openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /EuclideanEllipse.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open scoped Topology | |
| namespace CriticalPercolation.EuclideanEllipse | |
| def diagonal (a b : ℝ) (z : ℂ) : ℂ := ⟨a*z.re,b*z.im⟩ | |
| lemma diagonal_add (a b : ℝ) (z w : ℂ) : | |
| diagonal a b (z+w)=diagonal a b z+diagonal a b w := by | |
| apply Complex.ext <;> simp [diagonal,mul_add] | |
| lemma diagonal_sub (a b : ℝ) (z w : ℂ) : | |
| diagonal a b (z-w)=diagonal a b z-diagonal a b w := by | |
| apply Complex.ext <;> simp [diagonal,mul_sub] | |
| lemma diagonal_comp (a b c d : ℝ) (z : ℂ) : | |
| diagonal a b (diagonal c d z)=diagonal (a*c) (b*d) z := by | |
| apply Complex.ext <;> simp [diagonal,mul_assoc] | |
| lemma diagonal_one (z : ℂ) : diagonal 1 1 z=z := by apply Complex.ext <;> simp [diagonal] | |
| lemma diagonal_neg (a b : ℝ) (z : ℂ) : diagonal a b (-z)= -diagonal a b z := by | |
| apply Complex.ext <;> simp [diagonal] | |
| lemma diagonal_norm_le (a b M : ℝ) (hM : 0≤M) (ha : |a|≤M) (hb : |b|≤M) (z : ℂ) : | |
| ‖diagonal a b z‖≤M*‖z‖ := by | |
| have ha2 : a^2≤M^2 := by simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hM).mpr ha | |
| have hb2 : b^2≤M^2 := by simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hM).mpr hb | |
| have hsq : ‖diagonal a b z‖^2≤(M*‖z‖)^2 := by | |
| rw [Complex.sq_norm,Complex.normSq_apply,mul_pow,Complex.sq_norm,Complex.normSq_apply] | |
| dsimp [diagonal] | |
| nlinarith [mul_le_mul_of_nonneg_right ha2 (sq_nonneg z.re), | |
| mul_le_mul_of_nonneg_right hb2 (sq_nonneg z.im)] | |
| exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hM (norm_nonneg _))).mp hsq | |
| def normalize (R W : ℝ) (c : Circle) (z : ℂ) : ℂ := | |
| (c : ℂ)*diagonal (1/R) (1/W) ((c : ℂ)⁻¹*z) | |
| def expand (R W : ℝ) (c : Circle) (z : ℂ) : ℂ := | |
| (c : ℂ)*diagonal R W ((c : ℂ)⁻¹*z) | |
| lemma normalize_sub (R W : ℝ) (c : Circle) (z w : ℂ) : | |
| normalize R W c (z-w)=normalize R W c z-normalize R W c w := by | |
| simp only [normalize,mul_sub,diagonal_sub] | |
| lemma expand_normalize (R W : ℝ) (hR : R≠0) (hW : W≠0) (c : Circle) (z : ℂ) : | |
| expand R W c (normalize R W c z)=z := by | |
| simp only [normalize,expand,←mul_assoc,inv_mul_cancel₀ c.coe_ne_zero,one_mul, | |
| diagonal_comp,mul_one_div_cancel hR,mul_one_div_cancel hW,diagonal_one] | |
| rw [mul_assoc,mul_inv_cancel_left₀ c.coe_ne_zero] | |
| lemma normalize_expand (R W : ℝ) (hR : R≠0) (hW : W≠0) (c : Circle) (z : ℂ) : | |
| normalize R W c (expand R W c z)=z := by | |
| simp only [normalize,expand,←mul_assoc,inv_mul_cancel₀ c.coe_ne_zero,one_mul, | |
| diagonal_comp,one_div_mul_cancel hR,one_div_mul_cancel hW,diagonal_one] | |
| rw [mul_assoc,mul_inv_cancel_left₀ c.coe_ne_zero] | |
| lemma normalize_norm_le (R W : ℝ) (hW : 0<W) (hWR : W≤R) (c : Circle) (z : ℂ) : | |
| ‖normalize R W c z‖≤‖z‖/W := by | |
| have hR : 0<R := hW.trans_le hWR | |
| have h1 : |1/R|≤1/W := by rw [abs_of_pos (one_div_pos.mpr hR)]; exact one_div_le_one_div_of_le hW hWR | |
| have h2 : |1/W|≤1/W := (abs_of_pos (one_div_pos.mpr hW)).le | |
| have h := diagonal_norm_le (1/R) (1/W) (1/W) (by positivity) h1 h2 ((c : ℂ)⁻¹*z) | |
| simpa [normalize,norm_mul,Circle.norm_coe,div_eq_mul_inv,mul_comm] | |
| using h | |
| lemma expand_norm_le (R W : ℝ) (hW : 0≤W) (hWR : W≤R) (c : Circle) (z : ℂ) : | |
| ‖expand R W c z‖≤R*‖z‖ := by | |
| have hR := hW.trans hWR | |
| have h := diagonal_norm_le R W R hR (by simp [abs_of_nonneg hR]) | |
| (by simpa [abs_of_nonneg hW] using hWR) ((c : ℂ)⁻¹*z) | |
| simpa only [expand,norm_mul,Circle.norm_coe,norm_inv,inv_one,one_mul] using h | |
| lemma normalize_center (W : ℝ) (c : Circle) (z : ℂ) : | |
| normalize W W c z=(1/W : ℝ)•z := by | |
| have he : ∀ w : ℂ, diagonal (1/W) (1/W) w=(1/W : ℝ)•w := by intro w; apply Complex.ext <;> simp [diagonal] | |
| rw [normalize,he,mul_smul_comm,←mul_assoc,mul_inv_cancel₀ c.coe_ne_zero,one_mul] | |
| lemma normalize_neg (R W : ℝ) (c : Circle) (z : ℂ) : | |
| normalize R W (-c) z=normalize R W c z := by | |
| simp only [normalize,Circle.coe_neg,inv_neg,neg_mul,diagonal_neg,mul_neg,neg_neg] | |
| lemma normalize_components (R W : ℝ) (c : Circle) (z : ℂ) : | |
| (((c : ℂ)⁻¹*normalize R W c z).re=(1/R)*((c : ℂ)⁻¹*z).re) ∧ | |
| (((c : ℂ)⁻¹*normalize R W c z).im=(1/W)*((c : ℂ)⁻¹*z).im) := by | |
| simp [normalize,diagonal] | |
| lemma normalize_origin (R W : ℝ) (c : Circle) : normalize R W c 0=0 := by | |
| simp [normalize,diagonal,Complex.ext_iff] | |
| lemma normalize_real_linear (R W : ℝ) (c : Circle) (z w : ℂ) (a b : ℝ) : | |
| normalize R W c (a•z+b•w)=a•normalize R W c z+b•normalize R W c w := by | |
| have hd : ∀ q : ℂ, ∀ t : ℝ, diagonal (1/R) (1/W) (t•q)=t•diagonal (1/R) (1/W) q := by | |
| intro q t | |
| apply Complex.ext <;> simp [diagonal,Complex.mul_re,Complex.mul_im] <;> ring | |
| simp only [normalize,mul_add,mul_smul_comm,diagonal_add,hd] | |
| end CriticalPercolation.EuclideanEllipse | |
| end | |
| end | |
| end | |
| end OAI | |