openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Geometry /AssignmentNestedSplices.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.NatUnionCodes | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| def assignmentCut (levels alpha A : ZFSet.{u}) : ZFSet.{u} := | |
| ZFSet.sep (fun z => ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) A | |
| def assignmentUpper (levels alpha A : ZFSet.{u}) : ZFSet.{u} := | |
| ZFSet.sep (fun z => z ∉ assignmentCut levels alpha A) A | |
| def assignmentSplice (levels alpha Aq Ap : ZFSet.{u}) : ZFSet.{u} := | |
| assignmentCut levels alpha Aq ∪ assignmentUpper levels alpha Ap | |
| theorem mem_assignmentCut (levels alpha A z : ZFSet.{u}) : | |
| z ∈ assignmentCut levels alpha A ↔ | |
| z ∈ A ∧ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels := | |
| ZFSet.mem_sep | |
| theorem mem_assignmentUpper (levels alpha A z : ZFSet.{u}) : | |
| z ∈ assignmentUpper levels alpha A ↔ | |
| z ∈ A ∧ ¬ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels := by | |
| rw [assignmentUpper, ZFSet.mem_sep, mem_assignmentCut] | |
| constructor | |
| · rintro ⟨hz, outside⟩ | |
| exact ⟨hz, fun lower => outside ⟨hz, lower⟩⟩ | |
| · rintro ⟨hz, outside⟩ | |
| exact ⟨hz, fun lower => outside lower.2⟩ | |
| theorem mem_assignmentSplice (levels alpha Aq Ap z : ZFSet.{u}) : | |
| z ∈ assignmentSplice levels alpha Aq Ap ↔ | |
| (z ∈ Aq ∧ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) ∨ | |
| (z ∈ Ap ∧ ¬ ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) := by | |
| rw [assignmentSplice, ZFSet.mem_union, mem_assignmentCut, mem_assignmentUpper] | |
| theorem assignmentCut_subset (levels alpha A : ZFSet.{u}) : | |
| assignmentCut levels alpha A ⊆ A := | |
| fun _ hz => ((mem_assignmentCut levels alpha A _).1 hz).1 | |
| theorem assignmentUpper_subset (levels alpha A : ZFSet.{u}) : | |
| assignmentUpper levels alpha A ⊆ A := | |
| fun _ hz => ((mem_assignmentUpper levels alpha A _).1 hz).1 | |
| theorem assignmentSplice_subset (levels alpha Aq Ap : ZFSet.{u}) : | |
| assignmentSplice levels alpha Aq Ap ⊆ Aq ∪ Ap := by | |
| intro z hz | |
| rcases (mem_assignmentSplice levels alpha Aq Ap z).1 hz with lower | upper | |
| · exact ZFSet.mem_union.2 (Or.inl lower.1) | |
| · exact ZFSet.mem_union.2 (Or.inr upper.1) | |
| theorem assignmentCut_mono {levels alpha A B : ZFSet.{u}} (subset : A ⊆ B) : | |
| assignmentCut levels alpha A ⊆ assignmentCut levels alpha B := by | |
| intro z hz | |
| obtain ⟨hz, lower⟩ := (mem_assignmentCut levels alpha A z).1 hz | |
| exact (mem_assignmentCut levels alpha B z).2 ⟨subset hz, lower⟩ | |
| theorem assignmentCut_nested {levels alpha beta A : ZFSet.{u}} | |
| (below : alpha ⊆ beta) : | |
| assignmentCut levels alpha (assignmentCut levels beta A) = | |
| assignmentCut levels alpha A := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_assignmentCut, mem_assignmentCut, mem_assignmentCut] | |
| constructor | |
| · rintro ⟨⟨hz, _⟩, lower⟩ | |
| exact ⟨hz, lower⟩ | |
| · rintro ⟨hz, xi, hxi, entry⟩ | |
| exact ⟨⟨hz, xi, below hxi, entry⟩, xi, hxi, entry⟩ | |
| theorem assignmentCut_eq_of_lower {levels alpha A : ZFSet.{u}} | |
| (lower : ∀ z ∈ A, ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) : | |
| assignmentCut levels alpha A = A := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_assignmentCut] | |
| exact ⟨fun hz => hz.1, fun hz => ⟨hz, lower z hz⟩⟩ | |
| theorem assignmentCut_full {J eta levels A : ZFSet.{u}} | |
| (function : ZFSet.IsFunc J eta levels) (subset : A ⊆ J) : | |
| assignmentCut levels eta A = A := by | |
| apply assignmentCut_eq_of_lower | |
| intro z hz | |
| obtain ⟨xi, entry, _⟩ := function.2 z (subset hz) | |
| exact ⟨xi, (ZFSet.pair_mem_prod.1 (function.1 entry)).2, entry⟩ | |
| theorem assignmentCut_splice (levels alpha Aq Ap : ZFSet.{u}) : | |
| assignmentCut levels alpha (assignmentSplice levels alpha Aq Ap) = | |
| assignmentCut levels alpha Aq := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_assignmentCut, mem_assignmentSplice, mem_assignmentCut] | |
| constructor | |
| · rintro ⟨lower | upper, level⟩ | |
| · exact lower | |
| · exact False.elim (upper.2 level) | |
| · intro lower | |
