openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Transfinite /ExternalRequestRankBounds.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundInitialExternalGroup | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| open SetRealization | |
| universe u | |
| variable {o : Ordinal.{u}} (limit : Order.IsSuccLimit o) | |
| include limit | |
| theorem externalRectangleBound_rank_lt {P G : ZFSet.{u}} | |
| (points : ZFSet.rank P < o) (actors : ZFSet.rank G < o) : | |
| ZFSet.rank (externalRectangleBound P G) < o := | |
| rank_product_lt limit points (rank_product_lt limit points | |
| (rank_product_lt limit points (rank_product_lt limit actors actors))) | |
| theorem externalRectangles_rank_lt {P G : ZFSet.{u}} (r μ A : ZFSet.{u}) | |
| (points : ZFSet.rank P < o) (actors : ZFSet.rank G < o) : | |
| ZFSet.rank (externalRectangles P G r μ A) < o := | |
| (ZFSet.rank_mono (show externalRectangles P G r μ A ⊆ externalRectangleBound P G from | |
| fun _ hx => (ZFSet.mem_sep.1 hx).1)).trans_lt | |
| (externalRectangleBound_rank_lt limit points actors) | |
| theorem externalRequestCarrier_rank_lt {I P Rectangles : ZFSet.{u}} | |
| (indices : ZFSet.rank I < o) (points : ZFSet.rank P < o) | |
| (rectangles : ZFSet.rank Rectangles < o) : | |
| ZFSet.rank (externalRequestCarrier I P Rectangles) < o := by | |
| have tag := fun n => rank_singleton_lt limit (rank_natSetCode_lt limit n) | |
| rw [externalRequestCarrier, ZFSet.rank_union, ZFSet.rank_union] | |
| exact max_lt (max_lt (rank_product_lt limit (tag 0) indices) | |
| (rank_product_lt limit (tag 1) points)) (rank_product_lt limit (tag 2) rectangles) | |
| theorem externalChildLetters_rank_lt {I P Rectangles : ZFSet.{u}} | |
| (indices : ZFSet.rank I < o) (points : ZFSet.rank P < o) | |
| (rectangles : ZFSet.rank Rectangles < o) : | |
| ZFSet.rank (externalChildLetters I P Rectangles) < o := by | |
| have tag := fun n => rank_singleton_lt limit (rank_natSetCode_lt limit n) | |
| rw [externalChildLetters, ZFSet.rank_union] | |
| exact max_lt | |
| (rank_product_lt limit (externalRequestCarrier_rank_lt limit indices points rectangles) (tag 0)) | |
| (rank_product_lt limit (rank_product_lt limit (tag 2) rectangles) (tag 1)) | |
| theorem externalGeneratorAlphabet_rank_lt {G S : ZFSet.{u}} | |
| (actors : ZFSet.rank G < o) (letters : ZFSet.rank S < o) : | |
| ZFSet.rank (externalGeneratorAlphabet G S) < o := by | |
| rw [externalGeneratorAlphabet, ZFSet.rank_union] | |
| exact max_lt | |
| (rank_product_lt limit (rank_singleton_lt limit (rank_natSetCode_lt limit 0)) actors) | |
| (rank_product_lt limit (rank_singleton_lt limit (rank_natSetCode_lt limit 1)) | |
| (rank_product_lt limit letters (rank_natSetCode_lt limit 2))) | |
| theorem externalRequestAlphabet_rank_lt {I P G : ZFSet.{u}} (r μ A : ZFSet.{u}) | |
| (indices : ZFSet.rank I < o) (points : ZFSet.rank P < o) (actors : ZFSet.rank G < o) : | |
| ZFSet.rank (externalRequestAlphabet I P G r μ A) < o := | |
| externalGeneratorAlphabet_rank_lt limit actors | |
| (externalChildLetters_rank_lt limit indices points (externalRectangles_rank_lt limit r μ A points actors)) | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |