File size: 3,468 Bytes
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 | import OAI.Computability.DegreeRigidity.Syntax.SetSatisfactionBounded
import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleTupleSpace
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel
universe u
def prependValues (w : ℕ → ZFSet.{u}) : ℕ → (ℕ → ZFSet.{u}) → ℕ → ZFSet.{u}
| 0, e => e
| n+1, e => prependValues w n (cons (w n) e)
theorem prependValues_tail (w : ℕ → ZFSet.{u}) (n : ℕ) (e : ℕ → ZFSet.{u}) (i : ℕ) :
prependValues w n e (n+i) = e i := by
induction n generalizing e i with
| zero => simp [prependValues]
| succ n ih =>
change prependValues w n (cons (w n) e) (n+1+i) = e i
rw [show n+1+i = n+(i+1) by omega,ih]
rfl
theorem prependValues_lt (w : ℕ → ZFSet.{u}) (n : ℕ) (e : ℕ → ZFSet.{u})
(i : ℕ) (hi : i < n) : prependValues w n e i = w i := by
induction n generalizing e with
| zero => omega
| succ n ih =>
by_cases hin : i < n
· exact ih _ hin
· have he : i = n := by omega
subst i
exact prependValues_tail w n (cons (w n) e) 0
def readGraph : ℕ → ℕ → ℕ → (ℕ → ℕ) → Formula → Formula
| 0, _, _, _, ψ => ψ
| n+1, a, g, idx, ψ => .existsMem a
(.conj (.pairMem (idx n+1) 0 (g+1))
(readGraph n (a+1) (g+1) (fun i => idx i+1) ψ))
theorem readGraph_eval (n a g : ℕ) (idx : ℕ → ℕ) (ψ : Formula)
(e w : ℕ → ZFSet.{u}) (hw : ∀ i, i < n → w i ∈ e a)
(hg : ∀ i, i < n → ∀ y ∈ e a,
(ZFSet.pair (e (idx i)) y ∈ e g ↔ y = w i)) :
(readGraph n a g idx ψ).Eval e ↔ ψ.Eval (prependValues w n e) := by
induction n generalizing a g idx e with
| zero => rfl
| succ n ih =>
simp only [readGraph,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ]
constructor
· rintro ⟨x,hx,hpair,hbody⟩
have he : x = w n := (hg n (by omega) x hx).mp hpair
subst x
exact (ih (a+1) (g+1) (fun i => idx i+1) (cons (w n) e)
(fun i hi => hw i (by omega)) (fun i hi => hg i (by omega))).mp hbody
· intro h
refine ⟨w n,hw n (by omega),(hg n (by omega) _ (hw n (by omega))).mpr rfl,?_⟩
exact (ih (a+1) (g+1) (fun i => idx i+1) (cons (w n) e)
(fun i hi => hw i (by omega)) (fun i hi => hg i (by omega))).mpr h
def graphTruth (p : SentenceForm) (a g : ℕ) (idx : ℕ → ℕ) : Formula :=
readGraph p.bound a g idx (setBounded p (p.bound+a) id)
theorem graphTruth_spec (p : SentenceForm) (a g : ℕ) (idx : ℕ → ℕ)
(e : ℕ → ZFSet.{u}) {n : ℕ} (v : Fin n → ZFSet.{u})
(hv : ∀ i, v i ∈ e a) (hb : p.bound ≤ n) (hg : e g = tupleGraph v)
(hi : ∀ i, i < p.bound → e (idx i) = natSet i) :
(graphTruth p a g idx).Eval e ↔ p.Sat (e a : Set ZFSet) (tupleEnv v) := by
have hw (i : ℕ) (h : i < p.bound) : tupleEnv v i ∈ e a := by
simpa [tupleEnv,show i < n by omega] using hv ⟨i,by omega⟩
have hlookup (i : ℕ) (h : i < p.bound) (y : ZFSet.{u}) (_hy : y ∈ e a) :
ZFSet.pair (e (idx i)) y ∈ e g ↔ y = tupleEnv v i := by
rw [hg,hi i h]
simpa [tupleEnv,show i < n by omega] using tupleGraph_pair v ⟨i,by omega⟩ y
rw [graphTruth,readGraph_eval p.bound a g idx _ e (tupleEnv v) hw hlookup,setBounded_eval,
prependValues_tail]
exact p.finite_support _ _ _ (fun i h => prependValues_lt _ _ _ i h)
end TuringRigidity.RelativeConstructible
end OAI
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