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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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundBatchAttachment
namespace OAI
section
namespace PartitionPilot.Forcing.AttachmentBatchInputs
universe u
variable {P r μ J Qs rs μs As Es Fs Ss : ZFSet.{u}}
variable (d : AttachmentBatchInputs P r μ J Qs rs μs As Es Fs Ss)
theorem old_order_iff {x y : ZFSet.{u}} (hx : x ∈ P) (hy : y ∈ P) :
ZFSet.pair x y ∈ d.batchOrder ↔ ZFSet.pair x y ∈ r :=
d.coding.le_encode (.inl ⟨x, hx⟩) (.inl ⟨y, hy⟩)
theorem old_meet_entry {x y z : ZFSet.{u}}
(entry : ZFSet.pair (ZFSet.pair x y) z ∈ μ) :
ZFSet.pair (ZFSet.pair x y) z ∈ d.batchMeetGraph := by
have shape := ZFSet.pair_mem_prod.1 (d.oldOrder.operation.1 entry)
have inputs := ZFSet.pair_mem_prod.1 shape.1
exact d.coding.old_embedding_code.map_meet x y z x y z entry
((pair_mem_attachmentOldEmbeddingGraph P x x).2 ⟨inputs.1, rfl⟩)
((pair_mem_attachmentOldEmbeddingGraph P y y).2 ⟨inputs.2, rfl⟩)
((pair_mem_attachmentOldEmbeddingGraph P z z).2 ⟨shape.2, rfl⟩)
theorem old_meet_iff {x y z : ZFSet.{u}} (hx : x ∈ P) (hy : y ∈ P) :
ZFSet.pair (ZFSet.pair x y) z ∈ d.batchMeetGraph ↔
ZFSet.pair (ZFSet.pair x y) z ∈ μ := by
constructor
· intro entry
obtain ⟨w, old, _⟩ := d.oldOrder.operation.2 (ZFSet.pair x y)
(ZFSet.pair_mem_prod.2 ⟨hx, hy⟩)
have same : z = w :=
(show FunctionGraphOn (ZFSet.prod d.batchCarrier d.batchCarrier) d.batchMeetGraph from
⟨d.batchCarrier, d.batch_meetOrder.operation⟩).unique entry (d.old_meet_entry old)
exact same.symm ▸ old
· exact d.old_meet_entry
end PartitionPilot.Forcing.AttachmentBatchInputs
end
end OAI
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