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3277efa | 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 | import OAI.SetTheory.PartitionPrinciple.Definability.ExternalStarCarrierRanks
import OAI.SetTheory.PartitionPrinciple.Transfinite.ExternalBirthLayerSeparation
namespace OAI
namespace PartitionPilot.Forcing.GroundExternalBirthData
open SetFormula SetRealization 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 oldTimes oldBirth : ZFSet.{u}}
variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ}
variable (births : GroundExternalBirthData M initial I Sources Reps G m one T τ P0 r0 μ0 A0 oldTimes oldBirth completion)
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) (hP0 : P0 ∈ M.sets)
local notation "family" => completion.externalFamily group oldAction data.original
local notation "s" => completion.externalFamilyGroupPresentation group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
local notation "vpres" => data.externalStarVertexPresentation group oldAction pairing unions powers separation infinite hG
local notation "U" => completion.algebra.union.fields 0
local notation "B" => completion.externalPointCarrier P0 r0 μ0 A0
local notation "P" => completion.externalSeedCarrier P0 r0 μ0 A0
local notation "owner" => completion.externalStarOwner P0 r0 μ0 A0
local notation "levels" => completion.externalOwnerLevels P0 r0 μ0 A0
include births data group oldAction pairing unions powers separation infinite hG hP0
variable {HQs Hrs Hμs HTime : ZFSet.{u}}
variable (historySpaces : FunctionGraphOn HTime HQs)
variable (historyOrders : FunctionGraphOn HTime Hrs)
variable (historyMeets : FunctionGraphOn HTime Hμs)
variable (source : ZFSet.pair oldTimes (completion.algebra.union.fields 0) ∈ HQs ∧
ZFSet.pair oldTimes (completion.algebra.union.fields 1) ∈ Hrs ∧
ZFSet.pair oldTimes (completion.algebra.union.fields 2) ∈ Hμs)
include historySpaces historyOrders historyMeets source
theorem layerSeparatedCode (k : Nat) :
SeparatedLayerCode HQs Hrs Hμs B (completion.externalPointOrder P0 r0 μ0 A0)
(completion.externalPointMeet P0 r0 μ0 A0) births.birth (compressionStamp oldTimes k)
(births.layerOwners k) (births.layerIndices k) (births.layerMaps k)
(M.groundRelations U B) (births.layerComponents k) (births.layerFiber k) := by
refine ⟨births.layerIndices_function k,
(births.layerMaps_spec data group oldAction pairing unions powers separation infinite hG hP0 k).1,
rfl, births.layerComponents_function data group oldAction pairing unions powers separation infinite hG hP0 k,
?_, ?_, ?_⟩
· intro i _ t _ e Q R V indexEntry mapEntry spaceEntry orderEntry meetEntry
obtain rfl := ((pair_mem_constantFamilyGraph _ _ _ _).1 indexEntry).2
have qeq := historySpaces.unique spaceEntry source.1
have req := historyOrders.unique orderEntry source.2.1
have veq := historyMeets.unique meetEntry source.2.2
subst Q R V
exact births.layerMaps_embedding data group oldAction pairing unions powers separation infinite hG hP0 k mapEntry
· intro i _ t _ e Q x _ indexEntry mapEntry spaceEntry componentEntry
obtain rfl := ((pair_mem_constantFamilyGraph _ _ _ _).1 indexEntry).2
have qeq := historySpaces.unique spaceEntry source.1
subst Q
have owned := (pair_mem_graphRestriction.1 componentEntry).1
have hx := (pair_mem_graphRestriction.1 componentEntry).2
obtain ⟨_, _, _, _, covered⟩ := births.layerMaps_copy data group oldAction pairing unions powers separation infinite hG hP0 k hx owned mapEntry
exact covered
· intro x hx y hy i _ j _ first second different
exact Or.inl (births.layerComponents_earlier_meet data group oldAction pairing unions powers separation infinite hG hP0 k hx hy first second different)
theorem cutoffLayerSeparatedCode (k n : Nat) (before : k ≤ n) :
SeparatedLayerCode HQs Hrs Hμs (births.cutoff n) (births.cutoffOrder n)
(births.cutoffMeet n) (births.cutoffBirth n) (compressionStamp oldTimes k)
(births.layerOwners k) (births.layerIndices k) (births.layerMaps k)
(M.groundRelations U B) (births.layerComponents k) (births.layerFiber k) := by
have full := births.layerSeparatedCode data group oldAction pairing unions powers separation infinite hG hP0
historySpaces historyOrders historyMeets source k
refine ⟨full.indices, full.maps, births.layerFiber_cutoff k n before, full.components, ?_, ?_, ?_⟩
· intro i _ t _ e Q R V indexEntry mapEntry spaceEntry orderEntry meetEntry
obtain rfl := ((pair_mem_constantFamilyGraph _ _ _ _).1 indexEntry).2
have qeq := historySpaces.unique spaceEntry source.1
have req := historyOrders.unique orderEntry source.2.1
have veq := historyMeets.unique meetEntry source.2.2
subst Q R V
exact births.layerMaps_cutoff_embedding data group oldAction pairing unions powers separation infinite hG hP0 k n before mapEntry
· intro i hi t ht e Q x hx indexEntry mapEntry spaceEntry componentEntry
obtain ⟨_, u, hu, copied⟩ := (mem_relationImage _ _ _ _).1
(full.cover i hi t ht e Q x (births.cutoff_subset n hx) indexEntry mapEntry spaceEntry componentEntry)
exact (mem_relationImage _ _ _ _).2 ⟨hx, u, hu, copied⟩
· intro x hx y hy i _ j _ first second different
have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP0
obtain ⟨z, a, meetEntry, born, earlier⟩ := births.layerComponents_earlier_meet data group oldAction pairing unions powers separation infinite hG hP0 k hx hy first second different
have hxCut := ((mem_layerBirthFiber _ _ _ _).1 (births.layerFiber_cutoff k n before ▸ hx)).1
have hyCut := ((mem_layerBirthFiber _ _ _ _).1 (births.layerFiber_cutoff k n before ▸ hy)).1
have hz := (ZFSet.pair_mem_prod.1 (geometry.meet.operation.1 meetEntry)).2
have hzCut := births.cutoff_lower data group oldAction pairing unions powers separation infinite hG hP0 n z hz x hxCut
(geometry.meet.lower_left x y z meetEntry)
exact Or.inl ⟨z, a, pair_mem_graphRestriction.2 ⟨meetEntry, ZFSet.pair_mem_prod.2 ⟨hxCut, hyCut⟩⟩,
pair_mem_graphRestriction.2 ⟨born, hzCut⟩, earlier⟩
theorem layerWitness_spec (k : Nat) {N : ZFSet.{u}} :
births.layerWitness k ∈ M.sets ∧
GrowthLayerWitnessCode HQs Hrs Hμs B (completion.externalPointOrder P0 r0 μ0 A0)
(completion.externalPointMeet P0 r0 μ0 A0) births.birth (compressionStamp oldTimes k) N
(births.layerWitness k) :=
⟨births.layerWitness_ground data group oldAction pairing unions powers separation infinite hG hP0 k,
Or.inr (Or.inr ⟨_, _, _, _, _, _, rfl,
births.layerSeparatedCode data group oldAction pairing unions powers separation infinite hG hP0
historySpaces historyOrders historyMeets source k⟩)⟩
theorem cutoffLayerWitness_spec (k n : Nat) (before : k ≤ n) {N : ZFSet.{u}} :
births.layerWitness k ∈ M.sets ∧
GrowthLayerWitnessCode HQs Hrs Hμs (births.cutoff n) (births.cutoffOrder n)
(births.cutoffMeet n) (births.cutoffBirth n) (compressionStamp oldTimes k) N
(births.layerWitness k) :=
⟨births.layerWitness_ground data group oldAction pairing unions powers separation infinite hG hP0 k,
Or.inr (Or.inr ⟨_, _, _, _, _, _, rfl,
births.cutoffLayerSeparatedCode data group oldAction pairing unions powers separation infinite hG hP0
historySpaces historyOrders historyMeets source k n before⟩)⟩
end PartitionPilot.Forcing.GroundExternalBirthData
namespace PartitionPilot.Forcing.GroundExternalBirthData
open SetFormula SetRealization
universe u
variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}}
variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 oldTimes oldBirth : ZFSet.{u}}
variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ}
variable (births : GroundExternalBirthData M initial I Sources Reps G m one T τ P0 r0 μ0 A0 oldTimes oldBirth completion)
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) (action : 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) (hP0 : P0 ∈ M.sets)
variable {o : Ordinal.{u}} (limit : Order.IsSuccLimit o)
variable (omega : ZFSet.rank ZFSet.omega < o) (indices : ZFSet.rank I < o)
variable (points : ZFSet.rank P0 < o) (actors : ZFSet.rank G < o)
variable (completed : ZFSet.rank (completion.algebra.union.fields 0) < o)
variable (time : ZFSet.rank oldTimes < o)
include data group action pairing unions powers separation infinite hG hP0
limit omega indices points actors completed time
theorem layerWitness_rank_lt (k : Nat) : ZFSet.rank (births.layerWitness k) < o := by
have carrier := completion.externalPointCarrier_rank_lt_of_inputs limit r0 μ0 A0
omega indices points actors completed
have vertices := M.externalStarVertices_rank_lt_of_inputs limit r0 μ0 A0 m one
omega indices points actors
have fiber := (ZFSet.rank_mono (fun x hx => (births.mem_layerFiber k x).1 hx |>.1)).trans_lt carrier
have owners := (ZFSet.rank_mono (fun v hv => (births.layerOwners_cases (k := k) hv).1)).trans_lt vertices
have bound := M.groundRelations_rank_lt limit completed carrier
exact separatedLayerWitness_rank_lt limit owners
(rank_constantFamilyGraph_lt limit owners time)
(rank_functionGraph_lt limit
(births.layerMaps_spec data group action pairing unions powers separation infinite hG hP0 k).1
owners bound) bound
(rank_functionGraph_lt limit
(births.layerComponents_function data group action pairing unions powers separation infinite hG hP0 k)
fiber owners) fiber
variable {HQs Hrs Hμs HTime : ZFSet.{u}}
variable (historySpaces : FunctionGraphOn HTime HQs)
variable (historyOrders : FunctionGraphOn HTime Hrs)
variable (historyMeets : FunctionGraphOn HTime Hμs)
variable (source : ZFSet.pair oldTimes (completion.algebra.union.fields 0) ∈ HQs ∧
ZFSet.pair oldTimes (completion.algebra.union.fields 1) ∈ Hrs ∧
ZFSet.pair oldTimes (completion.algebra.union.fields 2) ∈ Hμs)
include historySpaces historyOrders historyMeets source
theorem ranked_layerWitness_spec (k : Nat) {N : ZFSet.{u}} :
births.layerWitness k ∈ M.sets ∧ ZFSet.rank (births.layerWitness k) < o ∧
GrowthLayerWitnessCode HQs Hrs Hμs (completion.externalPointCarrier P0 r0 μ0 A0)
(completion.externalPointOrder P0 r0 μ0 A0) (completion.externalPointMeet P0 r0 μ0 A0)
births.birth (compressionStamp oldTimes k) N (births.layerWitness k) := by
have code := births.layerWitness_spec data group action pairing unions powers separation infinite hG hP0
historySpaces historyOrders historyMeets source k (N := N)
exact ⟨code.1, births.layerWitness_rank_lt data group action pairing unions powers separation infinite
hG hP0 limit omega indices points actors completed time k, code.2⟩
theorem ranked_cutoffLayerWitness_spec (k n : Nat) (before : k ≤ n) {N : ZFSet.{u}} :
births.layerWitness k ∈ M.sets ∧ ZFSet.rank (births.layerWitness k) < o ∧
GrowthLayerWitnessCode HQs Hrs Hμs (births.cutoff n) (births.cutoffOrder n)
(births.cutoffMeet n) (births.cutoffBirth n) (compressionStamp oldTimes k) N
(births.layerWitness k) := by
have code := births.cutoffLayerWitness_spec data group action pairing unions powers separation infinite hG hP0
historySpaces historyOrders historyMeets source k n before (N := N)
exact ⟨code.1, births.layerWitness_rank_lt data group action pairing unions powers separation infinite
hG hP0 limit omega indices points actors completed time k, code.2⟩
end PartitionPilot.Forcing.GroundExternalBirthData
end OAI
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