openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /BoundedRelationImageEquality.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.GroundFlatCopyGraphs | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.BoundedFormula | |
| open SetFormula | |
| universe u | |
| def relationImageEqualityFormula {n} (output domain codomain graph : Fin n) : BoundedFormula n := | |
| .conj (.allMem output (.conj (.mem 0 codomain.succ) | |
| (imageMemberFormula 0 domain.succ graph.succ))) | |
| (.allMem codomain (imp (imageMemberFormula 0 domain.succ graph.succ) (.mem 0 output.succ))) | |
| theorem truth_relationImageEqualityFormula {n} (output domain codomain graph : Fin n) | |
| (env : Fin n → ZFSet.{u}) : | |
| Truth (relationImageEqualityFormula output domain codomain graph) env ↔ | |
| env output = relationImage (env domain) (env codomain) (env graph) := by | |
| simp only [relationImageEqualityFormula, Truth, truth_imageMemberFormula, truth_imp] | |
| change ((∀ y ∈ env output, y ∈ env codomain ∧ ∃ x ∈ env domain, | |
| ZFSet.pair x y ∈ env graph) ∧ | |
| (∀ y ∈ env codomain, (∃ x ∈ env domain, ZFSet.pair x y ∈ env graph) → y ∈ env output)) ↔ _ | |
| constructor | |
| · rintro ⟨forward, backward⟩ | |
| apply ZFSet.ext | |
| intro y | |
| rw [mem_relationImage] | |
| exact ⟨forward y, fun h => backward y h.1 h.2⟩ | |
| · intro equal | |
| constructor | |
| · intro y hy | |
| exact (mem_relationImage _ _ _ y).1 (equal ▸ hy) | |
| · intro y hy member | |
| exact equal ▸ (mem_relationImage _ _ _ y).2 ⟨hy, member⟩ | |
| end PartitionPilot.Forcing.BoundedFormula | |
| end | |
| end OAI | |