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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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundExternalLocalHeightSelection
namespace OAI
section
namespace PartitionPilot.Forcing.GroundExternalGeometryInputs
open SetRealization SetFormula 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 "U" => completion.algebra.union.fields 0
local notation "sGroup" => completion.externalFamilyGroupPresentation group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
local notation "qPoint" => data.externalPointPresentation group oldAction pairing unions powers separation infinite hG
local notation "letters" => externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0)
theorem externalCopyGraph_eq_graph :
letI := group.toGroup
let a := completion.algebra.union.action.toOrderAction group completion.algebra.union.meet
∀ w, completion.externalCopyGraph P0 r0 μ0 A0 ((sGroup).encode w) =
(SetMembers.presentation U).graph (qPoint) ((family).copy a w).toFun := by
let := group.toGroup
dsimp only
intro w
have classed := data.externalPointClassifier_entry_iff group oldAction pairing unions powers separation infinite hG
have graph := flatCopyEmbeddingGraph_eq_graph (family)
(completion.algebra.union.action.toOrderAction group completion.algebra.union.meet)
(sGroup) (SetMembers.presentation U) (qPoint) classed w
rw [data.externalPointPresentation_carrier] at graph
exact graph
include data group oldAction pairing unions powers separation infinite hG in
theorem externalCopy_ideal {w : ZFSet.{u}} (hw : w ∈ M.externalGroupCarrier G letters m one) :
IdealEmbeddingCode U (completion.externalPointCarrier P0 r0 μ0 A0)
(completion.algebra.union.fields 1) (completion.externalPointOrder P0 r0 μ0 A0)
(completion.algebra.union.fields 2) (completion.externalPointMeet P0 r0 μ0 A0)
(completion.externalCopyGraph P0 r0 μ0 A0 w) := by
let := group.toGroup
let a := completion.algebra.union.action.toOrderAction group completion.algebra.union.meet
let s := sGroup
let q := qPoint
have sCarrier := completion.externalFamilyGroupPresentation_carrier group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
obtain ⟨w, rfl⟩ := (s.mem_carrier w).1 (sCarrier.symm ▸ hw)
have entries := data.externalGeometry_entries group oldAction pairing unions powers separation infinite hG
have code := (SetMembers.presentation U).idealEmbeddingCode q ((family).copy a w)
(fun _ _ => Iff.rfl) entries.1
(fun x y z => (completion.algebra.union.meet.toSmallMeetOrder_meet_eq_iff x y z).symm) entries.2.1
rw [← data.externalCopyGraph_eq_graph group oldAction pairing unions powers separation infinite hG w] at code
dsimp only [q] at code
rw [data.externalPointPresentation_carrier] at code
exact code
include data group oldAction pairing unions powers separation infinite hG in
theorem externalIdentityCopy_ideal :
IdealEmbeddingCode U (completion.externalPointCarrier P0 r0 μ0 A0)
(completion.algebra.union.fields 1) (completion.externalPointOrder P0 r0 μ0 A0)
(completion.algebra.union.fields 2) (completion.externalPointMeet P0 r0 μ0 A0)
(completion.externalCopyGraph P0 r0 μ0 A0 (M.externalGroupIdentity G letters m one)) :=
data.externalCopy_ideal group oldAction pairing unions powers separation infinite hG
data.group.group_code.identity_mem
end PartitionPilot.Forcing.GroundExternalGeometryInputs
namespace PartitionPilot.Forcing.GroundExternalGeometryInputs
open SetRealization SetFormula 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 "U" => completion.algebra.union.fields 0
local notation "sGroup" => completion.externalFamilyGroupPresentation group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
local notation "qPoint" => data.externalPointPresentation group oldAction pairing unions powers separation infinite hG
local notation "letters" => externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0)
local notation "copyGraph" => completion.externalCopyGraph P0 r0 μ0 A0 (M.externalGroupIdentity G letters m one)
include data group oldAction pairing unions powers separation infinite hG in
theorem externalIdentityCopy_equivariant {g g' x y x' y' : ZFSet.{u}}
