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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 | import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundMainStageFromCodes
namespace OAI
section
namespace PartitionPilot.Forcing.BoundedFormula
open SetFormula
universe u
def meetOrderFormula {n} (P r μ : Fin n) : BoundedFormula n :=
.conj (partialOrderFormula P r) (.conj (binaryFunctionFormula P μ)
(.allMem P (.allMem P.succ (.allMem P.succ.succ
(imp (tripleMembershipFormula 2 1 0 μ.succ.succ.succ)
(.conj (pairMembershipFormula 0 2 r.succ.succ.succ)
(.conj (pairMembershipFormula 0 1 r.succ.succ.succ)
(.allMem P.succ.succ.succ (imp
(.conj (pairMembershipFormula 0 3 r.succ.succ.succ.succ)
(pairMembershipFormula 0 2 r.succ.succ.succ.succ))
(pairMembershipFormula 0 1 r.succ.succ.succ.succ))))))))))
theorem truth_meetOrderFormula {n} (P r μ : Fin n) (env : Fin n → ZFSet.{u}) :
Truth (meetOrderFormula P r μ) env ↔ MeetOrderCode (env P) (env r) (env μ) := by
simp only [meetOrderFormula, Truth, truth_partialOrderFormula,
truth_binaryFunctionFormula, truth_imp, truth_pairMembershipFormula,
truth_tripleMembershipFormula]
constructor
· rintro ⟨order, operation, laws⟩
have bounds (x y z : ZFSet.{u}) (entry : ZFSet.pair (ZFSet.pair x y) z ∈ env μ) :
x ∈ env P ∧ y ∈ env P ∧ z ∈ env P := by
have shape := ZFSet.pair_mem_prod.1 (operation.1 entry)
exact ⟨(ZFSet.pair_mem_prod.1 shape.1).1,
(ZFSet.pair_mem_prod.1 shape.1).2, shape.2⟩
refine ⟨order, operation, ?_, ?_, ?_⟩
· intro x y z entry
have h := bounds x y z entry
exact (laws x h.1 y h.2.1 z h.2.2 entry).1
· intro x y z entry
have h := bounds x y z entry
exact (laws x h.1 y h.2.1 z h.2.2 entry).2.1
· intro x y z w hw entry wx wy
have h := bounds x y z entry
exact (laws x h.1 y h.2.1 z h.2.2 entry).2.2 w hw ⟨wx, wy⟩
· intro meet
exact ⟨meet.order, meet.operation, fun x _ y _ z _ entry =>
⟨meet.lower_left x y z entry, meet.lower_right x y z entry,
fun w hw ⟨wx, wy⟩ => meet.greatest x y z w hw entry wx wy⟩⟩
theorem realize_meetOrderFormula (M : TransitiveGround.{u}) {n}
(P r μ : Fin n) (env : Fin n → ZFSet.{u}) (members : ∀ i, env i ∈ M.sets) :
Realize M.sets (meetOrderFormula P r μ).toFormula env ↔
MeetOrderCode (env P) (env r) (env μ) :=
(realize_toFormula M _ env members).trans (truth_meetOrderFormula P r μ env)
end PartitionPilot.Forcing.BoundedFormula
end
end OAI
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