| import Mathlib |
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| namespace OAI |
|
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| section |
| section |
| noncomputable section |
| open Set |
| open scoped Classical |
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| namespace CriticalPercolation.Relay |
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| def width (i : ℕ) : ℝ := max 2 (210-(i : ℝ)/2) |
| def selectedSign (u : ℝ) : ℝ := if 0 ≤ u then -1 else 1 |
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| def domain (J T σ : ℝ) : Set (ℝ×ℝ) := |
| {x | |x.1| ≤ J ∧ |x.2| ≤ J/4+630} ∪ |
| {x | |x.1-T| ≤ J ∧ |x.2-σ*T/4| ≤ J/4+630} ∪ |
| {x | 0 ≤ x.1 ∧ x.1 ≤ T ∧ |x.2-σ*x.1/4| ≤ 630} |
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| lemma width_lower (i : ℕ) : 2 ≤ width i := le_max_left _ _ |
| lemma width_upper (i : ℕ) : width i ≤ 210 := by |
| apply max_le (by norm_num) |
| linarith [Nat.cast_nonneg (α := ℝ) i] |
| lemma width_next (i : ℕ) : width i ≤ width (i+1)+1/2 := by |
| have h1 := width_lower (i+1) |
| have h2 : 210-((i+1 : ℕ) : ℝ)/2 ≤ width (i+1) := le_max_right _ _ |
| unfold width at h1 h2 ⊢ |
| push_cast at h1 h2 ⊢ |
| exact max_le (by linarith) (by linarith) |
| lemma width_last (L : ℕ) (hL : 416 ≤ L) : width L=2 := by |
| apply max_eq_left |
| have : (416 : ℝ) ≤ L := by exact_mod_cast hL |
| linarith |
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| lemma selectedSign_cases (u : ℝ) : selectedSign u=-1 ∨ selectedSign u=1 := by |
| unfold selectedSign |
| split <;> simp |
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| lemma selectedSign_abs (u : ℝ) : |selectedSign u|=1 := by |
| rcases selectedSign_cases u with h|h <;> rw [h] <;> norm_num |
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| lemma move_towards_axis (u b : ℝ) (hu : |u| ≤ b) : |
| |u+selectedSign u| ≤ max 1 (b-1) := by |
| rw [abs_le] at hu ⊢ |
| have h1 : 1 ≤ max 1 (b-1) := le_max_left _ _ |
| have h2 : b-1 ≤ max 1 (b-1) := le_max_right _ _ |
| unfold selectedSign |
| split <;> constructor <;> linarith |
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|
| lemma full_step_height (i : ℕ) (u σ d e ε : ℝ) (hσ : |σ|=1) |
| (hu : |u| ≤ width i+d) (hd₀ : 0 ≤ d) (hd : d+e/4 ≤ 1/4) |
| (hε : |ε| ≤ e/4) : |
| |u+selectedSign u-σ/4+ε| ≤ width (i+1) := by |
| have h1 := width_lower (i+1) |
| have h2 := width_next i |
| have hs : |σ/4|=1/4 := by rw [abs_div,hσ]; norm_num |
| calc |
| _ ≤ |u+selectedSign u|+|σ/4|+|ε| := |
| by linarith [abs_add_le (u+selectedSign u-σ/4) ε, abs_sub (u+selectedSign u) (σ/4)] |
