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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 | import OAI.SetTheory.PartitionPrinciple.Transfinite.StageTwoAssignmentSplices
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem assignmentCut_mem (separation : SeparationSchema M.sets)
{levels alpha A : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) (hA : A ∈ M.sets) :
assignmentCut levels alpha A ∈ M.sets := by
let formula : BoundedFormula 3 := BoundedFormula.exMem 1
(BoundedFormula.pairMembershipFormula 1 0 3)
let params := extend alpha (extend levels Fin.elim0)
have ground := M.bounded_separation_mem separation formula params
(Fin.cases hAlpha (Fin.cases hLevels (fun i => Fin.elim0 i))) hA
have equal : ZFSet.sep (fun z => BoundedFormula.Truth formula (extend z params)) A =
assignmentCut levels alpha A := by
apply ZFSet.ext
intro z
simp only [ZFSet.mem_sep, assignmentCut, formula,
BoundedFormula.truth_exMem, BoundedFormula.truth_pairMembershipFormula]
rfl
exact equal ▸ ground
theorem assignmentUpper_mem (separation : SeparationSchema M.sets)
{levels alpha A : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) (hA : A ∈ M.sets) :
assignmentUpper levels alpha A ∈ M.sets := by
exact M.bounded_separation_mem separation (.neg (.mem 0 1))
(extend (assignmentCut levels alpha A) Fin.elim0)
(Fin.cases (M.assignmentCut_mem separation hLevels hAlpha hA) (fun i => Fin.elim0 i)) hA
theorem assignmentSplice_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0) (separation : SeparationSchema M.sets)
{levels alpha Aq Ap : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets)
(hAq : Aq ∈ M.sets) (hAp : Ap ∈ M.sets) :
assignmentSplice levels alpha Aq Ap ∈ M.sets :=
M.binary_union_mem pairing unions
(M.assignmentCut_mem separation hLevels hAlpha hAq)
(M.assignmentUpper_mem separation hLevels hAlpha hAp)
theorem assignmentCut_small (separation : SeparationSchema M.sets)
{levels alpha A N : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets)
(hA : A ∈ M.sets) (hN : N ∈ M.sets) (small : M.groundEmbeddable A N) :
M.groundEmbeddable (assignmentCut levels alpha A) N :=
M.groundEmbeddable_restrict separation
(M.assignmentCut_mem separation hLevels hAlpha hA) hN
(assignmentCut_subset levels alpha A) small
theorem assignmentUpper_small (separation : SeparationSchema M.sets)
{levels alpha A N : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets)
(hA : A ∈ M.sets) (hN : N ∈ M.sets) (small : M.groundEmbeddable A N) :
M.groundEmbeddable (assignmentUpper levels alpha A) N :=
M.groundEmbeddable_restrict separation
(M.assignmentUpper_mem separation hLevels hAlpha hA) hN
(assignmentUpper_subset levels alpha A) small
theorem assignmentSplice_countable
(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)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{levels alpha Aq Ap : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets)
(hAq : Aq ∈ M.sets) (hAp : Ap ∈ M.sets)
(smallq : M.groundEmbeddable Aq ZFSet.omega)
(smallp : M.groundEmbeddable Ap ZFSet.omega) :
M.groundEmbeddable (assignmentSplice levels alpha Aq Ap) ZFSet.omega :=
M.groundEmbeddable_restrict separation
(M.assignmentSplice_mem pairing unions separation hLevels hAlpha hAq hAp)
(M.omega_mem separation infinite) (assignmentSplice_subset levels alpha Aq Ap)
(M.groundCountable_binary_union pairing unions powers separation infinite choice
hAq hAp smallq smallp)
theorem assignmentSplice_mem_groundSmallSubsets
(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)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{J levels alpha Aq Ap : ZFSet.{u}}
(hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets)
