import OAI.SetTheory.PartitionPrinciple.Definability.GroundInternalRowSetZF import OAI.SetTheory.PartitionPrinciple.Definability.GroundOutsideRowCodes import OAI.SetTheory.PartitionPrinciple.Forcing.GroundSeparatedRowConditions namespace OAI section namespace PartitionPilot.Forcing.TransitiveGround open SetFormula universe u variable (N : TransitiveGround.{u}) theorem exists_function_choosing_subsets (powers : Realize N.sets powerSet Fin.elim0) (separation : SeparationSchema N.sets) (choice : Realize N.sets SetFormula.choice Fin.elim0) {R : ZFSet.{u}} (hR : R ∈ N.sets) : ∃ f ∈ N.sets, ∀ S ∈ N.sets, S ⊆ R → S.Nonempty → ∃ x ∈ S, ZFSet.pair S x ∈ f ∧ ∀ z ∈ S, ZFSet.pair S z ∈ f → z = x := by obtain ⟨W, hW, subsets⟩ := N.power_witness powers hR let A := ZFSet.sep (fun S => S.Nonempty) W have hA : A ∈ N.sets := by have selected := N.bounded_separation_mem separation (BoundedFormula.nonemptyFormula 0) Fin.elim0 (fun i => Fin.elim0 i) hW have eq : ZFSet.sep (fun S => BoundedFormula.Truth (BoundedFormula.nonemptyFormula 0) (extend S Fin.elim0)) W = A := by apply ZFSet.ext intro S simp only [A, ZFSet.mem_sep, BoundedFormula.truth_nonemptyFormula] rfl exact eq ▸ selected obtain ⟨f, hf, code⟩ := N.choice_witness choice hA (fun S hS => (ZFSet.mem_sep.1 hS).2) refine ⟨f, hf, ?_⟩ intro S hS subset nonempty have member : S ∈ A := ZFSet.mem_sep.2 ⟨(subsets S hS).2 subset, nonempty⟩ obtain ⟨x, chosen, unique⟩ := code.function.2 S member exact ⟨x, code.chosen S x chosen, chosen, fun z _ hz => unique z hz⟩ end PartitionPilot.Forcing.TransitiveGround namespace PartitionPilot.Forcing.TransitiveGround open SetRealization SetFormula universe u variable (M : TransitiveGround.{u}) {L G : Type u} variable [Group G] [PartialOrder L] [OrderBot L] theorem internalRow_not_choice (a : PosetAction G L) (s : Presentation L) (b : Presentation G) (C : ZFSet.{u}) (directed : ∀ i j : L, ∃ k, i ≤ k ∧ j ≤ k) (unbounded : ∀ B : L, ∃ i : L, ¬ i ≤ B) (countableBound : ∀ A ∈ M.sets, A ⊆ s.carrier → M.groundEmbeddable A ZFSet.omega → ∃ D : L, ∀ i : L, s.encode i ∈ A → i ≤ D) (separate : ∀ B D : L, B ≤ D → ∃ g : G, g ∈ a.principalFix B ∧ ∀ x : L, x ≤ D → x ≤ a.iso g D → x ≤ B) (small : M.sets.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) (collection : CollectionSchema M.sets) (infinite : Realize M.sets infinity Fin.elim0) (groundChoice : Realize M.sets SetFormula.choice Fin.elim0) (hL : s.carrier ∈ M.sets) (hC : C ∈ M.sets) (hG : b.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets) (graph : a.actionGraph s b ∈ M.sets) (uncountable : ¬ M.groundEmbeddable C ZFSet.omega) : letI := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega let act := M.internalRowAction a s b C pairing unions powers separation infinite hL hC hG graph let c := SetMembers.presentation (M.smallPartialFunctions (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega) let F := a.principalSupportFamily (⊥ : L) directed ∀ H : ConditionFilter (SetMembers (M.smallPartialFunctions (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)), H.GenericFor (formulaFamily (act.symmetricDomain c (M.nameDomain c) F)) → ZFRealization ((act.symmetricDomain c (M.nameDomain c) F).values {q | H.mem q}) → ¬ Realize ((act.symmetricDomain c (M.nameDomain c) F).values {q | H.mem q}) SetFormula.choice Fin.elim0 := by let := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega intro act c F H generic zf allegedChoice let dom := act.symmetricDomain c (M.nameDomain c) F let N : TransitiveGround := ⟨dom.values {q | H.mem q}, dom.values_transitive {q | H.mem q}⟩ have hRows := M.nonbottom_carrier_mem s separation hL have hTwo := M.natSetCode_mem separation infinite 2 have carrier : c.carrier ∈ M.sets := M.countablePartialFunctions_mem pairing unions powers separation infinite (M.prod_mem pairing unions powers separation