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import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundStructuralFullIndexSize
namespace OAI
namespace PartitionPilot.Forcing
section
universe u
theorem assignmentOnIndices_indices_on {I T X A : ZFSet.{u}}
(shape : A ⊆ stageOneAssignments I T) :
assignmentOnIndices (assignmentIndices I T (assignmentOnIndices X T A)) T A =
assignmentOnIndices X T A := by
apply ZFSet.ext
intro z
constructor
· intro member
obtain ⟨hA, inside⟩ := (mem_assignmentOnIndices _ T A z).1 member
obtain ⟨t, ht, v, hv, w, hw, rfl⟩ := (mem_stageOneAssignments _ T z).1 inside
rw [assignmentIndices_on] at ht
exact (assignmentOnIndices_entry X T A t v w).2
⟨hA, (ZFSet.mem_inter.1 ht).2, hv, hw⟩
· intro member
have hA := ((mem_assignmentOnIndices X T A z).1 member).1
obtain ⟨t, ht, v, hv, w, hw, rfl⟩ := (mem_stageOneAssignments I T z).1 (shape hA)
exact (assignmentOnIndices_entry _ T A t v w).2
⟨hA, (mem_assignmentIndices I T (assignmentOnIndices X T A) t).2
⟨ht, v, hv, w, hw, member⟩, hv, hw⟩
end
namespace TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
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)
include pairing unions powers separation
theorem assignmentQueryRelation_mem {I T C D : ZFSet.{u}}
(hI : I ∈ M.sets) (hT : T ∈ M.sets) (hC : C ∈ M.sets) (hD : D ∈ M.sets) :
assignmentQueryRelation I T C D ∈ M.sets := by
let f := productFirst I (ZFSet.prod T T)
have hf : f ∈ M.sets := M.productFirst_mem pairing unions powers separation hI
(M.prod_mem pairing unions powers separation hT hT)
let params := extend I (extend f Fin.elim0)
have selected := M.bounded_separation_mem separation
(BoundedFormula.assignmentQueryRelationFormula 0 1 2) params
(Fin.cases hI (Fin.cases hf (fun j => Fin.elim0 j)))
(M.prod_mem pairing unions powers separation hC hD)
have equal : ZFSet.sep (fun z => BoundedFormula.Truth
(BoundedFormula.assignmentQueryRelationFormula 0 1 2) (extend z params))
(ZFSet.prod C D) = assignmentQueryRelation I T C D := by
apply ZFSet.ext
intro z
rw [ZFSet.mem_sep, assignmentQueryRelation, ZFSet.mem_sep]
exact and_congr Iff.rfl (BoundedFormula.truth_assignmentQueryRelationFormula 0 1 2
(extend z params) T rfl)
exact equal ▸ selected
theorem realizedAssignmentQueries_mem {I T C D : ZFSet.{u}}
(hI : I ∈ M.sets) (hT : T ∈ M.sets) (hC : C ∈ M.sets) (hD : D ∈ M.sets) :
realizedAssignmentQueries I T C D ∈ M.sets :=
M.relationImage_mem separation hD hC (M.relationConverse_mem pairing unions powers separation
hC hD (M.assignmentQueryRelation_mem pairing unions powers separation hI hT hC hD))
theorem exists_assignmentQuerySelector
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{I T C D : ZFSet.{u}} (hI : I ∈ M.sets) (hT : T ∈ M.sets)
(hC : C ∈ M.sets) (hD : D ∈ M.sets) :
∃ F ∈ M.sets, ZFSet.IsFunc (realizedAssignmentQueries I T C D) D F ∧
F ⊆ assignmentQueryRelation I T C D := by
apply M.uniformize_relation pairing unions powers separation choice
(M.realizedAssignmentQueries_mem pairing unions powers separation hI hT hC hD) hD
(M.assignmentQueryRelation_mem pairing unions powers separation hI hT hC hD)
intro q member
exact ((mem_realizedAssignmentQueries I T C D q).1 member).2
theorem exists_stageOneAssignmentQuerySelector
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{rows I T D : ZFSet.{u}} (hRows : rows ∈ M.sets) (hI : I ∈ M.sets)
(hT : T ∈ M.sets) (omega : ZFSet.omega ∈ M.sets) (hD : D ∈ M.sets) :
∃ F ∈ M.sets,
ZFSet.IsFunc (realizedAssignmentQueries I T (M.stageOneCandidateBound rows I T) D) D F ∧
F ⊆ assignmentQueryRelation I T (M.stageOneCandidateBound rows I T) D :=
M.exists_assignmentQuerySelector pairing unions powers separation choice hI hT
(M.stageOneCandidateBound_mem pairing unions powers separation hRows hI hT omega) hD
end TransitiveGround
namespace TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem exists_groundOmegaOneClosure
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(collection : CollectionSchema M.sets) (infinite : Realize M.sets infinity Fin.elim0)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{I K : ZFSet.{u}} (hK : K ∈ M.sets) (conditions : GroundInaccessibleConditions M K)
{n : Nat} {f : (Fin (n + 1) → ZFSet.{u}) → ZFSet.{u}}
(defined : GroundDefinableFunction M f) (params : Fin n → ZFSet.{u})
(members : ∀ j, params j ∈ M.sets)
(inflationary : ∀ X ∈ M.sets, X ⊆ I → X ⊆ f (extend X params))
(preserves : ∀ X ∈ M.sets, X ⊆ I → GroundBelow M X K →
f (extend X params) ⊆ I ∧ GroundBelow M (f (extend X params)) K) :
let C := M.groundOmegaOne pairing unions powers separation collection infinite
∃ history ∈ M.sets,
OrdinalRecursionTrace (fun alpha prior output =>
output = f (extend (refinementHistoryIndices alpha prior) params)) C history ∧
let X := refinementHistoryIndices C history
X ∈ M.sets ∧ X ⊆ I ∧ GroundBelow M X K ∧
(∀ alpha Y, ZFSet.pair alpha Y ∈ history →
Y ∈ M.sets ∧ Y ⊆ I ∧ GroundBelow M Y K) ∧
(∀ alpha beta Y Z, ZFSet.pair alpha Y ∈ history →
ZFSet.pair beta Z ∈ history → alpha ∈ beta → Y ⊆ Z) ∧
∀ B ∈ M.sets, M.groundEmbeddable B ZFSet.omega → B ⊆ X →
∃ beta ∈ C,
let U := refinementHistoryIndices beta history
U ∈ M.sets ∧ U ⊆ X ∧ GroundBelow M U K ∧ B ⊆ U ∧ f (extend U params) ⊆ X := by
let C := M.groundOmegaOne pairing unions powers separation collection infinite
have spec := M.groundOmegaOne_spec pairing unions powers separation collection infinite
have hC := spec.1
obtain ⟨history, hh, trace⟩ := M.exists_historyUnionTrace pairing unions separation collection
defined params members hC spec.2.1
have bounds := M.historyUnionTrace_below pairing unions powers separation choice hK conditions
(fun X => f (extend X params)) preserves C
(conditions.groundOmegaOne_mem pairing unions powers separation collection infinite hK) history hh trace
let X := refinementHistoryIndices C history
have hX : X ∈ M.sets := M.refinementHistoryIndices_mem unions separation hC hh
have step (beta Y : ZFSet.{u}) (entry : ZFSet.pair beta Y ∈ history) :
Y = f (extend (refinementHistoryIndices beta history) params) := by
have result := trace.2 beta (trace.1.pair_source entry) Y entry
change Y = f (extend (refinementHistoryIndices beta (graphRestriction beta history)) params) at result
rw [refinementHistoryIndices_restrict (fun _ hx => hx)] at result
exact result
have earlierUnion (beta : ZFSet.{u}) (member : beta ∈ C) :
refinementHistoryIndices beta history ∈ M.sets ∧
refinementHistoryIndices beta history ⊆ X ∧
GroundBelow M (refinementHistoryIndices beta history) K := by
have ground := M.refinementHistoryIndices_mem unions separation (M.transitive hC member) hh
have subset := refinementHistoryIndices_mono (history := history) (spec.2.1.1 beta member)
exact ⟨ground, subset, bounds.2.2.restrict separation ground hK subset⟩
