openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Geometry /GroundBelowExternalPoints.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.GroundBoundedExternalBirths | |
| import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundHistoryUnionBounds | |
| namespace OAI | |
| namespace PartitionPilot.Forcing | |
| namespace GroundBelow | |
| open SetFormula | |
| universe u | |
| variable {M : TransitiveGround.{u}} {A K : ZFSet.{u}} | |
| theorem finiteWords | |
| (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) | |
| (hA : A ∈ M.sets) (hK : K ∈ M.sets) | |
| (ordinal : ordinalSetProperty K) (strong : GroundStrongLimit M K) | |
| (omegaBelow : ZFSet.omega ∈ K) (alphabet : GroundBelow M A K) : | |
| GroundBelow M (M.finiteWords A) K := by | |
| have natural := GroundBelow.of_member pairing unions powers separation hK omegaBelow | |
| have relations := GroundBelow.relations pairing unions powers separation | |
| (M.omega_mem separation infinite) hA hK ordinal strong natural alphabet | |
| exact relations.restrict separation (M.finiteWords_mem pairing unions powers separation infinite hA) | |
| hK (fun _ member => (ZFSet.mem_sep.1 member).1) | |
| end GroundBelow | |
| section | |
| open SetFormula | |
| universe u | |
| namespace GroundBelow | |
| variable {M : TransitiveGround.{u}} {A E K : ZFSet.{u}} | |
| theorem equivalenceClasses | |
| (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) | |
| (hA : A ∈ M.sets) (hE : E ∈ M.sets) (hK : K ∈ M.sets) | |
| (strong : GroundStrongLimit M K) (small : GroundBelow M A K) : | |
| GroundBelow M (Forcing.equivalenceClasses A E) K := by | |
| have included : Forcing.equivalenceClasses A E ⊆ M.groundPower A := by | |
| intro X hX | |
| rw [M.equivalenceClasses_eq_sep_groundPower separation hA hE] at hX | |
| exact (ZFSet.mem_sep.1 hX).1 | |
| exact (strong.power_closed pairing unions powers separation hA hK small).restrict | |
| separation (M.equivalenceClasses_mem powers separation hA hE) hK included | |
| end GroundBelow | |
| namespace TransitiveGround | |
| variable (M : TransitiveGround.{u}) | |
| theorem externalGroupCarrier_below | |
| (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) | |
| {G S m one K : ZFSet.{u}} | |
| (hG : G ∈ M.sets) (hS : S ∈ M.sets) (hm : m ∈ M.sets) (hOne : one ∈ M.sets) (hK : K ∈ M.sets) | |
| (ordinal : ordinalSetProperty K) (strong : GroundStrongLimit M K) | |
| (omegaBelow : ZFSet.omega ∈ K) (alphabet : GroundBelow M (externalGeneratorAlphabet G S) K) : | |
| GroundBelow M (M.externalGroupCarrier G S m one) K := by | |
| have hAlphabet := M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS | |
| have hWords := M.finiteWords_mem pairing unions powers separation infinite hAlphabet | |
| have hRelation := M.externalWordEquivalence_mem pairing unions powers separation infinite hG hS hm hOne | |
| exact GroundBelow.equivalenceClasses pairing unions powers separation hWords hRelation hK strong | |
| (GroundBelow.finiteWords pairing unions powers separation infinite hAlphabet hK ordinal strong omegaBelow alphabet) | |
| end TransitiveGround | |
| end | |
| namespace TransitiveGround | |
| open SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{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 (infinite : Realize M.sets infinity Fin.elim0) | |
| variable (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| variable {K : ZFSet.{u}} (hK : K ∈ M.sets) (conditions : GroundInaccessibleConditions M K) | |
| include pairing unions powers separation hK conditions | |
| theorem externalRectangleBound_below {P G : ZFSet.{u}} | |
| (hP : P ∈ M.sets) (hG : G ∈ M.sets) | |
| (smallP : GroundBelow M P K) (smallG : GroundBelow M G K) : | |
| GroundBelow M (externalRectangleBound P G) K := by | |
| have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK | |
| exact (product ⟨hP, smallP⟩ (product ⟨hP, smallP⟩ | |
| (product ⟨hP, smallP⟩ (product ⟨hG, smallG⟩ ⟨hG, smallG⟩)))).2 | |
| theorem externalRectangles_below {P G r μ A : ZFSet.{u}} | |
| (hP : P ∈ M.sets) (hG : G ∈ M.sets) (hr : r ∈ M.sets) | |
| (hμ : μ ∈ M.sets) (hA : A ∈ M.sets) | |
| (smallP : GroundBelow M P K) (smallG : GroundBelow M G K) : | |
| GroundBelow M (externalRectangles P G r μ A) K := by | |
| exact (M.externalRectangleBound_below pairing unions powers separation hK conditions hP hG smallP smallG).restrict | |
| separation (M.externalRectangles_mem pairing unions powers separation hP hG hr hμ hA) hK | |
| (fun _ hz => (ZFSet.mem_sep.1 hz).1) | |
| include infinite choice | |
| theorem externalRequestCarrier_below {I P Rectangles : ZFSet.{u}} | |
| (hI : I ∈ M.sets) (hP : P ∈ M.sets) (hRectangles : Rectangles ∈ M.sets) | |
| (smallI : GroundBelow M I K) (smallP : GroundBelow M P K) | |
| (smallRectangles : GroundBelow M Rectangles K) : | |
| GroundBelow M (externalRequestCarrier I P Rectangles) K := by | |
| have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK | |
| have join := @GroundInaccessibleConditions.binary_union M K conditions pairing unions powers separation infinite choice hK | |
