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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 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 | import OAI.Dynamics.ConditionalShuffle.Trimming
namespace OAI
noncomputable section
namespace Revealed.Disintegration
open scoped Classical
open Thorp
lemma fairMass_comp_equiv {Ω X : Type*} [Fintype Ω] (f : Ω → X) (C : Equiv.Perm Ω) :
fairMass (f ∘ C) = fairMass f := by
funext x
unfold fairMass
congr 1
exact Equiv.sum_comp C (fun ω => if f ω = x then (1 : ℝ) else 0)
lemma conditional_injective {E G Ω : Type*} [Fintype E] [Fintype G] [Nonempty G]
[Fintype Ω] [Nonempty Ω] [DecidableEq G]
(a : Ω → E) (b : Ω → G) (ha : Function.Injective a) (e : E) :
(conditional (fairMass (fun ω => (a ω, b ω))) e = fun _ => (Fintype.card G : ℝ)⁻¹) ∨
∃ g, conditional (fairMass (fun ω => (a ω, b ω))) e = fun h => if h = g then 1 else 0 := by
by_cases he : ∃ ω, a ω = e
· obtain ⟨ω, rfl⟩ := he
right
refine ⟨b ω, ?_⟩
have hq : marginal (fairMass (fun ω => (a ω, b ω))) (a ω) = (Fintype.card Ω : ℝ)⁻¹ := by
rw [marginal_fairMass]
simp [fairMass, ha.eq_iff]
funext g
have hj : fairMass (fun ω => (a ω, b ω)) (a ω, g) =
(if b ω = g then 1 else 0) / (Fintype.card Ω : ℝ) := by
simp [fairMass, Prod.mk.injEq, ha.eq_iff, ite_and]
have hn : (Fintype.card Ω : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
rw [conditional, hq, ite_eq_right (inv_ne_zero hn), hj]
by_cases hg : g = b ω
· subst g
simp [hn]
· simp [hg, Ne.symm hg]
· left
have hq : marginal (fairMass (fun ω => (a ω, b ω))) e = 0 := by
rw [marginal_fairMass]
have hh : ∀ ω, a ω ≠ e := by simpa using he
simp [fairMass, hh]
funext g
simp [conditional, hq]
lemma conditional_mul_invariant {E G Ω : Type*} [Fintype E] [Fintype G] [Group G]
[Fintype Ω] (a : Ω → E) (b : Ω → G) (C : Equiv.Perm Ω) (τ : G)
(ha : ∀ ω, a (C ω) = a ω) (hb : ∀ ω, b (C ω) = b ω * τ) (e : E) (g : G) :
conditional (fairMass (fun ω => (a ω, b ω))) e (g * τ) =
conditional (fairMass (fun ω => (a ω, b ω))) e g := by
have hj : fairMass (fun ω => (a ω, b ω)) (e, g * τ) =
fairMass (fun ω => (a ω, b ω)) (e, g) := by
have h := congrFun (fairMass_comp_equiv (fun ω => (a ω, b ω)) C) (e, g * τ)
simp only [fairMass, Function.comp_apply, ha, hb, Prod.mk.injEq, mul_right_cancel_iff] at h
simpa only [fairMass, Prod.mk.injEq] using h.symm
unfold conditional
split_ifs <;> simp [hj]
end Revealed.Disintegration
namespace Thorp.Trim
open scoped Classical
open Revealed.Disintegration
variable {α : Type} [Fintype α] [DecidableEq α] [nontrivial_α : Nontrivial α]
lemma signDichotomy_uniform : signDichotomy (fun _ : Equiv.Perm α => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹) := by
left
rw [← Finset.mul_sum, Specht.complexSign_sum, mul_zero]
lemma sign_zero_of_right_invariant (μ : Equiv.Perm α → ℝ) (τ : Equiv.Perm α)
(hm : ∀ g, μ (g * τ) = μ g) (hτ : Specht.complexSign τ = -1) :
(∑ g, (μ g : ℂ) * Specht.complexSign g) = 0 := by
let retained_nontrivial_α := nontrivial_α
have h := Equiv.sum_comp (Equiv.mulRight τ) (fun g => (μ g : ℂ) * Specht.complexSign g)
change (∑ g, (μ (g * τ) : ℂ) * Specht.complexSign (g * τ)) = _ at h
simp only [hm, map_mul, hτ, mul_neg, mul_one, Finset.sum_neg_distrib] at h
linear_combination - (1 / 2 : ℂ) * h
lemma signDichotomy_conditional_injective {E Ω : Type*} [Fintype E] [Fintype Ω] [Nonempty Ω]
(a : Ω → E) (b : Ω → Equiv.Perm α) (ha : Function.Injective a) (e : E) :
signDichotomy (conditional (fairMass (fun ω => (a ω, b ω))) e) := by
rcases conditional_injective a b ha e with h | h
· rw [h]
exact signDichotomy_uniform
· exact Or.inr h
lemma signDichotomy_conditional_symmetry {E Ω : Type*} [Fintype E] [Fintype Ω]
(a : Ω → E) (b : Ω → Equiv.Perm α) (C : Equiv.Perm Ω) (τ : Equiv.Perm α)
(ha : ∀ ω, a (C ω) = a ω) (hb : ∀ ω, b (C ω) = b ω * τ)
(hτ : Specht.complexSign τ = -1) (e : E) :
signDichotomy (conditional (fairMass (fun ω => (a ω, b ω))) e) := by
left
exact sign_zero_of_right_invariant _ τ (conditional_mul_invariant a b C τ ha hb e) hτ
end Thorp.Trim
end
end OAI
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