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9dbc64e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | import OAI.Computability.DegreeRigidity.Representation.SourceTheory
import OAI.Computability.DegreeRigidity.Syntax.FiniteTerm
import OAI.Computability.DegreeRigidity.Syntax.BoundedFormulaRename
namespace OAI
namespace TuringRigidity.BoundedTruthRecursion
open TransitiveNameModel BoundedSetTheory
def Local (φ : Formula) (e : ℕ → ZFSet.{0}) : Prop :=
∀ D : ZFSet.{0}, Transitive D → ∀ t ∈ D, ∀ Y Z : ZFSet.{0},
(∀ s ∈ D, ZFSet.rank s < ZFSet.rank t → (s ∈ Y ↔ s ∈ Z)) →
(φ.Eval (cons t (cons Y e)) ↔ φ.Eval (cons t (cons Z e)))
def Partial (φ : Formula) (e : ℕ → ZFSet.{0}) (K D Y : ZFSet.{0}) : Prop :=
Transitive D ∧ D ⊆ K ∧ Y ⊆ D ∧
∀ t ∈ D, t ∈ Y ↔ φ.Eval (cons t (cons Y e))
theorem partial_sound (φ : Formula) (e : ℕ → ZFSet.{0}) (K D Y Z : ZFSet.{0})
(hl : Local φ e) (hp : Partial φ e K D Y)
(hz : ∀ t ∈ K, t ∈ Z ↔ φ.Eval (cons t (cons Z e))) :
∀ t ∈ D, t ∈ Y ↔ t ∈ Z := by
have wf : WellFounded (fun a b : ZFSet.{0} => ZFSet.rank a < ZFSet.rank b) :=
InvImage.wf ZFSet.rank Ordinal.lt_wf
intro t
induction t using wf.induction with
| h t ih =>
intro ht
exact (hp.2.2.2 t ht).trans ((hl D hp.1 t ht Y Z
(fun s hs hst => ih s hst hs)).trans (hz t (hp.2.1 ht)).symm)
def dropSlots : ℕ → ℕ
| 0 => 0
| 1 => 1
| n+2 => n+4
def partialFormula (φ : Formula) : Formula :=
.conj (.transitive 1) (.conj (.subset 1 2) (.conj (.subset 0 1)
(.allMem 1 (.iff (.member 0 1) (φ.rename dropSlots)))))
theorem partialFormula_spec (φ : Formula) (e : ℕ → ZFSet.{0}) (K D Y : ZFSet.{0}) :
(partialFormula φ).Eval (cons Y (cons D (cons K e))) ↔ Partial φ e K D Y := by
simp only [partialFormula,Formula.Eval,Formula.eval_transitive,Formula.eval_subset,
Formula.eval_allMem,Formula.eval_iff,Formula.eval_rename_comp,cons_zero,cons_succ,Partial]
have eq (t : ZFSet.{0}) : cons t (cons Y (cons D (cons K e))) ∘ dropSlots = cons t (cons Y e) := by
funext i; rcases i with _|_|i <;> rfl
simp only [eq]
noncomputable def domain (φ : Formula) (e : ℕ → ZFSet.{0}) (K Q : ZFSet.{0}) : ZFSet.{0} :=
K.sep (fun t => ∃ D ∈ Q, ∃ Y ∈ Q, Partial φ e K D Y ∧ t ∈ D)
noncomputable def truth (φ : Formula) (e : ℕ → ZFSet.{0}) (K Q : ZFSet.{0}) : ZFSet.{0} :=
K.sep (fun t => ∃ D ∈ Q, ∃ Y ∈ Q, Partial φ e K D Y ∧ t ∈ Y)
def candidateSlots : ℕ → ℕ
| 0 => 0
| 1 => 1
| n+2 => n+4
def unionFormula (φ : Formula) (which : ℕ) : Formula :=
.existsMem 1 (.existsMem 2 (.conj ((partialFormula φ).rename candidateSlots) (.member 2 which)))
theorem unionFormula_spec (φ : Formula) (e : ℕ → ZFSet.{0}) (K Q t : ZFSet.{0}) (which : ℕ) :
(unionFormula φ which).Eval (cons t (cons Q (cons K e))) ↔
∃ D ∈ Q, ∃ Y ∈ Q, Partial φ e K D Y ∧
t ∈ cons Y (cons D (cons t (cons Q (cons K e)))) which := by
simp only [unionFormula,Formula.Eval,Formula.eval_rename_comp,cons_zero,cons_succ]
have eq (D Y : ZFSet.{0}) :
cons Y (cons D (cons t (cons Q (cons K e)))) ∘ candidateSlots = cons Y (cons D (cons K e)) := by
funext i; rcases i with _|_|i <;> rfl
simp only [eq,partialFormula_spec]
theorem domain_truth_mem (M K Q : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hK : K ∈ M) (hQ : Q ∈ M) (φ : Formula) (e : ℕ → ZFSet.{0}) (he : ∀ i, e i ∈ M) :
domain φ e K Q ∈ M ∧ truth φ e K Q ∈ M := by
have henv : ∀ i, cons Q (cons K e) i ∈ M := by
intro i; rcases i with _|_|i; exact hQ; exact hK; exact he i
constructor
· have hs := sep_mem M hM hT.separation.finitePrefix.bounded (unionFormula φ 1) _ henv hK
simpa only [unionFormula_spec,cons_succ,cons_zero,domain] using hs
· have hs := sep_mem M hM hT.separation.finitePrefix.bounded (unionFormula φ 0) _ henv hK
simpa only [unionFormula_spec,cons_succ,cons_zero,truth] using hs
theorem domain_transitive (φ : Formula) (e : ℕ → ZFSet.{0}) (K Q : ZFSet.{0}) :
Transitive (domain φ e K Q) := by
intro t ht s hst
obtain ⟨_,D,hD,Y,hY,hp,htD⟩ := ZFSet.mem_sep.mp ht
have hsD := hp.1 t htD s hst
exact ZFSet.mem_sep.mpr ⟨hp.2.1 hsD,D,hD,Y,hY,hp,hsD⟩
theorem truth_subset_domain (φ : Formula) (e : ℕ → ZFSet.{0}) (K Q : ZFSet.{0}) :
truth φ e K Q ⊆ domain φ e K Q := by
intro t ht
obtain ⟨htK,D,hD,Y,hY,hp,htY⟩ := ZFSet.mem_sep.mp ht
exact ZFSet.mem_sep.mpr ⟨htK,D,hD,Y,hY,hp,hp.2.2.1 htY⟩
theorem truth_agrees (φ : Formula) (e : ℕ → ZFSet.{0}) (K Q Z : ZFSet.{0})
(hl : Local φ e) (hz : ∀ t ∈ K, t ∈ Z ↔ φ.Eval (cons t (cons Z e))) :
∀ t ∈ domain φ e K Q, t ∈ truth φ e K Q ↔ t ∈ Z := by
intro t ht
constructor
· intro h
obtain ⟨_,D,_,Y,_,hp,htY⟩ := ZFSet.mem_sep.mp h
exact (partial_sound φ e K D Y Z hl hp hz t (hp.2.2.1 htY)).mp htY
· intro h
obtain ⟨htK,D,hD,Y,hY,hp,htD⟩ := ZFSet.mem_sep.mp ht
exact ZFSet.mem_sep.mpr ⟨htK,D,hD,Y,hY,hp,(partial_sound φ e K D Y Z hl hp hz t htD).mpr h⟩
theorem partial_of_agrees (φ : Formula) (e : ℕ → ZFSet.{0}) (K D Y Z : ZFSet.{0})
(hl : Local φ e) (hz : ∀ t ∈ K, t ∈ Z ↔ φ.Eval (cons t (cons Z e)))
(hD : Transitive D) (hDK : D ⊆ K) (hYD : Y ⊆ D)
(ha : ∀ t ∈ D, t ∈ Y ↔ t ∈ Z) : Partial φ e K D Y := by
refine ⟨hD,hDK,hYD,?_⟩
intro t ht
exact (ha t ht).trans ((hz t (hDK ht)).trans
(hl D hD t ht Y Z (fun s hs _ => ha s hs)).symm)
theorem truth_internal (M K Z : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hKM : K ∈ M) (hK : Transitive K) (φ : Formula) (e : ℕ → ZFSet.{0})
(he : ∀ i, e i ∈ M) (hl : Local φ e)