| _ ≤ max 1 (width i+d-1)+1/4+e/4 := by |
| rw [hs] |
| linarith [move_towards_axis u (width i+d) hu] |
| _ ≤ width (i+1) := by |
| rcases le_total 1 (width i+d-1) with hh|hh |
| · rw [max_eq_right hh]; linarith |
| · rw [max_eq_left hh]; linarith |
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| lemma full_step_time (i : ℕ) (y z d e : ℝ) |
| (hy : |y-(i : ℝ)| ≤ (i+1)*e+d) (hz : |z-(y+1)| ≤ e/4) |
| (hde : d+e/4 ≤ e) : |z-((i+1 : ℕ) : ℝ)| ≤ (i+2)*e := by |
| rw [abs_le] at hy hz ⊢ |
| push_cast |
| constructor <;> nlinarith |
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| lemma path_axis_bound (L i : ℕ) (hi : i ≤ L) (y x : ℝ×ℝ) (d e σ : ℝ) |
| (hσ : |σ|=1) (he : 0 ≤ e) |
| (hyt : |y.1-(i : ℝ)| ≤ (i+1)*e+d) |
| (hyz : |y.2-σ*(i : ℝ)/4| ≤ width i+d) |
| (hxt : |x.1-y.1| ≤ 100) (hxz : |x.2-y.2| ≤ 100) : |
| |x.2-σ*x.1/4| ≤ 210+d+100+((L+1)*e+d+100)/4 := by |
| have ht : |x.1-(i : ℝ)| ≤ (L+1)*e+d+100 := by |
| have hc : (i : ℝ) ≤ L := by exact_mod_cast hi |
| calc |
| _ ≤ |x.1-y.1|+|y.1-(i : ℝ)| := abs_sub_le _ _ _ |
| _ ≤ 100+((i+1)*e+d) := add_le_add hxt hyt |
| _ ≤ _ := by nlinarith |
| have hsame : σ*(i : ℝ)/4-σ*x.1/4=σ*((i : ℝ)-x.1)/4 := by ring |
| calc |
| _ ≤ |x.2-y.2|+|y.2-σ*(i : ℝ)/4|+|σ*(i : ℝ)/4-σ*x.1/4| := by |
| linarith [abs_sub_le x.2 (σ*(i : ℝ)/4) (σ*x.1/4), |
| abs_sub_le x.2 y.2 (σ*(i : ℝ)/4)] |
| _ ≤ 100+(width i+d)+((L+1)*e+d+100)/4 := by |
| rw [hsame,abs_div,abs_mul,hσ,one_mul,abs_of_pos (by norm_num : (0:ℝ) < 4),abs_sub_comm (i : ℝ)] |
| exact add_le_add (add_le_add hxz hyz) (div_le_div_of_nonneg_right ht (by norm_num)) |
| _ ≤ _ := by linarith [width_upper i] |
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| lemma band_mem_domain (J T σ : ℝ) (hσ : |σ|=1) |
| (x : ℝ×ℝ) (ht : -J ≤ x.1 ∧ x.1 ≤ T+J) |
| (hz : |x.2-σ*x.1/4| ≤ 630) : x∈domain J T σ := by |
| have ha : |σ|=1 := hσ |
| by_cases h0 : x.1 < 0 |
| · apply Or.inl; apply Or.inl |
| have hx : |x.1| ≤ J := by rw [abs_of_neg h0]; linarith [ht.1] |
| refine ⟨hx,?_⟩ |
| calc |
| |x.2| ≤ |x.2-σ*x.1/4|+|σ*x.1/4| := by simpa using abs_sub_le x.2 (σ*x.1/4) 0 |
| _ ≤ 630+J/4 := by |
| rw [abs_div,abs_mul,ha,one_mul,abs_of_pos (by norm_num : (0:ℝ) < 4)] |
| exact add_le_add hz (div_le_div_of_nonneg_right hx (by norm_num)) |
| _ = _ := add_comm _ _ |
| · by_cases h1 : T < x.1 |
| · apply Or.inl; apply Or.inr |
| have hx : |x.1-T| ≤ J := by rw [abs_of_pos (by linarith)]; linarith [ht.2] |
| refine ⟨hx,?_⟩ |
| calc |
| _ ≤ |x.2-σ*x.1/4|+|σ*x.1/4-σ*T/4| := abs_sub_le _ _ _ |
| _ = |x.2-σ*x.1/4|+|x.1-T|/4 := by |
| rw [show σ*x.1/4-σ*T/4=σ*(x.1-T)/4 by ring,abs_div,abs_mul,ha,one_mul] |
