| import OAI.SetTheory.PartitionPrinciple.Geometry.FiniteIterationTraces |
|
|
| namespace OAI |
|
|
| namespace PartitionPilot.Forcing |
|
|
| open SetRealization |
| universe u |
|
|
| noncomputable def natSuccessorGraph : ZFSet.{u} := |
| natPresentation.graph natPresentation (fun n => ⟨n.down + 1⟩) |
|
|
| noncomputable def natAddGraph : ZFSet.{u} := |
| (natPresentation.product natPresentation).graph natPresentation |
| (fun n => ⟨n.1.down + n.2.down⟩) |
|
|
| noncomputable def natMulGraph : ZFSet.{u} := |
| (natPresentation.product natPresentation).graph natPresentation |
| (fun n => ⟨n.1.down * n.2.down⟩) |
|
|
| theorem pair_mem_natSuccessorGraph (a b : Nat) : |
| ZFSet.pair (natSetCode a) (natSetCode b) ∈ natSuccessorGraph.{u} ↔ b = a + 1 := by |
| change ZFSet.pair (natPresentation.encode ⟨a⟩) (natPresentation.encode ⟨b⟩) ∈ |
| natPresentation.graph natPresentation (fun n => ⟨n.down + 1⟩) ↔ _ |
| rw [Presentation.pair_mem_graph] |
| exact ⟨fun h => (congrArg ULift.down h).symm, fun h => ULift.ext _ _ h.symm⟩ |
|
|
| theorem pair_mem_natAddGraph (a b c : Nat) : |
| ZFSet.pair (ZFSet.pair (natSetCode a) (natSetCode b)) (natSetCode c) ∈ |
| natAddGraph.{u} ↔ c = a + b := by |
| change ZFSet.pair ((natPresentation.product natPresentation).encode (⟨a⟩, ⟨b⟩)) |
| (natPresentation.encode ⟨c⟩) ∈ (natPresentation.product natPresentation).graph |
| natPresentation (fun n => ⟨n.1.down + n.2.down⟩) ↔ _ |
| rw [Presentation.pair_mem_graph] |
| exact ⟨fun h => (congrArg ULift.down h).symm, fun h => ULift.ext _ _ h.symm⟩ |
|
|
| theorem pair_mem_natMulGraph (a b c : Nat) : |
| ZFSet.pair (ZFSet.pair (natSetCode a) (natSetCode b)) (natSetCode c) ∈ |
| natMulGraph.{u} ↔ c = a * b := by |
| change ZFSet.pair ((natPresentation.product natPresentation).encode (⟨a⟩, ⟨b⟩)) |
| (natPresentation.encode ⟨c⟩) ∈ (natPresentation.product natPresentation).graph |
| natPresentation (fun n => ⟨n.1.down * n.2.down⟩) ↔ _ |
| rw [Presentation.pair_mem_graph] |
| exact ⟨fun h => (congrArg ULift.down h).symm, fun h => ULift.ext _ _ h.symm⟩ |
|
|
| theorem natSuccessorGraph_isFunc : ZFSet.IsFunc ZFSet.omega ZFSet.omega natSuccessorGraph.{u} := |
| natPresentation.graph_isFunc natPresentation _ |
|
|
| theorem natAddGraph_isFunc : |
| ZFSet.IsFunc (ZFSet.prod ZFSet.omega ZFSet.omega) ZFSet.omega natAddGraph.{u} := |
| (natPresentation.product natPresentation).graph_isFunc natPresentation _ |
|
|
| theorem natMulGraph_isFunc : |
| ZFSet.IsFunc (ZFSet.prod ZFSet.omega ZFSet.omega) ZFSet.omega natMulGraph.{u} := |
| (natPresentation.product natPresentation).graph_isFunc natPresentation _ |
|
|
| noncomputable def natAddSection (a : Nat) : ZFSet.{u} := |
| natPresentation.graph natPresentation (fun n => ⟨a + n.down⟩) |
|
|
| theorem natAddSection_eq (a : Nat) : |
| natAddSection.{u} a = indexedRelationSection ZFSet.omega ZFSet.omega natAddGraph (natSetCode a) := by |
| apply ZFSet.ext |
| intro z |
| rw [natAddSection, Presentation.mem_graph, mem_indexedRelationSection] |
| constructor |
| · rintro ⟨n, rfl⟩ |
| exact ⟨natSetCode n.down, natPresentation.encode_mem n, natSetCode (a + n.down), |
| natPresentation.encode_mem ⟨a + n.down⟩, rfl, (pair_mem_natAddGraph a n.down _).2 rfl⟩ |
| · rintro ⟨x, hx, y, hy, rfl, entry⟩ |
| obtain ⟨b, rfl⟩ := (mem_omega_iff_natSetCode x).1 hx |
| obtain ⟨c, rfl⟩ := (mem_omega_iff_natSetCode y).1 hy |
| have eq := (pair_mem_natAddGraph a b c).1 entry |
| exact ⟨⟨b⟩, by simp only [natPresentation, eq]⟩ |
|
|
| theorem natIterate_successor (a b : Nat) : |
| natIterate (fun n : ULift.{u} Nat => ⟨n.down + 1⟩) ⟨a⟩ b = ⟨a + b⟩ := by |
| induction b with |
| | zero => rfl |
| | succ b ih => simp only [natIterate, ih, Nat.add_succ] |
|
|
| theorem natIterate_add (a b : Nat) : |
| natIterate (fun n : ULift.{u} Nat => ⟨a + n.down⟩) ⟨0⟩ b = ⟨a * b⟩ := by |
| induction b with |
| | zero => rfl |
| | succ b ih => simp only [natIterate, ih, Nat.mul_succ, Nat.add_comm] |
|
|
| def depthSlotIndex (n j : Nat) : Nat := 3 * n + j |
|
|
| theorem depthSlotIndex_lt_iff {n m j k : Nat} (hj : j < 3) (hk : k < 3) : |
| depthSlotIndex n j < depthSlotIndex m k ↔ n < m ∨ (n = m ∧ j < k) := by |
| unfold depthSlotIndex |
| omega |
|
|
| theorem depthSlotIndex_eq_iff {n m j k : Nat} (hj : j < 3) (hk : k < 3) : |
| depthSlotIndex n j = depthSlotIndex m k ↔ n = m ∧ j = k := by |
| unfold depthSlotIndex |
| omega |
|
|
| theorem depthSlotIndex_le_iff {n m j k : Nat} (hj : j < 3) (hk : k < 3) : |
| depthSlotIndex n j ≤ depthSlotIndex m k ↔ n < m ∨ (n = m ∧ j ≤ k) := by |
| unfold depthSlotIndex |
| omega |
|
|
| noncomputable def depthSlotGraph : ZFSet.{u} := |
| ZFSet.sep (fun z => ∃ n ∈ ZFSet.omega, ∃ j ∈ natSetCode 3, ∃ k ∈ ZFSet.omega, |
| ∃ t ∈ ZFSet.omega, z = ZFSet.pair (ZFSet.pair n j) k ∧ |
| ZFSet.pair (ZFSet.pair (natSetCode 3) n) t ∈ natMulGraph ∧ |
| ZFSet.pair (ZFSet.pair t j) k ∈ natAddGraph) |
| (ZFSet.prod (ZFSet.prod ZFSet.omega (natSetCode 3)) ZFSet.omega) |
|
|
| theorem mem_depthSlotGraph (z : ZFSet.{u}) : z ∈ depthSlotGraph ↔ |
| ∃ n ∈ ZFSet.omega, ∃ j ∈ natSetCode 3, ∃ k ∈ ZFSet.omega, |
| ∃ t ∈ ZFSet.omega, z = ZFSet.pair (ZFSet.pair n j) k ∧ |
| ZFSet.pair (ZFSet.pair (natSetCode 3) n) t ∈ natMulGraph ∧ |
| ZFSet.pair (ZFSet.pair t j) k ∈ natAddGraph := by |
| constructor |
| · exact fun hz => (ZFSet.mem_sep.1 hz).2 |
| · rintro ⟨n, hn, j, hj, k, hk, t, ht, rfl, mul, add⟩ |
| exact ZFSet.mem_sep.2 ⟨ZFSet.pair_mem_prod.2 ⟨ZFSet.pair_mem_prod.2 ⟨hn, hj⟩, hk⟩, |
| n, hn, j, hj, k, hk, t, ht, rfl, mul, add⟩ |
