File size: 16,724 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 | import OAI.SetTheory.PartitionPrinciple.Definability.GroundBinaryRowSets
namespace OAI
section
namespace PartitionPilot.Forcing.SetFormula
def binaryRowEntryFormula {n} (z c q i one bit : Fin n) : SetFormula n :=
ex (ex (ex (.conj (.mem 2 c.succ.succ.succ)
(.conj (.mem 1 q.succ.succ.succ)
(.conj (binaryApplication uniformCheckFormula (extend one.succ.succ.succ Fin.elim0) 2 0)
(.conj (BoundedFormula.tripleMembershipFormula i.succ.succ.succ 2 bit.succ.succ.succ 1).toFormula
(orderedPairFormula z.succ.succ.succ 0 1)))))))
def binaryRowCodeFormula : SetFormula 6 :=
let entry := reindex (extend 0 (extend 3 (extend 4 (extend 1
(extend 5 (extend 6 Fin.elim0)))))) (binaryRowEntryFormula 0 1 2 3 4 5)
.all (.conj (imp (.mem 0 2) entry) (imp entry (.mem 0 2)))
def binaryRowSetEntryFormula : SetFormula 6 :=
ex (ex (.conj (.mem 1 3)
(.conj (binaryApplication binaryRowCodeFormula
(extend 4 (extend 5 (extend 6 (extend 7 Fin.elim0)))) 1 0)
(orderedPairFormula 2 0 6))))
end PartitionPilot.Forcing.SetFormula
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem realize_binaryRowEntryFormula (I C : ZFSet.{u})
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(i : ZFSet.{u})
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0)
(separation : SeparationSchema M.sets) (collection : CollectionSchema M.sets)
(hC : C ∈ M.sets)
(hQ : M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega ∈ M.sets)
(hi : i ∈ M.sets) (hbit : (natSetCode 1 : ZFSet.{u}) ∈ M.sets)
(z : ZFSet.{u}) (hz : z ∈ M.sets) :
Realize M.sets (binaryRowEntryFormula 0 1 2 3 4 5)
(extend z (extend C (extend
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)
(extend i (extend (SetMembers.raw one) (extend (natSetCode 1) Fin.elim0)))))) ↔
z ∈ (M.binaryRowName I C one i).val := by
let Q := M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
let s := SetMembers.presentation Q
have hone : s.encode one ∈ M.sets := M.transitive hQ (SetMembers.raw_mem one)
have children (c : ZFSet.{u}) (hc : c ∈ C) : (Name.check s one c).val ∈ M.sets :=
M.check_mem s one pairing unions powers separation collection hone c (M.transitive hC hc)
have assignment (c p w : ZFSet.{u}) (hc : c ∈ M.sets) (hp : p ∈ M.sets) (hw : w ∈ M.sets) :
Realize M.sets (BoundedFormula.tripleMembershipFormula 6 2 8 1).toFormula
(extend w (extend p (extend c (extend z (extend C (extend Q
(extend i (extend (SetMembers.raw one) (extend (natSetCode 1) Fin.elim0))))))))) ↔
ZFSet.pair (ZFSet.pair i c) (natSetCode 1) ∈ p :=
(BoundedFormula.realize_toFormula M _ _
(Fin.cases hw (Fin.cases hp (Fin.cases hc (Fin.cases hz (Fin.cases hC
(Fin.cases hQ (Fin.cases hi (Fin.cases hone (Fin.cases hbit (fun j => Fin.elim0 j))))))))))).trans
(BoundedFormula.truth_tripleMembershipFormula 6 2 8 1 _)
rw [binaryRowEntryFormula, realize_ex]
constructor
· rintro ⟨c, hcM, rest⟩
obtain ⟨p, hpM, rest⟩ := (realize_ex M.sets _ _).1 rest
obtain ⟨w, hwM, hc, hp, checked, assigned, shape⟩ := (realize_ex M.sets _ _).1 rest
rw [realize_binaryApplication, map_extend, map_empty] at checked
have eq := (M.realize_uniformCheckFormula s one pairing unions powers separation collection
hone c w hcM hwM).1 checked
have pair := (M.realize_orderedPairFormula pairing 3 0 1 _ hz hwM hpM).1 shape
apply (M.mem_binaryRowName_code I C one i z).2
exact ⟨c, hc, p, hp,
pair.trans (congrArg (fun v => ZFSet.pair v p) eq), (assignment c p w hcM hpM hwM).1 assigned⟩
· intro member
obtain ⟨c, hc, p, hp, shape, assigned⟩ := (M.mem_binaryRowName_code I C one i z).1 member
have hcM := M.transitive hC hc
