| import Mathlib |
| import OAI.Probability.BinarySweep.FiniteLaws.UniformFiniteProbability |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| attribute [local instance] Classical.propDecidable |
| open Sparse |
|
|
| variable {b h : ℕ} {bits : Fin b → ℕ} |
|
|
| lemma gridLayer_at (g : GridChoices bits) (j : Fin b) (x : GridSlot bits) : |
| gridLayer bits g j x j = g j (fun i => x i) (x j) := by |
| simp [gridLayer,Equiv.piSplitAt_symm_apply] |
|
|
| @[simp] lemma independentPosition_zero (e : GridSlot bits × GridSlot bits) : |
| independentPosition e 0=e.1 := by ext i |
|
|
| @[simp] lemma independentPosition_last (e : GridSlot bits × GridSlot bits) : |
| independentPosition e (Fin.last b)=e.2 := by ext i |
|
|
| def endpointLine (e : GridSlot bits × GridSlot bits) (j : Fin b) : GridOutside bits j := |
| fun i => independentPosition e j.castSucc i |
|
|
| lemma endpoint_gridLayer_iff (g : GridChoices bits) (e : GridSlot bits × GridSlot bits) (j : Fin b) : |
| gridLayer bits g j (independentPosition e j.castSucc)=independentPosition e j.succ ↔ |
| g j (endpointLine e j) (e.1 j)=e.2 j := by |
| unfold endpointLine |
| constructor |
| · intro he |
| have hh := congrFun he j |
| simpa only [gridLayer_at,endpointLine,independentPosition,Fin.val_castSucc, |
| Fin.val_succ,lt_self_iff_false,ite_false,Nat.lt_succ_self,ite_true] using hh |
| · intro he |
| funext i |
| by_cases hi : i=j |
| · subst i |
| simpa only [gridLayer_at,endpointLine,independentPosition,Fin.val_castSucc, |
| Fin.val_succ,lt_self_iff_false,ite_false,Nat.lt_succ_self,ite_true] using he |
| · rw [gridLayer_other bits g j i hi] |
| exact (independentPosition_changes e j i hi).symm |
|
|
| lemma gridSweep_endpoint_iff (g : GridChoices bits) (e : GridSlot bits × GridSlot bits) : |
| gridSweep bits g e.1=e.2 ↔ ∀j, g j (endpointLine e j) (e.1 j)=e.2 j := by |
| constructor |
| · intro he j |
| have hp (t : Fin (b+1)) : gridPartialSweep bits g t e.1=independentPosition e t := by |
| have hh := independentPosition_of_path (fun t => gridPartialSweep bits g t e.1) |
| (gridPartialSweep_changes bits g e.1) t |
| simpa only [gridPartialSweep_zero,Equiv.Perm.one_apply,gridPartialSweep_last,he] using hh.symm |
| apply (endpoint_gridLayer_iff g e j).mp |
| rw [← hp j.castSucc,← hp j.succ,gridPartialSweep_succ,Equiv.Perm.mul_apply] |
| · intro he |
| have hh := chronological_trajectory b (gridLayer bits g) (independentPosition e) |
| (fun j => (endpoint_gridLayer_iff g e j).mpr (he j)) |
| simpa only [independentPosition_zero,independentPosition_last,gridSweep] using hh |
|
|
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| def localEndpointEvent (A : Finset I) (e : I → GridSlot bits × GridSlot bits) |
| (j : Fin b) (y : GridOutside bits j) (σ : Equiv.Perm (Slot (bits j))) : Prop := |
| ∀i ∈ A, endpointLine (e i) j=y → σ ((e i).1 j)=(e i).2 j |
| |
| def endpointMass (H : PathFamily bits h) (z : ℝ) (A : Finset I) |
| (e : I → GridSlot bits × GridSlot bits) : ℝ := |
| ∑g : GridChoices bits, if pathEvent H g ∧ ∀i ∈ A, gridSweep bits g (e i).1=(e i).2 then |
| gridWeight bits z g else 0 |
|
|
| def localEndpointMass (H : PathFamily bits h) (z : ℝ) (A : Finset I) |
| (e : I → GridSlot bits × GridSlot bits) (j : Fin b) (y : GridOutside bits j) : ℝ := |
| ∑σ, if Assigns (lineInput H j y) (lineOutput H j y) σ ∧ localEndpointEvent A e j y σ then |
| lineLaw (bits j) z σ else 0 |
|
|
| lemma endpointMass_factor (H : PathFamily bits h) (z : ℝ) (A : Finset I) |
| (e : I → GridSlot bits × GridSlot bits) : |
| endpointMass H z A e = ∏j, ∏y, localEndpointMass H z A e j y := by |
| have he (g : GridChoices bits) : |
| (pathEvent H g ∧ ∀i ∈ A, gridSweep bits g (e i).1=(e i).2) ↔ |
| ∀j y, Assigns (lineInput H j y) (lineOutput H j y) (g j y) ∧ |
| localEndpointEvent A e j y (g j y) := by |
| rw [pathEvent_iff_assigns] |
| simp_rw [gridSweep_endpoint_iff] |
| constructor |
| · rintro ⟨hg,hp⟩ j y |
| refine ⟨hg j y,?_⟩ |
| intro i hi hiy |
| rw [← hiy] |
| exact hp i hi j |
| · intro hh |
| refine ⟨fun j y => (hh j y).1,?_⟩ |
| intro i hi j |
| exact (hh j (endpointLine (e i) j)).2 i hi rfl |
| unfold endpointMass localEndpointMass |
| rw [← sum_prod_grid] |
| apply Finset.sum_congr rfl |
| intro g _ |
| simp only [Fintype.prod_ite_zero] |
| simp only [he,gridWeight] |
| split_ifs < |
|
|
| omit [Fintype I] in |
| lemma localEndpointEvent_insert (A : Finset I) (e : I → GridSlot bits × GridSlot bits) |
| (i : I) (j : Fin b) (y : GridOutside bits j) (σ : Equiv.Perm (Slot (bits j))) : |
| localEndpointEvent (insert i A) e j y σ ↔ |
| (endpointLine (e i) j=y → σ ((e i).1 j)=(e i).2 j) ∧ localEndpointEvent A e j y σ := by |
| simp only [localEndpointEvent,Finset.mem_insert,forall_eq_or_imp] |
|
|
| omit [Fintype I] [DecidableEq I] in |
| lemma localEndpointEvent_empty_line (A : Finset I) (e : I → GridSlot bits × GridSlot bits) |
| (j : Fin b) (y : GridOutside bits j) (ha : ∀i ∈ A, endpointLine (e i) j≠y) |
| (σ : Equiv.Perm (Slot (bits j))) : localEndpointEvent A e j y σ := by |
| intro i hi hiy |
| exact (ha i hi hiy).elim |
|
|
| lemma assigns_empty_line (H : PathFamily bits h) (j : Fin b) (y : GridOutside bits j) |
| (hn : ∀k, (fun i : {i : Fin b // i≠j} => H.position j.castSucc k i) ≠ y) |
| (σ : Equiv.Perm (Slot (bits j))) : Assigns (lineInput H j y) (lineOutput H j y) σ := by |
| intro k |
| exact (hn k.val k.property).elim |
|
|
| end BinaryCoordinateSweeps |
| |
| end |
|
|
| end OAI |
|
|
