openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /AssignmentQueryFormulas.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.RepresentativeIndexSteps | |
| namespace OAI | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| noncomputable def assignmentQueryRelation (I T C D : ZFSet.{u}) : ZFSet.{u} := | |
| ZFSet.sep (fun z => ∃ row L A, | |
| z = ZFSet.pair (ZFSet.pair row L) (ZFSet.pair row A) ∧ | |
| assignmentOnIndices (assignmentIndices I T L) T A = L) (ZFSet.prod C D) | |
| theorem assignmentQueryRelation_entry (I T C D row L other A : ZFSet.{u}) : | |
| ZFSet.pair (ZFSet.pair row L) (ZFSet.pair other A) ∈ assignmentQueryRelation I T C D ↔ | |
| ZFSet.pair row L ∈ C ∧ ZFSet.pair other A ∈ D ∧ row = other ∧ | |
| assignmentOnIndices (assignmentIndices I T L) T A = L := by | |
| rw [assignmentQueryRelation, ZFSet.mem_sep, ZFSet.pair_mem_prod] | |
| constructor | |
| · rintro ⟨⟨hq, hp⟩, r, L', A', eq, matchList⟩ | |
| obtain ⟨first, second⟩ := ZFSet.pair_inj.1 eq | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 first | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 second | |
| exact ⟨hq, hp, rfl, matchList⟩ | |
| · rintro ⟨hq, hp, rfl, matchList⟩ | |
| exact ⟨⟨hq, hp⟩, row, L, A, rfl, matchList⟩ | |
| noncomputable def realizedAssignmentQueries (I T C D : ZFSet.{u}) : ZFSet.{u} := | |
| relationImage D C (relationConverse C D (assignmentQueryRelation I T C D)) | |
| theorem mem_realizedAssignmentQueries (I T C D q : ZFSet.{u}) : | |
| q ∈ realizedAssignmentQueries I T C D ↔ | |
| q ∈ C ∧ ∃ p ∈ D, ZFSet.pair q p ∈ assignmentQueryRelation I T C D := by | |
| rw [realizedAssignmentQueries, mem_relationImage] | |
| constructor | |
| · rintro ⟨hq, p, hp, entry⟩ | |
| exact ⟨hq, p, hp, ((pair_mem_relationConverse C D _ q p).1 entry).2.2⟩ | |
| · rintro ⟨hq, p, hp, entry⟩ | |
| exact ⟨hq, p, hp, (pair_mem_relationConverse C D _ q p).2 ⟨hq, hp, entry⟩⟩ | |
| theorem assignmentIndexOverlap_iff (I T L z : ZFSet.{u}) : | |
| (∃ i ∈ I, ∃ w ∈ L, | |
| ZFSet.pair z i ∈ productFirst I (ZFSet.prod T T) ∧ | |
| ZFSet.pair w i ∈ productFirst I (ZFSet.prod T T)) ↔ | |
| z ∈ stageOneAssignments (assignmentIndices I T L) T := by | |
| constructor | |
| · rintro ⟨i, hi, w, hw, first, second⟩ | |
| obtain ⟨j, _, rest, hr, equal⟩ := (mem_productFirst I (ZFSet.prod T T) _).1 first | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal | |
| exact ZFSet.pair_mem_prod.2 | |
| ⟨(mem_relationImage L I _ _).2 ⟨hi, w, hw, second⟩, hr⟩ | |
| · intro member | |
| obtain ⟨i, hi, rest, hr, rfl⟩ := ZFSet.mem_prod.1 member | |
| obtain ⟨hi, w, hw, second⟩ := (mem_relationImage L I _ i).1 hi | |
| exact ⟨i, hi, w, hw, (mem_productFirst I (ZFSet.prod T T) _).2 | |
| ⟨i, hi, rest, hr, rfl⟩, second⟩ | |
| namespace BoundedFormula | |
| open SetFormula | |
| def assignmentQueryOverlapFormula {n} (z I L f : Fin n) : BoundedFormula n := | |
| exMem I (exMem L.succ (.conj (pairMembershipFormula z.succ.succ 1 f.succ.succ) | |
| (pairMembershipFormula 0 1 f.succ.succ))) | |
| theorem truth_assignmentQueryOverlapFormula {n} (z I L f : Fin n) | |
| (env : Fin n → ZFSet.{u}) : | |
| Truth (assignmentQueryOverlapFormula z I L f) env ↔ | |
| ∃ i ∈ env I, ∃ w ∈ env L, | |
| ZFSet.pair (env z) i ∈ env f ∧ ZFSet.pair w i ∈ env f := by | |
| simp only [assignmentQueryOverlapFormula, truth_exMem, Truth, truth_pairMembershipFormula] | |
| rfl | |
| def assignmentQueryMatchFormula {n} (I L A f : Fin n) : BoundedFormula n := | |
| .conj (.allMem L (.conj (.mem 0 A.succ) | |
| (assignmentQueryOverlapFormula 0 I.succ L.succ f.succ))) | |
| (.allMem A (.imp (assignmentQueryOverlapFormula 0 I.succ L.succ f.succ) (.mem 0 L.succ))) | |
| theorem truth_assignmentQueryMatchFormula {n} (I L A f : Fin n) | |
| (env : Fin n → ZFSet.{u}) (T : ZFSet.{u}) | |
| (projection : env f = productFirst (env I) (ZFSet.prod T T)) : | |
| Truth (assignmentQueryMatchFormula I L A f) env ↔ | |
| assignmentOnIndices (assignmentIndices (env I) T (env L)) T (env A) = env L := by | |
| simp only [assignmentQueryMatchFormula, Truth, truth_imp, truth_assignmentQueryOverlapFormula] | |
| change ((∀ z ∈ env L, z ∈ env A ∧ ∃ i ∈ env I, ∃ w ∈ env L, | |
| ZFSet.pair z i ∈ env f ∧ ZFSet.pair w i ∈ env f) ∧ | |
| (∀ z ∈ env A, (∃ i ∈ env I, ∃ w ∈ env L, | |
| ZFSet.pair z i ∈ env f ∧ ZFSet.pair w i ∈ env f) → z ∈ env L)) ↔ _ | |
| simp only [projection, assignmentIndexOverlap_iff] | |
| constructor | |
| · intro h | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_assignmentOnIndices] | |
| exact ⟨fun hz => h.2 z hz.1 hz.2, fun hz => h.1 z hz⟩ | |
| · intro equal | |
| constructor | |
| · intro z hz | |
| exact (mem_assignmentOnIndices _ _ _ z).1 (equal.symm ▸ hz) | |
| · intro z hz hi | |
| exact equal ▸ (mem_assignmentOnIndices _ _ _ z).2 ⟨hz, hi⟩ | |
| def assignmentQueryRelationFormula {n} (z I f : Fin n) : BoundedFormula n := | |
| exPairFields z (exPairFields 1 (exPairFields 2 | |
| (.conj (.equal 3 1) (assignmentQueryMatchFormula I.succ.succ.succ.succ.succ.succ | |
| 2 0 f.succ.succ.succ.succ.succ.succ)))) | |
| theorem truth_assignmentQueryRelationFormula {n} (z I f : Fin n) | |
| (env : Fin n → ZFSet.{u}) (T : ZFSet.{u}) | |
| (projection : env f = productFirst (env I) (ZFSet.prod T T)) : | |
| Truth (assignmentQueryRelationFormula z I f) env ↔ | |
| ∃ row L A, env z = ZFSet.pair (ZFSet.pair row L) (ZFSet.pair row A) ∧ | |
| assignmentOnIndices (assignmentIndices (env I) T L) T A = L := by | |
| simp only [assignmentQueryRelationFormula, truth_exPairFields] | |
| constructor | |
| · rintro ⟨q, p, hz, row, L, hq, other, A, hp, same, matchList⟩ | |
| change q = ZFSet.pair row L at hq | |
| change p = ZFSet.pair other A at hp | |
| change row = other at same | |
| have eq := (truth_assignmentQueryMatchFormula I.succ.succ.succ.succ.succ.succ 2 0 | |
| f.succ.succ.succ.succ.succ.succ | |
| (extend A (extend other (extend L (extend row (extend p (extend q env)))))) T projection).1 matchList | |
| exact ⟨row, L, A, by rw [hz, hq, hp, ← same], eq⟩ | |
| · rintro ⟨row, L, A, hz, matchList⟩ | |
| exact ⟨ZFSet.pair row L, ZFSet.pair row A, hz, row, L, rfl, row, A, rfl, | |
| rfl, (truth_assignmentQueryMatchFormula I.succ.succ.succ.succ.succ.succ 2 0 | |
| f.succ.succ.succ.succ.succ.succ | |
| (extend A (extend row (extend L (extend row | |
| (extend (ZFSet.pair row A) (extend (ZFSet.pair row L) env)))))) T projection).2 matchList⟩ | |
| end BoundedFormula | |
| end PartitionPilot.Forcing | |
| end OAI | |