openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Geometry /GroundAssignmentIndexLevels.lean
| import OAI.SetTheory.PartitionPrinciple.Transfinite.StageTwoAssignmentSplices | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) | |
| theorem assignmentCut_mem (separation : SeparationSchema M.sets) | |
| {levels alpha A : ZFSet.{u}} | |
| (hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) (hA : A ∈ M.sets) : | |
| assignmentCut levels alpha A ∈ M.sets := by | |
| let formula : BoundedFormula 3 := BoundedFormula.exMem 1 | |
| (BoundedFormula.pairMembershipFormula 1 0 3) | |
| let params := extend alpha (extend levels Fin.elim0) | |
| have ground := M.bounded_separation_mem separation formula params | |
| (Fin.cases hAlpha (Fin.cases hLevels (fun i => Fin.elim0 i))) hA | |
| have equal : ZFSet.sep (fun z => BoundedFormula.Truth formula (extend z params)) A = | |
| assignmentCut levels alpha A := by | |
| apply ZFSet.ext | |
| intro z | |
| simp only [ZFSet.mem_sep, assignmentCut, formula, | |
| BoundedFormula.truth_exMem, BoundedFormula.truth_pairMembershipFormula] | |
| rfl | |
| exact equal ▸ ground | |
| theorem assignmentUpper_mem (separation : SeparationSchema M.sets) | |
| {levels alpha A : ZFSet.{u}} | |
| (hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) (hA : A ∈ M.sets) : | |
| assignmentUpper levels alpha A ∈ M.sets := by | |
| exact M.bounded_separation_mem separation (.neg (.mem 0 1)) | |
| (extend (assignmentCut levels alpha A) Fin.elim0) | |
| (Fin.cases (M.assignmentCut_mem separation hLevels hAlpha hA) (fun i => Fin.elim0 i)) hA | |
| theorem assignmentSplice_mem | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) (separation : SeparationSchema M.sets) | |
| {levels alpha Aq Ap : ZFSet.{u}} | |
| (hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) | |
| (hAq : Aq ∈ M.sets) (hAp : Ap ∈ M.sets) : | |
| assignmentSplice levels alpha Aq Ap ∈ M.sets := | |
| M.binary_union_mem pairing unions | |
| (M.assignmentCut_mem separation hLevels hAlpha hAq) | |
| (M.assignmentUpper_mem separation hLevels hAlpha hAp) | |
| theorem assignmentCut_small (separation : SeparationSchema M.sets) | |
| {levels alpha A N : ZFSet.{u}} | |
| (hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) | |
| (hA : A ∈ M.sets) (hN : N ∈ M.sets) (small : M.groundEmbeddable A N) : | |
| M.groundEmbeddable (assignmentCut levels alpha A) N := | |
| M.groundEmbeddable_restrict separation | |
| (M.assignmentCut_mem separation hLevels hAlpha hA) hN | |
| (assignmentCut_subset levels alpha A) small | |
| theorem assignmentUpper_small (separation : SeparationSchema M.sets) | |
| {levels alpha A N : ZFSet.{u}} | |
| (hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) | |
| (hA : A ∈ M.sets) (hN : N ∈ M.sets) (small : M.groundEmbeddable A N) : | |
| M.groundEmbeddable (assignmentUpper levels alpha A) N := | |
| M.groundEmbeddable_restrict separation | |
| (M.assignmentUpper_mem separation hLevels hAlpha hA) hN | |
| (assignmentUpper_subset levels alpha A) small | |
| theorem assignmentSplice_countable | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| {levels alpha Aq Ap : ZFSet.{u}} | |
| (hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) | |
| (hAq : Aq ∈ M.sets) (hAp : Ap ∈ M.sets) | |
| (smallq : M.groundEmbeddable Aq ZFSet.omega) | |
| (smallp : M.groundEmbeddable Ap ZFSet.omega) : | |
| M.groundEmbeddable (assignmentSplice levels alpha Aq Ap) ZFSet.omega := | |
| M.groundEmbeddable_restrict separation | |
| (M.assignmentSplice_mem pairing unions separation hLevels hAlpha hAq hAp) | |
| (M.omega_mem separation infinite) (assignmentSplice_subset levels alpha Aq Ap) | |
| (M.groundCountable_binary_union pairing unions powers separation infinite choice | |
| hAq hAp smallq smallp) | |
| theorem assignmentSplice_mem_groundSmallSubsets | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| {J levels alpha Aq Ap : ZFSet.{u}} | |
| (hLevels : levels ∈ M.sets) (hAlpha : alpha ∈ M.sets) | |
