| import Mathlib |
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| namespace OAI |
|
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| section |
| section |
| noncomputable section |
| open scoped BigOperators |
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| namespace CriticalPercolation.DiskWinding |
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| def increment (a b : Real.Angle) : ℝ := (b-a).toReal |
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| lemma increment_add_of_short {a b c : Real.Angle} |
| (hab : |increment a b|<Real.pi/4) (hbc : |increment b c|<Real.pi/4) |
| (hac : |increment a c|<Real.pi/4) : |
| increment a b+increment b c=increment a c := by |
| let z := increment a b+increment b c-increment a c |
| have hco : (z : Real.Angle)=0 := by |
| simp only [z,increment,Real.Angle.coe_sub,Real.Angle.coe_add,Real.Angle.coe_toReal] |
| abel |
| have hbounded : -Real.pi<z ∧ z≤Real.pi := by |
| obtain ⟨habl,habu⟩ := abs_lt.mp hab |
| obtain ⟨hbcl,hbcu⟩ := abs_lt.mp hbc |
| obtain ⟨hacl,hacu⟩ := abs_lt.mp hac |
| dsimp [z] |
| constructor <;> linarith [Real.pi_pos] |
| have hz := Real.Angle.toReal_coe_eq_self_iff.mpr hbounded |
| rw [hco,Real.Angle.toReal_zero] at hz |
| dsimp [z] at hz |
| linarith |
|
|
| lemma sum_successive_sub (f : ℕ → ℝ) (K : ℕ) : |
| ∑ k∈Finset.range K, (f (k+1)-f k)=f K-f 0 := by |
| induction K with |
| | zero => simp |
| | succ K ih => rw [Finset.sum_range_succ,ih]; ring |
|
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| lemma cell_balance {a b c d : Real.Angle} |
| (hab : |increment a b|<Real.pi/4) (hbc : |increment b c|<Real.pi/4) |
| (hac : |increment a c|<Real.pi/4) (had : |increment a d|<Real.pi/4) |
| (hdc : |increment d c|<Real.pi/4) : |
| increment a b+increment b c=increment a d+increment d c := |
| (increment_add_of_short hab hbc hac).trans (increment_add_of_short had hdc hac).symm |
|
|
| theorem boundary_sum_zero (F : ℕ → ℕ → Real.Angle) (N K : ℕ) |
| (hcenter : ∀ k≤K, F 0 k=F 0 0) |
| (hseam : ∀ i≤N, F i K=F i 0) |
| (hhoriz : ∀ i≤N, ∀ k<K, |increment (F i k) (F i (k+1))|<Real.pi/4) |
| (hvert : ∀ i<N, ∀ k≤K, |increment (F i k) (F (i+1) k)|<Real.pi/4) |
| (hdiag : ∀ i<N, ∀ k<K, |increment (F i k) (F (i+1) (k+1))|<Real.pi/4) : |
| ∑ k∈Finset.range K, increment (F N k) (F N (k+1))=0 := by |
| have step : ∀ i<N, |
| (∑ k∈Finset.range K, increment (F (i+1) k) (F (i+1) (k+1)))= |
| ∑ k∈Finset.range K, increment (F i k) (F i (k+1)) := by |
| intro i hi |
| let v : ℕ → ℝ := fun k => increment (F i k) (F (i+1) k) |
| have hver : ∑ k∈Finset.range K, (v (k+1)-v k)=0 := by |
| rw [sum_successive_sub] |
| dsimp [v] |
| rw [hseam i (by omega),hseam (i+1) (by omega),sub_self] |
| have heach : ∀ k∈Finset.range K, |
| increment (F (i+1) k) (F (i+1) (k+1))= |
| increment (F i k) (F i (k+1))+(v (k+1)-v k) := by |
| intro k hk |
| have hk' := Finset.mem_range.mp hk |
| have h := cell_balance (hhoriz i (by omega) k hk') |
| (hvert i hi (k+1) (by omega)) (hdiag i hi k hk') |
| (hvert i hi k (by omega)) (hhoriz (i+1) (by omega) k hk') |
| dsimp [v] |
| linarith |
| rw [Finset.sum_congr rfl heach,Finset.sum_add_distrib,hver,add_zero] |
| have all : ∀ i≤N, ∑ k∈Finset.range K, increment (F i k) (F i (k+1))=0 := by |
| intro i |
| induction i with |
| | zero => |
| intro _ |
| apply Finset.sum_eq_zero |
| intro k hk |
| rw [hcenter k (by have := Finset.mem_range.mp hk; omega), |
| hcenter (k+1) (by have := Finset.mem_range.mp hk; omega)] |
| simp [increment] |
| | succ i ih => |
| intro hi |
| rw [step i (by omega),ih (by omega)] |
| exact all N le_rfl |
|
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| lemma boundary_sum_lift (F : ℕ → Real.Angle) (b : ℕ → ℝ) (K : ℕ) |
| (hlift : ∀ k≤K, F k=(b k : Real.Angle)) |
| (hstep : ∀ k<K, |b (k+1)-b k|<Real.pi) : |
| ∑ k∈Finset.range K, increment (F k) (F (k+1))=b K-b 0 := by |
| rw [←sum_successive_sub b K] |
| apply Finset.sum_congr rfl |
| intro k hk |
| have hk' := Finset.mem_range.mp hk |
| rw [increment,hlift k (by omega),hlift (k+1) (by omega),←Real.Angle.coe_sub] |
| exact Real.Angle.toReal_coe_eq_self_iff.mpr ⟨(abs_lt.mp (hstep k hk')).1, |
| (abs_lt.mp (hstep k hk')).2.le⟩ |
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| theorem exists_large_edge (F : ℕ → ℕ → Real.Angle) (N K : ℕ) (b : ℕ → ℝ) |
| (hcenter : ∀ k≤K, F 0 k=F 0 0) |
| (hseam : ∀ i≤N, F i K=F i 0) |
| (hlift : ∀ k≤K, F N k=(b k : Real.Angle)) |
| (hstep : ∀ k<K, |b (k+1)-b k|<Real.pi) |
| (hturn : b K-b 0=2*Real.pi) : |
| (∃ i≤N, ∃ k<K, Real.pi/4≤|increment (F i k) (F i (k+1))|) ∨ |
| (∃ i<N, ∃ k≤K, Real.pi/4≤|increment (F i k) (F (i+1) k)|) ∨ |
| (∃ i<N, ∃ k<K, Real.pi/4≤|increment (F i k) (F (i+1) (k+1))|) := by |
| by_contra hn |
| push Not at hn |
| have hz := boundary_sum_zero F N K hcenter hseam hn.1 hn.2.1 hn.2.2 |
| have ht := boundary_sum_lift (F N) b K hlift hstep |
| rw [hz,hturn] at ht |
| linarith [Real.pi_pos] |
|
|
| lemma boundary_mesh_lifts (φ e : ℕ → ℝ) (K : ℕ) |
| (hmesh : ∀ k<K, |φ (k+1)-φ k|<Real.pi/32) |
| (herr : ∀ k≤K, |e k|<Real.pi/8) : |
| ∀ k<K, |(2*φ (k+1)+Real.pi+e (k+1))-(2*φ k+Real.pi+e k)|<Real.pi := by |
| intro k hk |
| obtain ⟨hma,hmb⟩ := abs_lt.mp (hmesh k hk) |
| obtain ⟨hea,heb⟩ := abs_lt.mp (herr k (by omega)) |
| obtain ⟨hfa,hfb⟩ := abs_lt.mp (herr (k+1) (by omega)) |
| apply abs_lt.mpr |
| constructor <;> linarith [Real.pi_pos] |
|
|
| end CriticalPercolation.DiskWinding |
| end |
| end |
| end |
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| end OAI |
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