File size: 3,301 Bytes
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 | 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 <;> rfl
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; exact isEmptyElim a
simp only [ha,ite_true]
exact lineLaw_sign_zero hj z
end BinaryCoordinateSweeps
end
end OAI
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