|
|
| theorem pair_mem_depthSlotGraph (n j k : Nat) : |
| ZFSet.pair (ZFSet.pair (natSetCode n) (natSetCode j)) (natSetCode k) ∈ depthSlotGraph.{u} ↔ |
| j < 3 ∧ k = depthSlotIndex n j := by |
| rw [mem_depthSlotGraph] |
| constructor |
| · rintro ⟨n', _, j', hj, k', _, t, ht, equal, mul, add⟩ |
| obtain ⟨left, rfl⟩ := ZFSet.pair_inj.1 equal |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 left |
| obtain ⟨t, rfl⟩ := (mem_omega_iff_natSetCode t).1 ht |
| have product := (pair_mem_natMulGraph 3 n t).1 mul |
| have sum := (pair_mem_natAddGraph t j k).1 add |
| exact ⟨(natSetCode_mem_iff j 3).1 hj, by simp only [depthSlotIndex, product] at sum ⊢; exact sum⟩ |
| · rintro ⟨hj, rfl⟩ |
| exact ⟨natSetCode n, natSetCode_mem_omega n, natSetCode j, (natSetCode_mem_iff j 3).2 hj, |
| natSetCode (depthSlotIndex n j), natSetCode_mem_omega _, natSetCode (3 * n), |
| natSetCode_mem_omega _, rfl, (pair_mem_natMulGraph 3 n _).2 rfl, |
| (pair_mem_natAddGraph (3 * n) j _).2 rfl⟩ |
| |
| theorem depthSlotGraph_eq : depthSlotGraph.{u} = |
| (natPresentation.product (natInitialPresentation 3)).graph natPresentation |
| (fun p => ⟨depthSlotIndex p.1.down p.2.down.val⟩) := by |
| apply ZFSet.ext |
| intro z |
| constructor |
| · intro hz |
| obtain ⟨n, hn, j, hj, k, hk, _, _, rfl, _⟩ := (mem_depthSlotGraph z).1 hz |
| obtain ⟨n, rfl⟩ := (mem_omega_iff_natSetCode n).1 hn |
| obtain ⟨j, bound, rfl⟩ := (mem_natSetCode j 3).1 hj |
| obtain ⟨k, rfl⟩ := (mem_omega_iff_natSetCode k).1 hk |
| have equal := ((pair_mem_depthSlotGraph n j k).1 hz).2 |
| exact ((natPresentation.product (natInitialPresentation 3)).mem_graph natPresentation _ _).2 |
| ⟨(⟨n⟩, ⟨⟨j, bound⟩⟩), by simp only [Presentation.product, natPresentation, natInitialPresentation, equal]⟩ |
| · intro hz |
| obtain ⟨p, rfl⟩ := ((natPresentation.product (natInitialPresentation 3)).mem_graph natPresentation _ z).1 hz |
| exact (pair_mem_depthSlotGraph p.1.down p.2.down.val _).2 ⟨p.2.down.isLt, rfl⟩ |
| |
| theorem depthSlotGraph_function : |
| ZFSet.IsFunc (ZFSet.prod ZFSet.omega (natSetCode 3)) ZFSet.omega depthSlotGraph.{u} := by |
| rw [depthSlotGraph_eq] |
| exact (natPresentation.product (natInitialPresentation 3)).graph_isFunc natPresentation _ |
| |
| theorem depthSlotGraph_injection : |
| InjectionCode (ZFSet.prod ZFSet.omega (natSetCode 3)) ZFSet.omega depthSlotGraph.{u} := by |
| rw [depthSlotGraph_eq] |
| apply (natPresentation.product (natInitialPresentation 3)).graph_injection natPresentation _ |
| intro p q equal |
| have h := (depthSlotIndex_eq_iff p.2.down.isLt q.2.down.isLt).1 (congrArg ULift.down equal) |
| exact Prod.ext (ULift.ext _ _ h.1) (ULift.ext _ _ (Fin.ext h.2)) |
| |
| end PartitionPilot.Forcing |
| |
| end OAI |
| |
