openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /GroundBinaryRowStructuralDataBoundActions.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.GroundSemanticAssignmentLabelDomainTotality | |
| namespace OAI | |
| open scoped ZFSet | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) {L G : Type u} | |
| variable [PartialOrder L] [OrderBot L] [Group G] | |
| variable (a : PosetAction G L) (s : Presentation L) (b : Presentation G) (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 (hL : s.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets) | |
| variable (hC : C ∈ M.sets) (hG : b.carrier ∈ M.sets) | |
| variable (graph : a.actionGraph s b ∈ M.sets) | |
| variable (nodeMaps : a.conditionMapFamily s b ∈ M.sets) | |
| variable (supports : supportBasisCode s b a.principalFix ∈ M.sets) | |
| variable (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z) | |
| include collection order nodeMaps supports | |
| theorem binaryRowStructuralAssignmentImage_dataBound {mu : ZFSet.{u}} (hmu : mu ∈ M.sets) | |
| (muOrdinal : ordinalSetProperty mu) : | |
| letI := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega | |
| letI := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega | |
| ⟨s.carrier, hL⟩ | |
| let act := M.internalRowAction a s b C pairing unions powers separation infinite hL hC hG graph | |
| let c := SetMembers.presentation (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega) | |
| let rows := M.binaryRowNameGraph s.nonbottom.carrier C ⊤ | |
| let S := M.rowPositionSequences s.carrier s.conditionOrder s.nonbottom.carrier mu | |
| let T := rowTupleTerms (c.encode ⊤) mu s.nonbottom.carrier (⋃₀ ⋃₀ rows) rows S | |
| ∀ valid : ∀ z ∈ T, IsName c z, | |
| let lifted := Name.renamedFamily c.pairEmpty id T valid | |
| let squared := orderedPairNameFamily (c.pairEmpty.encode ⊤) lifted lifted | |
| let tuples := liftedRowTupleFamily c (c.encode ⊤) mu s.nonbottom.carrier (⋃₀ ⋃₀ rows) rows S valid | |
| ∀ (_hLifted : lifted ∈ M.sets) (_hSquared : squared ∈ M.sets) | |
| (_valid1 : ∀ z ∈ lifted, IsName c.pairEmpty z) | |
| (_valid2 : ∀ z ∈ squared, IsName c.pairEmpty z) | |
| (_closed1 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ lifted → (act.act c.pairEmpty g tau).val ∈ lifted) | |
| (_closed2 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ squared → (act.act c.pairEmpty g tau).val ∈ squared) | |
| (_children1 : ∀ tau : Name c.pairEmpty, tau.val ∈ lifted → | |
| HereditarilySymmetric act c.pairEmpty (a.principalSupportFamily ⊥ directed) tau) | |
| (_children2 : ∀ tau : Name c.pairEmpty, tau.val ∈ squared → | |
| HereditarilySymmetric act c.pairEmpty (a.principalSupportFamily ⊥ directed) tau) | |
| (stages history : ZFSet.{u}) (_hs : stages ∈ M.sets) (_hh : history ∈ M.sets) | |
| (_stagesOrdinal : ordinalSetProperty stages) | |
| (_trace : OrdinalRecursionTrace (fun level prior stage => stage = | |
| M.structuralForcingStage c.carrier c.conditionOrder (c.encode ⊤) (act.actionGraph c b) | |
| b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared level prior) stages history), | |
| let coordinates := M.prod_mem pairing unions powers separation (M.nonbottom_carrier_mem s separation hL) hC | |
| let two := M.natSetCode_mem separation infinite 2 | |
| let _carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite coordinates two | |
| let _forcingOrder := M.countableFunctionOrder_mem pairing unions powers separation infinite coordinates two | |
| let _forcingAction := M.internalRowActionGraph_mem a s b C pairing unions powers separation infinite hL hC hG graph | |
| let _omega := M.omega_mem separation infinite | |
| let _topMem := c.encode_mem ⊤ | |
| let _topRows : ∀ r ∈ c.carrier, ZFSet.pair r (c.encode ⊤) ∈ c.conditionOrder := | |
| fun r hr => by | |
| obtain ⟨r, rfl⟩ := (c.mem_carrier r).1 hr | |
| exact (c.pair_mem_conditionOrder r ⊤).2 le_top | |
| let packed := M.structuralRefinementHistory b.carrier s.carrier s.conditionOrder | |
| (supportBasisCode s b a.principalFix) lifted squared stages history | |
| let I := refinementDataHistoryIndices stages packed | |
| let anchors := refinementIndexFieldGraph I s.carrier 2 | |
| ∀ (g : G) (D : L) (A : ZFSet.{u}), A ∈ M.sets → | |
| (∀ z ∈ A, ZFSet.pair z (s.encode D) ∈ stageOneDataBounds A I lifted S s.carrier mu | |
| s.nonbottom.carrier s.conditionOrder anchors tuples) → | |
| let movedA := M.structuralAssignmentImage b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared stages history (b.encode g) A | |
| movedA ∈ M.sets ∧ | |
| stageOneDataBounds movedA I lifted S s.carrier mu s.nonbottom.carrier s.conditionOrder anchors tuples ∈ M.sets ∧ | |
| ∀ z ∈ movedA, ZFSet.pair z (s.encode (a.iso g D)) ∈ | |
| stageOneDataBounds movedA I lifted S s.carrier mu s.nonbottom.carrier s.conditionOrder anchors tuples := by | |
| let := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega | |
| let := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega | |
| ⟨s.carrier, hL⟩ | |
| intro act c rows S T valid lifted squared tuples hLifted hSquared valid1 valid2 closed1 closed2 | |
| children1 children2 stages history hs hh stagesOrdinal trace coordinates two carrier forcingOrder | |
| forcingAction omega topMem topRows packed I anchors | |
| have hNonbottom := M.nonbottom_carrier_mem s separation hL | |
| have hS := M.rowPositionSequences_mem pairing unions powers separation hL order hNonbottom hmu | |
| have hTuples := M.binaryLiftedRowTupleFamily_mem pairing unions powers separation collection | |
| infinite hL order hNonbottom hC hmu ⊤ valid | |
| exact M.semanticStructuralAssignmentImage_dataBound c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hs hh stagesOrdinal trace S mu tuples hS hmu hTuples | |
| (fun f z entry => ((pair_mem_relationCompose S _ lifted _ _ f z).1 entry).2.1) | |
| (fun g f hf tau entry => ⟨M.rowPositionSequenceAction_mem a s b pairing unions powers separation | |
| hL hG graph hmu muOrdinal g hf, | |
| M.binaryLiftedRowTupleFamily_action a s b C pairing unions powers separation infinite | |
| hL hC hG graph hmu muOrdinal valid g f tau entry⟩) | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end OAI | |