(actor : ZFSet.pair g g' ∈ M.externalOldInclusion G letters m one)
(first : ZFSet.pair x x' ∈ copyGraph) (second : ZFSet.pair y y' ∈ copyGraph)
(acted : ZFSet.pair (ZFSet.pair g x) y ∈ completion.algebra.union.fields 12) :
ZFSet.pair (ZFSet.pair g' x') y' ∈ completion.externalPointAction P0 r0 μ0 A0 := by
let := group.toGroup
let a := completion.algebra.union.action.toOrderAction group completion.algebra.union.meet
rw [completion.externalFamilyGroupPresentation_old group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground] at actor
obtain ⟨g, equal⟩ := ((SetMembers.presentation G).mem_graph (sGroup) _ _).1 actor
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal
have identity := completion.externalFamilyGroupPresentation_identity group oldAction data.original
pairing unions powers separation infinite hG data.words.letters_ground
rw [identity, data.externalCopyGraph_eq_graph group oldAction pairing unions powers separation infinite hG 1] at first second
obtain ⟨x, equal⟩ := ((SetMembers.presentation U).mem_graph (qPoint) _ _).1 first
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal
obtain ⟨y, equal⟩ := ((SetMembers.presentation U).mem_graph (qPoint) _ _).1 second
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal
have actionEq := (completion.algebra.union.action.toOrderAction_eq_iff group completion.algebra.union.meet g x y).2 acted
have entries := data.externalGeometry_entries group oldAction pairing unions powers separation infinite hG
apply (entries.2.2.1 _ _ _).2
change (((family).externalAction a).iso (old (SetMembers G) (family).Letters g)).toFun
(((family).copy a 1).toFun x) = ((family).copy a 1).toFun y
rw [(family).extends_old_action, actionEq]
include data group oldAction pairing unions powers separation infinite hG in
theorem externalIdentityCopy_agreement (hP : P0 ∈ M.sets)
{g h g' h' v v' : ZFSet.{u}}
(firstActor : ZFSet.pair g g' ∈ M.externalOldInclusion G letters m one)
(secondActor : ZFSet.pair h h' ∈ M.externalOldInclusion G letters m one)
(image : ZFSet.pair v v' ∈ copyGraph)
(agree : ActionAgreementBelow U (completion.algebra.union.fields 1)
(completion.algebra.union.fields 12) g h v) :
ActionAgreementBelow (completion.externalPointCarrier P0 r0 μ0 A0)
(completion.externalPointOrder P0 r0 μ0 A0) (completion.externalPointAction P0 r0 μ0 A0) g' h' v' := by
have ideal := data.externalIdentityCopy_ideal group oldAction pairing unions powers separation infinite hG
have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP
have firstShape := ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 firstActor)
have secondShape := ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 secondActor)
exact ideal.actionAgreementBelow completion.algebra.union.action geometry.action
firstShape.1 firstShape.2 secondShape.2 image agree
(fun _ _ _ _ _ _ first second acted => data.externalIdentityCopy_equivariant group oldAction
pairing unions powers separation infinite hG firstActor first second acted)
(fun _ _ _ _ _ _ first second acted => data.externalIdentityCopy_equivariant group oldAction
pairing unions powers separation infinite hG secondActor first second acted)
include data group oldAction pairing unions powers separation infinite hG in
theorem externalOriginal_agreement (hP : P0 ∈ M.sets)
{g h g' h' v v' : ZFSet.{u}}
(firstActor : ZFSet.pair g g' ∈ M.externalOldInclusion G letters m one)
(secondActor : ZFSet.pair h h' ∈ M.externalOldInclusion G letters m one)
(hv : v ∈ P0) (image : ZFSet.pair v v' ∈ copyGraph)
(agree : ActionAgreementBelow P0 r0 A0 g h v) :
ActionAgreementBelow (completion.externalPointCarrier P0 r0 μ0 A0)
(completion.externalPointOrder P0 r0 μ0 A0) (completion.externalPointAction P0 r0 μ0 A0) g' h' v' := by
have hg := (ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 firstActor)).1
have hh := (ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 secondActor)).1
exact data.externalIdentityCopy_agreement group oldAction pairing unions powers separation infinite hG hP
firstActor secondActor image (data.original.agreement g hg h hh v hv agree)
end PartitionPilot.Forcing.GroundExternalGeometryInputs
end
end OAI
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