| exact ⟨Or.inl lower, lower.2⟩ | |
| theorem assignmentUpper_splice (levels alpha Aq Ap : ZFSet.{u}) : | |
| assignmentUpper levels alpha (assignmentSplice levels alpha Aq Ap) = | |
| assignmentUpper levels alpha Ap := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_assignmentUpper, mem_assignmentSplice, mem_assignmentUpper] | |
| constructor | |
| · rintro ⟨lower | upper, outside⟩ | |
| · exact False.elim (outside lower.2) | |
| · exact upper | |
| · intro upper | |
| exact ⟨Or.inr upper, upper.2⟩ | |
| theorem assignmentSplice_self (levels alpha A : ZFSet.{u}) : | |
| assignmentSplice levels alpha A A = A := by | |
| classical | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_assignmentSplice] | |
| constructor | |
| · exact fun h => h.elim And.left And.left | |
| · intro hz | |
| by_cases lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels | |
| · exact Or.inl ⟨hz, lower⟩ | |
| · exact Or.inr ⟨hz, lower⟩ | |
| theorem assignmentSplice_of_lower {levels alpha Aq Ap : ZFSet.{u}} | |
| (lower : ∀ z ∈ Aq, ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels) : | |
| assignmentSplice levels alpha Aq Ap = Aq ∪ assignmentUpper levels alpha Ap ∧ | |
| assignmentCut levels alpha (assignmentSplice levels alpha Aq Ap) = Aq := by | |
| have equal := assignmentCut_eq_of_lower lower | |
| exact ⟨by rw [assignmentSplice, equal], by rw [assignmentCut_splice, equal]⟩ | |
| theorem assignmentSplice_extends {levels alpha Aq Ap : ZFSet.{u}} | |
| (contains : assignmentCut levels alpha Ap ⊆ assignmentCut levels alpha Aq) : | |
| Ap ⊆ assignmentSplice levels alpha Aq Ap := by | |
| classical | |
| intro z hz | |
| by_cases lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels | |
| · exact ZFSet.mem_union.2 (Or.inl (contains | |
| ((mem_assignmentCut levels alpha Ap z).2 ⟨hz, lower⟩))) | |
| · exact (mem_assignmentSplice levels alpha Aq Ap z).2 (Or.inr ⟨hz, lower⟩) | |
| end PartitionPilot.Forcing | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| variable {levels alpha beta : ZFSet.{u}} (below : alpha ⊆ beta) | |
| include below | |
| theorem assignmentCut_splice_below (Aq Ap : ZFSet.{u}) : | |
| assignmentCut levels alpha (assignmentSplice levels beta Aq Ap) = | |
| assignmentCut levels alpha Aq := by | |
| calc | |
| _ = assignmentCut levels alpha | |
| (assignmentCut levels beta (assignmentSplice levels beta Aq Ap)) := | |
| (assignmentCut_nested below).symm | |
| _ = assignmentCut levels alpha (assignmentCut levels beta Aq) := by rw [assignmentCut_splice] | |
| _ = _ := assignmentCut_nested below | |
| theorem assignmentUpper_splice_above (Aq Ap : ZFSet.{u}) : | |
| assignmentUpper levels beta (assignmentSplice levels alpha Aq Ap) = | |
| assignmentUpper levels beta Ap := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_assignmentUpper, mem_assignmentSplice, mem_assignmentUpper] | |
| constructor | |
| · rintro ⟨lower | upper, outside⟩ | |
| · obtain ⟨xi, hxi, entry⟩ := lower.2 | |
| exact False.elim (outside ⟨xi, below hxi, entry⟩) | |
| · exact ⟨upper.1, outside⟩ | |
| · rintro ⟨member, outside⟩ | |
| exact ⟨Or.inr ⟨member, fun ⟨xi, hxi, entry⟩ => outside ⟨xi, below hxi, entry⟩⟩, outside⟩ | |
| theorem assignmentSplice_nested (Ar Aq Ap : ZFSet.{u}) : | |
| assignmentSplice levels alpha Ar (assignmentSplice levels beta Aq Ap) = | |
| assignmentSplice levels beta (assignmentSplice levels alpha Ar Aq) Ap := by | |
| classical | |
| apply ZFSet.ext | |
| intro z | |
| simp only [mem_assignmentSplice] | |
| by_cases lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels | |
| · obtain ⟨xi, hxi, entry⟩ := lower | |
| have lower : ∃ xi ∈ alpha, ZFSet.pair z xi ∈ levels := ⟨xi, hxi, entry⟩ | |
| have higher : ∃ xi ∈ beta, ZFSet.pair z xi ∈ levels := ⟨xi, below hxi, entry⟩ | |
| simp only [lower, higher, and_true, not_true_eq_false, and_false, or_false] | |
| · by_cases higher : ∃ xi ∈ beta, ZFSet.pair z xi ∈ levels | |
| · simp only [lower, higher, and_true, not_true_eq_false, and_false, or_false, | |
| not_false_eq_true, false_or] | |
| · simp only [lower, higher, and_true, and_false, not_false_eq_true, false_or] | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |