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8df23f4 | 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 | import OAI.Computability.DegreeRigidity.SetModels.NameEncoding
namespace OAI
namespace TuringRigidity.AtomicForcing
open Set RecursiveNames CountableForcing
universe u
variable {P : Type u} [Preorder P]
def Below (p t : P) (A : P → Prop) : Prop :=
∀ q, q ≤ p → q ≤ t → ∃ r, r ≤ q ∧ A r
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))
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
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⟩
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)
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⟩⟩
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⟩
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)⟩
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⟩
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⟩
end TuringRigidity.AtomicForcing
end OAI
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