openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /GroundCountableFunctionForcing.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundNaturalNumbers | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) | |
| theorem countablePartialFunctions_mem | |
| (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) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {X Y : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ M.sets) : | |
| M.smallPartialFunctions X Y ZFSet.omega ∈ M.sets := | |
| M.smallPartialFunctions_mem pairing unions powers separation hX hY | |
| (M.omega_mem separation infinite) | |
| theorem countablePartialFunction_countable {X Y p : ZFSet.{u}} | |
| (hp : p ∈ M.smallPartialFunctions X Y ZFSet.omega) : Countable (SetMembers p) := | |
| M.smallPartialFunction_countable hp | |
| theorem countableFunctionOrder_mem | |
| (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) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {X Y : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ M.sets) : | |
| letI := M.smallFunctionOrder X Y ZFSet.omega | |
| (SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega)).conditionOrder ∈ M.sets := | |
| M.smallFunctionOrder_mem pairing unions powers separation hX hY | |
| (M.omega_mem separation infinite) | |
| theorem countablePartialFunctions_restrict (separation : SeparationSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {X Y p q : ZFSet.{u}} (hp : p ∈ M.smallPartialFunctions X Y ZFSet.omega) | |
| (hq : q ∈ M.sets) (subset : q ⊆ p) : q ∈ M.smallPartialFunctions X Y ZFSet.omega := | |
| M.smallPartialFunctions_restrict separation hp hq (M.omega_mem separation infinite) subset | |
| theorem exists_countableFunction_generic | |
| (small : M.sets.Countable) | |
| (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) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {X Y : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ M.sets) | |
| (p : SetMembers (M.smallPartialFunctions X Y ZFSet.omega)) : | |
| letI := M.smallFunctionOrder X Y ZFSet.omega | |
| ∃ H : ConditionFilter (SetMembers (M.smallPartialFunctions X Y ZFSet.omega)), H.mem p ∧ | |
| H.GenericFor (M.codedFamily (SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega))) ∧ | |
| H.GenericFor (formulaFamily (M.nameDomain | |
| (SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega)))) := | |
| M.exists_smallFunction_generic small pairing unions powers separation hX hY | |
| (M.omega_mem separation infinite) p | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |