openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /LevelInlinedDefinition.lean
| import OAI.Computability.DegreeRigidity.Constructibility.ConstructionStageBound | |
| import OAI.Computability.DegreeRigidity.Constructibility.RelativeModelPower | |
| import OAI.Computability.DegreeRigidity.Constructibility.PersistentConstruction | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SetModelArithmetic | |
| open SetDegreeDecoding PersistentRestrictions | |
| universe u | |
| theorem Construction.singleInputs_relative (t : Construction.{u}) (M : ZFSet.{u}) | |
| (hM : Transitive M) (hT : SourceT M) (P : Oracle) | |
| (hP : P ∈ modelReals M) (hi : t.Indexed M) : | |
| ∀ i, t.singleInputs (groundReals M) P i ∈ relativeModel M (groundReals M) := by | |
| let R := groundReals M | |
| let N := relativeModel M R | |
| have hR := groundReals_mem M hM hT | |
| have hN := relativeModel_transitive M R hM | |
| have hp := relativeModel_pairing M R hM hT hR | |
| have hu := relativeModel_union M R hM hT hR | |
| have hpow := relativeModel_power_set M R hM hT hR | |
| have hs := relativeModel_bounded_separation M R hM hT hR | |
| have hinf := ground_relativeModel_infinity M hM hT | |
| have hw := ground_relativeModel_omega M hM hT | |
| intro i | |
| rcases i with _|_|_|_|_|i | |
| · exact (mem_relativeModel M R R hM hT hR).mpr (parameter_in_relativeModel M R hM hT) | |
| · exact (relativeModel_ground_reals M hM hT (realCode P) (realCode_subset P)).mpr hP | |
| · exact hw | |
| · exact additionSet_mem N hN hp hu hpow hs hinf | |
| · exact product_mem N hN hp hu hpow hs hw hw | |
| · have hid := t.inputData_indexed M hi i | |
| change t.stageInputs R i ∈ N | |
| unfold Construction.stageInputs | |
| cases hd : t.inputData i with | |
| | reals => exact (mem_relativeModel M R R hM hT hR).mpr (parameter_in_relativeModel M R hM hT) | |
| | realPath path => exact (mem_relativeModel M R R hM hT hR).mpr (parameter_in_relativeModel M R hM hT) | |
| | stage o => | |
| rw [hd] at hid | |
| exact (mem_relativeModel M R _ hM hT hR).mpr (level_in_relativeModel M R hM hT o hid) | |
| theorem Construction.inlined_at_level (t : Construction.{u}) (M : ZFSet.{u}) | |
| (hM : Transitive M) (hT : SourceT M) (P : Oracle) | |
| (hP : P ∈ modelReals M) (ht : t.Certified (groundReals M) P) (hi : t.Indexed M) : | |
| ∃ γ : Ordinal.{u}, γ.toZFSet ∈ M ∧ | |
| t.value (groundReals M) P ∈ level (groundReals M) γ ∧ | |
| (∀ i, t.singleInputs (groundReals M) P i ∈ level (groundReals M) γ) ∧ | |
| ∀ z ∈ level (groundReals M) γ, | |
| t.singleMembership.Sat (level (groundReals M) γ : Set ZFSet) | |
| (cons z (t.singleInputs (groundReals M) P)) ↔ z ∈ t.value (groundReals M) P := by | |
| let R := groundReals M | |
| have hR := groundReals_mem M hM hT | |
| have hinputs := t.singleInputs_relative M hM hT P hP hi | |
| let e := cons (t.value R P) (t.singleInputs R P) | |
| have he (i : ℕ) (_hin : i < 6) : InRelativeModel M R (e i) := by | |
| cases i with | |
| | zero => exact t.value_in_relativeModel M R hM hT P ht hi | |
| | succ i => exact (mem_relativeModel M R _ hM hT hR).mp (hinputs i) | |
| obtain ⟨o,ho,hbound⟩ := finite_parameters_in_relative_level M R hM hT e 6 he | |
| let γ := max o t.stageBound | |
| have hγ : γ.toZFSet ∈ M := internal_ordinal_max M ho (t.stageBound_mem M hM hT hi) | |
| have hv : t.value R P ∈ level R γ := level_mono R (le_max_left _ _) (hbound 0 (by decide)) | |
| have hin (i : ℕ) : t.singleInputs R P i ∈ level R γ := by | |
| by_cases hi5 : i < 5 | |
| · exact level_mono R (le_max_left _ _) (hbound (i+1) (by omega)) | |
| · obtain ⟨k,rfl⟩ : ∃ k, i = k+5 := ⟨i-5,by omega⟩ | |
| change t.stageInputs R k ∈ level R γ | |
| exact level_mono R (le_max_right _ _) (t.stageInputs_bounded R k) | |
| refine ⟨γ,hγ,hv,hin,?_⟩ | |
| intro z hz | |
| exact t.singleMembership_of_inputs (level R γ) R (level_transitive R γ) P ht hin hv z hz | |
| theorem persistent_graph_inlined_at_level (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) | |
| (ρ : modelIdeal M hM hT ≃o modelIdeal M hM hT) | |
| (hρ : Persistent (modelIdeal M hM hT) ρ) : | |
| ∃ (t : Construction.{u}) (P : Oracle) (γ : Ordinal.{u}), | |
| P ∈ modelReals M ∧ t.Indexed M ∧ t.Certified (groundReals M) P ∧ | |
| γ.toZFSet ∈ M ∧ t.value (groundReals M) P = automorphismSet ρ ∧ | |
| automorphismSet ρ ∈ level (groundReals M) γ ∧ | |
| (∀ i, t.singleInputs (groundReals M) P i ∈ level (groundReals M) γ) ∧ | |
| ∀ z ∈ level (groundReals M) γ, | |
| t.singleMembership.Sat (level (groundReals M) γ : Set ZFSet) | |
| (cons z (t.singleInputs (groundReals M) P)) ↔ z ∈ automorphismSet ρ := by | |
| obtain ⟨t,P,hP,hi,ht,hval⟩ := persistent_graph_construction M hM hT ρ hρ | |
| obtain ⟨γ,hγ,hv,he,hf⟩ := t.inlined_at_level M hM hT P hP ht hi | |
| refine ⟨t,P,γ,hP,hi,ht,hγ,hval,hval ▸ hv,he,?_⟩ | |
| intro z hz | |
| rw [← hval] | |
| exact hf z hz | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |