File size: 3,664 Bytes
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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.NumericFamilySupport
import OAI.Computability.DegreeRigidity.Effective.TupleSystemCertificate
namespace OAI
namespace TuringRigidity.NumericSyntax
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding RelativeConstructible
open BoundedDefinability SetModelFunctions SetModelReals
universe u
theorem tupleLookup_variable (G t i x : ℕ) :
Numeric.{u} (fun e => TupleLookup (e G) (e t) (e i) (e x)) :=
(((defOrderedPair (t+2) 1 0).and (defPairMem (i+2) (x+2) 0)).existsMem (G+1)).existsNat
theorem tuplePrefix_variable (a G t x s : ℕ) :
Numeric.{u} (fun e => TuplePrefix (e a) (e G) (e t) (e x) (e s)) := by
let r : ℕ → ℕ := fun i => match i with
| 0 => 2*a | 1 => 1 | 2 => 2*G | 3 => 2*t | 4 => 2*x | 5 => 2*s | _ => 5
refine ⟨(tuplePrefixFormula 0 1 2 3 4 5 6).rename r,?_⟩
intro B _ e
rw [Formula.eval_rename_comp]
have he : mix e (params B) ∘ r = cons (e a) (cons ZFSet.omega (cons (e G)
(cons (e t) (cons (e x) (cons (e s) (fun _ => natSet 0)))))) := by
funext i
rcases i with _|_|_|_|_|_|i <;> simp only [r,params,Function.comp_apply,mix_even,cons_zero,cons_succ] <;> rfl
rw [he]
rfl
theorem setTruthStep_variable (a G T S c t : ℕ) :
Numeric.{u} (fun e => SetTruthStep (e a) (e G) (e T) (e S) (e c) (e t)) := by
have ha (tag : ℕ) (hr : Numeric.{u} (fun e => e 1 = e 0)) :=
(((((binaryGraph_definable _ (atomic_primrec tag) 3 2 (c+4)).and
((tupleLookup_variable (G+4) (t+4) 3 1).and
((tupleLookup_variable (G+4) (t+4) 2 0).and hr))).existsMem (a+3)).existsMem (a+2)).existsNat).existsNat
have he := ha 0 (equal_definable 1 0)
have hm := (((((binaryGraph_definable _ (atomic_primrec 1) 3 2 (c+4)).and
((tupleLookup_variable (G+4) (t+4) 3 1).and
((tupleLookup_variable (G+4) (t+4) 2 0).and (member_definable 1 0)))).existsMem (a+3)).existsMem
(a+2)).existsNat).existsNat
have hc := (((binaryGraph_definable _ (binary_primrec 2) 1 0 (c+2)).and
((defPairMem 1 (t+2) (S+2)).and (defPairMem 0 (t+2) (S+2)))).existsNat).existsNat
have hn := ((unaryGraph_definable _ (unary_primrec 3) 0 (c+1)).and
(defPairMem 0 (t+1) (S+1)).neg).existsNat
have hbody := (((tuplePrefix_variable (a+3) (G+3) (t+3) 1 0).and
(defPairMem 2 0 (S+3))).existsMem (T+2)).existsMem (a+1)
have hx := ((unaryGraph_definable _ (unary_primrec 4) 0 (c+1)).and hbody).existsNat
exact defOr he (defOr hm (defOr hc (defOr hn hx)))
theorem definesSubset_variable (a G T Z c t S : ℕ) :
Numeric.{u} (fun e => DefinesSubset (e a) (e G) (e T) (e Z) (e c) (e t) (e S)) :=
(defSubset S a).and (defAllMem (defIff (member_definable 0 (S+1))
(((tuplePrefix_variable (a+2) (G+2) (t+2) 1 0).and
(defPairMem (c+2) 0 (Z+2))).existsMem (T+1))) a)
theorem tupleSystem_definable (a G T : ℕ) :
Numeric.{u} (fun e => TupleSystem (e a) (e G) (e T)) := by
have hfg (f n a : ℕ) : Numeric.{u} (fun e => FunctionGraph (e n) (e a) (e f)) := by
refine ⟨.functionGraph (2*f) (2*n) (2*a),?_⟩
intro B hB e
simp only [Formula.eval_functionGraph,mix_even,FunctionGraph]
have hn := defAllMem ((((defOrderedPair 2 1 0).and
(hfg 0 1 (a+3))).existsMem (G+2)).existsNat) T
have hz : Numeric.{u} (fun e => ZFSet.pair (natSet 0) (natSet 0) ∈ e T) := (((equalZero 0).and
(defPairMem 0 0 (T+1))).existsNat).congr (fun e => by simp [show natSet.{u} 0 ∈ ZFSet.omega from (mem_omega _).mpr ⟨0,rfl⟩])
have hc := defAllMem (defAllMem
((tuplePrefix_variable (a+3) (G+3) 2 1 0).existsMem (T+2)) (a+1)) T
exact hn.and (hz.and hc)
end TuringRigidity.NumericSyntax
end OAI
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