| have tag := fun n => conditions.singleton pairing unions powers separation infinite hK | |
| (M.natSetCode_mem separation infinite n) | |
| exact (join (join (product (tag 0) ⟨hI, smallI⟩) (product (tag 1) ⟨hP, smallP⟩)) | |
| (product (tag 2) ⟨hRectangles, smallRectangles⟩)).2 | |
| theorem externalChildLetters_below {I P Rectangles : ZFSet.{u}} | |
| (hI : I ∈ M.sets) (hP : P ∈ M.sets) (hRectangles : Rectangles ∈ M.sets) | |
| (smallI : GroundBelow M I K) (smallP : GroundBelow M P K) | |
| (smallRectangles : GroundBelow M Rectangles K) : | |
| GroundBelow M (externalChildLetters I P Rectangles) K := by | |
| have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK | |
| have join := @GroundInaccessibleConditions.binary_union M K conditions pairing unions powers separation infinite choice hK | |
| have tag := fun n => conditions.singleton pairing unions powers separation infinite hK | |
| (M.natSetCode_mem separation infinite n) | |
| have requests := M.externalRequestCarrier_below pairing unions powers separation infinite choice hK conditions | |
| hI hP hRectangles smallI smallP smallRectangles | |
| exact (join (product ⟨M.externalRequestCarrier_mem pairing unions powers separation hI hP hRectangles, | |
| requests⟩ (tag 0)) (product (product (tag 2) ⟨hRectangles, smallRectangles⟩) (tag 1))).2 | |
| theorem externalGeneratorAlphabet_below {G S : ZFSet.{u}} | |
| (hG : G ∈ M.sets) (hS : S ∈ M.sets) | |
| (smallG : GroundBelow M G K) (smallS : GroundBelow M S K) : | |
| GroundBelow M (externalGeneratorAlphabet G S) K := by | |
| have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK | |
| have join := @GroundInaccessibleConditions.binary_union M K conditions pairing unions powers separation infinite choice hK | |
| have tag := fun n => conditions.singleton pairing unions powers separation infinite hK | |
| (M.natSetCode_mem separation infinite n) | |
| exact (join (product (tag 0) ⟨hG, smallG⟩) | |
| (product (tag 1) (product ⟨hS, smallS⟩ (conditions.natural pairing unions powers separation infinite hK 2)))).2 | |
| theorem externalGroupCarrier_below_of_carriers {G S m one : ZFSet.{u}} | |
| (hG : G ∈ M.sets) (hS : S ∈ M.sets) (hm : m ∈ M.sets) (hOne : one ∈ M.sets) | |
| (smallG : GroundBelow M G K) (smallS : GroundBelow M S K) : | |
| GroundBelow M (M.externalGroupCarrier G S m one) K := | |
| M.externalGroupCarrier_below pairing unions powers separation infinite hG hS hm hOne hK | |
| conditions.1 conditions.2.2.2 conditions.2.1 | |
| (M.externalGeneratorAlphabet_below pairing unions powers separation infinite choice hK conditions hG hS smallG smallS) | |
| end TransitiveGround | |
| namespace GroundExternalGeometryInputs | |
| open SetFormula | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m inv one T τ P r μ A K : ZFSet.{u}} | |
| variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ} | |
| variable (data : GroundExternalGeometryInputs M initial I Sources Reps G m inv one T τ P r μ A completion) | |
| 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 (infinite : Realize M.sets infinity Fin.elim0) | |
| variable (hP : P ∈ M.sets) (hG : G ∈ M.sets) (hK : K ∈ M.sets) | |
| local notation "letters" => externalChildLetters I P (externalRectangles P G r μ A) | |
| local notation "actors" => M.externalGroupCarrier G letters m one | |
| local notation "U" => completion.algebra.union.fields 0 | |
| include data pairing unions powers separation infinite hP hG hK | |
| theorem externalPointCarrier_below | |
| (ordinal : ordinalSetProperty K) (strong : GroundStrongLimit M K) | |
| (groupSmall : GroundBelow M actors K) (pointsSmall : GroundBelow M U K) : | |
| GroundBelow M (completion.externalPointCarrier P r μ A) K := by | |
| have quotient := data.point_quotient_ground pairing unions powers separation infinite hP hG | |
| have hProduct := M.prod_mem pairing unions powers separation | |
| data.group.carrier_mem (completion.algebra.union.ground 0) | |
| have smallProduct := GroundBelow.product pairing unions powers separation | |
| data.group.carrier_mem (completion.algebra.union.ground 0) hK ordinal strong groupSmall pointsSmall | |
| exact GroundBelow.equivalenceClasses pairing unions powers separation | |
| hProduct quotient.1 hK strong smallProduct | |
| theorem externalBoundedCarrier_below | |
| (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| (conditions : GroundInaccessibleConditions M K) | |
| (groupSmall : GroundBelow M actors K) (pointsSmall : GroundBelow M U K) : | |
| GroundBelow M (completion.externalBoundedCarrier P r μ A) K := by | |
| have hQ := (data.point_quotient_ground pairing unions powers separation infinite hP hG).2.2.1 | |
| exact GroundBelow.insert pairing unions powers separation infinite choice hQ hQ hK conditions.2.1 | |
| conditions.1 conditions.2.2.1 conditions.2.2.2 | |
| (data.externalPointCarrier_below pairing unions powers separation infinite hP hG hK | |
| conditions.1 conditions.2.2.2 groupSmall pointsSmall) | |
| end GroundExternalGeometryInputs | |
| end PartitionPilot.Forcing | |
| end OAI | |