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a2ee19a | 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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
/-!
# Information restrictions of the one-sample simulation
These pointwise statements make explicit the information restrictions used in
`sections/setup.tex`, Lemma `lem:simulation`: unused samples and online values
of sacrificed labels cannot affect the execution. The final two replay
identities also support the prefix/suffix construction in `sections/secretary.tex`.
No independence, positivity, or feasibility assumptions are needed for these
deterministic identities.
-/
namespace MatroidProphet
lemma simulationObserved_eq_of_eq_on_mask {n bits : ℕ} (A : HiddenRule n bits)
(r : Seed bits) (s s' : Weights n)
(hs : ∀ e ∈ A.mask r, s e = s' e) :
observed A r s = observed A r s' := by
classical
funext e
by_cases he : e ∈ A.mask r
· simp [observed, he, hs e he]
· simp [observed, he]
lemma simulationHistory_eq_of_eq_on_mask {n bits : ℕ} (A : HiddenRule n bits)
(r : Seed bits) (s s' : Weights n) {k : Fin n} (h : History n k)
(hs : ∀ e ∈ A.mask r, s e = s' e) :
simulationHistory A r s h = simulationHistory A r s' h := by
classical
funext j
by_cases he : (h j).1 ∈ A.mask r
· simp [simulationHistory, he, hs _ he]
· simp [simulationHistory, he]
/-- The simulator's current decision ignores every sample outside its mask. -/
theorem sampleSimulation_decide_eq_of_eq_on_mask {n bits : ℕ}
(A : HiddenRule n bits) (r : Seed bits) (s s' : Weights n)
(k : Fin n) (h : History n k)
(hs : ∀ e ∈ A.mask r, s e = s' e) :
(sampleSimulation A).decide k r s h =
(sampleSimulation A).decide k r s' h := by
classical
simp only [sampleSimulation]
rw [simulationObserved_eq_of_eq_on_mask A r s s' hs,
simulationHistory_eq_of_eq_on_mask A r s s' h hs]
lemma glue_eq_of_eq_on_regions {n : ℕ} (mask : Finset (Fin n))
(s s' v v' : Weights n)
(hs : ∀ e ∈ mask, s e = s' e)
(hv : ∀ e ∉ mask, v e = v' e) :
glue mask s v = glue mask s' v' := by
classical
funext e
by_cases he : e ∈ mask
· simp [glue, he, hs e he]
· simp [glue, he, hv e he]
/-- Neither unused samples nor sacrificed online values change any accepted prefix. -/
theorem sampleSimulation_acceptedThrough_noninterference {n bits : ℕ}
(A : HiddenRule n bits) (r : Seed bits) (s s' v v' : Weights n)
(π : ArrivalOrder n) (t : ℕ)
(hs : ∀ e ∈ A.mask r, s e = s' e)
(hv : ∀ e ∉ A.mask r, v e = v' e) :
acceptedThrough (sampleSimulation A) r s v π t =
acceptedThrough (sampleSimulation A) r s' v' π t := by
rw [sampleSimulation_acceptedThrough, sampleSimulation_acceptedThrough,
glue_eq_of_eq_on_regions (A.mask r) s s' v v' hs hv]
/-- The same information invariance holds for the reward in actual online values. -/
theorem sampleSimulation_reward_noninterference {n bits : ℕ}
(A : HiddenRule n bits) (r : Seed bits) (s s' v v' : Weights n)
(π : ArrivalOrder n)
(hs : ∀ e ∈ A.mask r, s e = s' e)
(hv : ∀ e ∉ A.mask r, v e = v' e) :
reward (sampleSimulation A) r s v π =
reward (sampleSimulation A) r s' v' π := by
rw [sampleSimulation_reward, sampleSimulation_reward,
glue_eq_of_eq_on_regions (A.mask r) s s' v v' hs hv]
/-- A sacrificed label is rejected at every prefix, irrespective of its value. -/
theorem sampleSimulation_rejects_mask {n bits : ℕ} (A : HiddenRule n bits)
(r : Seed bits) (s v : Weights n) (π : ArrivalOrder n) (t : ℕ)
(e : Fin n) (he : e ∈ A.mask r) :
e ∉ acceptedThrough (sampleSimulation A) r s v π t := by
rw [sampleSimulation_acceptedThrough]
exact fun h => (Finset.mem_sdiff.mp h).2 he
lemma glue_observed_left {n bits : ℕ} (A : HiddenRule n bits)
(r : Seed bits) (w : Weights n) :
glue (A.mask r) (observed A r w) w = w := by
classical
funext e
by_cases he : e ∈ A.mask r <;> simp [glue, observed, he]
/-- Replaying the full arrived history after supplying only the observed mask
agrees exactly with the hidden rule, at every prefix including zero. -/
theorem sampleSimulation_replay_acceptedThrough {n bits : ℕ}
(A : HiddenRule n bits) (r : Seed bits) (w : Weights n)
(π : ArrivalOrder n) (t : ℕ) :
acceptedThrough (sampleSimulation A) r (observed A r w) w π t =
hiddenAcceptedThrough A r w π t := by
rw [sampleSimulation_acceptedThrough, glue_observed_left]
/-- The replay identity preserves the actual reward without a numerical loss. -/
theorem sampleSimulation_replay_reward {n bits : ℕ}
(A : HiddenRule n bits) (r : Seed bits) (w : Weights n)
(π : ArrivalOrder n) :
reward (sampleSimulation A) r (observed A r w) w π = hiddenReward A r w π := by
rw [sampleSimulation_reward, glue_observed_left]
end MatroidProphet
end OAI
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