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namespace OAI
namespace PartitionPilot.Forcing
section
open SetFormula
universe u
theorem CountableLayerCode.of_literal_extension
{A B old born t D N injection : ZFSet.{u}}
(code : CountableLayerCode A old t D N injection)
(fiber : ∀ x ∈ B, ZFSet.pair x t ∈ born → x ∈ A ∧ ZFSet.pair x t ∈ old) :
CountableLayerCode B born t D N injection :=
⟨code.small, fun x hx entry => code.cover x (fiber x hx entry).1 (fiber x hx entry).2⟩
theorem CopyLayerCode.of_literal_extension
{Qs rs μs A B rA rB μA μB old born t I parts Indices Maps W N injection : ZFSet.{u}}
(code : CopyLayerCode Qs rs μs A rA μA old t I parts Indices Maps W N injection)
(carrier : A ⊆ B) (order : rA ⊆ rB) (meets : μA ⊆ μB)
(fiber : ∀ x ∈ B, ZFSet.pair x t ∈ born → x ∈ A ∧ ZFSet.pair x t ∈ old) :
CopyLayerCode Qs rs μs B rB μB born t I
(ZFSet.sep (fun p => p ∈ ZFSet.prod I A) parts) Indices Maps W N injection := by
refine ⟨code.small, code.indices, code.maps, ?_, ?_, ?_⟩
· intro i hi s hs e P R V index map qField rField mField
exact (code.embedding i hi s hs e P R V index map qField rField mField).enlarge_target
carrier order meets
· intro x hx entry
have retained := fiber x hx entry
obtain ⟨i, hi, part⟩ := code.cover x retained.1 retained.2
exact ⟨i, hi, ZFSet.mem_sep.2 ⟨part, ZFSet.pair_mem_prod.2 ⟨hi, retained.1⟩⟩⟩
· intro i hi s hs e P x _ index map qField part
have retained := ZFSet.mem_sep.1 part
exact relationImage_enlarge_target carrier
(code.inCopy i hi s hs e P x (ZFSet.pair_mem_prod.1 retained.2).2 index map qField retained.1)
theorem SeparatedLayerCode.of_literal_extension
{Qs rs μs A B rA rB μA μB old born t I Indices Maps W Components F : ZFSet.{u}}
(code : SeparatedLayerCode Qs rs μs A rA μA old t I Indices Maps W Components F)
(carrier : A ⊆ B) (order : rA ⊆ rB) (meets : μA ⊆ μB) (births : old ⊆ born)
(fiber : ∀ x ∈ B, ZFSet.pair x t ∈ born → x ∈ A ∧ ZFSet.pair x t ∈ old) :
SeparatedLayerCode Qs rs μs B rB μB born t I Indices Maps W Components F := by
refine ⟨code.indices, code.maps, ?_, code.components, ?_, ?_, ?_⟩
· rw [code.fiber]
apply ZFSet.ext
intro x
rw [mem_layerBirthFiber, mem_layerBirthFiber]
exact ⟨fun h => ⟨carrier h.1, births h.2⟩, fun h => fiber x h.1 h.2⟩
· intro i hi s hs e P R V index map qField rField mField
exact (code.embedding i hi s hs e P R V index map qField rField mField).enlarge_target
carrier order meets
· intro i hi s hs e P x _ index map qField component
have inF := (ZFSet.pair_mem_prod.1 (code.components.1 component)).1
have hx : x ∈ A := ((mem_layerBirthFiber A old t x).1 (code.fiber ▸ inF)).1
exact relationImage_enlarge_target carrier (code.cover i hi s hs e P x hx index map qField component)
· intro x hx y hy i hi j hj first second different
exact (code.separated x hx y hy i hi j hj first second different).of_literal_extension
carrier order meets births
namespace GrowthLayerWitnessCode
theorem exists_literal_extension {M : TransitiveGround.{u}}
{Qs rs μs A B rA rB μA μB old born t N z : ZFSet.{u}}
(code : GrowthLayerWitnessCode Qs rs μs A rA μA old t N z)
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0) (separation : SeparationSchema M.sets)
(hA : A ∈ M.sets) (hz : z ∈ M.sets)
(carrier : A ⊆ B) (order : rA ⊆ rB) (meets : μA ⊆ μB) (births : old ⊆ born)
(fiber : ∀ x ∈ B, ZFSet.pair x t ∈ born → x ∈ A ∧ ZFSet.pair x t ∈ old) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode Qs rs μs B rB μB born t N z' := by
rcases code with ⟨D, injection, rfl, code⟩ |
⟨I, parts, Indices, Maps, W, injection, rfl, code⟩ |
⟨I, Indices, Maps, W, Components, F, rfl, code⟩
· exact ⟨countableLayerWitness D injection, hz, Or.inl ⟨D, injection, rfl, code.of_literal_extension fiber⟩⟩
· have fields := M.nonemptyTupleCode_field_mem 6 _ hz
let retained := ZFSet.sep (fun p => p ∈ ZFSet.prod I A) parts
have ground : retained ∈ M.sets := M.injection_restriction_mem separation (fields 1) hA (fields 2)
exact ⟨copyLayerWitness I retained Indices Maps W injection,
M.copyLayerWitness_mem pairing unions (fields 1) ground (fields 3) (fields 4) (fields 5) (fields 6),
Or.inr (Or.inl ⟨I, retained, Indices, Maps, W, injection, rfl,
code.of_literal_extension carrier order meets fiber⟩)⟩
· exact ⟨separatedLayerWitness I Indices Maps W Components F, hz,
Or.inr (Or.inr ⟨I, Indices, Maps, W, Components, F, rfl,
code.of_literal_extension carrier order meets births fiber⟩)⟩
end GrowthLayerWitnessCode
theorem GrowthLayerWitnessCode.exists_fixedTop_earlier {M : TransitiveGround.{u}}
{Qs rs μs P r μ T birth s t N z : ZFSet.{u}}
(code : GrowthLayerWitnessCode Qs rs μs P r μ birth t N z)
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0) (separation : SeparationSchema M.sets)
(hP : P ∈ M.sets) (hz : z ∈ M.sets)
(meet : MeetOrderCode P r μ) (born : ZFSet.IsFunc P T birth) (before : t ∈ s) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode Qs rs μs (fixedTopCarrier P)
(fixedTopOrder P r) (fixedTopMeet P r) (fixedTopBirth P birth s) t N z' := by
apply code.exists_literal_extension pairing unions separation hP hz
(fixedTopCarrier_subset P) (fixedTopOrder_subset P r) (fixedTopMeet_subset meet)
(fixedTopBirth_subset P birth s)
intro x _ entry
have old := (fixedTopBirth_earlier_iff before).1 entry
exact ⟨(ZFSet.pair_mem_prod.1 (born.1 old)).1, old⟩
noncomputable def fixedTopLayerWitness (P : ZFSet.{u}) : ZFSet.{u} :=
countableLayerWitness {P} {ZFSet.pair P ∅}
theorem fixedTopLayerWitness_code {Qs rs μs P r birth s : ZFSet.{u}}
(earlier : ∀ x t, ZFSet.pair x t ∈ birth → t ∈ s) :
GrowthLayerWitnessCode Qs rs μs (fixedTopCarrier P) (fixedTopOrder P r)
(fixedTopMeet P r) (fixedTopBirth P birth s) s ZFSet.omega (fixedTopLayerWitness P) := by
refine Or.inl ⟨{P}, {ZFSet.pair P ∅}, rfl, fixedTopBirth_singleton_injection P, ?_⟩
intro x _ entry
exact ZFSet.mem_singleton.2 ((fixedTopBirth_new_time_iff earlier).1 entry)
theorem TransitiveGround.fixedTopLayerWitness_mem (M : TransitiveGround.{u})
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
{P : ZFSet.{u}} (hP : P ∈ M.sets) : fixedTopLayerWitness P ∈ M.sets := by
have singleton (x : ZFSet.{u}) (hx : x ∈ M.sets) : ({x} : ZFSet.{u}) ∈ M.sets := by
simpa only [ZFSet.pair_eq_singleton] using M.unorderedPair_mem pairing hx hx
exact M.countableLayerWitness_mem pairing (singleton P hP)
(singleton _ (M.orderedPair_mem pairing hP (M.empty_mem ⟨P, hP⟩)))
end
namespace GrowthLayerWitnessCode
open SetFormula
universe u
theorem exists_literal_extension_rank_lt {M : TransitiveGround.{u}}
{Qs rs μs A B rA rB μA μB old born t N z : ZFSet.{u}}
(code : GrowthLayerWitnessCode Qs rs μs A rA μA old t N z)
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0) (separation : SeparationSchema M.sets)
(hA : A ∈ M.sets) (hz : z ∈ M.sets)
{o : Ordinal.{u}} (limit : Order.IsSuccLimit o) (rank : ZFSet.rank z < o)
(carrier : A ⊆ B) (order : rA ⊆ rB) (meets : μA ⊆ μB) (births : old ⊆ born)
(fiber : ∀ x ∈ B, ZFSet.pair x t ∈ born → x ∈ A ∧ ZFSet.pair x t ∈ old) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode Qs rs μs B rB μB born t N z' ∧
ZFSet.rank z' < o := by
rcases code with ⟨D, injection, rfl, code⟩ |
⟨I, parts, Indices, Maps, W, injection, rfl, code⟩ |
⟨I, Indices, Maps, W, Components, F, rfl, code⟩
· exact ⟨countableLayerWitness D injection, hz,
Or.inl ⟨D, injection, rfl, code.of_literal_extension fiber⟩, rank⟩
· have fields := M.nonemptyTupleCode_field_mem 6 _ hz
have ranks := fun j => (rank_nonemptyTupleCode_field_le 6 _ j).trans_lt rank
let retained := ZFSet.sep (fun p => p ∈ ZFSet.prod I A) parts
have ground : retained ∈ M.sets :=
M.injection_restriction_mem separation (fields 1) hA (fields 2)
have retainedRank : ZFSet.rank retained < o :=
(ZFSet.rank_mono (show retained ⊆ parts from fun _ hp => (ZFSet.mem_sep.1 hp).1)).trans_lt (ranks 2)
exact ⟨copyLayerWitness I retained Indices Maps W injection,
M.copyLayerWitness_mem pairing unions (fields 1) ground (fields 3) (fields 4) (fields 5) (fields 6),
Or.inr (Or.inl ⟨I, retained, Indices, Maps, W, injection, rfl,
code.of_literal_extension carrier order meets fiber⟩),
copyLayerWitness_rank_lt limit (ranks 1) retainedRank (ranks 3) (ranks 4) (ranks 5) (ranks 6)⟩
· exact ⟨separatedLayerWitness I Indices Maps W Components F, hz,
Or.inr (Or.inr ⟨I, Indices, Maps, W, Components, F, rfl,
code.of_literal_extension carrier order meets births fiber⟩), rank⟩
end GrowthLayerWitnessCode
section
open SetFormula SetRealization
universe u
theorem GrowthLayerWitnessCode.exists_fixedTop_earlier_rank_lt
{M : TransitiveGround.{u}} {Qs rs μs P r μ T birth s t N z : ZFSet.{u}}
(code : GrowthLayerWitnessCode Qs rs μs P r μ birth t N z)
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0) (separation : SeparationSchema M.sets)
(hP : P ∈ M.sets) (hz : z ∈ M.sets)
{o : Ordinal.{u}} (limit : Order.IsSuccLimit o) (rank : ZFSet.rank z < o)
(meet : MeetOrderCode P r μ) (born : ZFSet.IsFunc P T birth) (before : t ∈ s) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode Qs rs μs (fixedTopCarrier P)
(fixedTopOrder P r) (fixedTopMeet P r) (fixedTopBirth P birth s) t N z' ∧
ZFSet.rank z' < o := by
apply code.exists_literal_extension_rank_lt pairing unions separation hP hz limit rank
(fixedTopCarrier_subset P) (fixedTopOrder_subset P r) (fixedTopMeet_subset meet)
(fixedTopBirth_subset P birth s)
intro x _ entry
have old := (fixedTopBirth_earlier_iff before).1 entry
exact ⟨(ZFSet.pair_mem_prod.1 (born.1 old)).1, old⟩
theorem fixedTopLayerWitness_rank_lt {P : ZFSet.{u}} {o : Ordinal.{u}}
(limit : Order.IsSuccLimit o) (points : ZFSet.rank P < o) :
ZFSet.rank (fixedTopLayerWitness P) < o :=
countableLayerWitness_rank_lt limit (rank_singleton_lt limit points)
(rank_singleton_lt limit (rank_orderedPair_lt limit points
(by simpa only [ZFSet.rank_empty] using limit.pos)))
end
namespace AttachmentBatchInputs
open SetFormula
universe u
variable {M : TransitiveGround.{u}}
variable {P r μ I Qs rs μs As Es Fs Ss : ZFSet.{u}}
variable (d : AttachmentBatchInputs P r μ I Qs rs μs As Es Fs Ss)
theorem exists_batchEarlierLayerWitness
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0) (separation : SeparationSchema M.sets)
{HQs Hrs Hμs T birth stamp t N z : ZFSet.{u}} (hP : P ∈ M.sets) (hz : z ∈ M.sets)
(born : ZFSet.IsFunc P T birth) (before : t ∈ stamp)
(code : GrowthLayerWitnessCode HQs Hrs Hμs P r μ birth t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
d.batchCarrier d.batchOrder d.batchMeetGraph (d.batchBirth birth stamp) t N z' := by
have meetSubset : μ ⊆ d.batchMeetGraph := by
intro z hz
obtain ⟨xy, hxy, v, hv, rfl⟩ := ZFSet.mem_prod.1 (d.oldOrder.operation.1 hz)
obtain ⟨x, hx, y, hy, rfl⟩ := ZFSet.mem_prod.1 hxy
exact d.old_meet_entry hz
apply code.exists_literal_extension pairing unions separation hP hz
(fun _ => old_mem_taggedExtension) ?_ meetSubset (taggedBirth_subset _ _ _ _)
· intro x _ entry
have old := (taggedBirth_earlier_iff before).1 entry
exact ⟨(ZFSet.pair_mem_prod.1 (born.1 old)).1, old⟩
· intro z hz
obtain ⟨x, hx, y, hy, rfl⟩ := ZFSet.mem_prod.1 (d.oldOrder.order.relation hz)
exact (d.old_order_iff hx hy).2 hz
end AttachmentBatchInputs
namespace GroundCompressionState
open SetFormula
universe u
variable {M : TransitiveGround.{u}}
variable {P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters : ZFSet.{u}}
variable {seed : FixedCompressionSeed P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters}
variable (state : GroundCompressionState M seed)
variable (data : GroundCompressionUnionData M state.data I Sources Reps G m one T)
variable {base old HQs Hrs Hμs t N z : ZFSet.{u}}
variable (births : GroundCompressionBirthData M state.data I Sources Reps G m one T data base old)
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 (hI : I ∈ M.sets) (hSources : Sources ∈ M.sets) (hReps : Reps ∈ M.sets)
variable (hG : G ∈ M.sets) (hT : T ∈ M.sets)
variable {o : Ordinal.{u}} (limit : Order.IsSuccLimit o)
include pairing unions powers separation hI hSources hReps hG hT limit
theorem exists_birthWitness_to_finite_rank_lt (n k : Nat) (before : n ≤ k)
(earlier : t ∈ compressionStamp base n) (hz : z ∈ M.sets) (rank : ZFSet.rank z < o)
(code : GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T n).D
(state.data.iterate M I Sources Reps G T n).rD
(state.data.iterate M I Sources Reps G T n).μD
(graphRestriction (state.data.iterate M I Sources Reps G T n).D births.birth) t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T k).D
(state.data.iterate M I Sources Reps G T k).rD
(state.data.iterate M I Sources Reps G T k).μD
(graphRestriction (state.data.iterate M I Sources Reps G T k).D births.birth) t N z' ∧ ZFSet.rank z' < o := by
have mono := fun n k h => (state.canonical_iterate_graphs_mono pairing unions powers separation
hI hSources hReps hG hT (n := n) (k := k) h).1
have growth := state.canonical_iterate_graphs_mono pairing unions powers separation
hI hSources hReps hG hT before
obtain ⟨current, equal⟩ := state.canonical_iterate_valid pairing unions powers separation
hI hSources hReps hG hT n
have ground : (state.data.iterate M I Sources Reps G T n).D ∈ M.sets := equal ▸ current.ground.carrier
exact code.exists_literal_extension_rank_lt pairing unions separation ground hz limit rank growth.1 growth.2.1 growth.2.2.1
(NatBirthExtensionCode.restriction_mono mono before)
(fun _ _ entry => births.extension.earlier_fiber mono births.initial_function
(fun _ _ h => compressionStamp_subset base h) n earlier
(pair_mem_graphRestriction.1 entry).1)
theorem exists_birthWitness_to_union_rank_lt (n : Nat) (before : t ∈ compressionStamp base n)
(hz : z ∈ M.sets) (rank : ZFSet.rank z < o)
(code : GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T n).D
(state.data.iterate M I Sources Reps G T n).rD
(state.data.iterate M I Sources Reps G T n).μD
(graphRestriction (state.data.iterate M I Sources Reps G T n).D births.birth) t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(data.fields 0) (data.fields 1) (data.fields 2) births.birth t N z' ∧ ZFSet.rank z' < o := by
have mono := fun n k h => (state.canonical_iterate_graphs_mono pairing unions powers separation
hI hSources hReps hG hT (n := n) (k := k) h).1
obtain ⟨current, equal⟩ := state.canonical_iterate_valid pairing unions powers separation
hI hSources hReps hG hT n
have ground : (state.data.iterate M I Sources Reps G T n).D ∈ M.sets := equal ▸ current.ground.carrier
exact code.exists_literal_extension_rank_lt pairing unions separation ground hz limit rank
((data.fieldUnion 0).subset n) ((data.fieldUnion 1).subset n) ((data.fieldUnion 2).subset n)
(graphRestriction_subset (state.data.iterate M I Sources Reps G T n).D births.birth)
(fun _ _ entry => births.extension.earlier_fiber mono births.initial_function
