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92416a2 | 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 | import OAI.Dynamics.ConditionalShuffle.Rates
namespace OAI
noncomputable section
namespace Revealed.Physical
open scoped Classical
open Thorp Thorp.Conditional Revealed.Split Revealed.Instrument Revealed.Disintegration
variable {ι α : Type} [Fintype ι] [Fintype α] [DecidableEq α]
lemma block_weighted_cost (d t : ℕ) (e : Outside ι α (Position d))
(c : (Equiv.Perm α → ℝ) → ℝ) :
(∑ p, probability (block (instrument d) t) e p * c (kernel (block (instrument d) t) e p)) =
∑ p, marginal (frameJoint (sectionFrame e) t) p * c (frameKernel (sectionFrame e) t p) := by
have h := weighted_conditional_map (observed (instrument d) e t)
(groupRun (instrument d) e t) (fullPath e) (fullPath_injective e) c
simp_rw [physical_path_frame, physical_group_frame] at h
simpa only [probability, kernel, joint, block, frameKernel, frameJoint, physical_group_frame] using h.symm
lemma block_discarded_le (d : ℕ) (e : Outside ι (Position (d + 1 + 2)) (Position (d + 2 + 2))) :
(∑ p, probability (block (instrument (d+2+2)) (400*(d+2+2))) e p *
Trim.discarded (Block.embedding (Block.eightEquiv d))
(kernel (block (instrument (d+2+2)) (400*(d+2+2))) e p)) ≤ 16 * conditionalRate d := by
rw [block_weighted_cost]
exact average_discarded_le d (sectionFrame e)
end Revealed.Physical
end
end OAI
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