(memberq : Aq ∈ M.groundSmallSubsets J ZFSet.omega)
(memberp : Ap ∈ M.groundSmallSubsets J ZFSet.omega) :
assignmentSplice levels alpha Aq Ap ∈ M.groundSmallSubsets J ZFSet.omega := by
obtain ⟨hAq, subsetq, smallq⟩ := (M.mem_groundSmallSubsets J ZFSet.omega Aq).1 memberq
obtain ⟨hAp, subsetp, smallp⟩ := (M.mem_groundSmallSubsets J ZFSet.omega Ap).1 memberp
refine (M.mem_groundSmallSubsets J ZFSet.omega _).2
⟨M.assignmentSplice_mem pairing unions separation hLevels hAlpha hAq hAp, ?_,
M.assignmentSplice_countable pairing unions powers separation infinite choice
hLevels hAlpha hAq hAp smallq smallp⟩
intro z hz
exact (ZFSet.mem_union.1 (assignmentSplice_subset levels alpha Aq Ap hz)).elim
(fun member => subsetq member) (fun member => subsetp member)
end PartitionPilot.Forcing.TransitiveGround
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
variable (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)
include pairing unions powers separation
theorem assignmentIndexLevels_mem {I T eta levels : ZFSet.{u}}
(hI : I ∈ M.sets) (hT : T ∈ M.sets) (hEta : eta ∈ M.sets)
(hLevels : levels ∈ M.sets) : assignmentIndexLevels I T eta levels ∈ M.sets :=
M.relationCompose_mem pairing unions powers separation
(M.stageOneAssignments_mem pairing unions powers separation hI hT) hI hEta
(M.productFirst_mem pairing unions powers separation hI
(M.prod_mem pairing unions powers separation hT hT)) hLevels
theorem stageTwoAssignmentLevels_mem {I0 I1 T : ZFSet.{u}}
(hI0 : I0 ∈ M.sets) (hI1 : I1 ∈ M.sets) (hT : T ∈ M.sets) :
stageTwoAssignmentLevels I0 I1 T ∈ M.sets := by
have hz := M.empty_mem ⟨I0, hI0⟩
have ho : natSetCode 1 ∈ M.sets := M.insert_mem pairing unions hz hz
have hEta := M.unorderedPair_mem pairing hz ho
have hI := M.binary_union_mem pairing unions hI0 hI1
exact M.assignmentIndexLevels_mem pairing unions powers separation hI hT hEta
(M.refinementIndexFieldGraph_mem pairing unions powers separation hI hEta 0)
theorem stageTwoAssignmentCuts_mem {I0 I1 T A : ZFSet.{u}}
(hI0 : I0 ∈ M.sets) (hI1 : I1 ∈ M.sets) (hT : T ∈ M.sets) (hA : A ∈ M.sets) :
assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 0) A ∈ M.sets ∧
assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 1) A ∈ M.sets ∧
assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 2) A ∈ M.sets := by
have hz := M.empty_mem ⟨I0, hI0⟩
have ho : natSetCode 1 ∈ M.sets := M.insert_mem pairing unions hz hz
have ht : natSetCode 2 ∈ M.sets := M.insert_mem pairing unions ho ho
have graph := M.stageTwoAssignmentLevels_mem pairing unions powers separation hI0 hI1 hT
exact ⟨M.assignmentCut_mem separation graph hz hA,
M.assignmentCut_mem separation graph ho hA, M.assignmentCut_mem separation graph ht hA⟩
theorem stageTwoAssignmentCut_one_mem_groundSmallSubsets
(infinite : Realize M.sets infinity Fin.elim0)
{I0 I1 T A : ZFSet.{u}}
(first : RefinementIndexLevelCode I0 ∅)
(second : RefinementIndexLevelCode I1 (natSetCode 1))
(hI0 : I0 ∈ M.sets) (hI1 : I1 ∈ M.sets) (hT : T ∈ M.sets)
(member : A ∈ M.groundSmallSubsets (stageOneAssignments (I0 ∪ I1) T) ZFSet.omega) :
assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 1) A ∈
M.groundSmallSubsets (stageOneAssignments I0 T) ZFSet.omega := by
obtain ⟨hA, _, small⟩ := (M.mem_groundSmallSubsets _ _ A).1 member
have ground := (M.stageTwoAssignmentCuts_mem pairing unions powers separation hI0 hI1 hT hA).2.1
exact (M.mem_groundSmallSubsets _ _ _).2 ⟨ground,
stageTwoAssignmentCut_one_subset first second A,
M.groundEmbeddable_restrict separation ground (M.omega_mem separation infinite)
(assignmentCut_subset _ _ A) small⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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