| import OAI.Dynamics.ConditionalShuffle.Rates |
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| namespace OAI |
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| 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] |
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| 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 |
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| 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) |
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| end Revealed.Physical |
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| end |
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| end OAI |
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