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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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.SetTheory.PartitionPrinciple.Forcing.ExternalAction
namespace OAI
section
namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.AllGeometry
open Attachment CosetGeometry
universe u
variable {A J : Type u} [Group J] {P : MeetOrder A}
variable (T : A) (greatest : ∀ x, P.le x T) (a : OrderAction J P)
noncomputable def rootEmbedding : IdealEmbedding P ((family T greatest).gluedOrder a) :=
(family T greatest).copy a 1
noncomputable def letter (r : Request P) (j : r.Child) :
MeetIso ((family T greatest).gluedOrder a) ((family T greatest).gluedOrder a) :=
((family T greatest).externalAction a).iso (generator J (Letters T greatest) ⟨r, j⟩)
theorem letter_root (r : Request P) (j : r.Child) (x : A) :
(letter T greatest a r j).toFun ((rootEmbedding T greatest a).toFun x) =
((family T greatest).copy a (generator J (Letters T greatest) ⟨r, j⟩)).toFun x :=
(family T greatest).generator_on_root a ⟨r, j⟩ x
theorem separation_letter_fixes (W x : A) (below : P.le x W) :
(letter T greatest a (.separation W) ⟨()⟩).toFun ((rootEmbedding T greatest a).toFun x) =
(rootEmbedding T greatest a).toFun x := by
erw [letter_root]
simpa only [one_mul, rootEmbedding] using separation_fixes T greatest a 1 W x below
theorem separation_letter_meet (W D X : A) (belowD : P.le W D) (belowX : P.le W X) :
((family T greatest).gluedOrder a).meet
((letter T greatest a (.separation W) ⟨()⟩).toFun ((rootEmbedding T greatest a).toFun D))
((rootEmbedding T greatest a).toFun X) = (rootEmbedding T greatest a).toFun W := by
erw [letter_root]
simpa only [one_mul, rootEmbedding] using separation_meet T greatest a 1 W D X belowD belowX
theorem compression_letter_restriction (K : A) (S : MeetIso P (idealOrder P K)) (x : A) :
(letter T greatest a (.compression K S) ⟨()⟩).toFun ((rootEmbedding T greatest a).toFun x) =
(rootEmbedding T greatest a).toFun (S.toFun x).val := by
erw [letter_root]
simpa only [one_mul, rootEmbedding] using compression_restriction T greatest a 1 K S x
theorem compression_letter_top (K : A) (S : MeetIso P (idealOrder P K)) :
(letter T greatest a (.compression K S) ⟨()⟩).toFun ((rootEmbedding T greatest a).toFun T) =
(rootEmbedding T greatest a).toFun K := by
erw [letter_root]
simpa only [one_mul, rootEmbedding] using compression_top T greatest a 1 K S
theorem compression_letter_agrees (K C : A) (S : MeetIso P (idealOrder P K)) (gamma : MeetIso P P)
(agreement : ∀ x, P.le x C → (S.toFun x).val = gamma.toFun x) (x : A) (below : P.le x C) :
(letter T greatest a (.compression K S) ⟨()⟩).toFun ((rootEmbedding T greatest a).toFun x) =
(rootEmbedding T greatest a).toFun (gamma.toFun x) := by
rw [compression_letter_restriction, agreement x below]
theorem rectangle_letters_root (r : Rectangle.Input P) (x : A) (below : P.le x r.C) :
(letter T greatest a (.rectangle r) ⟨false⟩).toFun ((rootEmbedding T greatest a).toFun x) =
(rootEmbedding T greatest a).toFun (r.g1.toFun x) ∧
(letter T greatest a (.rectangle r) ⟨true⟩).toFun ((rootEmbedding T greatest a).toFun x) =
(rootEmbedding T greatest a).toFun (r.g2.toFun x) := by
constructor
· erw [letter_root]
simpa only [one_mul, rootEmbedding] using rectangle_first_agrees T greatest a 1 r x below
· erw [letter_root]
simpa only [one_mul, rootEmbedding] using rectangle_second_agrees T greatest a 1 r x below
theorem rectangle_letters_agree (r : Rectangle.Input P) (x : A) (below : P.le x r.B) :
(letter T greatest a (.rectangle r) ⟨false⟩).toFun ((rootEmbedding T greatest a).toFun x) =
(letter T greatest a (.rectangle r) ⟨true⟩).toFun ((rootEmbedding T greatest a).toFun x) := by
erw [letter_root, letter_root]
simpa only [one_mul] using (rectangle_children_intersection T greatest a 1 r x x).mpr ⟨below, rfl⟩
theorem rectangle_letters_meet_root (r : Rectangle.Input P) (x : A) :
((family T greatest).gluedOrder a).meet
((letter T greatest a (.rectangle r) ⟨false⟩).toFun ((rootEmbedding T greatest a).toFun r.D))
((rootEmbedding T greatest a).toFun x) =
(rootEmbedding T greatest a).toFun (P.meet (r.g1.toFun r.C) x) ∧
((family T greatest).gluedOrder a).meet
((letter T greatest a (.rectangle r) ⟨true⟩).toFun ((rootEmbedding T greatest a).toFun r.D))
((rootEmbedding T greatest a).toFun x) =
(rootEmbedding T greatest a).toFun (P.meet (r.g2.toFun r.C) x) := by
constructor
· erw [letter_root]
simpa only [one_mul, rootEmbedding] using rectangle_first_meet_root T greatest a 1 r x
· erw [letter_root]
simpa only [one_mul, rootEmbedding] using rectangle_second_meet_root T greatest a 1 r x
theorem rectangle_letters_meet (r : Rectangle.Input P) :
((family T greatest).gluedOrder a).meet
((letter T greatest a (.rectangle r) ⟨false⟩).toFun ((rootEmbedding T greatest a).toFun r.D))
((letter T greatest a (.rectangle r) ⟨true⟩).toFun ((rootEmbedding T greatest a).toFun r.D)) =
(letter T greatest a (.rectangle r) ⟨false⟩).toFun ((rootEmbedding T greatest a).toFun r.B) ∧
(letter T greatest a (.rectangle r) ⟨false⟩).toFun ((rootEmbedding T greatest a).toFun r.B) =
(letter T greatest a (.rectangle r) ⟨true⟩).toFun ((rootEmbedding T greatest a).toFun r.B) := by
constructor
· erw [letter_root, letter_root, letter_root]
simpa only [one_mul] using rectangle_children_meet T greatest a 1 r
· exact rectangle_letters_agree T greatest a r r.B (P.refl _)
end PartitionPilot.Compression.RecursiveMaps.TileRequests.AllGeometry
end
end OAI
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