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e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundCodedBinaryHeights
namespace OAI
section
namespace PartitionPilot.Forcing
open SetRealization Compression.RecursiveMaps
universe u
theorem successorColumnGraph_injection {C base sequence : ZFSet.{u}}
(function : ZFSet.IsFunc ZFSet.omega C sequence)
(entries : ∀ n y, ZFSet.pair (natSetCode n) y ∈ sequence ↔ y = compressionStamp base n) :
InjectionCode ZFSet.omega C sequence := by
refine ⟨function, ?_⟩
intro x y z first second
obtain ⟨n, rfl⟩ := (mem_omega_iff_natSetCode x).1 (ZFSet.pair_mem_prod.1 (function.1 first)).1
obtain ⟨m, rfl⟩ := (mem_omega_iff_natSetCode y).1 (ZFSet.pair_mem_prod.1 (function.1 second)).1
have equal := ((entries n z).1 first).symm.trans ((entries m z).1 second)
have same : n = m := by
by_contra different
rcases Nat.lt_or_gt_of_ne different with lower | greater
· have member := (compressionStamp_mem_iff base n m).2 lower
rw [equal] at member
exact ZFSet.mem_irrefl _ member
· have member := (compressionStamp_mem_iff base m n).2 greater
rw [equal] at member
exact ZFSet.mem_irrefl _ member
exact congrArg natSetCode same
structure ExternalColumnCode (C K base sequence cap : ZFSet.{u}) : Prop where
function : ZFSet.IsFunc ZFSet.omega C sequence
entries : ∀ n y, ZFSet.pair (natSetCode n) y ∈ sequence ↔ y = compressionStamp base n
base_mem : base ∈ C
cap_mem : cap ∈ C
old_bound : ∀ a ∈ K, a ∈ base
cap_bound : ∀ n, compressionStamp base n ∈ cap
namespace ExternalColumnCode
variable {C K base sequence cap : ZFSet.{u}} (code : ExternalColumnCode C K base sequence cap)
include code
theorem value_mem (n : Nat) : compressionStamp base n ∈ C :=
(ZFSet.pair_mem_prod.1 (code.function.1 ((code.entries n _).2 rfl))).2
theorem old_below {a : ZFSet.{u}} (ha : a ∈ K) (n : Nat) : a ∈ compressionStamp base n :=
compressionStamp_old_mem (Or.inl (code.old_bound a ha)) n
theorem injection : InjectionCode ZFSet.omega C sequence :=
successorColumnGraph_injection code.function code.entries
noncomputable def smallColumn (n : Nat) : SetMembers C :=
SetMembers.ofMem (compressionStamp base n) (code.value_mem n)
theorem smallColumn_graph :
natPresentation.graph (SetMembers.presentation C) (fun n => code.smallColumn n.down) = sequence := by
apply ZFSet.ext
intro z
constructor
· intro hz
obtain ⟨n, rfl⟩ := (natPresentation.mem_graph (SetMembers.presentation C) _ z).1 hz
exact (code.entries n.down _).2 (SetMembers.raw_ofMem _ _)
· intro hz
obtain ⟨x, hx, y, _, rfl⟩ := ZFSet.mem_prod.1 (code.function.1 hz)
obtain ⟨n, rfl⟩ := (mem_omega_iff_natSetCode x).1 hx
have equal := (code.entries n y).1 hz
exact (natPresentation.mem_graph (SetMembers.presentation C) _ _).2
⟨⟨n⟩, by change ZFSet.pair (natSetCode n) (SetMembers.raw (code.smallColumn n)) = _
simp only [smallColumn, SetMembers.raw_ofMem, equal]⟩
theorem smallColumn_increasing (n : Nat) :
SetMembers.raw (code.smallColumn n) ∈ SetMembers.raw (code.smallColumn (n + 1)) := by
simp only [smallColumn, SetMembers.raw_ofMem]
exact (compressionStamp_mem_iff base n (n + 1)).2 (Nat.lt_succ_self n)
theorem smallColumn_bounded (n : Nat) : SetMembers.raw (code.smallColumn n) ∈ cap := by
simpa only [smallColumn, SetMembers.raw_ofMem] using code.cap_bound n
theorem old_height_below {P D old : ZFSet.{u}}
(function : ZFSet.IsFunc P (ZFSet.prod C D) old)
(columns : heightColumns P C D old ⊆ K)
{x a q : ZFSet.{u}} (entry : ZFSet.pair x (ZFSet.pair a q) ∈ old) (n : Nat) :
a ∈ SetMembers.raw (code.smallColumn n) := by
simpa only [smallColumn, SetMembers.raw_ofMem] using
code.old_below (columns (mem_heightColumns_of_entry function entry)) n
end ExternalColumnCode
structure GroundExternalColumnData (M : TransitiveGround.{u}) (C K : ZFSet.{u}) where
base : ZFSet.{u}
sequence : ZFSet.{u}
cap : ZFSet.{u}
sequence_ground : sequence ∈ M.sets
code : ExternalColumnCode C K base sequence cap
range_ground : relationImage ZFSet.omega C sequence ∈ M.sets
inverse_ground : relationConverse ZFSet.omega C sequence ∈ M.sets
inverse_code : InjectionCode (relationImage ZFSet.omega C sequence) ZFSet.omega
(relationConverse ZFSet.omega C sequence)
end PartitionPilot.Forcing
end
end OAI
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