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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 | import OAI.SetTheory.PartitionPrinciple.Geometry.CompressionExternalTileOverlaps
namespace OAI
namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
open Attachment CosetGeometry
universe u
variable {A J R : Type u} [Group J] {P : MeetOrder A}
variable (f : Family P R) (a : OrderAction J P)
theorem generator_root_of_tile (r : R) (j : f.Child r) (x y : A)
(equal : ((f.tile r).child j).toFun x = (f.tile r).root.toFun y) :
f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun x) =
(f.copy a 1).toFun y := by
rw [f.translate_copy, mul_one]
simpa only [one_mul] using f.transfer_child_root a 1 r j x y equal
theorem generator_root_meet_of_tile (r : R) (j : f.Child r) (x y z : A)
(equal : (f.tile r).order.meet (((f.tile r).child j).toFun x)
((f.tile r).root.toFun y) = (f.tile r).root.toFun z) :
(f.gluedOrder a).meet
(f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun x))
((f.copy a 1).toFun y) = (f.copy a 1).toFun z := by
rw [f.translate_copy, mul_one]
simpa only [one_mul] using f.transfer_child_root_meet a 1 r j x y z equal
theorem generator_children_meet_of_tile (r : R) (j k : f.Child r) (x y z : A)
(equal : (f.tile r).order.meet (((f.tile r).child j).toFun x)
(((f.tile r).child k).toFun y) = ((f.tile r).child j).toFun z) :
(f.gluedOrder a).meet
(f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun x))
(f.translate a (generator J f.Letters ⟨r, k⟩) ((f.copy a 1).toFun y)) =
f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun z) := by
simp only [f.translate_copy, mul_one]
simpa only [one_mul] using f.transfer_child_child_meet a 1 r j k x y z equal
theorem generator_fixes_ideal_of_tile (r : R) (j : f.Child r) (W : A)
(equal : ∀ x, P.le x W → ((f.tile r).child j).toFun x = (f.tile r).root.toFun x)
(z : f.Node a) (lower : (f.gluedOrder a).le z ((f.copy a 1).toFun W)) :
f.translate a (generator J f.Letters ⟨r, j⟩) z = z := by
obtain ⟨x, rfl⟩ := (f.copy a 1).ideal W z lower
exact f.generator_root_of_tile a r j x x (equal x (((f.copy a 1).le_iff x W).1 lower))
theorem identityCopy_preservesProfiles : PreservesProfiles (f.copy a 1).toMeetEmbedding :=
(f.copy a 1).preservesProfiles f.top f.greatest
end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
namespace PartitionPilot.Forcing.GroundCompressionCompletion
open SetRealization Compression Compression.RecursiveMaps Attachment CosetGeometry
universe u
variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}}
variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 : ZFSet.{u}}
variable (completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ)
variable (group : GroupCode G m inv one) (oldAction : PosetActionCode G P0 m one r0 A0)
variable (compression : CompressionOriginalData P0 r0 μ0 A0 G
(completion.algebra.union.fields 0) (completion.algebra.union.fields 1)
(completion.algebra.union.fields 2) (completion.algebra.union.fields 12))
local notation "U" => completion.algebra.union.fields 0
local notation "P" => completion.algebra.union.meet.toSmallMeetOrder
local notation "family" => completion.externalFamily group oldAction compression
local notation "action" => completion.algebra.union.action.toOrderAction group completion.algebra.union.meet
theorem external_separation_fixes (W : SetMembers P0) :
letI := group.toGroup
let w : SetMembers U := SetMembers.ofMem (SetMembers.raw W) (compression.subset (SetMembers.raw_mem W))
∀ z, ((family).gluedOrder (action)).le z (((family).copy (action) 1).toFun w) →
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inl W), ⟨()⟩⟩) z = z := by
let := group.toGroup
dsimp only
exact (family).generator_fixes_ideal_of_tile (action) (.inr (.inl W)) ⟨()⟩ _
(fun x lower => TileRequests.Request.separation_agrees _ _ _ x lower)
theorem external_separation_meet (W : SetMembers P0) (x y : SetMembers U)
(wx : ZFSet.pair (SetMembers.raw W) (SetMembers.raw x) ∈ completion.algebra.union.fields 1)
(wy : ZFSet.pair (SetMembers.raw W) (SetMembers.raw y) ∈ completion.algebra.union.fields 1) :
letI := group.toGroup
((family).gluedOrder (action)).meet
((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inl W), ⟨()⟩⟩)
(((family).copy (action) 1).toFun x)) (((family).copy (action) 1).toFun y) =
((family).copy (action) 1).toFun
(SetMembers.ofMem (SetMembers.raw W) (compression.subset (SetMembers.raw_mem W))) := by
let := group.toGroup
exact (family).generator_root_meet_of_tile (action) (.inr (.inl W)) ⟨()⟩ x y _
(BinaryTiles.separation_cross_meet P _ x y
(by simpa only [MeetOrderCode.toSmallMeetOrder, SetMembers.raw_ofMem] using wx)
(by simpa only [MeetOrderCode.toSmallMeetOrder, SetMembers.raw_ofMem] using wy))
theorem external_compression_restriction (i : SetMembers I) (x : SetMembers U) :
letI := group.toGroup