refine ⟨history, hh, trace, hX, bounds.2.1, bounds.2.2, ?_, ?_, ?_⟩
· intro alpha Y entry
exact ⟨M.transitive (M.union_mem unions (M.union_mem unions hh))
(pair_second_mem_doubleUnion entry), bounds.1 alpha Y entry⟩
· intro alpha beta Y Z first second before x inside
have earlier := earlierUnion beta (trace.1.pair_source second)
rw [step beta Z second]
exact inflationary _ earlier.1 (fun _ hx => bounds.2.1 (earlier.2.1 hx))
((mem_refinementHistoryIndices beta history x).2 ⟨alpha, before, Y, first, inside⟩)
· intro B hB small subset
obtain ⟨beta, member, captured⟩ := M.historyUnion_countable_capture
pairing unions powers separation collection infinite choice hh hB small subset
have earlier := earlierUnion beta member
refine ⟨beta, member, earlier.1, earlier.2.1, earlier.2.2, captured, ?_⟩
obtain ⟨Y, entry⟩ := trace.1.total member
rw [← step beta Y entry]
intro x inside
exact (mem_refinementHistoryIndices C history x).2 ⟨beta, member, Y, entry, inside⟩
theorem assignmentQueryClosure_representatives
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{rows I T D c X : ZFSet.{u}} (hI : I ∈ M.sets) (hT : T ∈ M.sets)
(omega : ZFSet.omega ∈ M.sets) (hX : X ∈ M.sets)
(shape : D ⊆ M.stageOneCandidateBound rows I T)
(selector : ZFSet.IsFunc
(realizedAssignmentQueries I T (M.stageOneCandidateBound rows I T) D) D c)
(selected : c ⊆ assignmentQueryRelation I T (M.stageOneCandidateBound rows I T) D)
(capture : ∀ B ∈ M.sets, M.groundEmbeddable B ZFSet.omega → B ⊆ X →
∃ U, B ⊆ U ∧ representativeIndexStep I T
(realizedAssignmentQueries I T (M.stageOneCandidateBound rows I T) D) c U ⊆ X) :
∀ row A, ZFSet.pair row A ∈ D →
∃ C, ZFSet.pair row C ∈ D ∧ assignmentIndices I T C ⊆ X ∧
assignmentOnIndices (assignmentIndices I T (assignmentOnIndices X T A)) T C =
assignmentOnIndices X T A := by
intro row A member
let L := assignmentOnIndices X T A
let B := assignmentIndices I T L
let Q := realizedAssignmentQueries I T (M.stageOneCandidateBound rows I T) D
have data := (M.pair_mem_stageOneCandidateBound rows I T row A).1 (shape member)
have hL : L ∈ M.sets := M.assignmentOnIndices_mem pairing unions powers separation hX hT data.2.1
have smallL := M.assignmentOnIndices_small pairing unions powers separation hX hT data.2.1
omega data.2.2.2
have listShape : L ⊆ stageOneAssignments I T :=
fun _ h => data.2.2.1 ((mem_assignmentOnIndices X T A _).1 h).1
have queryCandidate : ZFSet.pair row L ∈ M.stageOneCandidateBound rows I T :=
(M.pair_mem_stageOneCandidateBound rows I T row L).2 ⟨data.1, hL, listShape, smallL⟩
have query : ZFSet.pair row L ∈ Q :=
(mem_realizedAssignmentQueries I T (M.stageOneCandidateBound rows I T) D _).2
⟨queryCandidate, ZFSet.pair row A, member,
(assignmentQueryRelation_entry I T (M.stageOneCandidateBound rows I T) D row L row A).2
⟨queryCandidate, member, rfl, assignmentOnIndices_indices_on data.2.2.1⟩⟩
have hB : B ∈ M.sets := M.assignmentIndices_mem pairing unions powers separation hI hT hL
have smallB := M.assignmentIndices_small pairing unions powers separation choice hI hT hL omega smallL
have subsetB : B ⊆ X := by
intro i hi
change i ∈ assignmentIndices I T (assignmentOnIndices X T A) at hi
rw [assignmentIndices_on] at hi
exact (ZFSet.mem_inter.1 hi).2
obtain ⟨U, captured, closed⟩ := capture B hB smallB subsetB
obtain ⟨p, entry, _⟩ := selector.2 (ZFSet.pair row L) query
