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namespace OAI
namespace PartitionPilot.Forcing.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)
private noncomputable def storedClassFormula (side : Fin 2) : SetFormula 4 :=
let w := prefixIndex 4 15
let old := suffixIndex 4 15
exBlock 15 (.ex (.conj
(BoundedFormula.nonemptyTupleFormula 14 0 (fun j => (w j).succ)).toFormula
(.conj (BoundedFormula.pairMembershipFormula (old 0).succ 0 (old 2).succ).toFormula
(reindex (extend (old 1).succ (extend (old 3).succ (fun j => (w j).succ)))
(M.definable_storedAssignmentClassGraph pairing unions powers separation side).formula))))
private theorem realize_storedClassFormula (side : Fin 2)
(env : Fin 4 → ZFSet.{u}) (members : ∀ j, env j ∈ M.sets) :
Realize M.sets (storedClassFormula M pairing unions powers separation side) env ↔
∃ d : Fin 15 → ZFSet.{u}, (∀ j, d j ∈ M.sets) ∧
ZFSet.pair (env 0) (refinementFormulaDataCode d) ∈ env 2 ∧
env 1 = M.storedAssignmentClassGraph (env 3) d side := by
unfold storedClassFormula
rw [realize_exBlock]
apply exists_congr
intro d
apply and_congr_right
intro ground
have args (code : ZFSet.{u}) :
(fun j => extend code (extendMany 15 d env)
(extend (suffixIndex 4 15 1).succ
(extend (suffixIndex 4 15 3).succ (fun j => (prefixIndex 4 15 j).succ)) j)) =
extend (env 1) (extend (env 3) d) := by
rw [map_extend, map_extend]
simp only [extend, Fin.cases_succ, extendMany_suffix, extendMany_prefix]
simp only [realize_ex]
constructor
· rintro ⟨code, hc, tuple, stored, output⟩
have all : ∀ j, (extend code (extendMany 15 d env)) j ∈ M.sets :=
Fin.cases hc (extendMany_members 15 d env ground members)
have eqCode := (BoundedFormula.truth_nonemptyTupleFormula 14 0
(fun j => (prefixIndex 4 15 j).succ) _).1
((BoundedFormula.realize_toFormula M _ _ all).1 tuple)
change code = refinementFormulaDataCode d at eqCode
have edge := (BoundedFormula.truth_pairMembershipFormula
(suffixIndex 4 15 0).succ 0 (suffixIndex 4 15 2).succ _).1
((BoundedFormula.realize_toFormula M _ _ all).1 stored)
change ZFSet.pair (env 0) code ∈ env 2 at edge
rw [realize_reindex, args] at output
have value := ((M.definable_storedAssignmentClassGraph pairing unions powers separation side).correct
(extend (env 3) d) (Fin.cases (members 3) ground) (env 1) (members 1)).1 output
exact ⟨eqCode ▸ edge, value⟩
· rintro ⟨edge, value⟩
let code := refinementFormulaDataCode d
have hc := M.refinementFormulaDataCode_mem pairing d ground
have all : ∀ j, (extend code (extendMany 15 d env)) j ∈ M.sets :=
Fin.cases hc (extendMany_members 15 d env ground members)
refine ⟨code, hc, ?_, ?_, ?_⟩
· apply (BoundedFormula.realize_toFormula M _ _ all).2
exact (BoundedFormula.truth_nonemptyTupleFormula 14 0
(fun j => (prefixIndex 4 15 j).succ) _).2 rfl
· apply (BoundedFormula.realize_toFormula M _ _ all).2
exact (BoundedFormula.truth_pairMembershipFormula
(suffixIndex 4 15 0).succ 0 (suffixIndex 4 15 2).succ _).2 edge
· rw [realize_reindex, args]
exact ((M.definable_storedAssignmentClassGraph pairing unions powers separation side).correct
(extend (env 3) d) (Fin.cases (members 3) ground) (env 1) (members 1)).2 value
include pairing unions powers separation in
theorem exists_storedAssignmentClassFamily
(collection : CollectionSchema M.sets) (side : Fin 2)
{levels stages history A T : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hHistory : history ∈ M.sets) (hA : A ∈ M.sets)
(inside : levels ⊆ stages) (valid : RefinementDataHistory stages history T) :
∃ graph ∈ M.sets, FunctionGraphOn levels graph ∧
∀ alpha (d : Fin 15 → ZFSet.{u}),
ZFSet.pair alpha (refinementFormulaDataCode d) ∈ history →
∀ y, ZFSet.pair alpha y ∈ graph ↔
alpha ∈ levels ∧ y = M.storedAssignmentClassGraph A d side := by
let params : Fin 2 → ZFSet.{u} := ![history, A]
have members : ∀ j, params j ∈ M.sets := Fin.cases hHistory (Fin.cases hA (fun j => Fin.elim0 j))
let formula := storedClassFormula M pairing unions powers separation side
have meaning (alpha y : ZFSet.{u}) (ha : alpha ∈ M.sets) (hy : y ∈ M.sets) :=
realize_storedClassFormula M pairing unions powers separation side
(extend alpha (extend y params)) (Fin.cases ha (Fin.cases hy members))
have unique (alpha y z : ZFSet.{u}) (ha : alpha ∈ M.sets) (hy : y ∈ M.sets) (hz : z ∈ M.sets)
(first : Realize M.sets formula (extend alpha (extend y params)))
(second : Realize M.sets formula (extend alpha (extend z params))) : y = z := by
obtain ⟨d, _, de, dy⟩ := (meaning alpha y ha hy).1 first
obtain ⟨e, _, ee, ez⟩ := (meaning alpha z ha hz).1 second
have same := refinementFormulaDataCode_injective (valid.function.unique de ee)
exact dy.trans (same ▸ ez.symm)
obtain ⟨B, hB, bounded⟩ := M.collection_witness collection formula params members hLevels (by
intro alpha ha
obtain ⟨code, entry⟩ := valid.function.total (inside ha)
obtain ⟨d, rfl⟩ := valid.decoded alpha code entry
have fields := M.refinementFormulaDataCode_fields_mem _ (M.pair_right (M.transitive hHistory entry))
have output := (M.definable_storedAssignmentClassGraph pairing unions powers separation side).closed
(extend A d) (Fin.cases hA fields)
exact ⟨_, output, (meaning alpha _ (M.transitive hLevels ha) output).2 ⟨d, fields, entry, rfl⟩⟩)
let graph := M.definableRelation formula params levels B
have hGraph := M.definableRelation_mem pairing unions powers separation formula params members hLevels hB
refine ⟨graph, hGraph, ⟨B, fun _ h => (ZFSet.mem_sep.1 h).1, ?_⟩, ?_⟩
· intro alpha ha
obtain ⟨y, hy, rel⟩ := bounded alpha ha
refine ⟨y, (M.pair_mem_definableRelation formula params levels B alpha y).2 ⟨ha, hy, rel⟩, ?_⟩
intro z hz
obtain ⟨_, hz, rz⟩ := (M.pair_mem_definableRelation formula params levels B alpha z).1 hz
exact unique alpha z y (M.transitive hLevels ha) (M.transitive hB hz) (M.transitive hB hy) rz rel
· intro alpha d entry y
rw [M.pair_mem_definableRelation]
constructor
· rintro ⟨ha, hy, rel⟩
obtain ⟨e, _, ee, ey⟩ := (meaning alpha y (M.transitive hLevels ha) (M.transitive hB hy)).1 rel
have same := refinementFormulaDataCode_injective (valid.function.unique ee entry)
exact ⟨ha, same ▸ ey⟩
· rintro ⟨ha, rfl⟩
obtain ⟨z, hz, rel⟩ := bounded alpha ha
obtain ⟨e, _, ee, ez⟩ := (meaning alpha z (M.transitive hLevels ha) (M.transitive hB hz)).1 rel
have same := refinementFormulaDataCode_injective (valid.function.unique ee entry)
have eq : z = M.storedAssignmentClassGraph A d side := same ▸ ez
exact eq ▸ ⟨ha, hz, rel⟩
end PartitionPilot.Forcing.TransitiveGround
namespace PartitionPilot.Forcing.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)
variable (collection : CollectionSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0)
variable (choice : Realize M.sets SetFormula.choice Fin.elim0)
include pairing unions powers separation collection infinite choice
open scoped ZFSet
theorem exists_assignmentRequestBatch {stages history T A : ZFSet.{u}}
(hStages : stages ∈ M.sets) (hHistory : history ∈ M.sets) (hT : T ∈ M.sets)
(hA : A ∈ M.sets) (smallA : M.groundEmbeddable A ZFSet.omega)
(valid : RefinementDataHistory stages history T)
(shape : A ⊆ stageOneAssignments (refinementDataHistoryIndices stages history) T)
(localFunctions : ∀ alpha (d : Fin 15 → ZFSet.{u}),
ZFSet.pair alpha (refinementFormulaDataCode d) ∈ history →
∀ side, FunctionGraphOn (assignmentOnIndices (d 0) T A)
(M.storedAssignmentClassGraph A d side)) :
let levels := relationImage A stages (refinementDataHistoryLevels stages history T)
levels ∈ M.sets ∧ M.groundEmbeddable levels ZFSet.omega ∧ levels ⊆ stages ∧
∃ families : Fin 2 → ZFSet.{u},
(∀ side, families side ∈ M.sets ∧ FunctionGraphOn levels (families side)) ∧
(∀ side alpha (d : Fin 15 → ZFSet.{u}),
ZFSet.pair alpha (refinementFormulaDataCode d) ∈ history →
∀ graph, ZFSet.pair alpha graph ∈ families side ↔
alpha ∈ levels ∧ graph = M.storedAssignmentClassGraph A d side) ∧
let flat := fun side => ⋃₀ functionGraphRange levels (families side)
let requests := fun side => functionGraphRange A (flat side)
(∀ side, flat side ∈ M.sets ∧ FunctionGraphOn A (flat side)) ∧
(requests 0 ∪ requests 1) ∈ M.sets ∧
M.groundEmbeddable (requests 0 ∪ requests 1) ZFSet.omega := by
let levels := relationImage A stages (refinementDataHistoryLevels stages history T)
have hOmega := M.omega_mem separation infinite
have hLevelGraph := M.refinementDataHistoryLevels_mem pairing unions powers separation hStages hHistory hT
have hLevels := M.relationImage_mem separation hA hStages hLevelGraph
have smallLevels := M.groundEmbeddable_function_image pairing unions powers separation choice
hA hStages hOmega hLevelGraph valid.levels_function smallA
have inside : levels ⊆ stages := fun _ h => (mem_relationImage _ _ _ _).1 h |>.1
have family (side : Fin 2) := M.exists_storedAssignmentClassFamily pairing unions powers separation
collection side hLevels hHistory hA inside valid
choose families ground functions entries using family
let flat := fun side => ⋃₀ functionGraphRange levels (families side)
have flatEntry (side : Fin 2) (z : ZFSet.{u}) : z ∈ flat side ↔
∃ alpha ∈ levels, ∃ d : Fin 15 → ZFSet.{u},
ZFSet.pair alpha (refinementFormulaDataCode d) ∈ history ∧
z ∈ M.storedAssignmentClassGraph A d side := by
rw [ZFSet.mem_sUnion]
constructor
· rintro ⟨g, hg, hz⟩
obtain ⟨alpha, ha, stored⟩ := (mem_functionGraphRange levels (families side) g).1 hg
obtain ⟨code, entry⟩ := valid.function.total (inside ha)
obtain ⟨d, rfl⟩ := valid.decoded alpha code entry
exact ⟨alpha, ha, d, entry, ((entries side alpha d entry g).1 stored).2 ▸ hz⟩
· rintro ⟨alpha, ha, d, entry, hz⟩
exact ⟨_, (mem_functionGraphRange levels (families side) _).2
⟨alpha, ha, (entries side alpha d entry _).2 ⟨ha, rfl⟩⟩, hz⟩
have levelOf {alpha : ZFSet.{u}} {d : Fin 15 → ZFSet.{u}}
(entry : ZFSet.pair alpha (refinementFormulaDataCode d) ∈ history)
{x : ZFSet.{u}} (hx : x ∈ assignmentOnIndices (d 0) T A) :
ZFSet.pair x alpha ∈ refinementDataHistoryLevels stages history T := by
obtain ⟨i, hi, u, hu, w, hw, rfl⟩ := (mem_stageOneAssignments _ _ _).1
((mem_assignmentOnIndices _ _ _ _).1 hx).2
have terms := valid.terms alpha d entry
exact valid.coverage alpha (valid.function.pair_source entry) d entry i hi u
(terms.symm ▸ hu) w (terms.symm ▸ hw)
have cover (x : ZFSet.{u}) (hx : x ∈ A) :
∃ alpha ∈ levels, ∃ d : Fin 15 → ZFSet.{u},
ZFSet.pair alpha (refinementFormulaDataCode d) ∈ history ∧
x ∈ assignmentOnIndices (d 0) T A := by
obtain ⟨i, hi, u, hu, w, hw, rfl⟩ := (mem_stageOneAssignments _ _ _).1 (shape hx)
obtain ⟨alpha, ha, d, entry, idx⟩ := (mem_refinementDataHistoryIndices _ _ _).1 hi
have selected := (mem_assignmentOnIndices (d 0) T A _).2
⟨hx, (stageOneAssignment_mem_assignments _ _ _ _ _).2 ⟨idx, hu, hw⟩⟩
exact ⟨alpha, (mem_relationImage _ _ _ _).2 ⟨ha, _, hx, levelOf entry selected⟩,
d, entry, selected⟩
have flatGround (side : Fin 2) : flat side ∈ M.sets :=
M.union_mem unions (M.functionGraphRange_mem unions separation hLevels (ground side))
have flatFunction (side : Fin 2) : FunctionGraphOn A (flat side) := by
refine ⟨⋃₀ ⋃₀ flat side, ?_, ?_⟩
· intro z hz
obtain ⟨alpha, _, d, entry, member⟩ := (flatEntry side z).1 hz
obtain ⟨x, hx, y, rfl⟩ := (localFunctions alpha d entry side).shape z member
exact ZFSet.pair_mem_prod.2
⟨((mem_assignmentOnIndices _ _ _ _).1 hx).1, pair_second_mem_doubleUnion hz⟩
· intro x hx
obtain ⟨alpha, ha, d, entry, selected⟩ := cover x hx
obtain ⟨y, hy⟩ := (localFunctions alpha d entry side).total selected
refine ⟨y, (flatEntry side _).2 ⟨alpha, ha, d, entry, hy⟩, ?_⟩
intro z hz
obtain ⟨beta, _, e, ee, ez⟩ := (flatEntry side _).1 hz
have betaSelected := (localFunctions beta e ee side).pair_source ez
have sameLevel := (functionCode_partial valid.levels_function).unique x beta alpha
(levelOf ee betaSelected) (levelOf entry selected)
subst beta
have same := refinementFormulaDataCode_injective (valid.function.unique ee entry)
subst e
exact (localFunctions alpha d entry side).unique ez hy
let requests := fun side => functionGraphRange A (flat side)
have requestGround (side : Fin 2) : requests side ∈ M.sets :=
M.functionGraphRange_mem unions separation hA (flatGround side)
have requestSmall (side : Fin 2) : M.groundEmbeddable (requests side) ZFSet.omega :=
M.groundEmbeddable_function_image pairing unions powers separation choice hA
(M.union_mem unions (M.union_mem unions (flatGround side))) hOmega (flatGround side)
(flatFunction side).to_isFunc smallA
exact ⟨hLevels, smallLevels, inside, families, fun side => ⟨ground side, functions side⟩,
entries, fun side => ⟨flatGround side, flatFunction side⟩,
M.binary_union_mem pairing unions (requestGround 0) (requestGround 1),
M.groundCountable_binary_union pairing unions powers separation infinite choice
(requestGround 0) (requestGround 1) (requestSmall 0) (requestSmall 1)⟩
end PartitionPilot.Forcing.TransitiveGround
end OAI
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