c p) : EqForces a c p := by |
| induction a generalizing b c p with |
| | mk ι a s ih => |
| cases b with |
| | mk κ b t => |
| cases c with |
| | mk μ c v => |
| constructor |
| · intro i q hq hs |
| obtain ⟨r,hr,j,hj,he⟩ := hab.1 i q hq hs |
| obtain ⟨w,hw,k,hk,hf⟩ := hbc.1 j r (hr.trans hq) hj |
| exact ⟨w,hw.trans hr,k,hk,ih i (b j) (c k) w (eq_mono _ _ hw he) hf⟩ |
| · intro k q hq hv |
| obtain ⟨r,hr,j,hj,he⟩ := hbc.2 k q hq hv |
| obtain ⟨w,hw,i,hi,hf⟩ := hab.2 j r (hr.trans hq) hj |
| exact ⟨w,hw.trans hr,i,hi,ih i (b j) (c k) w hf (eq_mono _ _ hw he)⟩ |
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| theorem eq_of_dense (a b : Name P) (p : P) |
| (h : ∀ q, q ≤ p → ∃ r, r ≤ q ∧ EqForces a b r) : EqForces a b p := by |
| cases a with |
| | mk ι a s => |
| cases b with |
| | mk κ b t => |
| constructor |
| · intro i q hq hs |
| obtain ⟨r,hr,he⟩ := h q hq |
| obtain ⟨w,hw,j,hj,hf⟩ := he.1 i r le_rfl (hr.trans hs) |
| exact ⟨w,hw.trans hr,j,hj,hf⟩ |
| · intro j q hq ht |
| obtain ⟨r,hr,he⟩ := h q hq |
| obtain ⟨w,hw,i,hi,hf⟩ := he.2 j r le_rfl (hr.trans ht) |
| exact ⟨w,hw.trans hr,i,hi,hf⟩ |
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| theorem mem_of_dense (a b : Name P) (p : P) |
| (h : ∀ q, q ≤ p → ∃ r, r ≤ q ∧ MemForces a b r) : MemForces a b p := by |
| cases b with |
| | mk κ b t => |
| intro q hq |
| obtain ⟨r,hr,he⟩ := h q hq |
| obtain ⟨w,hw,j,hj,hf⟩ := he r le_rfl |
| exact ⟨w,hw.trans hr,j,hj,hf⟩ |
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| end TuringRigidity.AtomicForcing |
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| end OAI |
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