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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 | import OAI.Computability.DegreeRigidity.CohenForcing.BoundedForcing
import OAI.Computability.DegreeRigidity.Syntax.AtomicCertificate
namespace OAI
namespace TuringRigidity.BoundedForcing
open Set RecursiveNames TransitiveNameModel BoundedSetTheory AtomicForcing
universe u
def code : BoundedSetTheory.Formula → ℕ → ℕ → ℕ → ℕ → ℕ → (ℕ → ℕ) → ℕ → BoundedSetTheory.Formula
| .equal i j, _d,_c,k,_o,f,v,p => AtomicFormula.triple k f (v i) (v j) p
| .member i j, d,c,k,o,f,v,p => AtomicFormula.membership d c k o f (v i) (v j) p
| .conj φ ψ, d,c,k,o,f,v,p => .conj (code φ d c k o f v p) (code ψ d c k o f v p)
| .neg φ, d,c,k,o,f,v,p => Formula.allMem c
(Formula.imp (Formula.pairMem 0 (p+1) (o+1))
(.neg (code φ (d+1) (c+1) (k+1) (o+1) (f+1) (fun i => v i+1) 0)))
| .existsMem i φ, d,c,k,o,f,v,p => .existsMem d (.existsMem (c+1)
(.conj (Formula.pairMem 1 0 (v i+2)) (.conj (Formula.pairMem (p+2) 0 (o+2))
(code φ (d+2) (c+2) (k+2) (o+2) (f+2) (push 1 (fun i => v i+2)) (p+2)))))
variable {c : ZFSet.{u}} [Preorder (Conditions c)]
theorem eval_code (φ : BoundedSetTheory.Formula) {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) (e : ℕ → Name (Conditions c))
(he : ∀ i, (e i).encode (label c) ∈ d) (p : Conditions c)
(env : ℕ → ZFSet.{u}) (di ci ki oi fi : ℕ) (v : ℕ → ℕ) (pi : ℕ)
(slots : env di = d ∧ env ci = c ∧ env ki = ZFSet.prod d d ∧ env oi = o ∧ env fi = f ∧ env pi = label c p)
(hv : ∀ i, env (v i) = (e i).encode (label c)) :
(code φ di ci ki oi fi v pi).Eval env ↔ Forces e φ p := by
induction φ generalizing e p env di ci ki oi fi v pi with
| equal i j =>
simp only [code,AtomicFormula.eval_triple,slots.2.2.1,slots.2.2.2.2.1,slots.2.2.2.2.2,hv,Forces]
exact (triple_pair (he i) (he j)).trans (hf.correct hd ho _ _ (he i) (he j) p)
| member i j =>
exact eval_membership hd ho hf _ _ (he i) (he j) p env di ci ki oi fi (v i) (v j) pi
⟨slots.1,slots.2.1,slots.2.2.1,slots.2.2.2.1,slots.2.2.2.2.1,hv i,hv j,slots.2.2.2.2.2⟩
| conj φ ψ ihφ ihψ =>
exact and_congr (ihφ e he p env di ci ki oi fi v pi slots hv)
(ihψ e he p env di ci ki oi fi v pi slots hv)
| neg φ ih =>
simp only [code,Formula.eval_allMem,Formula.eval_imp,Formula.eval_pairMem,Formula.Eval,
cons_zero,cons_succ,slots.2.1,slots.2.2.2.1,slots.2.2.2.2.2]
have sub (q : Conditions c) := ih e he q (cons (label c q) env)
(di+1) (ci+1) (ki+1) (oi+1) (fi+1) (fun i => v i+1) 0
⟨slots.1,slots.2.1,slots.2.2.1,slots.2.2.2.1,slots.2.2.2.2.1,rfl⟩ hv
constructor
· intro h q hq hforce
exact h _ (label_mem c q) ((ho q p).mpr hq) ((sub q).mpr hforce)
· intro h q hq hqp hforce
obtain ⟨r,rfl⟩ := label_surjective c hq
exact h r ((ho r p).mp hqp) ((sub r).mp hforce)
| existsMem i φ ih =>
simp only [code,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ,
slots.1,slots.2.1,slots.2.2.2.1,slots.2.2.2.2.2,hv]
have hej (j) : ∀ n, (push ((e i).child j) e n).encode (label c) ∈ d := by
intro n
cases n with
| zero =>
exact names_childClosed d c hd (e i) (he i) _ (child_relation (e i) j)
| succ n => exact he n
have sub (j) (s : Conditions c) := ih (push ((e i).child j) e) (hej j) p
(cons (label c s) (cons (((e i).child j).encode (label c)) env))
(di+2) (ci+2) (ki+2) (oi+2) (fi+2) (push 1 (fun n => v n+2)) (pi+2)
⟨slots.1,slots.2.1,slots.2.2.1,slots.2.2.2.1,slots.2.2.2.2.1,slots.2.2.2.2.2⟩
(by intro n; cases n <;> simp only [push_zero,push_succ,cons_zero,cons_succ,hv])
constructor
· rintro ⟨x,hx,s,hs,hxs,hps,hφ⟩
obtain ⟨j,hj,hjs⟩ := (pair_mem_encode _ _ _ _).mp hxs
subst x
subst s
exact ⟨j,(ho p ((e i).tag j)).mp hps,(sub j ((e i).tag j)).mp hφ⟩
· rintro ⟨j,hj,hφ⟩
exact ⟨_,hej j 0,_,label_mem c ((e i).tag j),
(pair_mem_encode _ _ _ _).mpr ⟨j,rfl,rfl⟩,(ho p ((e i).tag j)).mpr hj,
(sub j ((e i).tag j)).mpr hφ⟩
end TuringRigidity.BoundedForcing
end OAI
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