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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | import OAI.SetTheory.PartitionPrinciple.Definability.GroundBoundedExternalBirths
import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundHistoryUnionBounds
namespace OAI
namespace PartitionPilot.Forcing
namespace GroundBelow
open SetFormula
universe u
variable {M : TransitiveGround.{u}} {A K : ZFSet.{u}}
theorem finiteWords
(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)
(infinite : Realize M.sets infinity Fin.elim0)
(hA : A ∈ M.sets) (hK : K ∈ M.sets)
(ordinal : ordinalSetProperty K) (strong : GroundStrongLimit M K)
(omegaBelow : ZFSet.omega ∈ K) (alphabet : GroundBelow M A K) :
GroundBelow M (M.finiteWords A) K := by
have natural := GroundBelow.of_member pairing unions powers separation hK omegaBelow
have relations := GroundBelow.relations pairing unions powers separation
(M.omega_mem separation infinite) hA hK ordinal strong natural alphabet
exact relations.restrict separation (M.finiteWords_mem pairing unions powers separation infinite hA)
hK (fun _ member => (ZFSet.mem_sep.1 member).1)
end GroundBelow
section
open SetFormula
universe u
namespace GroundBelow
variable {M : TransitiveGround.{u}} {A E K : ZFSet.{u}}
theorem equivalenceClasses
(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)
(hA : A ∈ M.sets) (hE : E ∈ M.sets) (hK : K ∈ M.sets)
(strong : GroundStrongLimit M K) (small : GroundBelow M A K) :
GroundBelow M (Forcing.equivalenceClasses A E) K := by
have included : Forcing.equivalenceClasses A E ⊆ M.groundPower A := by
intro X hX
rw [M.equivalenceClasses_eq_sep_groundPower separation hA hE] at hX
exact (ZFSet.mem_sep.1 hX).1
exact (strong.power_closed pairing unions powers separation hA hK small).restrict
separation (M.equivalenceClasses_mem powers separation hA hE) hK included
end GroundBelow
namespace TransitiveGround
variable (M : TransitiveGround.{u})
theorem externalGroupCarrier_below
(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)
(infinite : Realize M.sets infinity Fin.elim0)
{G S m one K : ZFSet.{u}}
(hG : G ∈ M.sets) (hS : S ∈ M.sets) (hm : m ∈ M.sets) (hOne : one ∈ M.sets) (hK : K ∈ M.sets)
(ordinal : ordinalSetProperty K) (strong : GroundStrongLimit M K)
(omegaBelow : ZFSet.omega ∈ K) (alphabet : GroundBelow M (externalGeneratorAlphabet G S) K) :
GroundBelow M (M.externalGroupCarrier G S m one) K := by
have hAlphabet := M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS
have hWords := M.finiteWords_mem pairing unions powers separation infinite hAlphabet
have hRelation := M.externalWordEquivalence_mem pairing unions powers separation infinite hG hS hm hOne
exact GroundBelow.equivalenceClasses pairing unions powers separation hWords hRelation hK strong
(GroundBelow.finiteWords pairing unions powers separation infinite hAlphabet hK ordinal strong omegaBelow alphabet)
end TransitiveGround
end
namespace TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
variable (unions : Realize M.sets SetFormula.union Fin.elim0)
variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0)
variable (choice : Realize M.sets SetFormula.choice Fin.elim0)
variable {K : ZFSet.{u}} (hK : K ∈ M.sets) (conditions : GroundInaccessibleConditions M K)
include pairing unions powers separation hK conditions
theorem externalRectangleBound_below {P G : ZFSet.{u}}
(hP : P ∈ M.sets) (hG : G ∈ M.sets)
(smallP : GroundBelow M P K) (smallG : GroundBelow M G K) :
GroundBelow M (externalRectangleBound P G) K := by
have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK
exact (product ⟨hP, smallP⟩ (product ⟨hP, smallP⟩
(product ⟨hP, smallP⟩ (product ⟨hG, smallG⟩ ⟨hG, smallG⟩)))).2
theorem externalRectangles_below {P G r μ A : ZFSet.{u}}
(hP : P ∈ M.sets) (hG : G ∈ M.sets) (hr : r ∈ M.sets)
(hμ : μ ∈ M.sets) (hA : A ∈ M.sets)
(smallP : GroundBelow M P K) (smallG : GroundBelow M G K) :
GroundBelow M (externalRectangles P G r μ A) K := by
exact (M.externalRectangleBound_below pairing unions powers separation hK conditions hP hG smallP smallG).restrict
separation (M.externalRectangles_mem pairing unions powers separation hP hG hr hμ hA) hK
(fun _ hz => (ZFSet.mem_sep.1 hz).1)
include infinite choice
theorem externalRequestCarrier_below {I P Rectangles : ZFSet.{u}}
(hI : I ∈ M.sets) (hP : P ∈ M.sets) (hRectangles : Rectangles ∈ M.sets)
(smallI : GroundBelow M I K) (smallP : GroundBelow M P K)
(smallRectangles : GroundBelow M Rectangles K) :
GroundBelow M (externalRequestCarrier I P Rectangles) K := by
have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK
have join := @GroundInaccessibleConditions.binary_union M K conditions pairing unions powers separation infinite choice hK
have tag := fun n => conditions.singleton pairing unions powers separation infinite hK
