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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 | import Mathlib
namespace OAI
universe u_A u_I u_X u_K
noncomputable section
namespace Problem310.AddressInjectivity
/-- Two descendants of different children cannot be the same leaf. -/
theorem child_eq_of_common_descendant {A : Type u_A} {U L : List A} {i j : A}
(hi : U ++ [i] <+: L) (hj : U ++ [j] <+: L) : i = j := by
have hi' := List.prefix_iff_eq_take.mp hi
have hj' := List.prefix_iff_eq_take.mp hj
have heq : U ++ [i] = U ++ [j] := by
rw [hi', hj']
simp
simpa using List.append_cancel_left heq
/-- A pointwise-selected terminal leaf still gives distinct terminal table addresses.
The selected leaf may depend arbitrarily on the selector outcome: only its child
subtree and the refinement of its terminal grid matter. -/
theorem terminalAddress_injective {I : Type u_I} {A : Type u_A} {X : Type u_X} {K : Type u_K}
(U : List A) (child : I → A) (point : I → X)
(leaf : I → List A) (base : A → ℕ) (level : List A → ℕ)
(key : ℕ → X → K)
(hprefix : ∀ i, U ++ [child i] <+: leaf i)
(hlevel : ∀ i, base (child i) ≤ level (leaf i))
(hrefine : ∀ {b B : ℕ} {x y : X}, b ≤ B →
key B x = key B y → key b x = key b y)
(hseparate : ∀ {i j : I}, child i = child j → i ≠ j →
key (base (child i)) (point i) ≠ key (base (child i)) (point j)) :
Function.Injective (fun i => (leaf i, key (level (leaf i)) (point i))) := by
intro i j hij
have hleaf : leaf i = leaf j := congrArg Prod.fst hij
have hchild : child i = child j :=
child_eq_of_common_descendant (hprefix i) (hleaf.symm ▸ hprefix j)
have hkey : key (level (leaf i)) (point i) = key (level (leaf i)) (point j) := by
have h := congrArg Prod.snd hij
simpa only [hleaf] using h
by_contra hne
exact hseparate hchild hne (hrefine (hlevel i) hkey)
/-- Selector trials in different child windows use different tables; within a
window coarse-key separation gives different entries. -/
theorem selectorAddress_injective {I : Type u_I} {A : Type u_A} {X : Type u_X} {K : Type u_K}
(child : I → A) (point : I → X) (key : A → X → K)
(hseparate : ∀ {i j : I}, child i = child j → i ≠ j →
key (child i) (point i) ≠ key (child i) (point j)) :
Function.Injective (fun i => (child i, key (child i) (point i))) := by
intro i j hij
have hchild : child i = child j := congrArg Prod.fst hij
have hkey : key (child i) (point i) = key (child i) (point j) := by
have h := congrArg Prod.snd hij
simpa only [hchild] using h
by_contra hne
exact hseparate hchild hne hkey
/-- Freshness is against the entire exposed center-address set, not merely the
entries on the center's actual route. -/
theorem selectorAddress_ne_centerAddress {I : Type u_I} {A : Type u_A} {X : Type u_X} {K : Type u_K}
(child : I → A) (point : I → X) (key : A → X → K) (x : X)
(hfresh : ∀ i, key (child i) (point i) ≠ key (child i) x)
(i : I) (a : A) :
(child i, key (child i) (point i)) ≠ (a, key a x) := by
intro h
have hchild : child i = a := congrArg Prod.fst h
have hkey := congrArg Prod.snd h
apply hfresh i
simpa only [hchild] using hkey
/-- Consequently all selected trial addresses are disjoint from the exposed
center addresses, as required by finite-product conditioning. -/
theorem disjoint_selector_center_ranges {I : Type u_I} {A : Type u_A} {X : Type u_X} {K : Type u_K}
(child : I → A) (point : I → X) (key : A → X → K) (x : X)
(hfresh : ∀ i, key (child i) (point i) ≠ key (child i) x) :
Disjoint (Set.range (fun i => (child i, key (child i) (point i))))
(Set.range (fun a => (a, key a x))) := by
rw [Set.disjoint_left]
rintro z ⟨i, rfl⟩ ⟨a, ha⟩
exact selectorAddress_ne_centerAddress child point key x hfresh i a ha.symm
end Problem310.AddressInjectivity
end
end OAI
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