openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /EnumerationTransport.lean
| import OAI.Computability.DegreeRigidity.SetModels.EnumerationConstruction | |
| import OAI.Computability.DegreeRigidity.Representation.GlobalLocality | |
| namespace OAI | |
| namespace TuringRigidity.EnumerationTransport | |
| open SetCoding RelationCoding BoundedDecoding ArithmeticModelDecoding EnumerationDecoding | |
| open IdealInterpretation GlobalLocality | |
| noncomputable section | |
| def elem (a : Degree) : fullIdeal := ⟨a,Set.mem_univ _⟩ | |
| def liftAntichain (p : AntichainCode) : ACode fullIdeal := ⟨elem p.bound,elem p.left,elem p.right⟩ | |
| def liftSet (p : SetCode) : SCode fullIdeal := ⟨elem p.bound,liftAntichain p.tags,liftAntichain p.decorated⟩ | |
| def liftRelation {n : ℕ} (p : RelationCode n) : RCode fullIdeal n := | |
| ⟨fun i => liftSet (p.coordinates i),liftAntichain p.tuples⟩ | |
| @[simp] theorem lift_external {n : ℕ} (p : RelationCode n) : (liftRelation p).external = p := by cases p; rfl | |
| def mapRelation {n : ℕ} (π : Degree ≃o Degree) (p : RelationCode n) : RelationCode n := | |
| ((liftRelation p).map (GlobalLocality.lift π)).external | |
| theorem map_below {n : ℕ} (π : Degree ≃o Degree) (p : RelationCode n) (b : Degree) | |
| (hp : RelationBelow p b) : RelationBelow (mapRelation π p) (π b) := by | |
| exact (liftRelation p).map_below (GlobalLocality.lift π) (elem b) (by simpa only [lift_external,elem] using hp) | |
| theorem map_two (π : Degree ≃o Degree) (p : RelationCode 2) (a b : Degree) : | |
| (mapRelation π p).Holds (two (π a) (π b)) ↔ p.Holds (two a b) := by | |
| have hv (x : Degree) : ((GlobalLocality.lift π) (elem x)).val = π x := rfl | |
| have he (x : Degree) : (elem x).val = x := rfl | |
| simpa only [lift_external,hv,he,mapRelation] using (liftRelation p).transport_two (GlobalLocality.lift π) (elem a) (elem b) | |
| theorem inverse_eq (π : Degree ≃o Degree) (a b : Degree) : π.symm a = b ↔ a = π b := by | |
| constructor | |
| · intro h | |
| have hh := congrArg π h | |
| simpa only [π.apply_symm_apply] using hh | |
| · intro h | |
| rw [h,π.symm_apply_apply] | |
| def mapEnumeration (π : Degree ≃o Degree) (c : Enumeration) : Enumeration where | |
| node := fun n => π (c.node n) | |
| injective := π.injective.comp c.injective | |
| output := fun n => π (c.output n) | |
| successor := mapRelation π c.successor | |
| payload := mapRelation π c.payload | |
| successor_spec := by | |
| intro a b | |
| have hh := map_two π c.successor (π.symm a) (π.symm b) | |
| simp only [π.apply_symm_apply] at hh | |
| rw [hh,c.successor_spec] | |
| exact exists_congr (fun k => and_congr (inverse_eq π a (c.node k)) (inverse_eq π b (c.node (k+1)))) | |
| payload_spec := by | |
| intro a b | |
| have hh := map_two π c.payload (π.symm a) (π.symm b) | |
| simp only [π.apply_symm_apply] at hh | |
| rw [hh,c.payload_spec] | |
| exact exists_congr (fun k => and_congr (inverse_eq π a (c.node k)) (inverse_eq π b (c.output k))) | |
| theorem mapEnumeration_below (π : Degree ≃o Degree) (c : Enumeration) (b : Degree) | |
| (hS : RelationBelow c.successor b) (hP : RelationBelow c.payload b) : | |
| RelationBelow (mapEnumeration π c).successor (π b) ∧ RelationBelow (mapEnumeration π c).payload (π b) := | |
| ⟨map_below π _ b hS,map_below π _ b hP⟩ | |
| end | |
| end TuringRigidity.EnumerationTransport | |
| end OAI | |