(M.natSetCode_mem separation infinite n)
exact (join (join (product (tag 0) ⟨hI, smallI⟩) (product (tag 1) ⟨hP, smallP⟩))
(product (tag 2) ⟨hRectangles, smallRectangles⟩)).2
theorem externalChildLetters_below {I P Rectangles : ZFSet.{u}}
(hI : I ∈ M.sets) (hP : P ∈ M.sets) (hRectangles : Rectangles ∈ M.sets)
(smallI : GroundBelow M I K) (smallP : GroundBelow M P K)
(smallRectangles : GroundBelow M Rectangles K) :
GroundBelow M (externalChildLetters I P Rectangles) K := by
have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK
have join := @GroundInaccessibleConditions.binary_union M K conditions pairing unions powers separation infinite choice hK
have tag := fun n => conditions.singleton pairing unions powers separation infinite hK
(M.natSetCode_mem separation infinite n)
have requests := M.externalRequestCarrier_below pairing unions powers separation infinite choice hK conditions
hI hP hRectangles smallI smallP smallRectangles
exact (join (product ⟨M.externalRequestCarrier_mem pairing unions powers separation hI hP hRectangles,
requests⟩ (tag 0)) (product (product (tag 2) ⟨hRectangles, smallRectangles⟩) (tag 1))).2
theorem externalGeneratorAlphabet_below {G S : ZFSet.{u}}
(hG : G ∈ M.sets) (hS : S ∈ M.sets)
(smallG : GroundBelow M G K) (smallS : GroundBelow M S K) :
GroundBelow M (externalGeneratorAlphabet G S) K := by
have product := @GroundInaccessibleConditions.product M K conditions pairing unions powers separation hK
have join := @GroundInaccessibleConditions.binary_union M K conditions pairing unions powers separation infinite choice hK
have tag := fun n => conditions.singleton pairing unions powers separation infinite hK
(M.natSetCode_mem separation infinite n)
exact (join (product (tag 0) ⟨hG, smallG⟩)
(product (tag 1) (product ⟨hS, smallS⟩ (conditions.natural pairing unions powers separation infinite hK 2)))).2
theorem externalGroupCarrier_below_of_carriers {G S m one : ZFSet.{u}}
(hG : G ∈ M.sets) (hS : S ∈ M.sets) (hm : m ∈ M.sets) (hOne : one ∈ M.sets)
(smallG : GroundBelow M G K) (smallS : GroundBelow M S K) :
GroundBelow M (M.externalGroupCarrier G S m one) K :=
M.externalGroupCarrier_below pairing unions powers separation infinite hG hS hm hOne hK
conditions.1 conditions.2.2.2 conditions.2.1
(M.externalGeneratorAlphabet_below pairing unions powers separation infinite choice hK conditions hG hS smallG smallS)
end TransitiveGround
namespace GroundExternalGeometryInputs
open SetFormula
universe u
variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}}
variable {I Sources Reps G m inv one T τ P r μ A K : ZFSet.{u}}
variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ}
variable (data : GroundExternalGeometryInputs M initial I Sources Reps G m inv one T τ P r μ A completion)
variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
variable (unions : Realize M.sets SetFormula.union Fin.elim0)
variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0)
variable (hP : P ∈ M.sets) (hG : G ∈ M.sets) (hK : K ∈ M.sets)
local notation "letters" => externalChildLetters I P (externalRectangles P G r μ A)
local notation "actors" => M.externalGroupCarrier G letters m one
local notation "U" => completion.algebra.union.fields 0
include data pairing unions powers separation infinite hP hG hK
theorem externalPointCarrier_below
(ordinal : ordinalSetProperty K) (strong : GroundStrongLimit M K)
(groupSmall : GroundBelow M actors K) (pointsSmall : GroundBelow M U K) :
GroundBelow M (completion.externalPointCarrier P r μ A) K := by
have quotient := data.point_quotient_ground pairing unions powers separation infinite hP hG
have hProduct := M.prod_mem pairing unions powers separation
data.group.carrier_mem (completion.algebra.union.ground 0)
have smallProduct := GroundBelow.product pairing unions powers separation
data.group.carrier_mem (completion.algebra.union.ground 0) hK ordinal strong groupSmall pointsSmall
exact GroundBelow.equivalenceClasses pairing unions powers separation
hProduct quotient.1 hK strong smallProduct
theorem externalBoundedCarrier_below
(choice : Realize M.sets SetFormula.choice Fin.elim0)
(conditions : GroundInaccessibleConditions M K)
(groupSmall : GroundBelow M actors K) (pointsSmall : GroundBelow M U K) :
GroundBelow M (completion.externalBoundedCarrier P r μ A) K := by
have hQ := (data.point_quotient_ground pairing unions powers separation infinite hP hG).2.2.1
exact GroundBelow.insert pairing unions powers separation infinite choice hQ hQ hK conditions.2.1
conditions.1 conditions.2.2.1 conditions.2.2.2
(data.externalPointCarrier_below pairing unions powers separation infinite hP hG hK
conditions.1 conditions.2.2.2 groupSmall pointsSmall)
end GroundExternalGeometryInputs
end PartitionPilot.Forcing
end OAI
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