openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /CompressionRequestRealization.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.CompressionExternalTileOverlaps | |
| namespace OAI | |
| namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.Family | |
| open Attachment CosetGeometry | |
| universe u | |
| variable {A J R : Type u} [Group J] {P : MeetOrder A} | |
| variable (f : Family P R) (a : OrderAction J P) | |
| theorem generator_root_of_tile (r : R) (j : f.Child r) (x y : A) | |
| (equal : ((f.tile r).child j).toFun x = (f.tile r).root.toFun y) : | |
| f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun x) = | |
| (f.copy a 1).toFun y := by | |
| rw [f.translate_copy, mul_one] | |
| simpa only [one_mul] using f.transfer_child_root a 1 r j x y equal | |
| theorem generator_root_meet_of_tile (r : R) (j : f.Child r) (x y z : A) | |
| (equal : (f.tile r).order.meet (((f.tile r).child j).toFun x) | |
| ((f.tile r).root.toFun y) = (f.tile r).root.toFun z) : | |
| (f.gluedOrder a).meet | |
| (f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun x)) | |
| ((f.copy a 1).toFun y) = (f.copy a 1).toFun z := by | |
| rw [f.translate_copy, mul_one] | |
| simpa only [one_mul] using f.transfer_child_root_meet a 1 r j x y z equal | |
| theorem generator_children_meet_of_tile (r : R) (j k : f.Child r) (x y z : A) | |
| (equal : (f.tile r).order.meet (((f.tile r).child j).toFun x) | |
| (((f.tile r).child k).toFun y) = ((f.tile r).child j).toFun z) : | |
| (f.gluedOrder a).meet | |
| (f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun x)) | |
| (f.translate a (generator J f.Letters ⟨r, k⟩) ((f.copy a 1).toFun y)) = | |
| f.translate a (generator J f.Letters ⟨r, j⟩) ((f.copy a 1).toFun z) := by | |
| simp only [f.translate_copy, mul_one] | |
| simpa only [one_mul] using f.transfer_child_child_meet a 1 r j k x y z equal | |
| theorem generator_fixes_ideal_of_tile (r : R) (j : f.Child r) (W : A) | |
| (equal : ∀ x, P.le x W → ((f.tile r).child j).toFun x = (f.tile r).root.toFun x) | |
| (z : f.Node a) (lower : (f.gluedOrder a).le z ((f.copy a 1).toFun W)) : | |
| f.translate a (generator J f.Letters ⟨r, j⟩) z = z := by | |
| obtain ⟨x, rfl⟩ := (f.copy a 1).ideal W z lower | |
| exact f.generator_root_of_tile a r j x x (equal x (((f.copy a 1).le_iff x W).1 lower)) | |
| theorem identityCopy_preservesProfiles : PreservesProfiles (f.copy a 1).toMeetEmbedding := | |
| (f.copy a 1).preservesProfiles f.top f.greatest | |
| end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family | |
| namespace PartitionPilot.Forcing.GroundCompressionCompletion | |
| open SetRealization Compression Compression.RecursiveMaps Attachment CosetGeometry | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 : ZFSet.{u}} | |
| variable (completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ) | |
| variable (group : GroupCode G m inv one) (oldAction : PosetActionCode G P0 m one r0 A0) | |
| variable (compression : CompressionOriginalData P0 r0 μ0 A0 G | |
| (completion.algebra.union.fields 0) (completion.algebra.union.fields 1) | |
| (completion.algebra.union.fields 2) (completion.algebra.union.fields 12)) | |
| local notation "U" => completion.algebra.union.fields 0 | |
| local notation "P" => completion.algebra.union.meet.toSmallMeetOrder | |
| local notation "family" => completion.externalFamily group oldAction compression | |
| local notation "action" => completion.algebra.union.action.toOrderAction group completion.algebra.union.meet | |
| theorem external_separation_fixes (W : SetMembers P0) : | |
| letI := group.toGroup | |
| let w : SetMembers U := SetMembers.ofMem (SetMembers.raw W) (compression.subset (SetMembers.raw_mem W)) | |
| ∀ z, ((family).gluedOrder (action)).le z (((family).copy (action) 1).toFun w) → | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inl W), ⟨()⟩⟩) z = z := by | |
| let := group.toGroup | |
| dsimp only | |
| exact (family).generator_fixes_ideal_of_tile (action) (.inr (.inl W)) ⟨()⟩ _ | |
| (fun x lower => TileRequests.Request.separation_agrees _ _ _ x lower) | |
| theorem external_separation_meet (W : SetMembers P0) (x y : SetMembers U) | |
| (wx : ZFSet.pair (SetMembers.raw W) (SetMembers.raw x) ∈ completion.algebra.union.fields 1) | |
| (wy : ZFSet.pair (SetMembers.raw W) (SetMembers.raw y) ∈ completion.algebra.union.fields 1) : | |
| letI := group.toGroup | |
| ((family).gluedOrder (action)).meet | |
| ((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inl W), ⟨()⟩⟩) | |
| (((family).copy (action) 1).toFun x)) (((family).copy (action) 1).toFun y) = | |
| ((family).copy (action) 1).toFun | |
| (SetMembers.ofMem (SetMembers.raw W) (compression.subset (SetMembers.raw_mem W))) := by | |
| let := group.toGroup | |
| exact (family).generator_root_meet_of_tile (action) (.inr (.inl W)) ⟨()⟩ x y _ | |
| (BinaryTiles.separation_cross_meet P _ x y | |
| (by simpa only [MeetOrderCode.toSmallMeetOrder, SetMembers.raw_ofMem] using wx) | |
| (by simpa only [MeetOrderCode.toSmallMeetOrder, SetMembers.raw_ofMem] using wy)) | |
| theorem external_compression_restriction (i : SetMembers I) (x : SetMembers U) : | |
| letI := group.toGroup | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inl i, ⟨()⟩⟩) | |
| (((family).copy (action) 1).toFun x) = | |
| ((family).copy (action) 1).toFun ((completion.externalCopy i).toFun x).val := by | |
| let := group.toGroup | |
| exact (family).generator_root_of_tile (action) (.inl i) ⟨()⟩ x _ | |
| (TileRequests.Request.compression_agrees _ _ _ _ x) | |
| theorem external_compression_top (i : SetMembers I) : | |
| letI := group.toGroup | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inl i, ⟨()⟩⟩) | |
| (((family).copy (action) 1).toFun (SetMembers.ofMem τ completion.top_mem)) = | |
| ((family).copy (action) 1).toFun (completion.externalCap i) := by | |
| let := group.toGroup | |
| exact (family).generator_root_of_tile (action) (.inl i) ⟨()⟩ _ _ | |
| (BinaryTiles.compression_top P _ _ (family).greatest _) | |
| theorem external_rectangle_roots (z : SetMembers (externalRectangles P0 G r0 μ0 A0)) | |
| (x : SetMembers U) (below : (P).le x (completion.externalRectangle group oldAction compression z).C) : | |
| letI := group.toGroup | |
| let r := completion.externalRectangle group oldAction compression z | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩) | |
| (((family).copy (action) 1).toFun x) = | |
| ((family).copy (action) 1).toFun (((action).iso r.first).toFun x) ∧ | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩) | |
| (((family).copy (action) 1).toFun x) = | |
| ((family).copy (action) 1).toFun (((action).iso r.second).toFun x) := by | |
| let := group.toGroup | |
| dsimp only | |
| let r := (completion.externalRectangle group oldAction compression z).toInput | |
| (family).top (family).greatest | |
| exact ⟨(family).generator_root_of_tile (action) (.inr (.inr z)) ⟨false⟩ x _ (r.first_agrees x below), | |
