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8df23f4 | 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 | import OAI.Computability.DegreeRigidity.SetModels.InternalFiniteSubsets
import OAI.Computability.DegreeRigidity.CohenForcing.InternalCohenConditions
import OAI.Computability.DegreeRigidity.CohenForcing.CohenCoordinates
namespace OAI
namespace TuringRigidity.CohenConditionCode
open Set TransitiveNameModel BoundedSetTheory InternalFiniteSubsets
open CohenSymmetry CohenCoordinates
open InternalCohen (bitSet alphabet bitSet_injective mem_alphabet)
noncomputable def graph (A : ZFSet.{0}) (p : Condition (Conditions A)) : ZFSet.{0} :=
ZFSet.range (fun i : ↥(support p) =>
ZFSet.pair (label A i.val) (bitSet ((p.val i.val).getD false)))
theorem pair_mem_graph (A : ZFSet.{0}) (p : Condition (Conditions A))
(i : Conditions A) (b : Bool) :
ZFSet.pair (label A i) (bitSet b) ∈ graph A p ↔ p.val i = some b := by
classical
rw [graph,ZFSet.mem_range]
constructor
· rintro ⟨j,hj⟩
obtain ⟨he,hb⟩ := ZFSet.pair_inj.mp hj
have hji := label_injective A he
have hn := (mem_support p j.val).mp j.property
have hv : p.val j.val = some ((p.val j.val).getD false) := by
cases h : p.val j.val with
| none => exact False.elim (hn h)
| some c => rfl
rw [bitSet_injective hb,hji] at hv
exact hv
· intro hi
refine ⟨⟨i,(mem_support p i).mpr (by simp [hi])⟩,?_⟩
simp [hi]
theorem mem_graph (A : ZFSet.{0}) (p : Condition (Conditions A)) (z : ZFSet.{0}) :
z ∈ graph A p ↔ ∃ i b, p.val i = some b ∧ z = ZFSet.pair (label A i) (bitSet b) := by
constructor
· intro hz
obtain ⟨i,rfl⟩ := ZFSet.mem_range.mp hz
exact ⟨i.val,_,(pair_mem_graph A p _ _).mp (ZFSet.mem_range_self i),rfl⟩
· rintro ⟨i,b,hi,rfl⟩; exact (pair_mem_graph A p i b).mpr hi
theorem graph_subset_iff (A : ZFSet.{0}) (p q : Condition (Conditions A)) :
graph A q ⊆ graph A p ↔ p ≤ q := by
constructor
· intro h i b hb; exact (pair_mem_graph A p i b).mp (h ((pair_mem_graph A q i b).mpr hb))
· intro h z hz
obtain ⟨i,b,hi,rfl⟩ := (mem_graph A q z).mp hz
exact (pair_mem_graph A p i b).mpr (h i b hi)
theorem graph_injective (A : ZFSet.{0}) : Function.Injective (graph A) := by
intro p q h
apply le_antisymm
· exact (graph_subset_iff A p q).mp (by rw [h])
· exact (graph_subset_iff A q p).mp (by rw [h])
def SingleValued (A p : ZFSet.{0}) : Prop :=
∀ x ∈ A, ∀ b ∈ alphabet, ∀ c ∈ alphabet,
ZFSet.pair x b ∈ p → ZFSet.pair x c ∈ p → b = c
def IsCondition (A p : ZFSet.{0}) : Prop :=
p ∈ finiteSubsets (ZFSet.prod A alphabet) ∧ SingleValued A p
theorem graph_isCondition (A : ZFSet.{0}) (p : Condition (Conditions A)) :
IsCondition A (graph A p) := by
constructor
· apply (mem_finiteSubsets_iff _ _).mpr
constructor
· intro z hz
obtain ⟨i,b,_,rfl⟩ := (mem_graph A p z).mp hz
exact ZFSet.mem_prod.mpr ⟨_,label_mem A i,_,(mem_alphabet _).mpr ⟨b,rfl⟩,rfl⟩
· rw [graph,ZFSet.coe_range]; exact Set.finite_range _
· intro x hx b hb c hc hpb hpc
obtain ⟨b,rfl⟩ := (mem_alphabet _).mp hb
obtain ⟨c,rfl⟩ := (mem_alphabet _).mp hc
let i : Conditions A := equivShrink A ⟨x,hx⟩
have hi : label A i = x := by simp [i,label]
rw [←hi,pair_mem_graph] at hpb hpc
exact congrArg bitSet (Option.some.inj (hpb.symm.trans hpc))
theorem isCondition_iff_graph (A x : ZFSet.{0}) :
IsCondition A x ↔ ∃ p : Condition (Conditions A), graph A p = x := by
classical
constructor
· intro hx
have hsub := ((mem_finiteSubsets_iff _ _).mp hx.1).1
have hfin := ((mem_finiteSubsets_iff _ _).mp hx.1).2
let v (i : Conditions A) : Option Bool :=
if h : ∃ b, ZFSet.pair (label A i) (bitSet b) ∈ x then some h.choose else none
have hv (i : Conditions A) (b : Bool) :
v i = some b ↔ ZFSet.pair (label A i) (bitSet b) ∈ x := by
dsimp only [v]
split_ifs with h
· constructor
· intro he; exact Option.some.inj he ▸ h.choose_spec
· intro hb
have he := hx.2 _ (label_mem A i) _ ((mem_alphabet _).mpr ⟨h.choose,rfl⟩)
_ ((mem_alphabet _).mpr ⟨b,rfl⟩) h.choose_spec hb
exact congrArg some (bitSet_injective he)
· constructor
· intro he; cases he
· intro hb; exact False.elim (h ⟨b,hb⟩)
have hfinite : {i | v i ≠ none}.Finite := by
let f (i : Conditions A) := ZFSet.pair (label A i) (bitSet ((v i).getD false))
have hf : Function.Injective f := by
intro i j he; exact label_injective A (ZFSet.pair_inj.mp he).1
apply (hfin.preimage hf.injOn).subset
intro i hi
change f i ∈ x
apply (hv i _).mp
cases he : v i with
| none => exact False.elim (hi he)
| some b => rfl
let p : Condition (Conditions A) := ⟨v,hfinite⟩
refine ⟨p,ZFSet.ext (fun z => ?_)⟩
constructor
· intro hz
obtain ⟨i,b,hi,rfl⟩ := (mem_graph A p z).mp hz
exact (hv i b).mp hi
· intro hz
obtain ⟨i,hi,b,hb,rfl⟩ := ZFSet.mem_prod.mp (hsub hz)
obtain ⟨b,rfl⟩ := (mem_alphabet _).mp hb
let j : Conditions A := equivShrink A ⟨i,hi⟩
have hj : label A j = i := by simp [j,label]
rw [←hj]
exact (pair_mem_graph A p j b).mpr ((hv j b).mpr (hj ▸ hz))
· rintro ⟨p,rfl⟩; exact graph_isCondition A p
end TuringRigidity.CohenConditionCode
end OAI
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