openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /FlatLimitConditionCodes.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.AssignmentNestedSplices | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| def FlatHistoryCuts (stages history levels row A : ZFSet.{u}) : Prop := | |
| ∀ beta ∈ stages, ∃ Q, ZFSet.pair beta Q ∈ history ∧ | |
| ZFSet.pair row (assignmentCut levels beta A) ∈ Q | |
| def FlatLimitConditionCode (stages history levels z : ZFSet.{u}) : Prop := | |
| ∃ row A, z = ZFSet.pair row A ∧ FlatHistoryCuts stages history levels row A | |
| theorem flatLimitConditionCode_pair (stages history levels row A : ZFSet.{u}) : | |
| FlatLimitConditionCode stages history levels (ZFSet.pair row A) ↔ | |
| FlatHistoryCuts stages history levels row A := by | |
| constructor | |
| · rintro ⟨row', A', equal, valid⟩ | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_injective equal | |
| exact valid | |
| · intro valid | |
| exact ⟨row, A, rfl, valid⟩ | |
| theorem flatHistoryCuts_lookup_iff {stages history levels row A : ZFSet.{u}} | |
| (function : FunctionGraphOn stages history) : | |
| FlatHistoryCuts stages history levels row A ↔ | |
| ∀ beta ∈ stages, ∀ Q, ZFSet.pair beta Q ∈ history → | |
| ZFSet.pair row (assignmentCut levels beta A) ∈ Q := by | |
| constructor | |
| · intro valid beta hbeta Q entry | |
| obtain ⟨Q', entry', member⟩ := valid beta hbeta | |
| exact function.unique entry' entry ▸ member | |
| · intro valid beta hbeta | |
| obtain ⟨Q, entry⟩ := function.total hbeta | |
| exact ⟨Q, entry, valid beta hbeta Q entry⟩ | |
| theorem flatHistoryCuts_restrict {stages earlier history levels row A : ZFSet.{u}} | |
| (valid : FlatHistoryCuts stages history levels row A) (subset : earlier ⊆ stages) : | |
| FlatHistoryCuts earlier (graphRestriction earlier history) levels row A := by | |
| intro beta hbeta | |
| obtain ⟨Q, entry, member⟩ := valid beta (subset hbeta) | |
| exact ⟨Q, pair_mem_graphRestriction.2 ⟨entry, hbeta⟩, member⟩ | |
| theorem flatHistoryCuts_cut {stages earlier history levels alpha row A : ZFSet.{u}} | |
| (valid : FlatHistoryCuts stages history levels row A) (subset : earlier ⊆ stages) | |
| (nested : ∀ beta ∈ earlier, assignmentCut levels beta (assignmentCut levels alpha A) = | |
| assignmentCut levels beta A) : | |
| FlatHistoryCuts earlier (graphRestriction earlier history) levels row | |
| (assignmentCut levels alpha A) := by | |
| intro beta hbeta | |
| obtain ⟨Q, entry, member⟩ := flatHistoryCuts_restrict valid subset beta hbeta | |
| exact ⟨Q, entry, (nested beta hbeta).symm ▸ member⟩ | |
| theorem flatHistoryCuts_cut_of_subset {stages earlier history levels alpha row A : ZFSet.{u}} | |
| (valid : FlatHistoryCuts stages history levels row A) (subset : earlier ⊆ stages) | |
| (below : ∀ beta ∈ earlier, beta ⊆ alpha) : | |
| FlatHistoryCuts earlier (graphRestriction earlier history) levels row | |
| (assignmentCut levels alpha A) := | |
| flatHistoryCuts_cut valid subset (fun beta hbeta => assignmentCut_nested (below beta hbeta)) | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |