File size: 2,946 Bytes
3277efa | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | 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
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