openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Transfinite /GroundCompressionCarrierRankBounds.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundInitialCompressionOrigin | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| open SetRealization SetFormula | |
| universe u | |
| namespace GroundInaccessibleConditions | |
| variable {M : TransitiveGround.{u}} {K : ZFSet.{u}} | |
| theorem exists_rank_limit_above (conditions : GroundInaccessibleConditions M K) | |
| (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) | |
| (hK : K ∈ M.sets) {o : Ordinal.{u}} (bound : o < ZFSet.rank K) : | |
| ∃ L ∈ K, Order.IsSuccLimit (ZFSet.rank L) ∧ o < ZFSet.rank L := by | |
| have hBase : ordinalCode o ∈ K := by | |
| rw [ordinalSetProperty_eq_code_rank K conditions.1] | |
| exact (mem_ordinalCode_iff o (ZFSet.rank K)).2 bound | |
| obtain ⟨_, _, L, _, _, _, hL, limit, above⟩ := M.exists_groundCompressionStamps_below | |
| pairing unions powers separation collection infinite hK conditions hBase | |
| refine ⟨L, hL, limit, ?_⟩ | |
| simpa only [rank_ordinalCode] using ZFSet.rank_lt_of_mem above | |
| end GroundInaccessibleConditions | |
| namespace GroundCompressionUnionData | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m one T : ZFSet.{u}} | |
| variable (data : GroundCompressionUnionData M initial I Sources Reps G m one T) | |
| theorem carrier_rank_lt_inaccessible | |
| (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) | |
| {K P r μ : ZFSet.{u}} (hK : K ∈ M.sets) (conditions : GroundInaccessibleConditions M K) | |
| (constant : RecursiveTemplateConstancy P r μ initial) | |
| (points : ZFSet.rank initial.D < ZFSet.rank K) (indices : ZFSet.rank initial.J < ZFSet.rank K) | |
| (templates : ZFSet.rank P < ZFSet.rank K) (tasks : ZFSet.rank I < ZFSet.rank K) : | |
| ZFSet.rank (data.fields 0) < ZFSet.rank K := by | |
| let b := max (ZFSet.rank initial.D) (max (ZFSet.rank initial.J) (max (ZFSet.rank P) (ZFSet.rank I))) | |
| have bound : b < ZFSet.rank K := max_lt points (max_lt indices (max_lt templates tasks)) | |
| obtain ⟨L, hL, limit, above⟩ := conditions.exists_rank_limit_above | |
| pairing unions powers separation collection infinite hK bound | |
| have pointBound : ZFSet.rank initial.D ≤ b := le_max_left _ _ | |
| have indexBound : ZFSet.rank initial.J ≤ b := (le_max_left _ _).trans (le_max_right _ _) | |
| have templateBound : ZFSet.rank P ≤ b := | |
| (le_max_left _ _).trans ((le_max_right _ _).trans (le_max_right _ _)) | |
| have taskBound : ZFSet.rank I ≤ b := | |
| (le_max_right _ _).trans ((le_max_right _ _).trans (le_max_right _ _)) | |
| exact (data.carrier_rank_le constant limit (pointBound.trans_lt above) (indexBound.trans_lt above) | |
| (templateBound.trans_lt above) (taskBound.trans_lt above)).trans_lt (ZFSet.rank_lt_of_mem hL) | |
| end GroundCompressionUnionData | |
| namespace TransitiveGround | |
| variable (M : TransitiveGround.{u}) | |
| theorem initialCompressionUnion_rank_lt | |
| {P G r μ A Labels Sources Reps m one T K : ZFSet.{u}} | |
| (data : GroundCompressionUnionData M (M.initialCompressionData P G r μ A Labels) | |
| (M.groundEligibleTasks P G r μ A Labels) Sources Reps G m one T) | |
| (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) | |
| (hK : K ∈ M.sets) (conditions : GroundInaccessibleConditions M K) | |
| (points : ZFSet.rank P < ZFSet.rank K) (labels : ZFSet.rank Labels < ZFSet.rank K) : | |
| ZFSet.rank (data.fields 0) < ZFSet.rank K := by | |
| have limit := conditions.rank_isLimit pairing unions powers separation infinite hK | |
| have tasks := M.groundEligibleTasks_rank_lt limit G r μ A points labels | |
| exact data.carrier_rank_lt_inaccessible pairing unions powers separation collection infinite hK conditions | |
| (M.initialCompressionData_template_constancy P G r μ A Labels) points tasks points tasks | |
| end TransitiveGround | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |