| import OAI.Computability.DegreeRigidity.OrdinalCodes.NumericTruth |
| import OAI.Computability.DegreeRigidity.Syntax.VariableCertificateBounds |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.NumericSyntax |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding RelativeConstructible |
| open BoundedDefinability SetModelFunctions SetModelReals |
| universe u |
| theorem boundWitness_allBounds (Q B c b : ℕ) : |
| Numeric.{u} (fun e => BoundWitness (e Q) (e B) (e c) (e b)) := |
| (((leastFamily_definable (Q+2) (c+2) 1).and |
| ((setSupportRecursion_definable 1 0).and (defPairMem (c+2) (b+2) 0))).existsMem |
| (B+1)).existsMem Q |
|
|
| theorem tupleAdequate_allBounds (Q B G c t : ℕ) : |
| Numeric.{u} (fun e => TupleAdequate (e Q) (e B) (e G) (e c) (e t)) := |
| ((((defOrderedPair (t+3) 2 1).and |
| ((boundWitness_allBounds (Q+3) (B+3) (c+3) 0).and |
| (defSubset 0 2))).existsNat).existsMem (G+1)).existsNat |
|
|
| theorem setTruthRecursion_allBounds (Q B a G T f s : ℕ) : |
| Numeric.{u} (fun e => SetTruthRecursion (e Q) (e B) (e G) (e a) (e T) (e f) (e s)) := by |
| have hn := defAllMem ((((tupleAdequate_allBounds (Q+3) (B+3) (G+3) 1 0).and |
| (defOrderedPair 2 1 0)).existsMem (T+2)).existsMem (f+1)) s |
| have hr := defAllMem (defAllMem (defImp |
| (tupleAdequate_allBounds (Q+2) (B+2) (G+2) 1 0) |
| (defIff (defPairMem 1 0 (s+2)) |
| (setTruthStep_variable (a+2) (G+2) (T+2) (s+2) 1 0))) (T+1)) f |
| exact hn.and hr |
|
|
| theorem truthWitness_allBounds (Q B a G T c s : ℕ) : |
| Numeric.{u} (fun e => TruthWitness (e Q) (e B) (e G) (e a) (e T) (e c) (e s)) := |
| ((leastFamily_definable (Q+1) (c+1) 0).and |
| (setTruthRecursion_allBounds (Q+1) (B+1) (a+1) (G+1) (T+1) 0 (s+1))).existsMem Q |
|
|
| theorem satisfactionWitness_allBounds (Q B a G H T z : ℕ) : |
| Numeric.{u} (fun e => SatisfactionWitness (e Q) (e B) (e G) (e H) (e a) (e T) (e z)) := |
| ((((truthWitness_allBounds (Q+3) (B+3) (a+3) (G+3) (T+3) 2 0).and |
| ((defPairMem 2 1 0).and (defOrderedPair (z+3) 2 1))).existsMem (H+2)).existsMem |
| (T+1)).existsNat |
|
|
| theorem parameterAdequate_allBounds (Q B G c t : ℕ) : |
| Numeric.{u} (fun e => ParameterAdequate (e Q) (e B) (e G) (e c) (e t)) := |
| ((((defOrderedPair (t+3) 2 1).and |
| ((boundWitness_allBounds (Q+3) (B+3) (c+3) 0).and |
| (defAllMem (defOr (member_definable 0 3) (equal_definable 0 3)) 0))).existsNat).existsMem (G+1)).existsNat |
|
|
| theorem definitionWitness_allBounds (Q B a G T Z S : ℕ) : |
| Numeric.{u} (fun e => DefinitionWitness (e Q) (e B) (e G) (e a) (e T) (e Z) (e S)) := |
| (((parameterAdequate_allBounds (Q+2) (B+2) (G+2) 1 0).and |
| (definesSubset_variable (a+2) (G+2) (T+2) (Z+2) 1 0 (S+2))).existsMem (T+1)).existsNat |
|
|
| theorem satisfactionSystem_allBounds (Q B a G T H Z : ℕ) : |
| Numeric.{u} (fun e => SatisfactionSystem (e Q) (e B) (e a) (e G) (e T) (e H) (e Z)) := by |
| have ht := defAllNat (defAllMem (defImp (leastFamily_definable (Q+2) 1 0) |
| ((setTruthRecursion_allBounds (Q+3) (B+3) (a+3) (G+3) (T+3) 1 0).existsMem (H+2))) (Q+1)) |
| have hn := defAllMem (satisfactionWitness_allBounds (Q+1) (B+1) (a+1) (G+1) (H+1) (T+1) 0) Z |
| have hc := defAllNat (defAllMem (defAllMem |
| (defImp (truthWitness_allBounds (Q+3) (B+3) (a+3) (G+3) (T+3) 2 0) |
| (defImp (defPairMem 2 1 0) (defPairMem 2 1 (Z+3)))) (H+2)) (T+1)) |
| exact (tupleSystem_definable a G T).and (ht.and (hn.and hc)) |
|
|
| theorem defSystem_allBounds (Q B a G T H Z D : ℕ) : |
| Numeric.{u} (fun e => DefSystem (e Q) (e B) (e a) (e G) (e T) (e H) (e Z) (e D)) := by |
| have hn := defAllMem (definitionWitness_allBounds (Q+1) (B+1) (a+1) (G+1) (T+1) (Z+1) 0) D |
| have hc := defAllNat (defAllMem (defImp (parameterAdequate_allBounds (Q+2) (B+2) (G+2) 1 0) |
| ((definesSubset_variable (a+3) (G+3) (T+3) (Z+3) 2 1 0).existsMem (D+2))) (T+1)) |
| exact (satisfactionSystem_allBounds Q B a G T H Z).and (hn.and hc) |
|
|
| end TuringRigidity.NumericSyntax |
|
|
| end OAI |
|
|
