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9dbc64e | 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 | import OAI.Computability.DegreeRigidity.Syntax.AtomicGraphCorrect
import OAI.Computability.DegreeRigidity.Syntax.AtomicRequirements
namespace OAI
namespace TuringRigidity.AtomicForcing
open Set RecursiveNames TransitiveNameModel BoundedSetTheory
universe u
def CodeWitness (d c k o f x y p : ZFSet.{u}) : Prop :=
∃ b ∈ d, ∃ t ∈ c, ZFSet.pair b t ∈ y ∧ ZFSet.pair p t ∈ o ∧ Triple k f x b p
def CodeNeg (d c k o f x y p : ZFSet.{u}) : Prop :=
∀ q ∈ c, ZFSet.pair q p ∈ o → ¬ CodeWitness d c k o f x y q
def CodeRefute (d c k o f x y p : ZFSet.{u}) : Prop :=
∃ a ∈ d, ∃ s ∈ c, ZFSet.pair a s ∈ x ∧ ZFSet.pair p s ∈ o ∧ CodeNeg d c k o f a y p
namespace AtomicFormula
open Formula
def witness (d c k o f x y p : ℕ) : Formula :=
.existsMem d (.existsMem (c+1)
(.conj (pairMem 1 0 (y+2)) (.conj (pairMem (p+2) 0 (o+2))
(triple (k+2) (f+2) (x+2) 1 (p+2)))))
theorem eval_witness (d c k o f x y p : ℕ) (e : ℕ → ZFSet.{u}) :
(witness d c k o f x y p).Eval e ↔
CodeWitness (e d) (e c) (e k) (e o) (e f) (e x) (e y) (e p) := by
simp only [witness,Formula.Eval,eval_pairMem,eval_triple,cons_zero,cons_succ,CodeWitness]
def negMem (d c k o f x y p : ℕ) : Formula :=
allMem c (imp (pairMem 0 (p+1) (o+1))
(.neg (witness (d+1) (c+1) (k+1) (o+1) (f+1) (x+1) (y+1) 0)))
theorem eval_negMem (d c k o f x y p : ℕ) (e : ℕ → ZFSet.{u}) :
(negMem d c k o f x y p).Eval e ↔
CodeNeg (e d) (e c) (e k) (e o) (e f) (e x) (e y) (e p) := by
simp only [negMem,eval_allMem,eval_imp,Formula.Eval,eval_pairMem,eval_witness,
cons_zero,cons_succ,CodeNeg]
def refute (d c k o f x y p : ℕ) : Formula :=
.existsMem d (.existsMem (c+1)
(.conj (pairMem 1 0 (x+2)) (.conj (pairMem (p+2) 0 (o+2))
(negMem (d+2) (c+2) (k+2) (o+2) (f+2) 1 (y+2) (p+2)))))
theorem eval_refute (d c k o f x y p : ℕ) (e : ℕ → ZFSet.{u}) :
(refute d c k o f x y p).Eval e ↔
CodeRefute (e d) (e c) (e k) (e o) (e f) (e x) (e y) (e p) := by
simp only [refute,Formula.Eval,eval_pairMem,eval_negMem,cons_zero,cons_succ,CodeRefute]
def memTest (d c k o f x y p : ℕ) : Formula :=
disj (witness d c k o f x y p) (negMem d c k o f x y p)
def eqTest (d c k o f x y p : ℕ) : Formula :=
disj (triple k f x y p) (disj (refute d c k o f x y p) (refute d c k o f y x p))
end AtomicFormula
variable {c : ZFSet.{u}} [Preorder (Conditions c)]
omit [Preorder (Conditions c)] in
private theorem child_mem {d : ZFSet.{u}} (hd : Transitive d)
(a : Name (Conditions c)) (ha : a.encode (label c) ∈ d) (i : a.arity) :
(a.child i).encode (label c) ∈ d := by
apply names_childClosed d c hd a ha
cases a
exact ⟨i,rfl⟩
