| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.SquareBoundary |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation.Boundary |
|
|
| def forwardDir (d : Dir) : Prop := d=1 ∨ d=2 |
|
|
| def forwardIndices (e : ℕ → Edge) (n : ℕ) : Finset ℕ := |
| (Finset.range n).filter (fun i => forwardDir (e i).2) |
|
|
| def outsideSites (e : ℕ → Edge) (n : ℕ) : Finset Grid := |
| (forwardIndices e n).image (fun i => rightSite (e i)) |
|
|
| lemma forward_step (d : Dir) : (step d).1-(step d).2= |
| if forwardDir d then (-1 : ℤ) else 1 := by |
| fin_cases d <;> norm_num [step,forwardDir,Fin.ext_iff] |
|
|
| lemma twice_forward_card (e : ℕ → Edge) (n : ℕ) |
| (hchain : ∀ i, head (e i)=tail (e (i+1))) (hclose : e n=e 0) : |
| 2*(forwardIndices e n).card=n := by |
| have hv (i : ℕ) : (e (i+1)).1=(e i).1+step (e i).2 := (hchain i).symm |
| have hd (i : ℕ) : |
| ((e (i+1)).1.1-(e (i+1)).1.2)-((e i).1.1-(e i).1.2)= |
| if forwardDir (e i).2 then (-1 : ℤ) else 1 := by |
| rw [hv] |
| simpa only [Prod.fst_add,Prod.snd_add] using (by |
| rw [← forward_step] |
| ring : ((e i).1.1+(step (e i).2).1-((e i).1.2+(step (e i).2).2))- |
| ((e i).1.1-(e i).1.2)=if forwardDir (e i).2 then (-1 : ℤ) else 1) |
| have hz : (∑ i∈Finset.range n, (if forwardDir (e i).2 then (-1 : ℤ) else 1))=0 := by |
| simp_rw [← hd] |
| rw [Finset.sum_range_sub (fun i => (e i).1.1-(e i).1.2),hclose,sub_self] |
| have hf : ∑ i∈Finset.range n, (if forwardDir (e i).2 then (1 : ℤ) else 0)= |
| ((forwardIndices e n).card : ℤ) := by |
| simp [forwardIndices,Finset.sum_boole] |
| have he (i : ℕ) : (if forwardDir (e i).2 then (-1 : ℤ) else 1)= |
| 1-2*(if forwardDir (e i).2 then (1 : ℤ) else 0) := by split <;> norm_num |
| simp_rw [he] at hz |
| rw [Finset.sum_sub_distrib,← Finset.mul_sum,hf] at hz |
| simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul,mul_one] at hz |
| omega |
|
|
| lemma rightSite_direction_injective : |
| Function.Injective (fun e : Edge => (rightSite e,e.2)) := by |
| rintro ⟨p,d⟩ ⟨q,c⟩ h |
| have hd : d=c := congrArg Prod.snd h |
| subst c |
| have hp : p=q := add_right_cancel (congrArg Prod.fst h) |
| exact Prod.ext hp rfl |
|
|
| lemma forward_card_le_twice_outside (e : ℕ → Edge) (n : ℕ) |
| (hinj : Set.InjOn e (Set.Iio n)) : |
| (forwardIndices e n).card≤2*(outsideSites e n).card := by |
| have hb := Finset.card_le_card_of_injOn |
| (s := forwardIndices e n) (t := outsideSites e n ×ˢ ({1,2} : Finset Dir)) |
| (f := fun i => (rightSite (e i),(e i).2)) |
| (by |
| intro i hi |
| refine Finset.mem_product.mpr ⟨Finset.mem_image.mpr ⟨i,hi,rfl⟩,?_⟩ |
| simpa only [forwardDir,Finset.mem_insert,Finset.mem_singleton] using (Finset.mem_filter.mp hi).2) |
| (by |
| intro i hi j hj he |
| exact hinj (Finset.mem_range.mp (Finset.mem_filter.mp hi).1) |
| (Finset.mem_range.mp (Finset.mem_filter.mp hj).1) (rightSite_direction_injective he)) |
| have hc : ({1,2} : Finset Dir).card=2 := by decide |
| rw [Finset.card_product,hc] at hb |
| omega |
|
|
| theorem length_le_four_outside (e : ℕ → Edge) (n : ℕ) |
| (hchain : ∀ i, head (e i)=tail (e (i+1))) (hclose : e n=e 0) |
| (hinj : Set.InjOn e (Set.Iio n)) : n≤4*(outsideSites e n).card := by |
| have h := twice_forward_card e n hchain hclose |
| have h' := forward_card_le_twice_outside e n hinj |
| omega |
| |
| lemma forward_child (e : Edge) (hdir : forwardDir e.2) : |
| rightSite e=leftSite e+(1,0) ∨ rightSite e=leftSite e+(0,1) := by |
| rcases e with ⟨⟨x,y⟩,d⟩ |
| rcases hdir with h|h |
| · change d=1 at h |
| subst d |
| left |
| simp [rightSite,leftSite,rightOffset,leftOffset] |
| · change d=2 at h |
| subst d |
| right |
| simp [rightSite,leftSite,rightOffset,leftOffset] |
| |
| lemma outsideSites_failed_children (S : Set Grid) (e : ℕ → Edge) (n : ℕ) |
| (hmem : ∀ i, e i∈squareBoundary S) : |
| ∀ z∈outsideSites e n, z∉S ∧ ∃ y∈S, z=y+(1,0) ∨ z=y+(0,1) := by |
| intro z hz |
| obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp hz |
| exact ⟨(hmem i).2,leftSite (e i),(hmem i).1, |
| forward_child (e i) (Finset.mem_filter.mp hi).2⟩ |
| |
| end CriticalPercolation.Boundary |
| end |
| end |
| end |
| |
| end OAI |
| |
