File size: 15,207 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 | import OAI.SetTheory.PartitionPrinciple.Forcing.RootedSupportOwners
import OAI.SetTheory.PartitionPrinciple.Geometry.GroundExternalLocalHeightFamilies
import OAI.SetTheory.PartitionPrinciple.Geometry.GroundExternalStarOverlaps
namespace OAI
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 "c" => data.externalCosetPresentation group oldAction pairing unions powers separation infinite hG
local notation "vpres" => data.externalStarVertexPresentation group oldAction pairing unions powers separation infinite hG
local notation "Requests" => externalRequestCarrier I P0 (externalRectangles P0 G r0 μ0 A0)
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 "E" => M.externalGroupCarrier G (externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0)) m one
local notation "C" => M.externalCosetCarrier G (externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0)) m one
include data group oldAction pairing unions powers separation infinite hG
theorem externalStarSupports_iff :
letI := group.toGroup
∀ (v : TileStars.Vertex (SetMembers G) (family).Letters Index) (y : (family).Node (action)),
ZFSet.pair ((vpres).encode v) ((q).encode y) ∈ completion.externalStarSupports P0 r0 μ0 A0 ↔
∃ x : (family).Space v, (((family).diagram (action)).vertex v).toFun x = y := by
let := group.toGroup
intro v 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 cc := data.externalCosetPresentation_carrier group oldAction pairing unions powers separation infinite hG
have vc := (data.externalStarPresentations_carriers group oldAction pairing unions powers separation infinite hG).1
have cosets := data.externalCosetClassifier_entry_iff group oldAction pairing unions powers separation infinite hG
have copied := data.externalPointClassifier_entry_iff group oldAction pairing unions powers separation infinite hG
have acted := (data.externalGeometry_entries group oldAction pairing unions powers separation infinite hG).2.2.1
have localImage := data.localTileCarrier_image group oldAction pairing unions powers separation infinite hG
rw [GroundCompressionCompletion.externalStarSupports, pair_mem_externalStarSupportGraph]
constructor
· rintro ⟨_, _, body⟩
cases v with
| inl coset =>
rcases body with cosetBody | tileBody
· obtain ⟨c', _, w, hw, x, hx, equal, classified, image⟩ := cosetBody
change ZFSet.pair (natSetCode 0) ((c).encode coset) = ZFSet.pair (natSetCode 0) c' at equal
obtain rfl : (c).encode coset = c' := (ZFSet.pair_inj.1 equal).2
obtain ⟨w, rfl⟩ := ((s).mem_carrier w).1 (sc.symm ▸ hw)
have atCoset := (cosets w coset).1 classified
obtain rfl := atCoset
let x' := SetMembers.ofMem x hx
refine ⟨((action).chart w).toFun x', ?_⟩
exact (copied w x' y).1 (by simpa only [x', SetMembers.raw_ofMem] using image)
· obtain ⟨_, _, _, _, _, _, equal, _⟩ := tileBody
change ZFSet.pair (natSetCode 0) _ = ZFSet.pair (natSetCode 1) _ at equal
exact False.elim (Nat.zero_ne_one (natSetCode_injective (ZFSet.pair_inj.1 equal).1))
| inr wr =>
rcases wr with ⟨w, r⟩
rcases body with cosetBody | tileBody
· obtain ⟨_, _, _, _, _, _, equal, _⟩ := cosetBody
change ZFSet.pair (natSetCode 1) _ = ZFSet.pair (natSetCode 0) _ at equal
exact False.elim (Nat.zero_ne_one (natSetCode_injective (ZFSet.pair_inj.1 equal).1.symm))
· obtain ⟨w', _, r', _, x, hx, equal, localEntry, actionEntry⟩ := tileBody
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 ⟨x', rfl⟩ := (localImage r x).1 ((mem_relationFiber _ _ _ _).2 ⟨hx, localEntry⟩) have translated := (acted w (((family).placedTile (action) 1 r).toFun x') y).1 actionEntry
rw [(family).translate_tile, mul_one] at translated
exact ⟨x', translated⟩ · rintro ⟨x, image⟩ refine ⟨vc ▸ (vpres).encode_mem v, qc ▸ (q).encode_mem y, ?_⟩ cases v with | inl coset => obtain ⟨w, rfl⟩ := representative_exists (SetMembers G) (family).Letters coset apply Or.inl let z := ((action).chart w).inverse x refine ⟨(c).encode (point (SetMembers G) (family).Letters w), cc ▸ (c).encode_mem _, (s).encode w, sc ▸ (s).encode_mem w, SetMembers.raw z, SetMembers.raw_mem z, rfl, (cosets _ _).2 rfl, (copied w z y).2 ?_⟩ change (((family).diagram (action)).vertex (.inl (point (SetMembers G) (family).Letters w))).toFun (((action).chart w).toFun (((action).chart w).inverse x)) = y erw [((action).chart w).right_inverse]
exact image
| inr wr =>
rcases wr with ⟨w, r⟩
apply Or.inr
let z := ((family).placedTile (action) 1 r).toFun x
have localEntry := (mem_relationFiber _ _ _ _).1 ((localImage r ((q).encode z)).2 ⟨x, rfl⟩)
refine ⟨(s).encode w, sc ▸ (s).encode_mem w, externalRequestIndexCode r,
(externalRequestPresentation I P0 (externalRectangles P0 G r0 μ0 A0)).encode_mem r,
(q).encode z, qc ▸ (q).encode_mem z, rfl, localEntry.2, (acted w z y).2 ?_⟩
change (family).translate (action) w (((family).placedTile (action) 1 r).toFun x) = y
erw [(family).translate_tile, mul_one]
exact image
theorem externalStarSupport_iff :
letI := group.toGroup
∀ (v : TileStars.Vertex (SetMembers G) (family).Letters Index) (y : (family).Node (action)),
(q).encode y ∈ completion.externalStarSupport P0 r0 μ0 A0 ((vpres).encode v) ↔
∃ x : (family).Space v, (((family).diagram (action)).vertex v).toFun x = y := by
let := group.toGroup
intro v y
rw [GroundCompressionCompletion.externalStarSupport, mem_relationFiber, data.externalStarSupports_iff group oldAction pairing unions powers separation infinite hG]
exact and_iff_right (data.externalPointPresentation_carrier group oldAction pairing unions powers separation infinite hG ▸ (q).encode_mem y)
end PartitionPilot.Forcing.GroundExternalGeometryInputs
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 "c" => data.externalCosetPresentation group oldAction pairing unions powers separation infinite hG
local notation "vpres" => data.externalStarVertexPresentation group oldAction pairing unions powers separation infinite hG
local notation "Requests" => externalRequestCarrier I P0 (externalRectangles P0 G r0 μ0 A0)
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 "E" => M.externalGroupCarrier G (externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0)) m one
local notation "C" => M.externalCosetCarrier G (externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0)) m one
include data group oldAction pairing unions powers separation infinite hG
theorem externalStarOverlaps_iff :
letI := group.toGroup
∀ (v : TileStars.Vertex (SetMembers G) (family).Letters Index) (y : (family).Node (action)),
ZFSet.pair ((vpres).encode v) ((q).encode y) ∈ completion.externalStarOverlaps P0 r0 μ0 A0 ↔
(∃ x : (family).Space v, (((family).diagram (action)).vertex v).toFun x = y) ∧
(∃ x : (family).Space (TileStars.parent (SetMembers G) (family).Letters Index Sigma.fst v),
(((family).diagram (action)).vertex (TileStars.parent (SetMembers G) (family).Letters Index Sigma.fst v)).toFun x = y) := by
let := group.toGroup
intro v y
have parent := (data.externalStarParent_iff group oldAction pairing unions powers separation infinite hG v _).2 rfl
have function := (data.externalStarParent_ground_function group oldAction pairing unions powers separation infinite hG).2
rw [GroundCompressionCompletion.externalStarOverlaps, pair_mem_parentSupportGraph_of_parent function parent, data.externalStarSupports_iff group oldAction pairing unions powers separation infinite hG, data.externalStarSupports_iff group oldAction pairing unions powers separation infinite hG]