hRows hC) hTwo have hRname : M.internalRowSetName s C hL ∈ dom.names := ⟨M.internalRowSetName_mem s C pairing unions powers separation collection infinite hL hC, M.internalRowSetName_hs a s b C pairing unions powers separation infinite hL hC hG graph directed⟩ have hR : Name.eval {q | H.mem q} (M.internalRowSetName s C hL) ∈ N.sets := (dom.mem_values {q | H.mem q} _).2 ⟨_, hRname, rfl⟩ have nPowers : Realize N.sets powerSet Fin.elim0 := zf.1.2.2.2.2.2.1 obtain ⟨f, hf, selection⟩ := N.exists_function_choosing_subsets nPowers zf.2.1 allegedChoice hR obtain ⟨fname, hfname, rfl⟩ := (dom.mem_values {q | H.mem q} f).1 hf have hsF : HereditarilySymmetric act c F fname := hfname.2 obtain ⟨B, fixedF⟩ := hsF.symmetric let Sname := M.internalOutsideRowsName s C hL B have hSname : Sname ∈ dom.names := ⟨M.internalOutsideRowsName_mem s C pairing unions powers separation collection infinite hL hC order B, M.internalOutsideRowsName_hs a s b C pairing unions powers separation infinite hL hC hG graph directed B⟩ have hS : Name.eval {q | H.mem q} Sname ∈ N.sets := (dom.mem_values {q | H.mem q} _).2 ⟨Sname, hSname, rfl⟩ obtain ⟨x, hx, chosen, unique⟩ := selection _ hS (M.eval_internalOutsideRowsName_subset s C hL B H) (M.eval_internalOutsideRowsName_nonempty s C hL B (unbounded B) H) obtain ⟨i, outside, rowValue⟩ := (M.mem_eval_internalOutsideRowsName s C hL B H x).1 hx let rowName := M.internalRowName s C hL i have hRowName : rowName ∈ dom.names := ⟨M.internalRowName_mem s C pairing unions powers separation collection infinite hL hC i, M.internalRowName_hs a s b C pairing unions powers separation infinite hL hC hG graph directed i⟩ let sn : dom.names := ⟨Sname, hSname⟩ let xn : dom.names := ⟨rowName, hRowName⟩ let fn : dom.names := ⟨fname, hfname⟩ let env := extend sn (extend xn (extend fn Fin.elim0)) have rowEq : Name.eval {q | H.mem q} rowName = x := (M.eval_internalRowName s C hL i H).trans rowValue have truth : Realize (dom.values {q | H.mem q}) (BoundedFormula.uniqueFiberFormula 0 1 2).toFormula (evalAssignment {q | H.mem q} env) := by apply (realize_uniqueFiber_values dom {q | H.mem q} 0 1 2 env).2 change Name.eval {q | H.mem q} rowName ∈ Name.eval {q | H.mem q} Sname ∧ ZFSet.pair (Name.eval {q | H.mem q} Sname) (Name.eval {q | H.mem q} rowName) ∈ Name.eval {q | H.mem q} fname ∧ ∀ z ∈ Name.eval {q | H.mem q} Sname, ZFSet.pair (Name.eval {q | H.mem q} Sname) z ∈ Name.eval {q | H.mem q} fname → z = Name.eval {q | H.mem q} rowName rw [rowEq] exact ⟨hx, chosen, unique⟩ obtain ⟨p, _, forced⟩ := (forcing_truth dom H generic (BoundedFormula.uniqueFiberFormula 0 1 2).toFormula env).1 truth let used := rowUsedRows s.nonbottom.carrier C (natSetCode 2) (SetMembers.raw p) obtain ⟨hUsed, usedSmall⟩ := M.countablePartialFunctions_rowSupport pairing unions powers separation infinite groundChoice hRows hC hTwo (SetMembers.raw_mem p) have usedSubset : used ⊆ s.carrier := by intro z hz exact ((s.mem_nonbottom_carrier z).1 (rowUsedRows_subset s.nonbottom.carrier C (natSetCode 2) (SetMembers.raw p) hz)).1 obtain ⟨D₀, bound₀⟩ := countableBound used hUsed usedSubset usedSmall obtain ⟨D₁, first, containsB⟩ := directed D₀ B obtain ⟨D, second, containsI⟩ := directed D₁ i.val have bound : ∀ j : {j : L // j ≠ ⊥}, s.encode j.val ∈ used → j.val ≤ D := fun j hj => le_trans (bound₀ j.val hj) (le_trans first second) obtain ⟨g, fixed, overlap⟩ := separate B D (le_trans containsB second) obtain ⟨r, _, rp, rgp⟩ := M.internalRowAction_common_extension_of_separated a s b C pairing unions powers separation infinite hL hC hG graph groundChoice g B D fixed overlap p bound obtain ⟨K, kr, groundGeneric, _, symmetricGeneric⟩ := M.exists_joint_generic c act F small carrier r have maps := M.conditionMapFamily_mem_of_actionGraph act c b pairing unions powers separation collection carrier hG (M.internalRowActionGraph_mem a s b C pairing unions powers separation infinite hL hC hG graph) have groundClosed : ∀ g τ, τ ∈ (M.nameDomain c).names → act.act c g τ ∈ (M.nameDomain c).names := M.act_mem c act b pairing unions powers separation collection carrier maps have closed : ∀ g τ, τ ∈ dom.names → act.act c g τ ∈ dom.names := fun g τ hτ => act.symmetricDomain_translated c (M.nameDomain c) F groundClosed g hτ have equal := act.uniqueFiber_eq_of_fixed_names c dom closed g sn xn fn (M.internalOutsideRowsName_action a s b C pairing unions powers separation infinite hL hC hG graph B g fixed) (fixedF fixed) p forced K symmetricGeneric (K.upward kr rp) (K.upward kr rgp) change Name.eval {q | K.mem q} (M.internalRowName s C hL i) = Name.eval {q | K.mem q} (act.act c g (M.internalRowName s C hL i)) at equal rw [M.internalRowName_action a s b C pairing unions powers separation infinite hL hC hG graph, M.eval_internalRowName, M.eval_internalRowName] at equal have moved : a.iso g i.val ≠ i.val := a.moved_ne_of_separated g B D i.val overlap containsI outside have distinct : s.encode i.val ≠ s.encode (a.iso g i.val) := fun eq => moved (s.injective eq).symm exact M.binaryGenericRow_ne pairing unions powers separation infinite groundChoice hRows hC (s.nonbottom.encode_mem i) (s.nonbottom.encode_mem (a.nonbottom.iso g i)) distinct uncountable K groundGeneric equal theorem exists_internalRow_ZF_not_choice (a : PosetAction G L) (s : Presentation L) (b : Presentation G) (C : ZFSet.{u}) (directed : ∀ i j : L, ∃ k, i ≤ k ∧ j ≤ k) (unbounded : ∀ B : L, ∃ i : L, ¬ i ≤ B) (countableBound : ∀ A ∈ M.sets, A ⊆ s.carrier → M.groundEmbeddable A ZFSet.omega → ∃ D : L, ∀ i : L, s.encode i ∈ A → i ≤ D) (separate : ∀ B D : L, B ≤ D → ∃ g : G, g ∈ a.principalFix B ∧ ∀ x : L, x ≤ D → x ≤ a.iso g D → x ≤ B) (small : M.sets.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) (collection : CollectionSchema M.sets) (infinite : Realize M.sets infinity Fin.elim0) (groundChoice : Realize M.sets SetFormula.choice Fin.elim0) (hL : s.carrier ∈ M.sets) (hC : C ∈ M.sets) (hG : b.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets) (graph : a.actionGraph s b ∈ M.sets) (uncountable : ¬ M.groundEmbeddable C ZFSet.omega) (p : SetMembers (M.smallPartialFunctions (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)) : letI := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega let act := M.internalRowAction a s b C pairing unions powers separation infinite hL hC hG graph let c := SetMembers.presentation (M.smallPartialFunctions (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega) let F := a.principalSupportFamily (⊥ : L) directed ∃ H : ConditionFilter (SetMembers (M.smallPartialFunctions (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)), H.mem p ∧ H.GenericFor (M.codedFamily c) ∧ H.GenericFor (formulaFamily (M.nameDomain c)) ∧ H.GenericFor (formulaFamily (act.symmetricDomain c (M.nameDomain c) F)) ∧ M.sets ⊆ (act.symmetricDomain c (M.nameDomain c) F).values {q | H.mem q} ∧ ZFRealization ((act.symmetricDomain c (M.nameDomain c) F).values {q | H.mem q}) ∧ ¬ Realize ((act.symmetricDomain c (M.nameDomain c) F).values {q | H.mem q}) SetFormula.choice Fin.elim0 := by let := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega obtain ⟨H, hp, ground, ordinary, symmetric, includes, zf, _, _, _, _⟩ := M.exists_internalRow_ZF_with_rowSet a s b C directed small pairing unions powers separation collection infinite groundChoice hL hC hG order graph uncountable p exact ⟨H, hp, ground, ordinary, symmetric, includes, zf, M.internalRow_not_choice a s b C directed unbounded countableBound separate small pairing unions powers separation collection infinite groundChoice hL hC hG order graph uncountable H symmetric zf⟩ end PartitionPilot.Forcing.TransitiveGround end end OAI