| (family).generator_root_of_tile (action) (.inr (.inr z)) ⟨true⟩ x _ (r.second_agrees x below)⟩ | |
| theorem external_rectangle_children (z : SetMembers (externalRectangles P0 G r0 μ0 A0)) | |
| (x : SetMembers U) (below : (P).le x (completion.externalRectangle group oldAction compression z).B) : | |
| letI := group.toGroup | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩) | |
| (((family).copy (action) 1).toFun x) = | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩) | |
| (((family).copy (action) 1).toFun x) := by | |
| let := group.toGroup | |
| let r := (completion.externalRectangle group oldAction compression z).toInput | |
| (family).top (family).greatest | |
| simp only [TileRequests.Family.translate_copy, mul_one] | |
| have equal := congrArg ((family).placedTile (action) 1 (.inr (.inr z))).toFun | |
| ((r.children_intersection x x).2 ⟨below, rfl⟩) | |
| change ((family).placedTile (action) 1 (.inr (.inr z))).toFun | |
| ((((family).tile (.inr (.inr z))).child ⟨false⟩).toFun x) = | |
| ((family).placedTile (action) 1 (.inr (.inr z))).toFun | |
| ((((family).tile (.inr (.inr z))).child ⟨true⟩).toFun x) at equal | |
| simpa only [one_mul] using | |
| ((family).child_identification (action) 1 (.inr (.inr z)) ⟨false⟩ x).symm.trans | |
| (equal.trans ((family).child_identification (action) 1 (.inr (.inr z)) ⟨true⟩ x)) | |
| theorem external_rectangle_root_meets (z : SetMembers (externalRectangles P0 G r0 μ0 A0)) | |
| (x : SetMembers U) : | |
| letI := group.toGroup | |
| let r := completion.externalRectangle group oldAction compression z | |
| ((family).gluedOrder (action)).meet | |
| ((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩) | |
| (((family).copy (action) 1).toFun r.D)) (((family).copy (action) 1).toFun x) = | |
| ((family).copy (action) 1).toFun ((P).meet (((action).iso r.first).toFun r.C) x) ∧ | |
| ((family).gluedOrder (action)).meet | |
| ((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩) | |
| (((family).copy (action) 1).toFun r.D)) (((family).copy (action) 1).toFun x) = | |
| ((family).copy (action) 1).toFun ((P).meet (((action).iso r.second).toFun r.C) x) := by | |
| let := group.toGroup | |
| dsimp only | |
| let r := (completion.externalRectangle group oldAction compression z).toInput | |
| (family).top (family).greatest | |
| exact ⟨(family).generator_root_meet_of_tile (action) (.inr (.inr z)) ⟨false⟩ r.D x _ (r.first_meet_root x), | |
| (family).generator_root_meet_of_tile (action) (.inr (.inr z)) ⟨true⟩ r.D x _ (r.second_meet_root x)⟩ | |
| theorem external_rectangle_children_meet (z : SetMembers (externalRectangles P0 G r0 μ0 A0)) : | |
| letI := group.toGroup | |
| let r := completion.externalRectangle group oldAction compression z | |
| ((family).gluedOrder (action)).meet | |
| ((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩) | |
| (((family).copy (action) 1).toFun r.D)) | |
| ((family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨true⟩⟩) | |
| (((family).copy (action) 1).toFun r.D)) = | |
| (family).translate (action) (generator (SetMembers G) (family).Letters ⟨.inr (.inr z), ⟨false⟩⟩) | |
| (((family).copy (action) 1).toFun r.B) := by | |
| let := group.toGroup | |
| dsimp only | |
| let r := (completion.externalRectangle group oldAction compression z).toInput | |
| (family).top (family).greatest | |
| exact (family).generator_children_meet_of_tile (action) (.inr (.inr z)) ⟨false⟩ ⟨true⟩ r.D r.D r.B r.children_meet | |
| end PartitionPilot.Forcing.GroundCompressionCompletion | |
| end OAI | |