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e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundFinalInsertionExceptions
namespace OAI
section
namespace PartitionPilot.Forcing
universe u
namespace MeetEmbeddingCode
variable {P Q r R μ ν e F x x' : ZFSet.{u}}
variable (embedding : MeetEmbeddingCode P Q r R μ ν e)
variable (source : MeetOrderCode P r μ) (target : MeetOrderCode Q R ν)
include embedding source target in
theorem profilePositive_image_iff (subset : F ⊆ P) (image : ZFSet.pair x x' ∈ e) :
(∃ f' ∈ relationImage F Q e, ZFSet.pair f' x' ∈ R) ↔
∃ f ∈ F, ZFSet.pair f x ∈ r := by
constructor
· rintro ⟨f', member, lower⟩
obtain ⟨_, f, hf, entry⟩ := (mem_relationImage F Q e f').1 member
exact ⟨f, hf, (embedding.order_iff source target entry image).1 lower⟩
· rintro ⟨f, hf, lower⟩
obtain ⟨f', entry, _⟩ := embedding.injection.function.2 f (subset hf)
exact ⟨f', (mem_relationImage F Q e f').2
⟨(ZFSet.pair_mem_prod.1 (embedding.injection.function.1 entry)).2, f, hf, entry⟩,
embedding.monotone f x f' x' lower entry image⟩
end MeetEmbeddingCode
namespace CodedInsertionProfile
variable {P Q r R μ ν e F c E K c' E' K' x x' : ZFSet.{u}}
theorem at_embedding (profile : CodedInsertionProfile P r μ c E K F)
(embedding : MeetEmbeddingCode P Q r R μ ν e)
(source : MeetOrderCode P r μ) (target : MeetOrderCode Q R ν) (subset : F ⊆ P)
(imageC : ZFSet.pair c c' ∈ e) (imageE : ZFSet.pair E E' ∈ e)
(imageK : ZFSet.pair K K' ∈ e) (imageX : ZFSet.pair x x' ∈ e)
(above : ZFSet.pair E' x' ∈ R) :
((∃ f' ∈ relationImage F Q e, ZFSet.pair f' x' ∈ R) → ZFSet.pair K' x' ∈ R) ∧
(¬ (∃ f' ∈ relationImage F Q e, ZFSet.pair f' x' ∈ R) →
ZFSet.pair (ZFSet.pair K' x') c' ∈ ν) := by
have hx := (ZFSet.pair_mem_prod.1 (embedding.injection.function.1 imageX)).1
have old := profile x hx ((embedding.order_iff source target imageE imageX).1 above)
have positive := embedding.profilePositive_image_iff source target subset imageX
exact ⟨fun hp => embedding.monotone K x K' x' (old.1 (positive.1 hp)) imageK imageX,
fun hn => embedding.map_meet K x c K' x' c'
(old.2 (fun hp => hn (positive.2 hp))) imageK imageX imageC⟩
end CodedInsertionProfile
end PartitionPilot.Forcing
end
end OAI
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