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f205175 | 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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | 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
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