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inl i, ⟨()⟩⟩)
(((family).copy (action) 1).toFun x) =
((family).copy (action) 1).toFun ((completion.externalCopy i).toFun x).val := by
let := group.toGroup
exact (family).generator_root_of_tile (action) (.inl i) ⟨()⟩ x _
(TileRequests.Request.compression_agrees _ _ _ _ x)
theorem external_compression_top (i : SetMembers I) :
letI := group.toGroup
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inl i, ⟨()⟩⟩)
(((family).copy (action) 1).toFun (SetMembers.ofMem τ completion.top_mem)) =
((family).copy (action) 1).toFun (completion.externalCap i) := by
let := group.toGroup
exact (family).generator_root_of_tile (action) (.inl i) ⟨()⟩ _ _
(BinaryTiles.compression_top P _ _ (family).greatest _)
theorem external_rectangle_roots (z : SetMembers (externalRectangles P0 G r0 μ0 A0))
(x : SetMembers U) (below : (P).le x (completion.externalRectangle group oldAction compression z).C) :
letI := group.toGroup
let r := completion.externalRectangle group oldAction compression z
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩)
(((family).copy (action) 1).toFun x) =
((family).copy (action) 1).toFun (((action).iso r.first).toFun x) ∧
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩)
(((family).copy (action) 1).toFun x) =
((family).copy (action) 1).toFun (((action).iso r.second).toFun x) := by
let := group.toGroup
dsimp only
let r := (completion.externalRectangle group oldAction compression z).toInput
(family).top (family).greatest
exact ⟨(family).generator_root_of_tile (action) (.inr (.inr z)) ⟨false⟩ x _ (r.first_agrees x below),
(family).generator_root_of_tile (action) (.inr (.inr z)) ⟨true⟩ x _ (r.second_agrees x below)⟩
theorem external_rectangle_children (z : SetMembers (externalRectangles P0 G r0 μ0 A0))
(x : SetMembers U) (below : (P).le x (completion.externalRectangle group oldAction compression z).B) :
letI := group.toGroup
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩)
(((family).copy (action) 1).toFun x) =
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩)
(((family).copy (action) 1).toFun x) := by
let := group.toGroup
let r := (completion.externalRectangle group oldAction compression z).toInput
(family).top (family).greatest
simp only [TileRequests.Family.translate_copy, mul_one]
have equal := congrArg ((family).placedTile (action) 1 (.inr (.inr z))).toFun
((r.children_intersection x x).2 ⟨below, rfl⟩)
change ((family).placedTile (action) 1 (.inr (.inr z))).toFun
((((family).tile (.inr (.inr z))).child ⟨false⟩).toFun x) =
((family).placedTile (action) 1 (.inr (.inr z))).toFun
((((family).tile (.inr (.inr z))).child ⟨true⟩).toFun x) at equal
simpa only [one_mul] using
((family).child_identification (action) 1 (.inr (.inr z)) ⟨false⟩ x).symm.trans
(equal.trans ((family).child_identification (action) 1 (.inr (.inr z)) ⟨true⟩ x))
theorem external_rectangle_root_meets (z : SetMembers (externalRectangles P0 G r0 μ0 A0))
(x : SetMembers U) :
letI := group.toGroup
let r := completion.externalRectangle group oldAction compression z
((family).gluedOrder (action)).meet
((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩)
(((family).copy (action) 1).toFun r.D)) (((family).copy (action) 1).toFun x) =
((family).copy (action) 1).toFun ((P).meet (((action).iso r.first).toFun r.C) x) ∧
((family).gluedOrder (action)).meet
((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩)
(((family).copy (action) 1).toFun r.D)) (((family).copy (action) 1).toFun x) =
((family).copy (action) 1).toFun ((P).meet (((action).iso r.second).toFun r.C) x) := by
let := group.toGroup
dsimp only
let r := (completion.externalRectangle group oldAction compression z).toInput
(family).top (family).greatest
exact ⟨(family).generator_root_meet_of_tile (action) (.inr (.inr z)) ⟨false⟩ r.D x _ (r.first_meet_root x),
(family).generator_root_meet_of_tile (action) (.inr (.inr z)) ⟨true⟩ r.D x _ (r.second_meet_root x)⟩
theorem external_rectangle_children_meet (z : SetMembers (externalRectangles P0 G r0 μ0 A0)) :
letI := group.toGroup
let r := completion.externalRectangle group oldAction compression z
((family).gluedOrder (action)).meet
((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩)
(((family).copy (action) 1).toFun r.D))
((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩)
(((family).copy (action) 1).toFun r.D)) =
(family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩)
(((family).copy (action) 1).toFun r.B) := by
let := group.toGroup
dsimp only
let r := (completion.externalRectangle group oldAction compression z).toInput
(family).top (family).greatest
exact (family).generator_children_meet_of_tile (action) (.inr (.inr z)) ⟨false⟩ ⟨true⟩ r.D r.D r.B r.children_meet
end PartitionPilot.Forcing.GroundCompressionCompletion
end OAI
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