openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /GroundBinaryRowNameSupportBounds.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundBinaryRowRecordedPreparationBounds | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) {L : Type u} [PartialOrder L] [OrderBot L] | |
| variable (s : Presentation L) (C : ZFSet.{u}) | |
| variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| variable (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| variable (collection : CollectionSchema M.sets) | |
| variable (infinite : Realize M.sets infinity Fin.elim0) | |
| variable (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| variable (hL : s.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets) (hC : C ∈ M.sets) | |
| variable (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z) | |
| variable (countable_bound : ∀ W ∈ M.sets, W ⊆ s.carrier → M.groundEmbeddable W ZFSet.omega → | |
| ∃ D : L, ∀ x : L, s.encode x ∈ W → x ≤ D) | |
| include pairing unions powers separation collection infinite choice hL order hC directed countable_bound | |
| theorem binaryRowStructuralCondition_nameSupportBound {mu : ZFSet.{u}} (hmu : mu ∈ M.sets) | |
| (one : SetMembers (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)) : | |
| let c := SetMembers.presentation (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega) | |
| let rows := M.binaryRowNameGraph s.nonbottom.carrier C one | |
| let S := M.rowPositionSequences s.carrier s.conditionOrder s.nonbottom.carrier mu | |
| let T := rowTupleTerms (c.encode one) mu s.nonbottom.carrier (ZFSet.sUnion (ZFSet.sUnion rows)) rows S | |
| ∀ valid : ∀ z ∈ T, IsName c z, | |
| let lifted := Name.renamedFamily c.pairEmpty id T valid | |
| let tuples := liftedRowTupleFamily c (c.encode one) mu s.nonbottom.carrier | |
| (ZFSet.sUnion (ZFSet.sUnion rows)) rows S valid | |
| ∀ (_hLifted : lifted ∈ M.sets) (G basis T2 stages history : ZFSet.{u}), | |
| G ∈ M.sets → basis ∈ M.sets → T2 ∈ M.sets → stages ∈ M.sets → history ∈ M.sets → | |
| let I := refinementDataHistoryIndices stages | |
| (M.structuralRefinementHistory G s.carrier s.conditionOrder basis lifted T2 stages history) | |
| let anchors := refinementIndexFieldGraph I s.carrier 2 | |
| ∀ (p : SetMembers (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)) (A : ZFSet.{u}), | |
| ZFSet.pair (c.encode p) A ∈ | |
| M.structuralForcingCarrier c.carrier G s.carrier s.conditionOrder basis lifted T2 stages history → | |
| ∀ {G P : Type u} [Group G] [PartialOrder P] [OrderTop P] | |
| (lower : Presentation P) (action : PosetAction G P) (nodeAction : PosetAction G L) | |
| (base : L) (names : Fin 2 → Name lower), | |
| (∀ i, HereditarilySymmetric action lower (nodeAction.principalSupportFamily base directed) (names i)) → | |
| ∀ E : L, ∃ D : L, | |
| (∀ x : L, s.encode x ∈ rowUsedRows s.nonbottom.carrier C (natSetCode 2) (SetMembers.raw p) → x ≤ D) ∧ | |
| (∀ z ∈ A, ZFSet.pair z (s.encode D) ∈ | |
| stageOneDataBounds A I lifted S s.carrier mu s.nonbottom.carrier s.conditionOrder anchors tuples) ∧ | |
| E ≤ D ∧ ∀ i, nodeAction.principalFix D ≤ action.stabilizer lower (names i) := by | |
| intro c rows S T valid lifted tuples hLifted Gcode basis T2 stages history | |
| hG hb hT2 hs hh I anchors row A member G P instG instP instTop lower action nodeAction base names symmetric E | |
| obtain ⟨first, firstSupport⟩ := (symmetric 0).symmetric | |
| obtain ⟨second, secondSupport⟩ := (symmetric 1).symmetric | |
| obtain ⟨common, firstBelow, secondBelow⟩ := directed first second | |
| let extra : ZFSet.{u} := {s.encode E, s.encode common} | |
| have hE := M.transitive hL (s.encode_mem E) | |
| have hCommon := M.transitive hL (s.encode_mem common) | |
| have extraGround : extra ∈ M.sets := M.unorderedPair_mem pairing hE hCommon | |
| have extraSubset : extra ⊆ s.carrier := by | |
| intro x hx | |
| rcases ZFSet.mem_pair.1 hx with rfl | rfl | |
| · exact s.encode_mem E | |
| · exact s.encode_mem common | |
| have extraSmall : M.groundEmbeddable extra ZFSet.omega := | |
| M.unorderedPair_countable pairing unions powers separation infinite hE hCommon | |
| obtain ⟨D, rowBound, assignmentBound, extraBound⟩ := | |
| M.binaryRowStructuralConditionDataBound s C pairing unions powers separation collection | |
| infinite choice hL order hC directed countable_bound hmu one valid hLifted | |
| Gcode basis T2 stages history hG hb hT2 hs hh row A member extra extraGround extraSubset extraSmall | |
| have commonBelow : common ≤ D := extraBound common (ZFSet.mem_pair.2 (Or.inr rfl)) | |
| have atCommon : ∀ i : Fin 2, nodeAction.principalFix common ≤ action.stabilizer lower (names i) := | |
| Fin.cases ((nodeAction.principalFix_antitone firstBelow).trans firstSupport) | |
| (Fin.cases ((nodeAction.principalFix_antitone secondBelow).trans secondSupport) (fun j => Fin.elim0 j)) | |
| exact ⟨D, rowBound, assignmentBound, extraBound E (ZFSet.mem_pair.2 (Or.inl rfl)), | |
| fun i => (nodeAction.principalFix_antitone commonBelow).trans (atCommon i)⟩ | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |