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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 | import OAI.SetTheory.PartitionPrinciple.Geometry.ExternalOwnerCoordinateLaws
namespace OAI
namespace PartitionPilot.Forcing.GroundCompressionCompletion
universe u
variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}}
variable {I Sources Reps G m one T τ : ZFSet.{u}}
variable (completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ)
noncomputable def externalTileSlotPoints (P r μ A : ZFSet.{u}) : ZFSet.{u} :=
externalTileSlotSupports
(M.externalStarVertices G (externalChildLetters I P (externalRectangles P G r μ A))
(externalRequestCarrier I P (externalRectangles P G r μ A)) m one)
(M.externalGroupCarrier G (externalChildLetters I P (externalRectangles P G r μ A)) m one)
(externalRequestCarrier I P (externalRectangles P G r μ A))
(completion.algebra.union.fields 0) (completion.externalPointCarrier P r μ A)
(completion.externalPointAction P r μ A) (completion.localTileVertexImages P r μ A)
end PartitionPilot.Forcing.GroundCompressionCompletion
namespace PartitionPilot.Forcing.GroundExternalGeometryInputs
open SetRealization SetFormula Compression Compression.RecursiveMaps.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 (data : GroundExternalGeometryInputs M initial I Sources Reps G m inv one T τ P0 r0 μ0 A0 completion)
variable (group : GroupCode G m inv one) (oldAction : PosetActionCode G P0 m one r0 A0)
variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
variable (unions : Realize M.sets SetFormula.union Fin.elim0)
variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0) (hG : G ∈ M.sets)
local notation "family" => completion.externalFamily group oldAction data.original
local notation "action" => completion.algebra.union.action.toOrderAction group completion.algebra.union.meet
local notation "s" => completion.externalFamilyGroupPresentation group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
local notation "q" => data.externalPointPresentation group oldAction pairing unions powers separation infinite hG
local notation "vpres" => data.externalStarVertexPresentation group oldAction pairing unions powers separation infinite hG
local notation "Index" => ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0)
local notation "U" => completion.algebra.union.fields 0
local notation "Q" => completion.externalPointCarrier P0 r0 μ0 A0
local notation "vcode" => completion.externalFamilyVertexCode group oldAction data.original
local notation "slots" => completion.externalTileSlotPoints P0 r0 μ0 A0
include data group oldAction pairing unions powers separation infinite hG
theorem externalTileSlotPoints_entry :
letI := group.toGroup
∀ (w : External (SetMembers G) (family).Letters) (r : Index)
(i : Option ((family).Child r)) (y : (family).Node (action)),
ZFSet.pair (ZFSet.pair ((vpres).encode (.inr (w, r))) (vcode r i)) ((q).encode y) ∈ slots ↔
∃ x : SetMembers U,
((family).placedTile (action) w r).toFun (((family).flatVertex r i).toFun x) = y := by
let := group.toGroup
intro w r i y
have sc := completion.externalFamilyGroupPresentation_carrier group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
have qc := data.externalPointPresentation_carrier group oldAction pairing unions powers separation infinite hG
have vc := (data.externalStarPresentations_carriers group oldAction pairing unions powers separation infinite hG).1
have acted := (data.externalGeometry_entries group oldAction pairing unions powers separation infinite hG).2.2.1
rw [GroundCompressionCompletion.externalTileSlotPoints, pair_mem_externalTileSlotSupports]
constructor
· rintro ⟨_, _, _, w', _, r', _, x, hx, z, hz, equal, copied, moved⟩
change ZFSet.pair (natSetCode 1) (ZFSet.pair ((s).encode w) (externalRequestIndexCode r)) =
ZFSet.pair (natSetCode 1) (ZFSet.pair w' r') at equal
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 (ZFSet.pair_inj.1 equal).2
obtain ⟨z, rfl⟩ := ((q).mem_carrier z).1 (qc.symm ▸ hz)
let x' := SetMembers.ofMem x hx
have copied' := (data.localTileVertexImages_entry_iff group oldAction pairing unions powers separation infinite hG
r i x' z).1 (by simpa only [GroundCompressionCompletion.externalFamilyTileVertexPresentation_encode,
x', SetMembers.raw_ofMem] using copied)
have moved' := (acted w z y).1 moved
rw [← copied', ← (family).flatVertex_copy, (family).translate_tile, mul_one] at moved'
exact ⟨x', moved'⟩
· rintro ⟨x, rfl⟩
refine ⟨vc ▸ (vpres).encode_mem _, data.localTileVertexCode_mem group oldAction r i,
qc ▸ (q).encode_mem _, (s).encode w, sc ▸ (s).encode_mem _, externalRequestIndexCode r,