exact and_iff_right (data.externalPointPresentation_carrier group oldAction pairing unions powers separation infinite hG ▸ (q).encode_mem y)
theorem externalStarFresh_iff :
letI := group.toGroup
∀ (v : TileStars.Vertex (SetMembers G) (family).Letters Index) (y : (family).Node (action)),
ZFSet.pair ((vpres).encode v) ((q).encode y) ∈ completion.externalStarFresh P0 r0 μ0 A0 ↔
(∃ x : (family).Space v, (((family).diagram (action)).vertex v).toFun x = y) ∧
¬ (∃ x : (family).Space (TileStars.parent (SetMembers G) (family).Letters Index Sigma.fst v),
(((family).diagram (action)).vertex (TileStars.parent (SetMembers G) (family).Letters Index Sigma.fst v)).toFun x = y) := by
let := group.toGroup
intro v y
rw [GroundCompressionCompletion.externalStarFresh, pair_mem_freshSupportGraph]
change (ZFSet.pair ((vpres).encode v) ((q).encode y) ∈ completion.externalStarSupports P0 r0 μ0 A0 ∧
ZFSet.pair ((vpres).encode v) ((q).encode y) ∉ completion.externalStarOverlaps P0 r0 μ0 A0) ↔ _
rw [data.externalStarSupports_iff group oldAction pairing unions powers separation infinite hG, data.externalStarOverlaps_iff group oldAction pairing unions powers separation infinite hG]
tauto
theorem externalStarOverlaps_vertex_iff :
letI := group.toGroup
∀ (v : TileStars.Vertex (SetMembers G) (family).Letters Index)
(nonzero : ((family).diagram (action)).scheme.depth v ≠ 0) (x : (family).Space v),
ZFSet.pair ((vpres).encode v) ((q).encode ((((family).diagram (action)).vertex v).toFun x)) ∈
completion.externalStarOverlaps P0 r0 μ0 A0 ↔
(((family).diagram (action)).scheme.order v).le x (((family).diagram (action)).scheme.childAnchor v nonzero) := by
let := group.toGroup
intro v nonzero x
erw [data.externalStarOverlaps_iff group oldAction pairing unions powers separation infinite hG]
exact ⟨fun h => (((family).diagram (action)).scheme.supports_parent_iff v nonzero x).1 h.2,
fun h => ⟨⟨x, rfl⟩, (((family).diagram (action)).scheme.supports_parent_iff v nonzero x).2 h⟩⟩
theorem externalStarFresh_vertex_iff :
letI := group.toGroup
∀ (v : TileStars.Vertex (SetMembers G) (family).Letters Index)
(nonzero : ((family).diagram (action)).scheme.depth v ≠ 0) (x : (family).Space v),
ZFSet.pair ((vpres).encode v) ((q).encode ((((family).diagram (action)).vertex v).toFun x)) ∈
completion.externalStarFresh P0 r0 μ0 A0 ↔
¬ (((family).diagram (action)).scheme.order v).le x (((family).diagram (action)).scheme.childAnchor v nonzero) := by
let := group.toGroup
intro v nonzero x
rw [GroundCompressionCompletion.externalStarFresh, pair_mem_freshSupportGraph]
change (ZFSet.pair ((vpres).encode v) ((q).encode ((((family).diagram (action)).vertex v).toFun x)) ∈
completion.externalStarSupports P0 r0 μ0 A0 ∧
ZFSet.pair ((vpres).encode v) ((q).encode ((((family).diagram (action)).vertex v).toFun x)) ∉
completion.externalStarOverlaps P0 r0 μ0 A0) ↔ _
erw [data.externalStarSupports_iff group oldAction pairing unions powers separation infinite hG, data.externalStarOverlaps_vertex_iff group oldAction pairing unions powers separation infinite hG]
exact and_iff_right ⟨x, rfl⟩
theorem externalStarFresh_root :
letI := group.toGroup
∀ y : (family).Node (action),
ZFSet.pair ((vpres).encode (TileStars.root (SetMembers G) (family).Letters Index)) ((q).encode y) ∉
completion.externalStarFresh P0 r0 μ0 A0 := by
let := group.toGroup
intro y entry
have parts := (data.externalStarFresh_iff group oldAction pairing unions powers separation infinite hG _ y).1 entry
erw [TileStars.parent_root] at parts
exact parts.2 parts.1
end PartitionPilot.Forcing.GroundExternalGeometryInputs
end OAI
|