File size: 3,862 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 72 73 74 | 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
|