(externalRequestPresentation I P0 (externalRectangles P0 G r0 μ0 A0)).encode_mem _,
SetMembers.raw x, SetMembers.raw_mem x,
(q).encode (((family).placedTile (action) 1 r).toFun (((family).flatVertex r i).toFun x)),
qc ▸ (q).encode_mem _, rfl, ?_, ?_⟩
· apply (data.localTileVertexImages_entry_iff group oldAction pairing unions powers separation infinite hG r i x _).2
exact ((family).flatVertex_copy (action) 1 r i x).symm
· apply (acted w _ _).2
rw [(family).translate_tile, mul_one]
theorem externalTileSlotPoints_ground (hP0 : P0 ∈ M.sets) : slots ∈ M.sets := by
have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP0
have localGround := data.localTileGroundInputs group oldAction pairing unions powers separation infinite hG hP0
exact M.externalTileSlotSupports_mem pairing unions powers separation infinite
(data.externalStarCarriers_ground pairing unions powers separation hG).1 data.group.carrier_mem
data.words.requests_ground (completion.algebra.union.ground 0) geometry.carrier_ground
geometry.action_ground localGround.2.2.2.1
theorem externalTileSlotPoints_decode {v j x : ZFSet.{u}}
(supported : ZFSet.pair (ZFSet.pair v j) x ∈ slots) :
letI := group.toGroup
∃ (w : External (SetMembers G) (family).Letters) (r : Index)
(i : Option ((family).Child r)) (a : SetMembers U),
v = (vpres).encode (.inr (w, r)) ∧ j = vcode r i ∧
x = (q).encode (((family).placedTile (action) w r).toFun (((family).flatVertex r i).toFun a)) := by
let := group.toGroup
have sc := completion.externalFamilyGroupPresentation_carrier group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
have qc := data.externalPointPresentation_carrier group oldAction pairing unions powers separation infinite hG
have raw := (pair_mem_externalTileSlotSupports _ _ _ _ _ _ _ _ _ _).1 supported
obtain ⟨_, _, hx, w, hw, r, hr, a, _, z, _, equal, copied, _⟩ := raw
obtain ⟨w, rfl⟩ := ((s).mem_carrier w).1 (sc.symm ▸ hw)
obtain ⟨r, rfl⟩ := ((externalRequestPresentation I P0 (externalRectangles P0 G r0 μ0 A0)).mem_carrier r).1 hr
obtain ⟨x, rfl⟩ := ((q).mem_carrier x).1 (qc.symm ▸ hx)
have vertex : ZFSet.pair (ZFSet.pair (externalRequestIndexCode r) j) a ∈
completion.externalVertices P0 r0 μ0 A0 :=
((pair_mem_relationCompose _ _ _ _ _ _ _).1 copied).1
have shape := (triple_mem_externalTileVertices _ _ _ _ _ _).1 vertex
have range : j = natSetCode 0 ∨ j = natSetCode 1 ∨
(externalRequestIndexCode r ∈ ZFSet.prod {natSetCode 2} (externalRectangles P0 G r0 μ0 A0) ∧
j = natSetCode 2) := by tauto
obtain ⟨i, rfl⟩ := (completion.externalFamilyVertexCode_range group oldAction data.original r j).1 range
have ev : v = (vpres).encode (.inr (w, r)) := equal
subst v
obtain ⟨a, ha⟩ := (data.externalTileSlotPoints_entry group oldAction pairing unions powers separation infinite hG
w r i x).1 supported
exact ⟨w, r, i, a, rfl, rfl, congrArg (q).encode ha.symm⟩
theorem externalTileSlotPoints_owner_cover {x v : ZFSet.{u}}
(fresh : x ∉ completion.externalSeedCarrier P0 r0 μ0 A0)
(owned : ZFSet.pair x v ∈ completion.externalStarOwner P0 r0 μ0 A0) :
∃ j ∈ natSetCode 3, ZFSet.pair (ZFSet.pair v j) x ∈ slots := by
let := group.toGroup
obtain ⟨w, r, a, rfl, rfl, _⟩ := data.externalStarOwner_fresh_tile group oldAction pairing unions powers
separation infinite hG x v owned fresh
obtain ⟨i, b, rfl⟩ := (family).flatVertex_cover r a
exact ⟨vcode r i, data.localTileVertexCode_mem group oldAction r i,
(data.externalTileSlotPoints_entry group oldAction pairing unions powers separation infinite hG w r i _).2 ⟨b, rfl⟩⟩
theorem externalTileSlotPoints_lower {v j x y : ZFSet.{u}}
(hx : x ∈ Q) (lower : ZFSet.pair x y ∈ completion.externalPointOrder P0 r0 μ0 A0)
(supported : ZFSet.pair (ZFSet.pair v j) y ∈ slots) :
ZFSet.pair (ZFSet.pair v j) x ∈ slots := by
let := group.toGroup
obtain ⟨w, r, i, a, rfl, rfl, rfl⟩ :=
data.externalTileSlotPoints_decode group oldAction pairing unions powers separation infinite hG supported
have qc := data.externalPointPresentation_carrier group oldAction pairing unions powers separation infinite hG
obtain ⟨x, rfl⟩ := ((q).mem_carrier x).1 (qc.symm ▸ hx)
have below := ((data.externalGeometry_entries group oldAction pairing unions powers separation infinite hG).1 x _).1 lower
rw [(family).flatVertex_copy] at below
obtain ⟨b, equal⟩ := ((family).copy (action) ((family).flatWord w r i)).ideal a x below
apply (data.externalTileSlotPoints_entry group oldAction pairing unions powers separation infinite hG w r i x).2
exact ⟨b, ((family).flatVertex_copy (action) w r i b).trans equal⟩
end PartitionPilot.Forcing.GroundExternalGeometryInputs
end OAI
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