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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 | import OAI.SetTheory.PartitionPrinciple.Transfinite.StageOneDataBoundLaws
namespace OAI
section
namespace PartitionPilot.Forcing
universe u
noncomputable def assignmentIndexLevels (I T eta levels : ZFSet.{u}) : ZFSet.{u} :=
relationCompose (stageOneAssignments I T) I eta
(productFirst I (ZFSet.prod T T)) levels
theorem pair_mem_assignmentIndexLevels (I T eta levels z level : ZFSet.{u}) :
ZFSet.pair z level ∈ assignmentIndexLevels I T eta levels ↔
∃ i ∈ I, ∃ u ∈ T, ∃ w ∈ T,
z = stageOneAssignment i u w ∧ level ∈ eta ∧ ZFSet.pair i level ∈ levels := by
rw [assignmentIndexLevels, pair_mem_relationCompose]
constructor
· rintro ⟨_, hl, i, hi, projection, entry⟩
obtain ⟨j, _, rest, hr, equal⟩ := (mem_productFirst I (ZFSet.prod T T) _).1 projection
obtain ⟨hz, hji⟩ := ZFSet.pair_inj.1 equal
obtain ⟨u, hu, w, hw, rfl⟩ := ZFSet.mem_prod.1 hr
exact ⟨i, hi, u, hu, w, hw, hji ▸ hz, hl, entry⟩
· rintro ⟨i, hi, u, hu, w, hw, rfl, hl, entry⟩
refine ⟨(stageOneAssignment_mem_assignments I T i u w).2 ⟨hi, hu, hw⟩,
hl, i, hi, ?_, entry⟩
exact (mem_productFirst I (ZFSet.prod T T) _).2
⟨i, hi, ZFSet.pair u w, ZFSet.pair_mem_prod.2 ⟨hu, hw⟩, rfl⟩
theorem assignmentIndexLevels_entry (I T eta levels i u w level : ZFSet.{u}) :
ZFSet.pair (stageOneAssignment i u w) level ∈ assignmentIndexLevels I T eta levels ↔
i ∈ I ∧ u ∈ T ∧ w ∈ T ∧ level ∈ eta ∧ ZFSet.pair i level ∈ levels := by
rw [pair_mem_assignmentIndexLevels]
constructor
· rintro ⟨j, hj, v, hv, x, hx, equal, hl, entry⟩
obtain ⟨rfl, rfl, rfl⟩ := (stageOneAssignment_eq_iff i u w j v x).1 equal
exact ⟨hj, hv, hx, hl, entry⟩
· rintro ⟨hi, hu, hw, hl, entry⟩
exact ⟨i, hi, u, hu, w, hw, rfl, hl, entry⟩
theorem assignmentIndexLevels_function {I T eta levels : ZFSet.{u}}
(function : ZFSet.IsFunc I eta levels) :
ZFSet.IsFunc (stageOneAssignments I T) eta (assignmentIndexLevels I T eta levels) := by
refine ⟨?_, ?_⟩
· intro z hz
rw [assignmentIndexLevels, relationCompose] at hz
exact (ZFSet.mem_sep.1 hz).1
· intro z hz
obtain ⟨i, hi, u, hu, w, hw, rfl⟩ := (mem_stageOneAssignments I T z).1 hz
obtain ⟨level, entry, unique⟩ := function.2 i hi
have hl := (ZFSet.pair_mem_prod.1 (function.1 entry)).2
refine ⟨level, (assignmentIndexLevels_entry I T eta levels i u w level).2
⟨hi, hu, hw, hl, entry⟩, ?_⟩
intro other ho
exact unique other ((assignmentIndexLevels_entry I T eta levels i u w other).1 ho).2.2.2.2
end PartitionPilot.Forcing
end
end OAI
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