openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Transfinite /GroundCountableRowSequenceReflection.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.GroundSemanticStructuralPrefixPreparationInputs | |
| namespace OAI | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) {G L : Type u} [Group G] [PartialOrder L] | |
| variable (t : Presentation G) (b : Presentation L) (d : PosetAction G L) | |
| variable (base : L) (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z) | |
| theorem countableRow_semanticStructuralTrace_internally_closed | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| (collection : CollectionSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| (X : ZFSet.{u}) (hX : X ∈ M.sets) : | |
| letI := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega | |
| let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega) | |
| ∀ (a : PosetAction G (SetMembers (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega))) | |
| (T T2 rowTop : ZFSet.{u}), | |
| a.actionGraph c t ∈ M.sets → | |
| t.carrier ∈ M.sets → b.carrier ∈ M.sets → b.conditionOrder ∈ M.sets → | |
| d.actionGraph b t ∈ M.sets → d.conditionMapFamily b t ∈ M.sets → | |
| supportBasisCode b t d.principalFix ∈ M.sets → T ∈ M.sets → T2 ∈ M.sets → | |
| (∀ z ∈ T, IsName c.pairEmpty z) → (∀ z ∈ T2, IsName c.pairEmpty z) → | |
| (∀ g (tau : Name c.pairEmpty), tau.val ∈ T → (a.act c.pairEmpty g tau).val ∈ T) → | |
| (∀ g (tau : Name c.pairEmpty), tau.val ∈ T2 → (a.act c.pairEmpty g tau).val ∈ T2) → | |
| (∀ tau : Name c.pairEmpty, tau.val ∈ T → | |
| HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) → | |
| (∀ tau : Name c.pairEmpty, tau.val ∈ T2 → | |
| HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) → | |
| rowTop ∈ c.carrier → (∀ row ∈ c.carrier, ZFSet.pair row rowTop ∈ c.conditionOrder) → | |
| ∀ stages history : ZFSet.{u}, stages ∈ M.sets → history ∈ M.sets → | |
| ordinalSetProperty stages → | |
| OrdinalRecursionTrace (fun level prior stage => stage = | |
| M.structuralForcingStage c.carrier c.conditionOrder rowTop (a.actionGraph c t) | |
| t.carrier b.carrier b.conditionOrder (d.actionGraph b t) | |
| (supportBasisCode b t d.principalFix) T T2 level prior) stages history → | |
| let packed := M.structuralRefinementHistory t.carrier b.carrier b.conditionOrder | |
| (supportBasisCode b t d.principalFix) T T2 stages history | |
| letI := M.refinementHistoryPartialOrder c.conditionOrder_code T stages packed | |
| InternallyCountablyClosed M (SetMembers.presentation | |
| (M.refinementHistoryForcing c.carrier T stages packed)) := by | |
| let := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega | |
| let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega) | |
| dsimp only | |
| intro a T T2 rowTop action group nodes nodeOrder nodeAction nodeMaps supports hT hT2 | |
| valid1 valid2 closed1 closed2 children1 children2 topMem topRows stages history hs hh ordinal trace | |
| have two := M.natSetCode_mem separation infinite 2 | |
| have carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite hX two | |
| have order := M.countableFunctionOrder_mem pairing unions powers separation infinite hX two | |
| have rowClosed := M.countable_function_internally_closed pairing unions powers separation infinite choice hX two | |
| exact M.semanticStructuralTrace_internally_closed c a t b d base directed T T2 rowTop | |
| pairing unions powers separation collection carrier order action group nodes nodeOrder nodeAction | |
| nodeMaps supports hT hT2 (M.omega_mem separation infinite) valid1 valid2 closed1 closed2 | |
| children1 children2 topMem topRows hs hh ordinal trace infinite choice rowClosed | |
