| import Mathlib | |
| namespace OAI | |
| /-! | |
| # A robust four-row permanent inequality | |
| `Fin 4` labels four sites. A law is represented by real probability masses; | |
| `IsProbability` asserts nonnegativity and total mass one. All row norms use | |
| the uniform counting normalization. | |
| -/ | |
| noncomputable section | |
| namespace FourRow | |
| open scoped BigOperators | |
| abbrev Site := Fin 4 | |
| abbrev Perm := Equiv.Perm Site | |
| abbrev Law := Perm → ℝ | |
| abbrev Functions := Site → Site → ℝ | |
| def IsProbability (ν : Law) : Prop := | |
| (∀ π, 0 ≤ ν π) ∧ ∑ π, ν π = 1 | |
| def UniformMarginals (ν : Law) : Prop := | |
| ∀ i j : Site, ∑ π, (if π i = j then ν π else 0) = 1 / 4 | |
| def totalVariation (ν : Law) : ℝ := | |
| (∑ π, |ν π - 1 / 24|) / 2 | |
| def lpNorm (p : ℝ) (f : Site → ℝ) : ℝ := | |
| ((∑ j, (f j) ^ p) / 4) ^ (1 / p) | |
| def permanentExpectation (ν : Law) (f : Functions) : ℝ := | |
| ∑ π, ν π * ∏ i, f i (π i) | |
| end FourRow | |
| end | |
| end OAI | |