openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /CohenForcing /CohenAbsorptionTree.lean
| import OAI.Computability.DegreeRigidity.SetModels.InternalRegularTree | |
| namespace OAI | |
| namespace TuringRigidity.CohenAbsorptionTree | |
| open TransitiveNameModel BoundedSetTheory InternalRegularOperations InternalRegularAlgebra | |
| open InternalProjectedGeneric RegularBinaryTree RegularDecisionPartition | |
| theorem top_code (c : ZFSet.{0}) : IsCode c c := by | |
| refine ⟨(fun _ h => h),ZFSet.ext (fun p => ?_)⟩ | |
| rw [mem_regular] | |
| exact ⟨And.left,fun hp => ⟨hp,fun q hq _ => ⟨q,hq,fun _ h => h⟩⟩⟩ | |
| theorem internal_dense_tree (M q : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M) | |
| (hq : q ∈ M) (hct : InternallyCountable M q) : | |
| ∃ c ∈ M, | |
| (∀ z, z ∈ c ↔ ∃ p ∈ q, ∃ s ∈ CohenGroundPoset.conditions ZFSet.omega, | |
| z = TaggedProductConditions.code p s) ∧ | |
| InternalCollapse.orderSet c ∈ M ∧ InternallyCountable M c ∧ | |
| ∃ B ∈ M, (∀ U, U ∈ B ↔ U ∈ M ∧ IsCode c U) ∧ | |
| ∃ T ∈ M, ∃ E : ℕ → ZFSet.{0}, orbitGraph E ∈ M ∧ | |
| ∃ A ∈ M, FunctionGraph InternalCohen.conditions (positive B) A ∧ | |
| (∀ (s : List Bool) U, | |
| ZFSet.pair (InternalCohen.wordCode s) U ∈ A ↔ U = tree c B T E s) ∧ | |
| (∀ s : List Bool, tree c B T E s ∈ positive B) ∧ | |
| (∀ s : List Bool, Refines c (tree c B T E s) (E s.length) | |
| (tree c B T E (s ++ [false])) (tree c B T E (s ++ [true]))) ∧ | |
| (∀ s t : List Bool, s <+: t → tree c B T E t ⊆ tree c B T E s) ∧ | |
| (∀ s u v : List Bool, ∀ p ∈ tree c B T E (s ++ [false] ++ u), | |
| p ∉ tree c B T E (s ++ [true] ++ v)) ∧ | |
| (∀ D, IsCode c D → D ≠ ∅ → ∃ s : List Bool, tree c B T E s ⊆ D) := by | |
| obtain ⟨c,hc,hcs,hco,hcc,B,hBM,hB,S,hSM,hSc,hSP,hSs,hSd,T,hTM,hTf,hTs⟩ := | |
| InternalRegularSplitter.counted_cohen_presentation M q hM hT hq hct | |
| have hc0 : c ≠ ∅ := by | |
| obtain ⟨g,_,hg,_⟩ := hcc | |
| obtain ⟨p,hp,_,_⟩ := hg.2 (natSet 0) ((mem_omega _).mpr ⟨0,rfl⟩) | |
| exact fun he => ZFSet.notMem_empty p (he ▸ hp) | |
| have hcP : c ∈ positive B := ZFSet.mem_sep.mpr ⟨(hB c).mpr ⟨hc,top_code c⟩,hc0⟩ | |
| obtain ⟨g,hgM,hgf,hgo⟩ := hSc | |
| obtain ⟨E,hES,hEg⟩ := InternalCountableFamily.sequence_of_graph S g hgf | |
| have hE : ∀ n, E n ∈ B := fun n => (ZFSet.mem_sep.mp (hSP (hES n))).1 | |
| have hEM : orbitGraph E ∈ M := hEg.symm ▸ hgM | |
| have hDense : ∀ D, IsCode c D → D ≠ ∅ → ∃ n, E n ≠ ∅ ∧ E n ⊆ D := by | |
| intro D hD hD0 | |
| obtain ⟨V,hV,hVD⟩ := hSd D hD hD0 | |
| obtain ⟨n,hn,hnV⟩ := hgo V hV | |
| obtain ⟨n,rfl⟩ := (mem_omega n).mp hn | |
| have hVE : V = E n := (orbitGraph_pair E n V).mp (hEg.symm ▸ hnV) | |
| exact ⟨n,(ZFSet.mem_sep.mp (hSP (hES n))).2,hVE ▸ hVD⟩ | |
| obtain ⟨A,hAM,hAf,hAs⟩ := InternalRegularTree.tree_graph_internal M c B T E | |
| hM hT hBM hTM hcP hTf hE hEM | |
| refine ⟨c,hc,hcs,hco,hcc,B,hBM,hB,T,hTM,E,hEM,A,hAM,hAf,hAs, | |
| tree_positive c B T E hcP hTf hE, | |
| tree_refines c B T E hcP hTf hE hTs, | |
| tree_prefix c B T E hcP hTf hE hTs, | |
| tree_incompatible c B T E hcP hTf hE hTs,?_⟩ | |
| exact tree_dense c B T E hcP hTf hE (fun U hU => ((hB U).mp hU).2) hTs hDense | |
| end TuringRigidity.CohenAbsorptionTree | |
| end OAI | |