| theorem countableRow_semanticStructuralTrace_jointZF_sequenceReflection | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| (collection : CollectionSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| (X : ZFSet.{u}) (hX : X ∈ M.sets) (small : M.sets.Countable) : | |
| letI := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega | |
| let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega) | |
| ∀ (a : PosetAction G (SetMembers (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega))) | |
| (T T2 rowTop : ZFSet.{u}) | |
| (rowAction : a.actionGraph c t ∈ M.sets) | |
| (group : t.carrier ∈ M.sets) (nodes : b.carrier ∈ M.sets) | |
| (nodeOrder : b.conditionOrder ∈ M.sets) (nodeAction : d.actionGraph b t ∈ M.sets) | |
| (nodeMaps : d.conditionMapFamily b t ∈ M.sets) | |
| (supports : supportBasisCode b t d.principalFix ∈ M.sets) | |
| (hT : T ∈ M.sets) (hT2 : T2 ∈ M.sets) | |
| (valid1 : ∀ z ∈ T, IsName c.pairEmpty z) | |
| (valid2 : ∀ z ∈ T2, IsName c.pairEmpty z) | |
| (closed1 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ T → (a.act c.pairEmpty g tau).val ∈ T) | |
| (closed2 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ T2 → (a.act c.pairEmpty g tau).val ∈ T2) | |
| (symmetric1 : ∀ tau : Name c.pairEmpty, tau.val ∈ T → | |
| HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) | |
| (symmetric2 : ∀ tau : Name c.pairEmpty, tau.val ∈ T2 → | |
| HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) | |
| (topMem : rowTop ∈ c.carrier) | |
| (topRows : ∀ row ∈ c.carrier, ZFSet.pair row rowTop ∈ c.conditionOrder) | |
| (stages history : ZFSet.{u}) (hs : stages ∈ M.sets) (hh : history ∈ M.sets) | |
| (ordinal : ordinalSetProperty stages) | |
| (trace : OrdinalRecursionTrace (fun level prior stage => stage = | |
| M.structuralForcingStage c.carrier c.conditionOrder rowTop (a.actionGraph c t) | |
| t.carrier b.carrier b.conditionOrder (d.actionGraph b t) | |
| (supportBasisCode b t d.principalFix) T T2 level prior) stages history), | |
| let two := M.natSetCode_mem separation infinite 2 | |
| let carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite hX two | |
| let rowOrderGround := M.countableFunctionOrder_mem pairing unions powers separation infinite hX two | |
| let omega := M.omega_mem separation infinite | |
| let packed := M.structuralRefinementHistory t.carrier b.carrier b.conditionOrder | |
| (supportBasisCode b t d.principalFix) T T2 stages history | |
| let Q := M.refinementHistoryForcing c.carrier T stages packed | |
| let Cuts := {alpha : ZFSet // alpha ∈ M.sets ∧ ordinalSetProperty alpha ∧ alpha ⊆ stages} | |
| let lower := (fun j : Cuts => SetMembers (M.refinementHistoryForcing c.carrier T (Subtype.val j) (graphRestriction (Subtype.val j) packed))) | |
| let run := M.structuralForcingTrace_recursiveOrder c.carrier c.conditionOrder rowTop (a.actionGraph c t) | |
| t.carrier b.carrier b.conditionOrder (d.actionGraph b t) (supportBasisCode b t d.principalFix) T T2 ordinal trace | |
| let witnesses := M.structuralForcingTrace_witnesses c a t b d base directed T T2 rowTop pairing unions powers separation collection carrier rowOrderGround rowAction group nodes nodeOrder nodeAction nodeMaps supports hT hT2 omega valid1 valid2 closed1 closed2 symmetric1 symmetric2 topMem topRows hs hh ordinal trace | |
| let mono := M.structuralHistory_rawTests_mono t b d base directed T T2 stages history | |
| pairing unions powers separation collection group nodes supports hh hT witnesses | |
| let hPacked := M.structuralRefinementHistory_mem pairing unions powers separation collection | |
| ![t.carrier, b.carrier, b.conditionOrder, supportBasisCode b t d.principalFix, T, T2, stages, history] | |
| (Fin.cases group (Fin.cases nodes (Fin.cases nodeOrder (Fin.cases supports | |
| (Fin.cases hT (Fin.cases hT2 (Fin.cases hs (Fin.cases hh (fun j => Fin.elim0 j))))))))) | |