(fun _ _ h => compressionStamp_subset base h) n before entry)
theorem exists_priorWitness_to_finite_rank_lt (k : Nat) (before : t ∈ base ∨ t = base) (hz : z ∈ M.sets) (rank : ZFSet.rank z < o)
(code : GrowthLayerWitnessCode HQs Hrs Hμs state.data.D state.data.rD state.data.μD old t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T k).D
(state.data.iterate M I Sources Reps G T k).rD
(state.data.iterate M I Sources Reps G T k).μD
(graphRestriction (state.data.iterate M I Sources Reps G T k).D births.birth) t N z' ∧ ZFSet.rank z' < o := by
apply state.exists_birthWitness_to_finite_rank_lt data births pairing unions powers separation hI hSources hReps hG hT limit
0 k (Nat.zero_le k) (compressionStamp_old_mem before 0) hz rank
simpa only [RecursiveStateData.iterate_zero, births.original] using code
theorem exists_priorWitness_to_union_rank_lt (before : t ∈ base ∨ t = base) (hz : z ∈ M.sets) (rank : ZFSet.rank z < o)
(code : GrowthLayerWitnessCode HQs Hrs Hμs state.data.D state.data.rD state.data.μD old t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(data.fields 0) (data.fields 1) (data.fields 2) births.birth t N z' ∧ ZFSet.rank z' < o := by
apply state.exists_birthWitness_to_union_rank_lt data births pairing unions powers separation hI hSources hReps hG hT limit
0 (compressionStamp_old_mem before 0) hz rank
simpa only [RecursiveStateData.iterate_zero, births.original] using code
omit unions powers separation hI hSources hReps hG hT in
theorem exists_limitWitness_rank_lt :
∃ z ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(data.fields 0) (data.fields 1) (data.fields 2) births.birth births.limit N z ∧ ZFSet.rank z < o := by
have hEmpty := M.empty_mem ⟨data.fields 0, data.ground 0⟩
exact ⟨countableLayerWitness ∅ ∅, M.countableLayerWitness_mem pairing hEmpty hEmpty,
Or.inl ⟨∅, ∅, rfl, ⟨InjectionCode.empty N, fun x _ entry => (births.no_limit_birth x entry).elim⟩⟩,
countableLayerWitness_rank_lt limit (by simpa only [ZFSet.rank_empty] using limit.pos)
(by simpa only [ZFSet.rank_empty] using limit.pos)⟩
end GroundCompressionState
namespace GroundCompressionState
open SetFormula
universe u
variable {M : TransitiveGround.{u}}
variable {P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters : ZFSet.{u}}
variable {seed : FixedCompressionSeed P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters}
variable (state : GroundCompressionState M seed)
variable (data : GroundCompressionUnionData M state.data I Sources Reps G m one T)
variable {base old HQs Hrs Hμs t N z : ZFSet.{u}}
variable (births : GroundCompressionBirthData M state.data I Sources Reps G m one T data base old)
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 (hI : I ∈ M.sets) (hSources : Sources ∈ M.sets) (hReps : Reps ∈ M.sets)
variable (hG : G ∈ M.sets) (hT : T ∈ M.sets)
include pairing unions powers separation hI hSources hReps hG hT
theorem exists_birthWitness_to_finite (n k : Nat) (before : n ≤ k)
(earlier : t ∈ compressionStamp base n) (hz : z ∈ M.sets)
(code : GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T n).D
(state.data.iterate M I Sources Reps G T n).rD
(state.data.iterate M I Sources Reps G T n).μD
(graphRestriction (state.data.iterate M I Sources Reps G T n).D births.birth) t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T k).D
(state.data.iterate M I Sources Reps G T k).rD
(state.data.iterate M I Sources Reps G T k).μD
(graphRestriction (state.data.iterate M I Sources Reps G T k).D births.birth) t N z' := by
have mono := fun n k h => (state.canonical_iterate_graphs_mono pairing unions powers separation
hI hSources hReps hG hT (n := n) (k := k) h).1
have growth := state.canonical_iterate_graphs_mono pairing unions powers separation
