openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Geometry /AssignmentIndexLevels.lean
| 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 | |