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caec230 | 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 | import OAI.MeasureTheory.DyadicAvoidance.FirstDefault
import OAI.MeasureTheory.DyadicAvoidance.OrderedRouting
import OAI.MeasureTheory.DyadicAvoidance.RoutingPath
namespace OAI
universe u_Ω u_ι u_X u_Y
noncomputable section
namespace Problem310.FirstDefault
/-- A path cylinder is exactly the event that first-success routing selects
its prescribed child at each of its prescribed nodes. -/
theorem cylinder_iff_chooseChild {M d : ℕ} {Ω : Type u_Ω}
(S : Address M → Ω → Bool) (f : Path M d) (ω : Ω) :
ω ∈ cylinder S f ↔ ∀ k : Fin d,
OrderedRouting.chooseChild (fun j => S (pathPrefix f k, j) ω) = (f k).castSucc := by
constructor
· intro h k
apply (OrderedRouting.chooseChild_eq_nondefault_iff _ _).mpr
constructor
· have hi := h ⟨k, ⟨(f k).val, Nat.lt_succ_self _⟩⟩
simpa only [address, requiredBit, decide_true] using hi
· intro j hj
have hi := h ⟨k, ⟨j.val, lt_trans hj (Nat.lt_succ_self _)⟩⟩
have hne : j.val ≠ (f k).val := Nat.ne_of_lt hj
simpa only [address, requiredBit, hne, decide_false] using hi
· intro h i
obtain ⟨k, j⟩ := i
obtain ⟨hsuccess, hprev⟩ :=
(OrderedRouting.chooseChild_eq_nondefault_iff _ _).mp (h k)
by_cases heq : j.val = (f k).val
· have ha : address f ⟨k, j⟩ = (pathPrefix f k, f k) := by
apply Prod.ext
· rfl
· exact Fin.ext heq
change S (address f ⟨k, j⟩) ω = requiredBit f ⟨k, j⟩
simpa only [ha, requiredBit, heq, decide_true] using hsuccess
· have hlt : j.val < (f k).val := by omega
have hj := hprev ⟨j.val, lt_trans hlt (f k).isLt⟩ hlt
change S (address f ⟨k, j⟩) ω = requiredBit f ⟨k, j⟩
simpa only [address, requiredBit, heq, decide_false] using hj
/-- Canonical child choice from a family of selector bits at tree addresses. -/
def centerChoice {M : ℕ} {Ω : Type u_Ω} (S : Address M → Ω → Bool)
(P : List (Fin (M + 1))) (ω : Ω) : Fin (M + 1) :=
OrderedRouting.chooseChild (fun j => S (P, j) ω)
/-- The actual routed center avoids the default exactly on the finite union
of the path cylinders used by the exact first-default probability law. -/
theorem noDefault_iff_mem_cylinders {M d : ℕ} {Ω : Type u_Ω}
(S : Address M → Ω → Bool) (ω : Ω) :
(∀ k < d, centerChoice S (RoutingPath.routeFrom (centerChoice S) k [] ω) ω ≠
Fin.last M) ↔ ω ∈ ⋃ f : Path M d, cylinder S f := by
rw [RoutingPath.noDefault_iff_exists_choices]
simp only [Set.mem_iUnion, cylinder_iff_chooseChild]
constructor
· rintro ⟨f, hf⟩
refine ⟨f, fun k => ?_⟩
simpa only [centerChoice, pathPrefix_eq_take_ofFn] using hf k.val k.isLt
· rintro ⟨f, hf⟩
refine ⟨f, fun k hk => ?_⟩
simpa only [centerChoice, pathPrefix_eq_take_ofFn] using hf ⟨k, hk⟩
/-- The equality is also available as a set rewrite for probability measures. -/
theorem noDefault_event_eq_cylinders {M d : ℕ} {Ω : Type u_Ω}
(S : Address M → Ω → Bool) :
{ω | ∀ k < d, centerChoice S (RoutingPath.routeFrom (centerChoice S) k [] ω) ω ≠
Fin.last M} = ⋃ f : Path M d, cylinder S f := by
ext ω
exact noDefault_iff_mem_cylinders S ω
/-- Change the auxiliary parameter space without changing any node choices. -/
theorem routeFrom_change_parameter {ι : Type u_ι} {X : Type u_X} {Y : Type u_Y}
(f : List ι → X → ι) (g : List ι → Y → ι) (x : X) (y : Y)
(h : ∀ P, f P x = g P y) (n : ℕ) (P : List ι) :
RoutingPath.routeFrom f n P x = RoutingPath.routeFrom g n P y := by
induction n with
| zero => rfl
| succ n ih => simp only [RoutingPath.routeFrom_succ, ih, h]
/-- Equivalent leaf-membership formulation, convenient for concrete routers. -/
theorem default_not_mem_iff_mem_cylinders {M d : ℕ} {Ω : Type u_Ω}
(S : Address M → Ω → Bool) (ω : Ω) :
Fin.last M ∉ RoutingPath.routeFrom (centerChoice S) d [] ω ↔
ω ∈ ⋃ f : Path M d, cylinder S f := by
rw [← RoutingPath.noDefault_iff_not_mem]
exact noDefault_iff_mem_cylinders S ω
/-- Spatially parametrized selectors have the same no-default event as the
cylinders at a fixed center. This is directly usable with `selectorValue`. -/
theorem spatial_default_not_mem_iff {M d : ℕ} {Ω : Type u_Ω} {X : Type u_X}
(S : Ω → List (Fin (M + 1)) → Fin M → X → Bool) (ω : Ω) (x : X) :
Fin.last M ∉ RoutingPath.routeFrom
(fun P y => OrderedRouting.chooseChild (fun j => S ω P j y)) d [] x ↔
ω ∈ ⋃ f : Path M d,
cylinder (fun a ω' => S ω' a.1 a.2 x) f := by
have hr := routeFrom_change_parameter
(centerChoice (fun a ω' => S ω' a.1 a.2 x))
(fun P y => OrderedRouting.chooseChild (fun j => S ω P j y)) ω x
(fun _ => rfl) d []
rw [← hr]
exact default_not_mem_iff_mem_cylinders _ ω
end Problem310.FirstDefault
end
end OAI
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