have hpM := M.transitive hQ hp
refine ⟨c, hcM, ?_⟩
rw [realize_ex]
refine ⟨p, hpM, ?_⟩
rw [realize_ex]
refine ⟨(Name.check s one c).val, children c hc, hc, hp, ?_, ?_, ?_⟩
· rw [realize_binaryApplication, map_extend, map_empty]
exact (M.realize_uniformCheckFormula s one pairing unions powers separation collection
hone c _ hcM (children c hc)).2 rfl
· exact (assignment c p _ hcM hpM (children c hc)).2 assigned
· exact (M.realize_orderedPairFormula pairing 3 0 1 _ hz (children c hc) hpM).2 shape
theorem binaryRowName_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0)
(separation : SeparationSchema M.sets) (collection : CollectionSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{I C : ZFSet.{u}} (hI : I ∈ M.sets) (hC : C ∈ M.sets)
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(i : ZFSet.{u}) (hi : i ∈ M.sets) : (M.binaryRowName I C one i).val ∈ M.sets := by
let Q := M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
let s := SetMembers.presentation Q
have hQ : Q ∈ M.sets := M.countablePartialFunctions_mem pairing unions powers separation infinite
(M.prod_mem pairing unions powers separation hI hC) (M.natSetCode_mem separation infinite 2)
have hone : s.encode one ∈ M.sets := M.transitive hQ (SetMembers.raw_mem one)
have hcheck := M.check_mem s one pairing unions powers separation collection hone C hC
have childBound (c : ZFSet.{u}) (hc : c ∈ C) :
(Name.check s one c).val ∈ ZFSet.sUnion (ZFSet.sUnion (Name.check s one C).val) :=
Name.child_mem_doubleUnion ((Name.entry_check one C (Name.check s one c) one).2 ⟨c, hc, rfl, rfl⟩)
obtain ⟨B, hB, pairBound⟩ := M.exists_pair_bound pairing unions powers
(M.union_mem unions (M.union_mem unions hcheck)) hQ
have bound : (M.binaryRowName I C one i).val ⊆ B := by
intro z hz
obtain ⟨c, hc, p, hp, eq, _⟩ := (M.mem_binaryRowName_code I C one i z).1 hz
exact eq.symm ▸ pairBound _ (childBound c hc) p hp
let params := extend C (extend Q (extend i (extend (SetMembers.raw one)
(extend (natSetCode 1) Fin.elim0))))
have hbit := M.natSetCode_mem separation infinite 1
have selected := M.separation_mem separation (binaryRowEntryFormula 0 1 2 3 4 5) params
(Fin.cases hC (Fin.cases hQ (Fin.cases hi (Fin.cases hone
(Fin.cases hbit (fun j => Fin.elim0 j)))))) hB
have eq : ZFSet.sep (fun z => Realize M.sets (binaryRowEntryFormula 0 1 2 3 4 5)
(extend z params)) B = (M.binaryRowName I C one i).val := by
apply ZFSet.ext
intro z
rw [ZFSet.mem_sep]
constructor
· rintro ⟨hz, rel⟩
exact (M.realize_binaryRowEntryFormula I C one i pairing unions powers separation collection
hC hQ hi hbit z (M.transitive hB hz)).1 rel
· intro hz
exact ⟨bound hz, (M.realize_binaryRowEntryFormula I C one i pairing unions powers separation collection
hC hQ hi hbit z (M.transitive hB (bound hz))).2 hz⟩
exact eq ▸ selected
theorem realize_binaryRowCodeFormula
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0)
(separation : SeparationSchema M.sets) (collection : CollectionSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{I C : ZFSet.{u}} (hI : I ∈ M.sets) (hC : C ∈ M.sets)
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(i w : ZFSet.{u}) (hi : i ∈ M.sets) (hw : w ∈ M.sets) :
Realize M.sets binaryRowCodeFormula (extend i (extend w (extend C (extend
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)
(extend (SetMembers.raw one) (extend (natSetCode 1) Fin.elim0)))))) ↔
w = (M.binaryRowName I C one i).val := by
have hQ := M.countablePartialFunctions_mem pairing unions powers separation infinite
(M.prod_mem pairing unions powers separation hI hC) (M.natSetCode_mem separation infinite 2)
