File size: 8,266 Bytes
e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 | 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
|