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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import OAI.SetTheory.PartitionPrinciple.Definability.TaggedSumPresentations
namespace OAI
namespace PartitionPilot.Forcing
universe u
structure EquivalenceRelationCode (S E : ZFSet.{u}) : Prop where
shape : E ⊆ ZFSet.prod S S
refl : ∀ x ∈ S, ZFSet.pair x x ∈ E
symm : ∀ x y, ZFSet.pair x y ∈ E → ZFSet.pair y x ∈ E
trans : ∀ x y z, ZFSet.pair x y ∈ E → ZFSet.pair y z ∈ E → ZFSet.pair x z ∈ E
def canonicalRepresentativeProperty (S E c x : ZFSet.{u}) : Prop :=
∃ n ∈ ZFSet.omega, ZFSet.pair x n ∈ c ∧
∀ y ∈ S, ZFSet.pair x y ∈ E → ∀ m ∈ n, ZFSet.pair y m ∉ c
noncomputable def canonicalRepresentatives (S E c : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (canonicalRepresentativeProperty S E c) S
theorem mem_canonicalRepresentatives (S E c x : ZFSet.{u}) :
x ∈ canonicalRepresentatives S E c ↔ x ∈ S ∧ canonicalRepresentativeProperty S E c x :=
ZFSet.mem_sep
theorem canonicalRepresentatives_subset (S E c : ZFSet.{u}) :
canonicalRepresentatives S E c ⊆ S :=
fun _ h => (ZFSet.mem_sep.1 h).1
theorem canonical_exists {S E c x : ZFSet.{u}}
(equivalence : EquivalenceRelationCode S E) (injection : InjectionCode S ZFSet.omega c)
(hx : x ∈ S) : ∃ y ∈ canonicalRepresentatives S E c, ZFSet.pair x y ∈ E := by
classical
let valid : Nat → Prop := fun n => ∃ y ∈ S, ZFSet.pair x y ∈ E ∧ ZFSet.pair y (natSetCode n) ∈ c
have existsValid : ∃ n, valid n := by
obtain ⟨a, entry, _⟩ := injection.function.2 x hx
have aNat := (ZFSet.pair_mem_prod.1 (injection.function.1 entry)).2
obtain ⟨n, rfl⟩ := (mem_omega_iff_natSetCode a).1 aNat
exact ⟨n, x, hx, equivalence.refl x hx, entry⟩
obtain ⟨y, hy, related, entry⟩ := Nat.find_spec existsValid
refine ⟨y, (mem_canonicalRepresentatives S E c y).2
⟨hy, natSetCode (Nat.find existsValid), natSetCode_mem_omega _, entry, ?_⟩, related⟩
intro z hz yz m hm smaller
obtain ⟨k, lower, equal⟩ := (mem_natSetCode m (Nat.find existsValid)).1 hm
have validK : valid k := ⟨z, hz, equivalence.trans x y z related yz, equal ▸ smaller⟩
exact Nat.not_lt_of_ge (Nat.find_min' existsValid validK) lower
theorem canonical_equiv_eq {S E c x y : ZFSet.{u}}
(equivalence : EquivalenceRelationCode S E) (injection : InjectionCode S ZFSet.omega c)
(hx : x ∈ canonicalRepresentatives S E c) (hy : y ∈ canonicalRepresentatives S E c)
(related : ZFSet.pair x y ∈ E) : x = y := by
obtain ⟨xS, n, hn, xn, leastX⟩ := (mem_canonicalRepresentatives S E c x).1 hx
obtain ⟨yS, m, hm, ym, leastY⟩ := (mem_canonicalRepresentatives S E c y).1 hy
obtain ⟨k, rfl⟩ := (mem_omega_iff_natSetCode n).1 hn
obtain ⟨l, rfl⟩ := (mem_omega_iff_natSetCode m).1 hm
rcases lt_trichotomy k l with smaller | equal | larger
· exact False.elim (leastY x xS (equivalence.symm x y related) (natSetCode k)
((natSetCode_mem_iff k l).2 smaller) xn)
· subst l
exact injection.injective x y (natSetCode k) xn ym
· exact False.elim (leastX y yS related (natSetCode l)
((natSetCode_mem_iff l k).2 larger) ym)
theorem canonical_exists_unique {S E c x : ZFSet.{u}}
(equivalence : EquivalenceRelationCode S E) (injection : InjectionCode S ZFSet.omega c)
(hx : x ∈ S) : ∃! y, y ∈ canonicalRepresentatives S E c ∧ ZFSet.pair x y ∈ E := by
obtain ⟨y, hy, related⟩ := canonical_exists equivalence injection hx
refine ⟨y, ⟨hy, related⟩, ?_⟩
intro z hz
exact canonical_equiv_eq equivalence injection hz.1 hy
(equivalence.trans z x y (equivalence.symm x z hz.2) related)