have hname := M.binaryRowName_mem pairing unions powers separation collection infinite hI hC one i hi
have hbit := M.natSetCode_mem separation infinite 1
rw [binaryRowCodeFormula]
constructor
· intro spec
apply ZFSet.ext
intro z
constructor
· intro hz
have hzM := M.transitive hw hz
have rel := (realize_imp M.sets _ _ _).1 (spec z hzM).1 hz
rw [realize_reindex, map_extend, map_extend, map_extend, map_extend, map_extend,
map_extend, map_empty] at rel
exact (M.realize_binaryRowEntryFormula I C one i pairing unions powers separation collection
hC hQ hi hbit z hzM).1 rel
· intro hz
have hzM := M.transitive hname hz
apply (realize_imp M.sets _ _ _).1 (spec z hzM).2
rw [realize_reindex, map_extend, map_extend, map_extend, map_extend, map_extend,
map_extend, map_empty]
exact (M.realize_binaryRowEntryFormula I C one i pairing unions powers separation collection
hC hQ hi hbit z hzM).2 hz
· intro eq z hzM
constructor
· apply (realize_imp M.sets _ _ _).2
intro hz
rw [realize_reindex, map_extend, map_extend, map_extend, map_extend, map_extend,
map_extend, map_empty]
exact (M.realize_binaryRowEntryFormula I C one i pairing unions powers separation collection
hC hQ hi hbit z hzM).2 (eq ▸ hz)
· apply (realize_imp M.sets _ _ _).2
intro rel
rw [realize_reindex, map_extend, map_extend, map_extend, map_extend, map_extend,
map_extend, map_empty] at rel
exact eq.symm ▸ (M.realize_binaryRowEntryFormula I C one i pairing unions powers separation collection
hC hQ hi hbit z hzM).1 rel
theorem exists_binaryRowName_bound
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0)
(separation : SeparationSchema M.sets) (collection : CollectionSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{I C : ZFSet.{u}} (hI : I ∈ M.sets) (hC : C ∈ M.sets)
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) :
∃ W ∈ M.sets, ∀ i ∈ I, (M.binaryRowName I C one i).val ∈ W := by
let Q := M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
have hQ : Q ∈ M.sets := M.countablePartialFunctions_mem pairing unions powers separation infinite
(M.prod_mem pairing unions powers separation hI hC) (M.natSetCode_mem separation infinite 2)
have hone := M.transitive hQ (SetMembers.raw_mem one)
let params := extend C (extend Q (extend (SetMembers.raw one) (extend (natSetCode 1) Fin.elim0)))
have members : ∀ j, params j ∈ M.sets := Fin.cases hC (Fin.cases hQ
(Fin.cases hone (Fin.cases (M.natSetCode_mem separation infinite 1) (fun j => Fin.elim0 j))))
have witnesses : ∀ i ∈ I, ∃ w ∈ M.sets,
Realize M.sets binaryRowCodeFormula (extend i (extend w params)) := by
intro i hi
have hiM := M.transitive hI hi
have hname := M.binaryRowName_mem pairing unions powers separation collection infinite hI hC one i hiM
exact ⟨(M.binaryRowName I C one i).val, hname,
(M.realize_binaryRowCodeFormula pairing unions powers separation collection infinite
hI hC one i _ hiM hname).2 rfl⟩
obtain ⟨W, hW, covers⟩ := M.collection_witness collection binaryRowCodeFormula params members hI witnesses
refine ⟨W, hW, ?_⟩
intro i hi
obtain ⟨w, hw, code⟩ := covers i hi
have eq := (M.realize_binaryRowCodeFormula pairing unions powers separation collection infinite
hI hC one i w (M.transitive hI hi) (M.transitive hW hw)).1 code
exact eq ▸ hw
theorem realize_binaryRowSetEntryFormula
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0)
(separation : SeparationSchema M.sets) (collection : CollectionSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{I C : ZFSet.{u}} (hI : I ∈ M.sets) (hC : C ∈ M.sets)
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))
(z : ZFSet.{u}) (hz : z ∈ M.sets) :
Realize M.sets binaryRowSetEntryFormula (extend z (extend I (extend C (extend
