| import Mathlib |
| import OAI.Probability.BinarySweep.Conditional.Placement |
| import OAI.Probability.BinarySweep.FiniteLaws.AssignmentSign |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma realSign_gridLayer {b : ℕ} (bits : Fin b → ℕ) (g : GridChoices bits) (j : Fin b) : |
| realSign (gridLayer bits g j)=∏y, realSign (g j y) := by |
| simp [gridLayer,realSign,Equiv.Perm.sign_prodCongrLeft,map_prod] |
|
|
| lemma realSign_gridSweep {b : ℕ} (bits : Fin b → ℕ) (g : GridChoices bits) : |
| realSign (gridSweep bits g)=∏j, ∏y, realSign (g j y) := by |
| simp [gridSweep,map_list_prod,List.map_reverse,List.prod_reverse, |
| List.map_ofFn,List.prod_ofFn,realSign_gridLayer] |
|
|
| lemma signed_path_integrand_factor {b h : ℕ} {bits : Fin b → ℕ} |
| (H : PathFamily bits h) (z : ℝ) (g : GridChoices bits) : |
| (if pathEvent H g then gridWeight bits z g*realSign (gridSweep bits g) else 0) = |
| ∏j, ∏y, if Assigns (lineInput H j y) (lineOutput H j y) (g j y) then |
| lineLaw (bits j) z (g j y)*realSign (g j y) else 0 := by |
| rw [pathEvent_iff_assigns,realSign_gridSweep] |
| simp only [Fintype.prod_ite_zero,Finset.prod_mul_distrib,gridWeight] |
| split_ifs < |
|
|
| lemma signed_path_weight_factor {b h : ℕ} {bits : Fin b → ℕ} |
| (H : PathFamily bits h) (z : ℝ) : |
| (∑g : GridChoices bits, if pathEvent H g then |
| gridWeight bits z g*realSign (gridSweep bits g) else 0) = |
| ∏j, ∏y, ∑σ : Equiv.Perm (Slot (bits j)), |
| if Assigns (lineInput H j y) (lineOutput H j y) σ then |
| lineLaw (bits j) z σ*realSign σ else 0 := by |
| simp_rw [signed_path_integrand_factor H z] |
| exact sum_prod_grid bits (fun j y σ => |
| if Assigns (lineInput H j y) (lineOutput H j y) σ then |
| lineLaw (bits j) z σ*realSign σ else 0) |
|
|
| theorem signed_path_weight_zero_uniform {b h : ℕ} {bits : Fin b → ℕ} |
| (H : PathFamily bits h) (j : Fin b) (y : GridOutside bits j) |
| {u v : Slot (bits j)} (huv : u≠v) |
| (hu : u∉Set.range (lineOutput H j y)) (hv : v∉Set.range (lineOutput H j y)) : |
| (∑g : GridChoices bits, if pathEvent H g then |
| gridWeight bits 0 g*realSign (gridSweep bits g) else 0)=0 := by |
| rw [signed_path_weight_factor] |
| apply Finset.prod_eq_zero (Finset.mem_univ j) |
| apply Finset.prod_eq_zero (Finset.mem_univ y) |
| simpa only [lineLaw,sub_zero,one_mul,zero_mul,add_zero] using |
| uniform_sign_assigns_zero (lineInput H j y) (lineOutput H j y) huv hu hv |
|
|
| theorem signed_path_weight_zero_empty {b h : ℕ} {bits : Fin b → ℕ} |
| (H : PathFamily bits h) (z : ℝ) (j : Fin b) (hj : 0<bits j) |
| (y : GridOutside bits j) (hy : lineHoles H j y=0) : |
| (∑g : GridChoices bits, if pathEvent H g then |
| gridWeight bits z g*realSign (gridSweep bits g) else 0)=0 := by |
| rw [signed_path_weight_factor] |
| apply Finset.prod_eq_zero (Finset.mem_univ j) |
| apply Finset.prod_eq_zero (Finset.mem_univ y) |
| have hz : IsEmpty (LinePaths H j y) := Fintype.card_eq_zero_iff.mp hy |
| have ha (σ : Equiv.Perm (Slot (bits j))) : |
| Assigns (lineInput H j y) (lineOutput H j y) σ := by |
| intro a |
| simp only [ha,ite_true] |
| exact lineLaw_sign_zero hj z |
|
|
| end BinaryCoordinateSweeps |
| |
| end |
|
|
| end OAI |
|
|
