openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Geometry /ExternalTileVertexPresentation.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.CompressionExternalTileOverlaps | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.GroundCompressionCompletion | |
| open SetRealization | |
| 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)) | |
| theorem externalFamilyVertexCode_injective | |
| (r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0)) : | |
| Function.Injective (completion.externalFamilyVertexCode group oldAction compression r) := by | |
| rcases r with i | (x | z) | |
| all_goals | |
| intro v w equal | |
| cases v with | |
| | none => | |
| cases w with | |
| | none => rfl | |
| | some w => | |
| cases w with | up w => | |
| cases w <;> simp only [externalFamilyVertexCode_root, externalFamilyVertexCode_compression, | |
| externalFamilyVertexCode_separation, externalFamilyVertexCode_rectangle, | |
| natSetCode_injective.eq_iff] at equal <;> contradiction | |
| | some v => | |
| cases w with | |
| | none => | |
| cases v with | up v => | |
| cases v <;> simp only [externalFamilyVertexCode_root, externalFamilyVertexCode_compression, | |
| externalFamilyVertexCode_separation, externalFamilyVertexCode_rectangle, | |
| natSetCode_injective.eq_iff] at equal <;> contradiction | |
| | some w => | |
| cases v with | up v => | |
| cases w with | up w => | |
| cases v <;> cases w <;> simp_all only [externalFamilyVertexCode_compression, | |
| externalFamilyVertexCode_separation, externalFamilyVertexCode_rectangle, | |
| natSetCode_injective.eq_iff, Bool.false_eq_true] <;> simp_all | |
| theorem externalFamilyVertexCode_range | |
| (r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0)) (v : ZFSet.{u}) : | |
| (v = natSetCode 0 ∨ v = natSetCode 1 ∨ | |
| (externalRequestIndexCode r ∈ ZFSet.prod {natSetCode 2} (externalRectangles P0 G r0 μ0 A0) ∧ | |
| v = natSetCode 2)) ↔ | |
| ∃ i, completion.externalFamilyVertexCode group oldAction compression r i = v := by | |
| rcases r with j | (x | z) | |
| · simp only [externalRequestIndexCode, ZFSet.pair_mem_prod, ZFSet.mem_singleton, | |
| natSetCode_injective.eq_iff, show ¬ (0 : Nat) = 2 by decide, false_and, or_false] | |
| constructor | |
| · rintro (rfl | rfl) | |
| · exact ⟨none, rfl⟩ | |
| · exact ⟨some ⟨()⟩, rfl⟩ | |
| · rintro ⟨i, rfl⟩ | |
| cases i with | |
| | none => exact Or.inl rfl | |
| | some i => exact Or.inr rfl | |
| · simp only [externalRequestIndexCode, ZFSet.pair_mem_prod, ZFSet.mem_singleton, | |
| natSetCode_injective.eq_iff, show ¬ (1 : Nat) = 2 by decide, false_and, or_false] | |
| constructor | |
| · rintro (rfl | rfl) | |
| · exact ⟨none, rfl⟩ | |
| · exact ⟨some ⟨()⟩, rfl⟩ | |
| · rintro ⟨i, rfl⟩ | |
| cases i with | |
| | none => exact Or.inl rfl | |
| | some i => exact Or.inr rfl | |
| · simp only [externalRequestIndexCode, ZFSet.pair_mem_prod, ZFSet.mem_singleton, | |
| SetMembers.raw_mem, and_self, true_and] | |
| constructor | |
| · rintro (rfl | rfl | rfl) | |
| · exact ⟨none, rfl⟩ | |
| · exact ⟨some ⟨false⟩, rfl⟩ | |
| · exact ⟨some ⟨true⟩, rfl⟩ | |
| · rintro ⟨i, rfl⟩ | |
| cases i with | |
| | none => exact Or.inl rfl | |
| | some i => | |
| cases i with | up b => | |
| cases b | |
| · exact Or.inr (Or.inl rfl) | |
