| 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 |
|
|