| (memberq : Aq ∈ M.groundSmallSubsets J ZFSet.omega) | |
| (memberp : Ap ∈ M.groundSmallSubsets J ZFSet.omega) : | |
| assignmentSplice levels alpha Aq Ap ∈ M.groundSmallSubsets J ZFSet.omega := by | |
| obtain ⟨hAq, subsetq, smallq⟩ := (M.mem_groundSmallSubsets J ZFSet.omega Aq).1 memberq | |
| obtain ⟨hAp, subsetp, smallp⟩ := (M.mem_groundSmallSubsets J ZFSet.omega Ap).1 memberp | |
| refine (M.mem_groundSmallSubsets J ZFSet.omega _).2 | |
| ⟨M.assignmentSplice_mem pairing unions separation hLevels hAlpha hAq hAp, ?_, | |
| M.assignmentSplice_countable pairing unions powers separation infinite choice | |
| hLevels hAlpha hAq hAp smallq smallp⟩ | |
| intro z hz | |
| exact (ZFSet.mem_union.1 (assignmentSplice_subset levels alpha Aq Ap hz)).elim | |
| (fun member => subsetq member) (fun member => subsetp member) | |
| end PartitionPilot.Forcing.TransitiveGround | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) | |
| variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| include pairing unions powers separation | |
| theorem assignmentIndexLevels_mem {I T eta levels : ZFSet.{u}} | |
| (hI : I ∈ M.sets) (hT : T ∈ M.sets) (hEta : eta ∈ M.sets) | |
| (hLevels : levels ∈ M.sets) : assignmentIndexLevels I T eta levels ∈ M.sets := | |
| M.relationCompose_mem pairing unions powers separation | |
| (M.stageOneAssignments_mem pairing unions powers separation hI hT) hI hEta | |
| (M.productFirst_mem pairing unions powers separation hI | |
| (M.prod_mem pairing unions powers separation hT hT)) hLevels | |
| theorem stageTwoAssignmentLevels_mem {I0 I1 T : ZFSet.{u}} | |
| (hI0 : I0 ∈ M.sets) (hI1 : I1 ∈ M.sets) (hT : T ∈ M.sets) : | |
| stageTwoAssignmentLevels I0 I1 T ∈ M.sets := by | |
| have hz := M.empty_mem ⟨I0, hI0⟩ | |
| have ho : natSetCode 1 ∈ M.sets := M.insert_mem pairing unions hz hz | |
| have hEta := M.unorderedPair_mem pairing hz ho | |
| have hI := M.binary_union_mem pairing unions hI0 hI1 | |
| exact M.assignmentIndexLevels_mem pairing unions powers separation hI hT hEta | |
| (M.refinementIndexFieldGraph_mem pairing unions powers separation hI hEta 0) | |
| theorem stageTwoAssignmentCuts_mem {I0 I1 T A : ZFSet.{u}} | |
| (hI0 : I0 ∈ M.sets) (hI1 : I1 ∈ M.sets) (hT : T ∈ M.sets) (hA : A ∈ M.sets) : | |
| assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 0) A ∈ M.sets ∧ | |
| assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 1) A ∈ M.sets ∧ | |
| assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 2) A ∈ M.sets := by | |
| have hz := M.empty_mem ⟨I0, hI0⟩ | |
| have ho : natSetCode 1 ∈ M.sets := M.insert_mem pairing unions hz hz | |
| have ht : natSetCode 2 ∈ M.sets := M.insert_mem pairing unions ho ho | |
| have graph := M.stageTwoAssignmentLevels_mem pairing unions powers separation hI0 hI1 hT | |
| exact ⟨M.assignmentCut_mem separation graph hz hA, | |
| M.assignmentCut_mem separation graph ho hA, M.assignmentCut_mem separation graph ht hA⟩ | |
| theorem stageTwoAssignmentCut_one_mem_groundSmallSubsets | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {I0 I1 T A : ZFSet.{u}} | |
| (first : RefinementIndexLevelCode I0 ∅) | |
| (second : RefinementIndexLevelCode I1 (natSetCode 1)) | |
| (hI0 : I0 ∈ M.sets) (hI1 : I1 ∈ M.sets) (hT : T ∈ M.sets) | |
| (member : A ∈ M.groundSmallSubsets (stageOneAssignments (I0 ∪ I1) T) ZFSet.omega) : | |
| assignmentCut (stageTwoAssignmentLevels I0 I1 T) (natSetCode 1) A ∈ | |
| M.groundSmallSubsets (stageOneAssignments I0 T) ZFSet.omega := by | |
| obtain ⟨hA, _, small⟩ := (M.mem_groundSmallSubsets _ _ A).1 member | |
| have ground := (M.stageTwoAssignmentCuts_mem pairing unions powers separation hI0 hI1 hT hA).2.1 | |
| exact (M.mem_groundSmallSubsets _ _ _).2 ⟨ground, | |
| stageTwoAssignmentCut_one_subset first second A, | |
| M.groundEmbeddable_restrict separation ground (M.omega_mem separation infinite) | |
| (assignmentCut_subset _ _ A) small⟩ | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |