File size: 6,006 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 | import OAI.SetTheory.PartitionPrinciple.Definability.BatchAttachmentRealization
namespace OAI
open scoped ZFSet
section
namespace PartitionPilot.Forcing
universe u
namespace BoundedFormula
open SetFormula
def componentHeightEntryFormula {n}
(w I Qs B Hs Cs E C D boundQs boundHs boundE : Fin n) : BoundedFormula n :=
let lift (i : Fin n) : Fin (n + 8) := i.succ.succ.succ.succ.succ.succ.succ.succ
let body : BoundedFormula (n + 8) :=
.conj (pairOfPairsFormula (lift w) 7 3 1 0)
(.conj (pairMembershipFormula 7 6 (lift Qs))
(.conj (pairMembershipFormula 7 5 (lift Hs))
(.conj (pairMembershipFormula 7 4 (lift E))
(.conj (pairMembershipFormula 7 1 (lift Cs))
(.conj (pairMembershipFormula 3 2 5) (pairMembershipFormula 2 0 4))))))
exMem I (exMem boundQs.succ (exMem boundHs.succ.succ
(exMem boundE.succ.succ.succ (exMem 2
(exMem B.succ.succ.succ.succ.succ
(exMem C.succ.succ.succ.succ.succ.succ
(exMem D.succ.succ.succ.succ.succ.succ.succ body)))))))
theorem truth_componentHeightEntryFormula {n}
(w I Qs B Hs Cs E C D boundQs boundHs boundE : Fin n) (env : Fin n → ZFSet.{u}) :
Truth (componentHeightEntryFormula w I Qs B Hs Cs E C D boundQs boundHs boundE) env ↔
∃ i ∈ env I, ∃ Q ∈ env boundQs, ∃ g ∈ env boundHs, ∃ f ∈ env boundE,
∃ q ∈ Q, ∃ a ∈ env B, ∃ c ∈ env C, ∃ z ∈ env D,
env w = ZFSet.pair (ZFSet.pair i q) (ZFSet.pair c z) ∧
ZFSet.pair i Q ∈ env Qs ∧ ZFSet.pair i g ∈ env Hs ∧ ZFSet.pair i f ∈ env E ∧
ZFSet.pair i c ∈ env Cs ∧ ZFSet.pair q a ∈ g ∧ ZFSet.pair a z ∈ f := by
simp only [componentHeightEntryFormula, truth_exMem, Truth,
truth_pairOfPairsFormula, truth_pairMembershipFormula]
rfl
def taggedHeightEntryFormula {n} (w P N H fresh : Fin n) : BoundedFormula n :=
exMem N (exMem H.succ
(.conj (orderedTripleFormula w.succ.succ P.succ.succ 1 0)
(pairMembershipFormula 1 0 fresh.succ.succ)))
theorem truth_taggedHeightEntryFormula {n} (w P N H fresh : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (taggedHeightEntryFormula w P N H fresh) env ↔
∃ x ∈ env N, ∃ h ∈ env H,
env w = ZFSet.pair (ZFSet.pair (env P) x) h ∧ ZFSet.pair x h ∈ env fresh := by
simp only [taggedHeightEntryFormula, truth_exMem, Truth,
truth_orderedTripleFormula, truth_pairMembershipFormula]
rfl
end BoundedFormula
namespace TransitiveGround
open SetFormula
variable (M : TransitiveGround.{u})
theorem componentHeightGraph_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)
{I Qs B Hs Cs E C D : ZFSet.{u}}
(hI : I ∈ M.sets) (hQs : Qs ∈ M.sets) (hB : B ∈ M.sets) (hHs : Hs ∈ M.sets)
(hCs : Cs ∈ M.sets) (hE : E ∈ M.sets) (hC : C ∈ M.sets) (hD : D ∈ M.sets) :
componentHeightGraph I Qs B Hs Cs E C D ∈ M.sets := by
have qBound := M.union_mem unions (M.union_mem unions hQs)
have hBound := M.union_mem unions (M.union_mem unions hHs)
have eBound := M.union_mem unions (M.union_mem unions hE)
let params := extend I (extend Qs (extend B (extend Hs (extend Cs (extend E
(extend C (extend D (extend (⋃₀ ⋃₀ Qs) (extend (⋃₀ ⋃₀ Hs)
(extend (⋃₀ ⋃₀ E) Fin.elim0))))))))))
have selected := M.bounded_separation_mem separation
(BoundedFormula.componentHeightEntryFormula 0 1 2 3 4 5 6 7 8 9 10 11) params
(Fin.cases hI (Fin.cases hQs (Fin.cases hB (Fin.cases hHs (Fin.cases hCs
(Fin.cases hE (Fin.cases hC (Fin.cases hD (Fin.cases qBound
(Fin.cases hBound (Fin.cases eBound (fun i => Fin.elim0 i))))))))))))
(M.prod_mem pairing unions powers separation
(M.indexedAttachmentDomain_mem pairing unions powers separation hI hQs)
(M.prod_mem pairing unions powers separation hC hD))
have equal : ZFSet.sep (fun z => BoundedFormula.Truth
(BoundedFormula.componentHeightEntryFormula 0 1 2 3 4 5 6 7 8 9 10 11) (extend z params))
(ZFSet.prod (indexedAttachmentDomain I Qs) (ZFSet.prod C D)) =
componentHeightGraph I Qs B Hs Cs E C D := by
apply ZFSet.ext
intro z
rw [componentHeightGraph, ZFSet.mem_sep, ZFSet.mem_sep]
exact and_congr Iff.rfl
(BoundedFormula.truth_componentHeightEntryFormula 0 1 2 3 4 5 6 7 8 9 10 11 _)
exact equal ▸ selected
theorem taggedHeightGraph_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)
{P N H old fresh : ZFSet.{u}}
(hP : P ∈ M.sets) (hN : N ∈ M.sets) (hH : H ∈ M.sets)
(hold : old ∈ M.sets) (hfresh : fresh ∈ M.sets) :
taggedHeightGraph P N H old fresh ∈ M.sets := by
let params := extend P (extend N (extend H (extend fresh Fin.elim0)))
have selected := M.bounded_separation_mem separation
(BoundedFormula.taggedHeightEntryFormula 0 1 2 3 4) params
(Fin.cases hP (Fin.cases hN (Fin.cases hH (Fin.cases hfresh (fun i => Fin.elim0 i)))))
(M.prod_mem pairing unions powers separation
(M.taggedExtension_mem pairing unions powers separation hP hN) hH)
have equal : ZFSet.sep (fun z => BoundedFormula.Truth
(BoundedFormula.taggedHeightEntryFormula 0 1 2 3 4) (extend z params))
(ZFSet.prod (taggedExtension P N) H) =
ZFSet.sep (fun w => ∃ x ∈ N, ∃ h ∈ H,
w = ZFSet.pair (ZFSet.pair P x) h ∧ ZFSet.pair x h ∈ fresh)
(ZFSet.prod (taggedExtension P N) H) := by
apply ZFSet.ext
intro z
rw [ZFSet.mem_sep, ZFSet.mem_sep]
exact and_congr Iff.rfl (BoundedFormula.truth_taggedHeightEntryFormula 0 1 2 3 4 _)
exact M.binary_union_mem pairing unions hold (equal ▸ selected)
end TransitiveGround
end PartitionPilot.Forcing
end
end OAI
|