theorem codeWitness_iff {d o f : ZFSet.{u}} (hd : Transitive d)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(hf : Graph d c o f) (a b : Name (Conditions c))
(ha : a.encode (label c) ∈ d) (hb : b.encode (label c) ∈ d) (p : Conditions c) :
CodeWitness d c (ZFSet.prod d d) o f (a.encode (label c)) (b.encode (label c)) (label c p) ↔
Witness a b p := by
have hrel (j) := (triple_pair ha (child_mem hd b hb j)).trans
(hf.correct hd ho a (b.child j) ha (child_mem hd b hb j) p)
cases b with
| mk κ b t =>
constructor
· rintro ⟨x,hx,s,hs,hxs,hps,he⟩
obtain ⟨j,hj,hjs⟩ := (pair_mem_encode _ _ _ _).mp hxs
subst x
subst s
exact ⟨j,(ho p (t j)).mp hps,(hrel j).mp he⟩
· rintro ⟨j,hj,he⟩
exact ⟨_,child_mem hd _ hb j,_,label_mem c (t j),
(pair_mem_encode _ _ _ _).mpr ⟨j,rfl,rfl⟩,(ho p (t j)).mpr hj,(hrel j).mpr he⟩
theorem negMem_iff_no_witness (a b : Name (Conditions c)) (p : Conditions c) :
NegMem a b p ↔ ∀ q, q ≤ p → ¬ Witness a b q := by
constructor
· intro h q hq hw
exact h q hq (witness_mem a b q hw)
· intro h q hq hm
cases b with
| mk κ b t =>
obtain ⟨r,hr,hw⟩ := hm q le_rfl
exact h r (hr.trans hq) hw
theorem codeNeg_iff {d o f : ZFSet.{u}} (hd : Transitive d)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(hf : Graph d c o f) (a b : Name (Conditions c))
(ha : a.encode (label c) ∈ d) (hb : b.encode (label c) ∈ d) (p : Conditions c) :
CodeNeg d c (ZFSet.prod d d) o f (a.encode (label c)) (b.encode (label c)) (label c p) ↔
NegMem a b p := by
rw [negMem_iff_no_witness]
constructor
· intro h q hq hw
exact h _ (label_mem c q) ((ho q p).mpr hq) ((codeWitness_iff hd ho hf a b ha hb q).mpr hw)
· intro h q hq hqp hw
obtain ⟨r,rfl⟩ := label_surjective c hq
exact h r ((ho r p).mp hqp) ((codeWitness_iff hd ho hf a b ha hb r).mp hw)
theorem codeRefute_iff {d o f : ZFSet.{u}} (hd : Transitive d)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(hf : Graph d c o f) (a b : Name (Conditions c))
(ha : a.encode (label c) ∈ d) (hb : b.encode (label c) ∈ d) (p : Conditions c) :
CodeRefute d c (ZFSet.prod d d) o f (a.encode (label c)) (b.encode (label c)) (label c p) ↔
∃ i, p ≤ a.tag i ∧ NegMem (a.child i) b p := by
constructor
· rintro ⟨x,hx,s,hs,hxs,hps,hn⟩
obtain ⟨i,hi,his⟩ := (pair_mem_encode _ _ _ _).mp hxs
subst x
subst s
exact ⟨i,(ho p (a.tag i)).mp hps,
(codeNeg_iff hd ho hf _ b (child_mem hd a ha i) hb p).mp hn⟩
· rintro ⟨i,hi,hn⟩
exact ⟨_,child_mem hd a ha i,_,label_mem c (a.tag i),
(pair_mem_encode _ _ _ _).mpr ⟨i,rfl,rfl⟩,(ho p (a.tag i)).mpr hi,
(codeNeg_iff hd ho hf _ b (child_mem hd a ha i) hb p).mpr hn⟩
end TuringRigidity.AtomicForcing
end OAI
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