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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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundBinaryRows
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization
universe u
variable (M : TransitiveGround.{u})
noncomputable def binaryRowName (I C : ZFSet.{u})
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(i : ZFSet.{u}) :
Name (SetMembers.presentation (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) :=
Name.assemble
(fun z : {z : SetMembers C × SetMembers
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega) //
ZFSet.pair (ZFSet.pair i (SetMembers.raw z.1)) (natSetCode 1) ∈ SetMembers.raw z.2} =>
Name.check (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) one (SetMembers.raw z.val.1))
(fun z => z.val.2)
theorem entry_binaryRowName (I C : ZFSet.{u})
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(i : ZFSet.{u})
(σ : Name (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)))
(p : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) :
Name.Entry σ (M.binaryRowName I C one i) p ↔ ∃ c ∈ C,
Name.check (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) one c = σ ∧
ZFSet.pair (ZFSet.pair i c) (natSetCode 1) ∈ SetMembers.raw p := by
rw [binaryRowName, Name.entry_assemble]
constructor
· rintro ⟨z, eq, rfl⟩
exact ⟨SetMembers.raw z.val.1, SetMembers.raw_mem z.val.1, eq, z.property⟩
· rintro ⟨c, hc, eq, assigned⟩
refine ⟨⟨(SetMembers.ofMem c hc, p), ?_⟩, ?_, rfl⟩
· simpa only [SetMembers.raw_ofMem] using assigned
· simpa only [SetMembers.raw_ofMem] using eq
theorem mem_binaryRowName_code (I C : ZFSet.{u})
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(i z : ZFSet.{u}) :
z ∈ (M.binaryRowName I C one i).val ↔ ∃ c ∈ C,
∃ p ∈ M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega,
z = ZFSet.pair (Name.check (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) one c).val p ∧
ZFSet.pair (ZFSet.pair i c) (natSetCode 1) ∈ p := by
constructor
· intro hz
obtain ⟨σ, p, eq⟩ := Name.entries_cover (M.binaryRowName I C one i) hz
change z = ZFSet.pair σ.val (SetMembers.raw p) at eq
have entry : Name.Entry σ (M.binaryRowName I C one i) p := by
change ZFSet.pair σ.val (SetMembers.raw p) ∈ (M.binaryRowName I C one i).val
rw [← eq]
exact hz
obtain ⟨c, hc, child, assigned⟩ := (M.entry_binaryRowName I C one i σ p).1 entry
refine ⟨c, hc, SetMembers.raw p, SetMembers.raw_mem p, ?_, assigned⟩
rw [child]
exact eq
· rintro ⟨c, hc, p, hp, eq, assigned⟩
have entry := (M.entry_binaryRowName I C one i _ (SetMembers.ofMem p hp)).2
⟨c, hc, rfl, by simpa only [SetMembers.raw_ofMem] using assigned⟩
change ZFSet.pair (Name.check (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) one c).val
(SetMembers.raw (SetMembers.ofMem p hp)) ∈ (M.binaryRowName I C one i).val at entry
rw [SetMembers.raw_ofMem] at entry
exact eq.symm ▸ entry
theorem mem_eval_binaryRowName (I C : ZFSet.{u})
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(H : Set (SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)))
(hone : one ∈ H) (i x : ZFSet.{u}) :
x ∈ Name.eval H (M.binaryRowName I C one i) ↔ x ∈ C ∧
∃ p, p ∈ H ∧ ZFSet.pair (ZFSet.pair i x) (natSetCode 1) ∈ SetMembers.raw p := by
rw [Name.mem_eval]
constructor
· rintro ⟨σ, p, entry, hp, eq⟩
obtain ⟨c, hc, rfl, assigned⟩ := (M.entry_binaryRowName I C one i σ p).1 entry
rw [Name.eval_check H one hone] at eq
exact ⟨eq ▸ hc, p, hp, eq ▸ assigned⟩
· rintro ⟨hx, p, hp, assigned⟩
exact ⟨Name.check (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) one x, p,
(M.entry_binaryRowName I C one i _ p).2 ⟨x, hx, rfl, assigned⟩,
hp, Name.eval_check H one hone x⟩
theorem eval_binaryRowName (I C : ZFSet.{u})
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) :
letI := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
∀ H : ConditionFilter (SetMembers
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)),
H.mem one → ∀ i : ZFSet.{u},
Name.eval {p | H.mem p} (M.binaryRowName I C one i) = M.binaryGenericRow I C H i := by
let := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
intro H hone i
apply ZFSet.ext
intro x
rw [M.mem_eval_binaryRowName I C one {p | H.mem p} hone i x,
M.mem_binaryGenericRow I C H i x,
M.mem_countableGenericGraph (ZFSet.prod I C) (natSetCode 2) H]
rfl
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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