| import OAI.Dynamics.ConditionalShuffle.Trimming |
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| namespace OAI |
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| 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) |
|
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| 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] |
|
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| 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 |
|
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| 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 |
|
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| end Thorp.Trim |
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| end |
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| end OAI |
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