| import Mathlib |
| import OAI.Probability.BinarySweep.Conditional.Placement |
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|
| namespace OAI |
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|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| attribute [local instance] Classical.propDecidable |
| variable {b h : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
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| lemma realSign_ne_zero {X : Type*} [Fintype X] [DecidableEq X] (g : Equiv.Perm X) : |
| realSign g≠0 := by |
| simp [realSign] |
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|
| lemma realSign_inv {X : Type*} [Fintype X] [DecidableEq X] (f : Equiv.Perm X) : |
| realSign f⁻¹=(realSign f)⁻¹ := by |
| have hm : realSign f⁻¹*realSign f=1 := by rw [←map_mul,inv_mul_cancel,map_one] |
| calc |
| realSign f⁻¹ = (realSign f⁻¹*realSign f)*(realSign f)⁻¹ := by |
| rw [mul_assoc,mul_inv_cancel₀ (realSign_ne_zero f),mul_one] |
| _ = _ := by rw [hm,one_mul] |
|
|
| lemma remaining_relative_realSign (g g₀ : ConditionalChoices H) : |
| realSign ((remainingPerm H g₀.val g₀.property)⁻¹*remainingPerm H g.val g.property)= |
| realSign ((gridSweep bits g₀.val)⁻¹*gridSweep bits g.val) := by |
| let f := (gridSweep bits g₀.val)⁻¹*gridSweep bits g.val |
| have hfix (i : Fin h) : f (H.position 0 i)=H.position 0 i := by |
| change (gridSweep bits g₀.val).symm (gridSweep bits g.val (H.position 0 i))=_ |
| rw [gridSweep_path H g.val g.property,←gridSweep_path H g₀.val g₀.property, |
| Equiv.symm_apply_apply] |
| have hp (x : GridSlot bits) : f x∉Set.range (H.position 0) ↔ x∉Set.range (H.position 0) := by |
| constructor |
| · intro hx hi; obtain ⟨i,rfl⟩ := hi; exact hx ⟨i,hfix i |>.symm⟩ |
| · intro hx hi; obtain ⟨i,hi⟩ := hi |
| exact hx ⟨i,f.injective ((hfix i).trans hi)⟩ |
| have hs := Equiv.Perm.sign_subtypePerm f (p := fun x => x∉Set.range (H.position 0)) hp (by |
| intro x hx hi; obtain ⟨i,rfl⟩ := hi; exact hx (hfix i)) |
| have he : Equiv.Perm.subtypePerm f hp = |
| (remainingPerm H g₀.val g₀.property)⁻¹*remainingPerm H g.val g.property := by |
| apply Equiv.ext |
| intro x |
| apply Subtype.ext |
| change f x.val = |
| (gridSweep bits g₀.val).symm |
| ((freeIdentification H) ((freeIdentification H).symm |
| ((remainingBijection H g.val g.property) x))).val |
| rw [Equiv.apply_symm_apply] |
| rfl |
| rw [he] at hs |
| convert congrArg (fun u : ℤˣ => ((u:ℤ):ℝ)) hs using 1 |
| · unfold realSign; dsimp; congr 3 |
| congr 1; exact Subsingleton.elim _ _ |
| · rfl |
| |
| theorem remaining_realSign_factor (g g₀ : ConditionalChoices H) : |
| realSign (remainingPerm H g.val g.property) = |
| (realSign (remainingPerm H g₀.val g₀.property)/realSign (gridSweep bits g₀.val))* |
| realSign (gridSweep bits g.val) := by |
| have he := remaining_relative_realSign H g g₀ |
| simp only [map_mul,realSign_inv] at he |
| have h₀ := realSign_ne_zero (remainingPerm H g₀.val g₀.property) |
| have h₁ := realSign_ne_zero (gridSweep bits g₀.val) |
| field_simp [h₀,h₁] at he ⊢ |
| nlinarith [he] |
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| end BinaryCoordinateSweeps |
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| end |
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| end OAI |
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