hI hSources hReps hG hT before
obtain ⟨current, equal⟩ := state.canonical_iterate_valid pairing unions powers separation
hI hSources hReps hG hT n
have ground : (state.data.iterate M I Sources Reps G T n).D ∈ M.sets := equal ▸ current.ground.carrier
exact code.exists_literal_extension pairing unions separation ground hz growth.1 growth.2.1 growth.2.2.1
(NatBirthExtensionCode.restriction_mono mono before)
(fun _ _ entry => births.extension.earlier_fiber mono births.initial_function
(fun _ _ h => compressionStamp_subset base h) n earlier
(pair_mem_graphRestriction.1 entry).1)
theorem exists_birthWitness_to_union (n : Nat) (before : t ∈ compressionStamp base n)
(hz : z ∈ M.sets)
(code : GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T n).D
(state.data.iterate M I Sources Reps G T n).rD
(state.data.iterate M I Sources Reps G T n).μD
(graphRestriction (state.data.iterate M I Sources Reps G T n).D births.birth) t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(data.fields 0) (data.fields 1) (data.fields 2) births.birth t N z' := by
have mono := fun n k h => (state.canonical_iterate_graphs_mono pairing unions powers separation
hI hSources hReps hG hT (n := n) (k := k) h).1
obtain ⟨current, equal⟩ := state.canonical_iterate_valid pairing unions powers separation
hI hSources hReps hG hT n
have ground : (state.data.iterate M I Sources Reps G T n).D ∈ M.sets := equal ▸ current.ground.carrier
exact code.exists_literal_extension pairing unions separation ground hz
((data.fieldUnion 0).subset n) ((data.fieldUnion 1).subset n) ((data.fieldUnion 2).subset n)
(graphRestriction_subset (state.data.iterate M I Sources Reps G T n).D births.birth)
(fun _ _ entry => births.extension.earlier_fiber mono births.initial_function
(fun _ _ h => compressionStamp_subset base h) n before entry)
theorem exists_priorWitness_to_finite (k : Nat) (before : t ∈ base ∨ t = base) (hz : z ∈ M.sets)
(code : GrowthLayerWitnessCode HQs Hrs Hμs state.data.D state.data.rD state.data.μD old t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(state.data.iterate M I Sources Reps G T k).D
(state.data.iterate M I Sources Reps G T k).rD
(state.data.iterate M I Sources Reps G T k).μD
(graphRestriction (state.data.iterate M I Sources Reps G T k).D births.birth) t N z' := by
apply state.exists_birthWitness_to_finite data births pairing unions powers separation hI hSources hReps hG hT
0 k (Nat.zero_le k) (compressionStamp_old_mem before 0) hz
simpa only [RecursiveStateData.iterate_zero, births.original] using code
theorem exists_priorWitness_to_union (before : t ∈ base ∨ t = base) (hz : z ∈ M.sets)
(code : GrowthLayerWitnessCode HQs Hrs Hμs state.data.D state.data.rD state.data.μD old t N z) :
∃ z' ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(data.fields 0) (data.fields 1) (data.fields 2) births.birth t N z' := by
apply state.exists_birthWitness_to_union data births pairing unions powers separation hI hSources hReps hG hT
0 (compressionStamp_old_mem before 0) hz
simpa only [RecursiveStateData.iterate_zero, births.original] using code
omit unions powers separation hI hSources hReps hG hT in
theorem exists_limitWitness :
∃ z ∈ M.sets, GrowthLayerWitnessCode HQs Hrs Hμs
(data.fields 0) (data.fields 1) (data.fields 2) births.birth births.limit N z := by
have hEmpty := M.empty_mem ⟨data.fields 0, data.ground 0⟩
exact ⟨countableLayerWitness ∅ ∅, M.countableLayerWitness_mem pairing hEmpty hEmpty,
Or.inl ⟨∅, ∅, rfl, ⟨InjectionCode.empty N, fun x _ entry => (births.no_limit_birth x entry).elim⟩⟩⟩
end GroundCompressionState
end PartitionPilot.Forcing
end OAI
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