have inD := (ZFSet.pair_mem_prod.1 (selector.1 entry)).2
obtain ⟨other, _, C, _, _, _, rfl⟩ :=
(M.mem_stageOneCandidateBound rows I T p).1 (shape inD)
have matched := (assignmentQueryRelation_entry I T (M.stageOneCandidateBound rows I T) D
row L other C).1 (selected entry)
obtain rfl := matched.2.2.1
refine ⟨C, inD, ?_, matched.2.2.2⟩
have localQuery : AssignmentQueryWithin U T (ZFSet.pair row L) :=
⟨row, L, rfl, (assignment_subset_iff_indices_subset listShape).2 captured⟩
exact fun _ h => closed (representativeIndexStep_covers query localQuery entry h)
theorem exists_assignmentRepresentativeClosure
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(collection : CollectionSchema M.sets) (infinite : Realize M.sets infinity Fin.elim0)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{rows I T D K : ZFSet.{u}} (hRows : rows ∈ M.sets) (hI : I ∈ M.sets)
(hT : T ∈ M.sets) (hD : D ∈ M.sets) (hK : K ∈ M.sets)
(conditions : GroundInaccessibleConditions M K)
(smallRows : GroundBelow M rows K) (smallT : GroundBelow M T K)
(shape : D ⊆ M.stageOneCandidateBound rows I T) :
∃ X ∈ M.sets, X ⊆ I ∧ GroundBelow M X K ∧
∀ row A, ZFSet.pair row A ∈ D →
∃ C, ZFSet.pair row C ∈ D ∧ assignmentIndices I T C ⊆ X ∧
assignmentOnIndices (assignmentIndices I T (assignmentOnIndices X T A)) T C =
assignmentOnIndices X T A := by
have omega := M.omega_mem separation infinite
let Q := realizedAssignmentQueries I T (M.stageOneCandidateBound rows I T) D
have hQ : Q ∈ M.sets := M.realizedAssignmentQueries_mem pairing unions powers separation hI hT
(M.stageOneCandidateBound_mem pairing unions powers separation hRows hI hT omega) hD
have queries : Q ⊆ M.stageOneCandidateBound rows I T :=
fun _ h => ((mem_realizedAssignmentQueries I T (M.stageOneCandidateBound rows I T) D _).1 h).1
obtain ⟨c, hc, selector, selected⟩ := M.exists_stageOneAssignmentQuerySelector
pairing unions powers separation choice hRows hI hT omega hD
let params : Fin 4 → ZFSet.{u} := extend I (extend T (extend Q (extend c Fin.elim0)))
have members : ∀ j, params j ∈ M.sets :=
Fin.cases hI (Fin.cases hT (Fin.cases hQ (Fin.cases hc (fun j => Fin.elim0 j))))
have inflationary : ∀ X ∈ M.sets, X ⊆ I →
X ⊆ representativeIndexStep I T Q c X :=
fun X _ _ => representativeIndexStep_contains I T Q c X
have preserves : ∀ X ∈ M.sets, X ⊆ I → GroundBelow M X K →
representativeIndexStep I T Q c X ⊆ I ∧
GroundBelow M (representativeIndexStep I T Q c X) K := by
intro X hX subset smallX
exact ⟨representativeIndexStep_subset subset,
M.representativeIndexStep_below pairing unions powers separation infinite choice
hRows hI hT hQ hc hD hX hK conditions selector queries shape smallRows smallT smallX⟩
obtain ⟨history, _, _, hX, subsetX, smallX, _, _, capture⟩ :=
M.exists_groundOmegaOneClosure pairing unions powers separation collection infinite choice
hK conditions (M.definable_representativeIndexStep pairing unions separation)
params members inflationary preserves
refine ⟨_, hX, subsetX, smallX,
M.assignmentQueryClosure_representatives pairing unions powers separation choice
hI hT omega hX shape selector selected ?_⟩
intro B hB countable subset
obtain ⟨beta, _, _, _, _, bounded, closed⟩ := capture B hB countable subset
exact ⟨refinementHistoryIndices beta history, bounded, closed⟩
end TransitiveGround
end PartitionPilot.Forcing
end OAI
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