noncomputable def canonicalRepresentativesInjection (S E c : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => z ∈ ZFSet.prod (canonicalRepresentatives S E c) ZFSet.omega) c
theorem canonicalRepresentativesInjection_code {S E c : ZFSet.{u}}
(injection : InjectionCode S ZFSet.omega c) :
InjectionCode (canonicalRepresentatives S E c) ZFSet.omega (canonicalRepresentativesInjection S E c) :=
injection.restrict (canonicalRepresentatives_subset S E c)
namespace BoundedFormula
open SetFormula
def canonicalRepresentativeFormula {n} (x s e c omega : Fin n) : BoundedFormula n :=
exMem omega (.conj (pairMembershipFormula x.succ 0 c.succ)
(.allMem s.succ (imp (pairMembershipFormula x.succ.succ 0 e.succ.succ)
(.allMem 1 (.neg (pairMembershipFormula 1 0 c.succ.succ.succ))))))
theorem truth_canonicalRepresentativeFormula {n} (x s e c omega : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (canonicalRepresentativeFormula x s e c omega) env ↔
∃ k ∈ env omega, ZFSet.pair (env x) k ∈ env c ∧
∀ y ∈ env s, ZFSet.pair (env x) y ∈ env e → ∀ m ∈ k, ZFSet.pair y m ∉ env c := by
simp only [canonicalRepresentativeFormula, truth_exMem, Truth, truth_imp, truth_pairMembershipFormula]
rfl
end BoundedFormula
namespace TransitiveGround
open SetFormula
variable (M : TransitiveGround.{u})
theorem canonicalRepresentatives_mem (separation : SeparationSchema M.sets)
{S E c : ZFSet.{u}} (hS : S ∈ M.sets) (hE : E ∈ M.sets) (hc : c ∈ M.sets)
(hOmega : ZFSet.omega ∈ M.sets) : canonicalRepresentatives S E c ∈ M.sets := by
let params := extend S (extend E (extend c (extend ZFSet.omega Fin.elim0)))
have selected := M.bounded_separation_mem separation
(BoundedFormula.canonicalRepresentativeFormula 0 1 2 3 4) params
(Fin.cases hS (Fin.cases hE (Fin.cases hc (Fin.cases hOmega (fun i => Fin.elim0 i))))) hS
have eq : ZFSet.sep (fun x => BoundedFormula.Truth
(BoundedFormula.canonicalRepresentativeFormula 0 1 2 3 4) (extend x params)) S =
canonicalRepresentatives S E c := by
apply ZFSet.ext
intro x
rw [ZFSet.mem_sep, mem_canonicalRepresentatives]
exact and_congr Iff.rfl (BoundedFormula.truth_canonicalRepresentativeFormula 0 1 2 3 4 _)
exact eq ▸ selected
theorem canonicalRepresentativesInjection_mem (separation : SeparationSchema M.sets)
{S E c : ZFSet.{u}} (hS : S ∈ M.sets) (hE : E ∈ M.sets) (hc : c ∈ M.sets)
(hOmega : ZFSet.omega ∈ M.sets) : canonicalRepresentativesInjection S E c ∈ M.sets :=
M.injection_restriction_mem separation
(M.canonicalRepresentatives_mem separation hS hE hc hOmega) hOmega hc
theorem canonicalRepresentatives_spec (separation : SeparationSchema M.sets)
{S E c : ZFSet.{u}} (hS : S ∈ M.sets) (hE : E ∈ M.sets) (hc : c ∈ M.sets)
(hOmega : ZFSet.omega ∈ M.sets)
(equivalence : EquivalenceRelationCode S E) (injection : InjectionCode S ZFSet.omega c) :
canonicalRepresentatives S E c ∈ M.sets ∧ canonicalRepresentatives S E c ⊆ S ∧
canonicalRepresentativesInjection S E c ∈ M.sets ∧
InjectionCode (canonicalRepresentatives S E c) ZFSet.omega (canonicalRepresentativesInjection S E c) ∧
∀ x ∈ S, ∃! y, y ∈ canonicalRepresentatives S E c ∧ ZFSet.pair x y ∈ E :=
⟨M.canonicalRepresentatives_mem separation hS hE hc hOmega,
canonicalRepresentatives_subset S E c,
M.canonicalRepresentativesInjection_mem separation hS hE hc hOmega,
canonicalRepresentativesInjection_code injection,
fun _ hx => canonical_exists_unique equivalence injection hx⟩
end TransitiveGround
end PartitionPilot.Forcing
end OAI
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