| letI := M.refinementHistoryPartialOrder c.conditionOrder_code T stages packed | |
| let act := M.semanticStructuralAction c a t b d base directed T T2 rowTop pairing unions powers separation collection carrier rowOrderGround rowAction group nodes nodeOrder nodeAction nodeMaps supports hT hT2 omega valid1 valid2 closed1 closed2 symmetric1 symmetric2 topMem topRows hs hh ordinal trace | |
| letI : ∀ j : Cuts, PartialOrder (lower j) := fun j => | |
| M.refinementHistoryPartialOrder c.conditionOrder_code T j.val (graphRestriction j.val packed) | |
| let restrictions : ∀ j : Cuts, SetMembers Q →o lower j := fun j => | |
| (run).restriction ordinal j.property.2.1 j.property.2.2 c.conditionOrder_code | |
| let inclusions : ∀ j : Cuts, lower j ↪o SetMembers Q := fun j => | |
| (run).inclusion ordinal j.property.2.1 j.property.2.2 pairing unions powers separation infinite choice | |
| hs hPacked hT j.property.1 carrier rowTop topMem topRows mono c.conditionOrder_code | |
| let projections : ∀ j, CompleteProjection (restrictions j) (inclusions j) := fun j => | |
| M.semanticStructuralTrace_completeProjection c a t b d base directed T T2 rowTop pairing unions powers separation collection carrier rowOrderGround rowAction group nodes nodeOrder nodeAction nodeMaps supports hT hT2 omega valid1 valid2 closed1 closed2 symmetric1 symmetric2 topMem topRows hs hh ordinal trace j.val j.property.1 | |
| j.property.2.1 j.property.2.2 infinite choice | |
| let s := SetMembers.presentation Q | |
| let F := d.principalSupportFamily base directed | |
| ∀ p : SetMembers Q, ∃ H : ConditionFilter (SetMembers Q), | |
| H.mem p ∧ H.GenericFor (M.codedFamily s) ∧ | |
| H.GenericFor (formulaFamily (M.nameDomain s)) ∧ | |
| H.GenericFor (formulaFamily ((act).symmetricDomain s (M.nameDomain s) F)) ∧ | |
| (∀ j : Cuts, | |
| ((projections j).projectedFilter H).GenericFor | |
| (M.codedFamily (SetMembers.presentation (M.refinementHistoryForcing c.carrier T j.val (graphRestriction j.val packed)))) ∧ | |
| ((projections j).projectedFilter H).GenericFor | |
| (formulaFamily (M.nameDomain (SetMembers.presentation (M.refinementHistoryForcing c.carrier T j.val (graphRestriction j.val packed)))))) ∧ | |
| M.sets ⊆ ((act).symmetricDomain s (M.nameDomain s) F).values {q | H.mem q} ∧ | |
| ZFRealization (((act).symmetricDomain s (M.nameDomain s) F).values {q | H.mem q}) ∧ | |
| (∀ U ∈ M.sets, | |
| GroundSequenceReflection M (M.ordinaryExtension s H) U ∧ | |
| GroundSequenceReflection M (M.symmetricExtension s (act) F H) U) := by | |
| let := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega | |
| let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega) | |
| dsimp only | |
| intro a T T2 rowTop rowAction group nodes nodeOrder nodeAction nodeMaps supports hT hT2 | |
| valid1 valid2 closed1 closed2 symmetric1 symmetric2 topMem topRows stages history hs hh ordinal trace | |
| have two := M.natSetCode_mem separation infinite 2 | |
| have carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite hX two | |
| have rowOrderGround := M.countableFunctionOrder_mem pairing unions powers separation infinite hX two | |
| have rowClosed := M.countable_function_internally_closed pairing unions powers separation infinite choice hX two | |
| exact M.semanticStructuralTrace_jointZF_sequenceReflection c a t b d base directed T T2 rowTop | |
| pairing unions powers separation collection carrier rowOrderGround rowAction group nodes nodeOrder | |
| nodeAction nodeMaps supports hT hT2 (M.omega_mem separation infinite) valid1 valid2 closed1 closed2 | |
| symmetric1 symmetric2 topMem topRows hs hh ordinal trace infinite choice rowClosed small | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end OAI | |