| norm_num |
| _ ≤ 630+J/4 := add_le_add hz (div_le_div_of_nonneg_right hx (by norm_num)) |
| _ = _ := add_comm _ _ |
| · exact Or.inr ⟨by linarith,by linarith,hz⟩ |
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| lemma move_path_contained (J : ℝ) (L i : ℕ) (hi : i ≤ L) (hJ : 26200 < J) |
| (y x : ℝ×ℝ) (d e σ : ℝ) (hσ : |σ|=1) (he : 0 ≤ e) |
| (hsmall : (L+1)*e+d < 1) |
| (hyt : |y.1-(i : ℝ)| ≤ (i+1)*e+d) |
| (hyz : |y.2-σ*(i : ℝ)/4| ≤ width i+d) |
| (hxt : |x.1-y.1| ≤ 100) (hxz : |x.2-y.2| ≤ 100) : |
| x∈domain J (L+1/2) σ := by |
| have hi' : (i : ℝ) ≤ L := by exact_mod_cast hi |
| have hi₀ : (0 : ℝ) ≤ i := Nat.cast_nonneg _ |
| have heL : 0 ≤ (L+1)*e := mul_nonneg (by positivity) he |
| apply band_mem_domain J (L+1/2) σ hσ x |
| · rw [abs_le] at hyt hxt |
| constructor <;> nlinarith |
| · exact (path_axis_bound L i hi y x d e σ hσ he hyt hyz hxt hxz).trans (by linarith) |
|
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| lemma last_crossing_height (L : ℕ) (y x : ℝ×ℝ) (d σ : ℝ) (hσ : |σ|=1) |
| (hy : |y.2-σ*(L : ℝ)/4| ≤ 2+d) (hx : |x.2-y.2| ≤ 100) (hd : d < 1) : |
| |x.2-σ*(L+1/2)/4| < 210 := by |
| have he : σ*(L : ℝ)/4-σ*(L+1/2)/4=-σ/8 := by ring |
| calc |
| _ ≤ |x.2-y.2|+|y.2-σ*(L : ℝ)/4|+|σ*(L : ℝ)/4-σ*(L+1/2)/4| := |
| by linarith [abs_sub_le x.2 (σ*(L : ℝ)/4) (σ*(L+1/2)/4), |
| abs_sub_le x.2 y.2 (σ*(L : ℝ)/4)] |
| _ ≤ 100+(2+d)+1/8 := by |
| rw [he,abs_div,abs_neg,hσ] |
| norm_num |
| linarith |
| _ < _ := by linarith |
|
|
| end CriticalPercolation.Relay |
| namespace SimpleGraph.Walk |
| open scoped _root_.SimpleGraph _root_.SimpleGraph.Walk |
| variable {V : Type*} {H : SimpleGraph V} |
|
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| lemma exists_first_crossing (f : V → ℝ) (ρ T : ℝ) |
| (hf : ∀ ⦃x y⦄, H.Adj x y → |f x-f y| ≤ ρ) |
| {a b : V} (w : H.Walk a b) (ha : f a < T) (hb : T ≤ f b) : |
| ∃ x∈w.support, T ≤ f x ∧ f x ≤ T+ρ := by |
| induction w with |
| | nil => exact (not_le_of_gt ha hb).elim |
| | @cons a c b hac w ih => |
| by_cases hc : T ≤ f c |
| · refine ⟨c,?_,hc,?_⟩ |
| · simp only [_root_.SimpleGraph.Walk.support_cons,List.mem_cons] |
| exact Or.inr w.start_mem_support |
| · have hh := hf hac |
| rw [abs_le] at hh |
| linarith |
| · obtain ⟨x,hx,h1,h2⟩ := ih (lt_of_not_ge hc) hb |
| exact ⟨x,by simp only [_root_.SimpleGraph.Walk.support_cons,List.mem_cons]; exact Or.inr hx,h1,h2⟩ |
| end SimpleGraph.Walk |
| end |
| end |
| end |
|
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| end OAI |
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