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9b6e014 | 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 | 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; exact hu ⟨i,he⟩
· intro he; exact hv ⟨i,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 <;> ring
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 [uniformLaw]
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; exact Fin.elim0 i⟩
have hs := uniform_sign_assigns_zero E E huv
(by rintro ⟨i,_⟩; exact Fin.elim0 i) (by rintro ⟨i,_⟩; exact Fin.elim0 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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