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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 | import OAI.SetTheory.PartitionPrinciple.Forcing.GroundGraphValueDescendantWitnesses
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {P Q : Type u}
theorem rename_between_mem (s : Presentation P) (t : Presentation Q)
(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)
(hs : s.carrier ∈ M.sets) (ht : t.carrier ∈ M.sets)
(f : P → Q) (hf : s.graph t f ∈ M.sets)
(τ : Name s) (hτ : τ.val ∈ M.sets) : (Name.rename t f τ).val ∈ M.sets := by
let U := s.carrier ∪ t.carrier
let D := ZFSet.sep (fun x => x ∉ s.carrier) U
let F := s.graph t f
let G := F ∪ identityFunctionGraph D
have subS : s.carrier ⊆ U := fun _ hx => ZFSet.mem_union.2 (Or.inl hx)
have subT : t.carrier ⊆ U := fun _ hx => ZFSet.mem_union.2 (Or.inr hx)
have hU : U ∈ M.sets := M.binary_union_mem pairing unions hs ht
have hD : D ∈ M.sets := M.bounded_separation_mem separation (.neg (.mem 0 1))
(extend s.carrier Fin.elim0) (Fin.cases hs (fun i => Fin.elim0 i)) hU
have hG : G ∈ M.sets := M.binary_union_mem pairing unions hf
(M.identityFunctionGraph_mem pairing unions powers separation hD)
have ff := s.graph_isFunc t f
have full : ZFSet.IsFunc U U G := by
constructor
· intro z hz
rcases ZFSet.mem_union.1 hz with old | identity
· obtain ⟨x, hx, y, hy, eq⟩ := ZFSet.mem_prod.1 (ff.1 old)
exact ZFSet.mem_prod.2 ⟨x, subS hx, y, subT hy, eq⟩
· obtain ⟨x, hx, rfl⟩ := (mem_identityFunctionGraph D z).1 identity
exact ZFSet.pair_mem_prod.2 ⟨(ZFSet.mem_sep.1 hx).1, (ZFSet.mem_sep.1 hx).1⟩
· intro x hx
by_cases source : x ∈ s.carrier
· obtain ⟨y, entry, unique⟩ := ff.2 x source
refine ⟨y, ZFSet.mem_union.2 (Or.inl entry), ?_⟩
intro z hz
rcases ZFSet.mem_union.1 hz with old | identity
· exact unique z old
· exact False.elim ((ZFSet.mem_sep.1
((pair_mem_identityFunctionGraph D x z).1 identity).2).2 source)
· refine ⟨x, ZFSet.mem_union.2 (Or.inr
((pair_mem_identityFunctionGraph D x x).2 ⟨rfl, ZFSet.mem_sep.2 ⟨hx, source⟩⟩)), ?_⟩
intro z hz
rcases ZFSet.mem_union.1 hz with old | identity
· exact False.elim (source (ZFSet.pair_mem_prod.1 (ff.1 old)).1)
· exact ((pair_mem_identityFunctionGraph D x z).1 identity).1.symm
let u := SetMembers.presentation U
let es := s.intoSet U subS
let et := t.intoSet U subT
let h := u.decodeFunction u G full
have commute (p : P) : h (es p) = et (f p) := by
apply (u.decodeFunction_iff u G full (es p) (et (f p))).2
rw [s.encode_intoSet U subS, t.encode_intoSet U subT]
exact ZFSet.mem_union.2 (Or.inl ((s.pair_mem_graph t f p (f p)).2 rfl))
have sourceCode := Name.rename_val_of_encode u es (s.encode_intoSet U subS) τ
have ground := M.rename_mem u pairing unions powers separation collection hU h
((u.graph_decodeFunction u G full).symm ▸ hG)
(Name.rename u es τ) (sourceCode ▸ hτ)
have equation : Name.rename u h (Name.rename u es τ) =
Name.rename u et (Name.rename t f τ) := by
rw [Name.rename_comp, Name.rename_comp]
apply congrArg (fun k => Name.rename u k τ)
funext p
exact commute p
rw [equation, Name.rename_val_of_encode u et (t.encode_intoSet U subT)] at ground
exact ground
end PartitionPilot.Forcing.TransitiveGround
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {Lower Upper : Type u}
variable [PartialOrder Lower] [PartialOrder Upper]
variable (lower : Presentation Lower) (upper : Presentation Upper)
variable {restriction : Upper →o Lower} {inclusion : Lower ↪o Upper}
variable (projection : CompleteProjection restriction inclusion)
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)
variable (separation : SeparationSchema M.sets) (collection : CollectionSchema M.sets)
variable (lowerGround : lower.carrier ∈ M.sets) (upperGround : upper.carrier ∈ M.sets)
variable (inclusionGround : lower.graph upper inclusion ∈ M.sets)
variable (H : ConditionFilter Upper)
variable (ordinary : H.GenericFor (formulaFamily (M.nameDomain upper)))
include pairing unions powers separation collection lowerGround upperGround inclusionGround ordinary
theorem completeProjection_eval_eq_iff_forcesEq (sigma tau : Name lower)
(sigmaGround : sigma.val ∈ M.sets) (tauGround : tau.val ∈ M.sets) :
Name.eval {q | (projection.projectedFilter H).mem q} sigma =
Name.eval {q | (projection.projectedFilter H).mem q} tau ↔
∃ q, (projection.projectedFilter H).mem q ∧ Name.ForcesEq q sigma tau := by
have liftedSigma := M.rename_between_mem lower upper pairing unions powers separation collection
lowerGround upperGround inclusion inclusionGround sigma sigmaGround
have liftedTau := M.rename_between_mem lower upper pairing unions powers separation collection
lowerGround upperGround inclusion inclusionGround tau tauGround
have truth := Name.eval_eq_iff_forcesEq (M.nameDomain upper) H
(fun E member => ordinary E (Or.inl member)) liftedSigma liftedTau
simp only [Name.eval_rename] at truth
constructor
· intro equal
obtain ⟨p, member, forced⟩ := truth.1 equal
exact ⟨restriction p, (projection.projectedFilter_mem_iff H _).2 ⟨p, member, rfl⟩,
(Name.forcesEq_rename_completeProjection lower upper restriction inclusion projection p sigma tau).1 forced⟩
· rintro ⟨q, member, forced⟩
apply truth.2
refine ⟨inclusion q, member, ?_⟩
apply (Name.forcesEq_rename_completeProjection lower upper restriction inclusion projection
(inclusion q) sigma tau).2
simpa only [projection.retract] using forced
theorem completeProjection_eval_mem_iff_forcesMem (sigma tau : Name lower)
(sigmaGround : sigma.val ∈ M.sets) (tauGround : tau.val ∈ M.sets) :
Name.eval {q | (projection.projectedFilter H).mem q} sigma ∈
Name.eval {q | (projection.projectedFilter H).mem q} tau ↔
∃ q, (projection.projectedFilter H).mem q ∧ Name.ForcesMem q sigma tau := by
have liftedSigma := M.rename_between_mem lower upper pairing unions powers separation collection
lowerGround upperGround inclusion inclusionGround sigma sigmaGround
have liftedTau := M.rename_between_mem lower upper pairing unions powers separation collection
lowerGround upperGround inclusion inclusionGround tau tauGround
have truth := Name.eval_mem_iff_forcesMem (M.nameDomain upper) H
(fun E member => ordinary E (Or.inl member)) liftedSigma liftedTau
simp only [Name.eval_rename] at truth
constructor
· intro member
obtain ⟨p, hp, forced⟩ := truth.1 member
exact ⟨restriction p, (projection.projectedFilter_mem_iff H _).2 ⟨p, hp, rfl⟩,
(Name.forcesMem_rename_completeProjection lower upper restriction inclusion projection p sigma tau).1 forced⟩
· rintro ⟨q, member, forced⟩
apply truth.2
refine ⟨inclusion q, member, ?_⟩
apply (Name.forcesMem_rename_completeProjection lower upper restriction inclusion projection
(inclusion q) sigma tau).2
simpa only [projection.retract] using forced
theorem completeProjection_forcesNotEq_sound (sigma tau : Name lower)
(sigmaGround : sigma.val ∈ M.sets) (tauGround : tau.val ∈ M.sets)
{q : Lower} (member : (projection.projectedFilter H).mem q)
(negative : ∀ r, r ≤ q → ¬ Name.ForcesEq r sigma tau) :
Name.eval {r | (projection.projectedFilter H).mem r} sigma ≠
Name.eval {r | (projection.projectedFilter H).mem r} tau := by
intro equal
obtain ⟨r, hr, forced⟩ := (M.completeProjection_eval_eq_iff_forcesEq lower upper projection
pairing unions powers separation collection lowerGround upperGround inclusionGround H ordinary
sigma tau sigmaGround tauGround).1 equal
obtain ⟨p, _hp, pq, pr⟩ := (projection.projectedFilter H).directed member hr
exact negative p pq (Name.forcesEq_mono forced pr)
theorem completeProjection_forcesNotMem_sound (sigma tau : Name lower)
(sigmaGround : sigma.val ∈ M.sets) (tauGround : tau.val ∈ M.sets)
{q : Lower} (member : (projection.projectedFilter H).mem q)
(negative : Name.ForcesNotMem q sigma tau) :
Name.eval {r | (projection.projectedFilter H).mem r} sigma ∉
Name.eval {r | (projection.projectedFilter H).mem r} tau := by
intro belongs
obtain ⟨r, hr, forced⟩ := (M.completeProjection_eval_mem_iff_forcesMem lower upper projection
pairing unions powers separation collection lowerGround upperGround inclusionGround H ordinary
sigma tau sigmaGround tauGround).1 belongs
obtain ⟨p, _hp, pq, pr⟩ := (projection.projectedFilter H).directed member hr
exact negative p pq (Name.forcesMem_mono forced pr)
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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