| · exact Or.inr (Or.inr rfl) | |
| noncomputable def externalFamilyTileVertexPresentation : | |
| Presentation (Sigma fun r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0) => | |
| Option ((completion.externalFamily group oldAction compression).Child r) × | |
| SetMembers (completion.algebra.union.fields 0)) where | |
| carrier := externalTileVertices (externalRequestCarrier I P0 (externalRectangles P0 G r0 μ0 A0)) | |
| (externalRectangles P0 G r0 μ0 A0) (completion.algebra.union.fields 0) | |
| encode v := ZFSet.pair (ZFSet.pair (externalRequestIndexCode v.1) | |
| (completion.externalFamilyVertexCode group oldAction compression v.1 v.2.1)) (SetMembers.raw v.2.2) | |
| injective := by | |
| rintro ⟨r, i, x⟩ ⟨s, j, y⟩ equal | |
| have outer := ZFSet.pair_inj.1 equal | |
| have inner := ZFSet.pair_inj.1 outer.1 | |
| have same := externalRequestIndexCode_injective I P0 (externalRectangles P0 G r0 μ0 A0) inner.1 | |
| cases same | |
| have vertex := completion.externalFamilyVertexCode_injective group oldAction compression r inner.2 | |
| have point := SetMembers.raw_injective _ outer.2 | |
| cases vertex | |
| cases point | |
| rfl | |
| mem_carrier z := by | |
| constructor | |
| · intro hz | |
| have shape : ∃ q i x, z = ZFSet.pair (ZFSet.pair q i) x := by | |
| rcases ZFSet.mem_union.1 hz with h | h | |
| all_goals | |
| obtain ⟨qi, hqi, x, _, rfl⟩ := ZFSet.mem_prod.1 h | |
| obtain ⟨q, _, i, _, rfl⟩ := ZFSet.mem_prod.1 hqi | |
| exact ⟨q, i, x, rfl⟩ | |
| obtain ⟨q, i, x, rfl⟩ := shape | |
| obtain ⟨hqi, hx⟩ := (triple_mem_externalTileVertices _ _ _ _ _ _).1 hz | |
| have hq : q ∈ externalRequestCarrier I P0 (externalRectangles P0 G r0 μ0 A0) := by | |
| rcases hqi with ⟨hq, _⟩ | ⟨hq, _⟩ | |
| · exact hq | |
| · exact ZFSet.mem_union.2 (Or.inr hq) | |
| obtain ⟨r, rfl⟩ := ((externalRequestPresentation I P0 (externalRectangles P0 G r0 μ0 A0)).mem_carrier q).1 hq | |
| have hv : i = natSetCode 0 ∨ i = natSetCode 1 ∨ | |
| (externalRequestIndexCode r ∈ ZFSet.prod {natSetCode 2} (externalRectangles P0 G r0 μ0 A0) ∧ | |
| i = natSetCode 2) := by tauto | |
| obtain ⟨v, rfl⟩ := (completion.externalFamilyVertexCode_range group oldAction compression r i).1 hv | |
| exact ⟨⟨r, v, SetMembers.ofMem x hx⟩, by simp only [SetMembers.raw_ofMem]; rfl⟩ | |
| · rintro ⟨⟨r, i, x⟩, rfl⟩ | |
| apply (triple_mem_externalTileVertices _ _ _ _ _ _).2 | |
| refine ⟨?_, SetMembers.raw_mem x⟩ | |
| have hv := (completion.externalFamilyVertexCode_range group oldAction compression r _).2 ⟨i, rfl⟩ | |
| have hq := (externalRequestPresentation I P0 (externalRectangles P0 G r0 μ0 A0)).encode_mem r | |
| tauto | |
| theorem externalFamilyTileVertexPresentation_encode | |
| (r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0)) | |
| (i : Option ((completion.externalFamily group oldAction compression).Child r)) | |
| (x : SetMembers (completion.algebra.union.fields 0)) : | |
| (completion.externalFamilyTileVertexPresentation group oldAction compression).encode ⟨r, i, x⟩ = | |
| ZFSet.pair (ZFSet.pair (externalRequestIndexCode r) | |
| (completion.externalFamilyVertexCode group oldAction compression r i)) (SetMembers.raw x) := rfl | |
| end PartitionPilot.Forcing.GroundCompressionCompletion | |
| end | |
| end OAI | |