(hz : ∀ t ∈ K, t ∈ Z ↔ φ.Eval (cons t (cons Z e))) :
K.sep (fun t => t ∈ Z) ∈ M := by
classical
obtain ⟨Q,hQM,hQ⟩ := internal_power M hM hT.powerSet hKM
let D := domain φ e K Q
let Y := truth φ e K Q
obtain ⟨hDM,hYM⟩ := domain_truth_mem M K Q hM hT hKM hQM φ e he
have hD : Transitive D := domain_transitive φ e K Q
have hDK : D ⊆ K := fun _ h => (ZFSet.mem_sep.mp h).1
have hYD : Y ⊆ D := truth_subset_domain φ e K Q
have ha : ∀ t ∈ D, t ∈ Y ↔ t ∈ Z := truth_agrees φ e K Q Z hl hz
have hcover (t : ZFSet.{0}) : t ∈ K → t ∈ D := by
induction t using ZFSet.inductionOn with
| h t ih =>
intro htK
by_cases htD : t ∈ D
· exact htD
have hchildren : t ⊆ D := fun s hs => ih s hs (hK t htK s hs)
let D' := D ∪ ({t} : ZFSet.{0})
let Y' := if t ∈ Z then Y ∪ ({t} : ZFSet.{0}) else Y
have hDD' : D ⊆ D' := fun _ h => ZFSet.mem_union.mpr (Or.inl h)
have htD' : t ∈ D' := ZFSet.mem_union.mpr (Or.inr (ZFSet.mem_singleton.mpr rfl))
have hD' : Transitive D' := by
intro a ha b hb
rcases ZFSet.mem_union.mp ha with ha | ha
· exact hDD' (hD a ha b hb)
· have ha := ZFSet.mem_singleton.mp ha
subst a
exact hDD' (hchildren hb)
have hD'K : D' ⊆ K := by
intro a ha
rcases ZFSet.mem_union.mp ha with ha | ha
· exact hDK ha
· exact (ZFSet.mem_singleton.mp ha) ▸ htK
have hY'D' : Y' ⊆ D' := by
intro a ha
dsimp [Y'] at ha
split_ifs at ha
· rcases ZFSet.mem_union.mp ha with ha | ha
· exact hDD' (hYD ha)
· exact (ZFSet.mem_singleton.mp ha) ▸ htD'
· exact hDD' (hYD ha)
have ha' : ∀ a ∈ D', a ∈ Y' ↔ a ∈ Z := by
intro a haD'
by_cases hat : a = t
· subst a
have htY : t ∉ Y := fun h => htD (hYD h)
simp only [Y']
split_ifs with htZ <;> simp [htZ,htY]
· have haD : a ∈ D := (ZFSet.mem_union.mp haD').resolve_right (fun h => hat (ZFSet.mem_singleton.mp h))
have hae := ha a haD
dsimp [Y']
split_ifs <;> simpa only [ZFSet.mem_union,ZFSet.mem_singleton,hat,or_false] using hae
have htM := hM K hKM t htK
have hsM := singleton_mem M hM hT.pairing htM
have hD'M : D' ∈ M := binary_union_mem M hM hT.pairing hT.union hDM hsM
have hY'M : Y' ∈ M := by
dsimp [Y']; split_ifs
· exact binary_union_mem M hM hT.pairing hT.union hYM hsM
· exact hYM
exact ZFSet.mem_sep.mpr ⟨htK,D',(hQ _).mpr ⟨hD'M,hD'K⟩,
Y',(hQ _).mpr ⟨hY'M,fun _ h => hD'K (hY'D' h)⟩,
partial_of_agrees φ e K D' Y' Z hl hz hD' hD'K hY'D' ha',htD'⟩
have eq : Y = K.sep (fun t => t ∈ Z) := by
apply ZFSet.ext; intro t
constructor
· intro ht
exact ZFSet.mem_sep.mpr ⟨hDK (hYD ht),(ha t (hYD ht)).mp ht⟩
· intro ht
obtain ⟨htK,htZ⟩ := ZFSet.mem_sep.mp ht
exact (ha t (hcover t htK)).mpr htZ
exact eq ▸ hYM
end TuringRigidity.BoundedTruthRecursion
end OAI
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