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)
(extend (SetMembers.raw one) (extend (natSetCode 1) Fin.elim0)))))) ↔
z ∈ (M.binaryRowSetName I C one).val := by
have hQ := M.countablePartialFunctions_mem pairing unions powers separation infinite
(M.prod_mem pairing unions powers separation hI hC) (M.natSetCode_mem separation infinite 2)
have hone := M.transitive hQ (SetMembers.raw_mem one)
rw [binaryRowSetEntryFormula, realize_ex]
constructor
· rintro ⟨i, hiM, rest⟩
obtain ⟨w, hwM, hi, code, shape⟩ := (realize_ex M.sets _ _).1 rest
rw [realize_binaryApplication, map_extend, map_extend, map_extend, map_extend, map_empty] at code
have eq := (M.realize_binaryRowCodeFormula pairing unions powers separation collection infinite
hI hC one i w hiM hwM).1 code
have pair := (M.realize_orderedPairFormula pairing 2 0 6 _ hz hwM hone).1 shape
exact (M.mem_binaryRowSetName_code I C one z).2
⟨i, hi, pair.trans (congrArg (fun v => ZFSet.pair v (SetMembers.raw one)) eq)⟩
· intro member
obtain ⟨i, hi, shape⟩ := (M.mem_binaryRowSetName_code I C one z).1 member
have hiM := M.transitive hI hi
have hname := M.binaryRowName_mem pairing unions powers separation collection infinite hI hC one i hiM
refine ⟨i, hiM, ?_⟩
rw [realize_ex]
refine ⟨(M.binaryRowName I C one i).val, hname, hi, ?_, ?_⟩
· rw [realize_binaryApplication, map_extend, map_extend, map_extend, map_extend, map_empty]
exact (M.realize_binaryRowCodeFormula pairing unions powers separation collection infinite
hI hC one i _ hiM hname).2 rfl
· exact (M.realize_orderedPairFormula pairing 2 0 6 _ hz hname hone).2 shape
theorem binaryRowSetName_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0)
(separation : SeparationSchema M.sets) (collection : CollectionSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{I C : ZFSet.{u}} (hI : I ∈ M.sets) (hC : C ∈ M.sets)
(one : SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) :
(M.binaryRowSetName I C one).val ∈ M.sets := by
have hQ := M.countablePartialFunctions_mem pairing unions powers separation infinite
(M.prod_mem pairing unions powers separation hI hC) (M.natSetCode_mem separation infinite 2)
have hone := M.transitive hQ (SetMembers.raw_mem one)
obtain ⟨W, hW, inW⟩ := M.exists_binaryRowName_bound pairing unions powers separation collection infinite
hI hC one
obtain ⟨B, hB, pairBound⟩ := M.exists_pair_bound pairing unions powers hW
(M.unorderedPair_mem pairing hone hone)
have bound : (M.binaryRowSetName I C one).val ⊆ B := by
intro z hz
obtain ⟨i, hi, eq⟩ := (M.mem_binaryRowSetName_code I C one z).1 hz
exact eq.symm ▸ pairBound _ (inW i hi) _ (ZFSet.mem_pair.2 (Or.inl rfl))
let params := extend I (extend C (extend
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)
(extend (SetMembers.raw one) (extend (natSetCode 1) Fin.elim0))))
have selected := M.separation_mem separation binaryRowSetEntryFormula params
(Fin.cases hI (Fin.cases hC (Fin.cases hQ (Fin.cases hone
(Fin.cases (M.natSetCode_mem separation infinite 1) (fun j => Fin.elim0 j)))))) hB
have eq : ZFSet.sep (fun z => Realize M.sets binaryRowSetEntryFormula
(extend z params)) B = (M.binaryRowSetName I C one).val := by
apply ZFSet.ext
intro z
rw [ZFSet.mem_sep]
constructor
· rintro ⟨hz, rel⟩
exact (M.realize_binaryRowSetEntryFormula pairing unions powers separation collection infinite
hI hC one z (M.transitive hB hz)).1 rel
· intro hz
exact ⟨bound hz, (M.realize_binaryRowSetEntryFormula pairing unions powers separation collection infinite
hI hC one z (M.transitive hB (bound hz))).2 hz⟩
exact eq ▸ selected
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
|