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e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundExternalTileSlotSupports
namespace OAI
section
namespace PartitionPilot.Forcing
open SetFormula
universe u
noncomputable def externalSlotCopyGraph
(V E I U B action images j : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ v ∈ V, ∃ u ∈ U, ∃ y ∈ B, ∃ w ∈ E, ∃ r ∈ I, ∃ x ∈ B,
z = ZFSet.pair (ZFSet.pair v u) y ∧ v = ZFSet.pair (natSetCode 1) (ZFSet.pair w r) ∧
ZFSet.pair (ZFSet.pair (ZFSet.pair r j) u) x ∈ images ∧
ZFSet.pair (ZFSet.pair w x) y ∈ action)
(ZFSet.prod (ZFSet.prod V U) B)
theorem triple_mem_externalSlotCopyGraph (V E I U B action images j v u y : ZFSet.{u}) :
ZFSet.pair (ZFSet.pair v u) y ∈ externalSlotCopyGraph V E I U B action images j ↔
v ∈ V ∧ u ∈ U ∧ y ∈ B ∧ ∃ w ∈ E, ∃ r ∈ I, ∃ x ∈ B,
v = ZFSet.pair (natSetCode 1) (ZFSet.pair w r) ∧
ZFSet.pair (ZFSet.pair (ZFSet.pair r j) u) x ∈ images ∧
ZFSet.pair (ZFSet.pair w x) y ∈ action := by
rw [externalSlotCopyGraph, ZFSet.mem_sep]
constructor
· rintro ⟨_, v', hv, u', hu, y', hy, w, hw, r, hr, x, hx, equal, rest⟩
obtain ⟨left, rfl⟩ := ZFSet.pair_inj.1 equal
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 left
exact ⟨hv, hu, hy, w, hw, r, hr, x, hx, rest⟩
· rintro ⟨hv, hu, hy, w, hw, r, hr, x, hx, rest⟩
exact ⟨ZFSet.pair_mem_prod.2 ⟨ZFSet.pair_mem_prod.2 ⟨hv, hu⟩, hy⟩,
v, hv, u, hu, y, hy, w, hw, r, hr, x, hx, rfl, rest⟩
namespace BoundedFormula
def externalSlotCopyEntryFormula {n} (z V E I U B action images j one : Fin n) : BoundedFormula n :=
let lift (i : Fin n) := i.succ.succ.succ.succ.succ.succ
exMem V (exMem U.succ (exMem B.succ.succ (exMem E.succ.succ.succ
(exMem I.succ.succ.succ.succ (exMem B.succ.succ.succ.succ.succ
(.conj (orderedTripleFormula (lift z) 5 4 3)
(.conj (orderedRightTripleFormula 5 (lift one) 2 1)
(.conj (slotImageMembershipFormula 1 (lift j) 4 0 (lift images))
(tripleMembershipFormula 2 0 3 (lift action))))))))))
theorem truth_externalSlotCopyEntryFormula {n} (z V E I U B action images j one : Fin n)
(env : Fin n → ZFSet.{u}) (oneEq : env one = natSetCode 1) :
Truth (externalSlotCopyEntryFormula z V E I U B action images j one) env ↔
env z ∈ externalSlotCopyGraph (env V) (env E) (env I) (env U) (env B)
(env action) (env images) (env j) := by
simp only [externalSlotCopyEntryFormula, truth_exMem, Truth, truth_orderedTripleFormula,
truth_orderedRightTripleFormula, truth_slotImageMembershipFormula, truth_tripleMembershipFormula]
simp only [extend, Fin.cases_zero, Fin.cases_succ]
rw [oneEq, externalSlotCopyGraph, ZFSet.mem_sep]
refine ⟨fun h => ⟨?_, h⟩, And.right⟩
obtain ⟨v, hv, u, hu, y, hy, _, _, _, _, _, _, equal, _⟩ := h
rw [equal]
exact ZFSet.pair_mem_prod.2 ⟨ZFSet.pair_mem_prod.2 ⟨hv, hu⟩, hy⟩
end BoundedFormula
namespace TransitiveGround
variable (M : TransitiveGround.{u})
theorem externalSlotCopyGraph_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)
{V E I U B action images j : ZFSet.{u}}
(hV : V ∈ M.sets) (hE : E ∈ M.sets) (hI : I ∈ M.sets) (hU : U ∈ M.sets)
(hB : B ∈ M.sets) (hAction : action ∈ M.sets) (hImages : images ∈ M.sets) (hj : j ∈ M.sets) :
externalSlotCopyGraph V E I U B action images j ∈ M.sets := by
have zero := M.empty_mem ⟨U, hU⟩
have one : natSetCode 1 ∈ M.sets := M.insert_mem pairing unions zero zero
let params := extend V (extend E (extend I (extend U (extend B (extend action
(extend images (extend j (extend (natSetCode 1) Fin.elim0))))))))
have selected := M.bounded_separation_mem separation
(BoundedFormula.externalSlotCopyEntryFormula 0 1 2 3 4 5 6 7 8 9) params
(Fin.cases hV (Fin.cases hE (Fin.cases hI (Fin.cases hU (Fin.cases hB
(Fin.cases hAction (Fin.cases hImages (Fin.cases hj (Fin.cases one (fun i => Fin.elim0 i))))))))))
(M.prod_mem pairing unions powers separation (M.prod_mem pairing unions powers separation hV hU) hB)
have equal : ZFSet.sep (fun z => BoundedFormula.Truth
(BoundedFormula.externalSlotCopyEntryFormula 0 1 2 3 4 5 6 7 8 9) (extend z params))
(ZFSet.prod (ZFSet.prod V U) B) = externalSlotCopyGraph V E I U B action images j := by
apply ZFSet.ext
intro z
rw [ZFSet.mem_sep, BoundedFormula.truth_externalSlotCopyEntryFormula _ _ _ _ _ _ _ _ _ _ _ rfl]
exact ⟨And.right, fun hz => ⟨(ZFSet.mem_sep.1 hz).1, hz⟩⟩
exact equal ▸ selected
end TransitiveGround
end PartitionPilot.Forcing
end
end OAI
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