| import Mathlib |
| import OAI.Probability.BinarySweep.Processes.IndependentHistory |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| variable {I X : Type*} [Fintype I] [Fintype X] [DecidableEq X] |
|
|
| lemma realSign_swap {u v : X} (huv : u≠v) : realSign (Equiv.swap u v) = -1 := by |
| simp [realSign,Equiv.Perm.sign_swap huv] |
|
|
| omit [Fintype I] [Fintype X] in |
| lemma assigns_swap_left (A B : I ↪ X) {u v : X} |
| (hu : u∉Set.range B) (hv : v∉Set.range B) (σ : Equiv.Perm X) : |
| Assigns A B (Equiv.swap u v*σ) ↔ Assigns A B σ := by |
| have hf (i : I) : Equiv.swap u v (B i)=B i := by |
| apply Equiv.swap_apply_of_ne_of_ne |
| · intro he |
| · intro he |
| constructor |
| · intro hh i |
| apply (Equiv.swap u v).injective |
| exact (hh i).trans (hf i).symm |
| · intro hh i |
| change Equiv.swap u v (σ (A i))=B i |
| rw [hh i,hf] |
|
|
| omit [Fintype I] in |
| theorem sum_sign_assigns_zero (A B : I ↪ X) {u v : X} |
| (huv : u≠v) (hu : u∉Set.range B) (hv : v∉Set.range B) : |
| (∑σ : Equiv.Perm X, if Assigns A B σ then realSign σ else 0) = 0 := by |
| classical |
| let f (σ : Equiv.Perm X) := if Assigns A B σ then realSign σ else 0 |
| have ht (σ : Equiv.Perm X) : f (Equiv.swap u v*σ) = -f σ := by |
| dsimp [f] |
| rw [assigns_swap_left A B hu hv,map_mul,realSign_swap huv] |
| split_ifs < |
| have he := Equiv.sum_comp (Equiv.mulLeft (Equiv.swap u v)) f |
| change (∑σ, f (Equiv.swap u v*σ))=∑σ,f σ at he |
| simp_rw [ht] at he |
| rw [Finset.sum_neg_distrib] at he |
| dsimp [f] at he |
| linarith |
|
|
| omit [Fintype I] in |
| lemma uniform_sign_assigns_zero (A B : I ↪ X) {u v : X} |
| (huv : u≠v) (hu : u∉Set.range B) (hv : v∉Set.range B) : |
| (∑σ : Equiv.Perm X, if Assigns A B σ then |
| uniformLaw (Equiv.Perm X) σ * realSign σ else 0) = 0 := by |
| have ht (σ : Equiv.Perm X) : |
| (if Assigns A B σ then uniformLaw (Equiv.Perm X) σ*realSign σ else 0) = |
| (Fintype.card (Equiv.Perm X):ℝ)⁻¹*(if Assigns A B σ then realSign σ else 0) := by |
| split_ifs < |
| simp_rw [ht] |
| rw [←Finset.mul_sum,sum_sign_assigns_zero A B huv hu hv,mul_zero] |
| |
| lemma uniform_sign_zero [Nontrivial X] : |
| (∑σ : Equiv.Perm X, uniformLaw (Equiv.Perm X) σ*realSign σ)=0 := by |
| obtain ⟨u,v,huv⟩ := exists_pair_ne X |
| let E : Fin 0 ↪ X := ⟨Fin.elim0,by intro i |
| have hs := uniform_sign_assigns_zero E E huv |
| (by rintro ⟨i,_⟩ |
| simpa [Assigns] using hs |
|
|
| lemma lineLaw_sign_zero {d : ℕ} (hd : 0<d) (z : ℝ) : |
| (∑σ : Equiv.Perm (Slot d), lineLaw d z σ*realSign σ)=0 := by |
| have : Nonempty (Fin d) := ⟨⟨0,hd⟩⟩ |
| have : Nontrivial (Slot d) := inferInstance |
| simp_rw [lineLaw,add_mul,mul_assoc] |
| rw [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.mul_sum, |
| uniform_sign_zero,binary_sign_annihilated d hd,mul_zero,mul_zero,add_zero] |
| |
| end BinaryCoordinateSweeps |
| |
| end |
|
|
| end OAI |
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|
