| import OAI.Geometry.Relativity.CKS.ConeSupport |
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| namespace OAI |
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| noncomputable section |
| namespace CKSBending |
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
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| def phi (x : ℝ) : ℝ := |
| (1 - Real.smoothTransition ((x-6)/6)) + Real.smoothTransition ((x-6)/6) * (x⁻¹)^3 |
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| def xi (x : ℝ) : ℝ := 1 - Real.smoothTransition (x-1) |
| def zeta (x : ℝ) : ℝ := Real.smoothTransition (x-4) |
| def beta (x : ℝ) : ℝ := 1 - Real.smoothTransition (2*x-7) |
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| def bendingV (R r : ℝ) : ℝ := r * phi (r/R) * xi (r/R^2) |
| def bendingZeta (R r : ℝ) : ℝ := zeta (r/R) |
| def bendingBeta (R r : ℝ) : ℝ := beta (r/R) |
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| lemma phi_one {x : ℝ} (hx : x ≤ 6) : phi x = 1 := by |
| simp [phi,Real.smoothTransition.zero_of_nonpos (show (x-6)/6 ≤ 0 by linarith)] |
| lemma phi_tail {x : ℝ} (hx : 12 ≤ x) : phi x = (x⁻¹)^3 := by |
| simp [phi,Real.smoothTransition.one_of_one_le (show 1 ≤ (x-6)/6 by linarith)] |
| lemma phi_pos {x : ℝ} (hx : 0 < x) : 0 < phi x := by |
| have h0 := Real.smoothTransition.nonneg ((x-6)/6) |
| have h1 := Real.smoothTransition.le_one ((x-6)/6) |
| have hp : 0 < (x⁻¹)^3 := pow_pos (inv_pos.mpr hx) _ |
| unfold phi |
| by_cases he : Real.smoothTransition ((x-6)/6) = 0 |
| · simp [he] |
| · exact add_pos_of_nonneg_of_pos (sub_nonneg.mpr h1) (mul_pos (lt_of_le_of_ne h0 (Ne.symm he)) hp) |
| lemma phi_le_one {x : ℝ} (hx : 0 < x) : phi x ≤ 1 := by |
| by_cases h6 : x ≤ 6 |
| · rw [phi_one h6] |
| · have h1x : 1 ≤ x := by linarith |
| have hi : x⁻¹ ≤ 1 := (inv_le_one₀ hx).mpr h1x |
| have hp : (x⁻¹)^3 ≤ 1 := by simpa using pow_le_pow_left₀ (inv_nonneg.mpr hx.le) hi 3 |
| have h0 := Real.smoothTransition.nonneg ((x-6)/6) |
| have hprod := mul_le_mul_of_nonneg_left hp h0 |
| unfold phi |
| linarith |
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| lemma phi_smooth : ContDiffOn ℝ ∞ phi (Ioi 0) := by |
| unfold phi |
| have hc : ContDiff ℝ ∞ (fun x : ℝ => Real.smoothTransition ((x-6)/6)) := by fun_prop |
| exact (contDiffOn_const.sub hc.contDiffOn).add |
| (hc.contDiffOn.mul ((contDiffOn_id.inv (fun x hx => ne_of_gt hx)).pow 3)) |
| lemma xi_smooth : ContDiff ℝ ∞ xi := by unfold xi; fun_prop |
| lemma zeta_smooth : ContDiff ℝ ∞ zeta := by unfold zeta; fun_prop |
| lemma beta_smooth : ContDiff ℝ ∞ beta := by unfold beta; fun_prop |
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| lemma xi_nonneg (x : ℝ) : 0 ≤ xi x := sub_nonneg.mpr (Real.smoothTransition.le_one _) |
| lemma xi_le_one (x : ℝ) : xi x ≤ 1 := by unfold xi; linarith [Real.smoothTransition.nonneg (x-1)] |
| lemma xi_one {x : ℝ} (hx : x ≤ 1) : xi x = 1 := by |
| simp [xi,Real.smoothTransition.zero_of_nonpos (show x-1 ≤ 0 by linarith)] |
| lemma xi_zero {x : ℝ} (hx : 2 ≤ x) : xi x = 0 := by |
| simp [xi,Real.smoothTransition.one_of_one_le (show 1 ≤ x-1 by linarith)] |
| lemma zeta_nonneg (x : ℝ) : 0 ≤ zeta x := Real.smoothTransition.nonneg _ |
| lemma zeta_le_one (x : ℝ) : zeta x ≤ 1 := Real.smoothTransition.le_one _ |
| lemma zeta_zero {x : ℝ} (hx : x ≤ 4) : zeta x = 0 := |
| Real.smoothTransition.zero_of_nonpos (by linarith) |
| lemma zeta_one {x : ℝ} (hx : 5 ≤ x) : zeta x = 1 := |
| Real.smoothTransition.one_of_one_le (by linarith) |
| lemma beta_nonneg (x : ℝ) : 0 ≤ beta x := sub_nonneg.mpr (Real.smoothTransition.le_one _) |
| lemma beta_le_one (x : ℝ) : beta x ≤ 1 := by unfold beta; linarith [Real.smoothTransition.nonneg (2*x-7)] |
| lemma beta_one {x : ℝ} (hx : x ≤ 3) : beta x = 1 := by |
| simp [beta,Real.smoothTransition.zero_of_nonpos (show 2*x-7 ≤ 0 by linarith)] |
| lemma beta_one_near {x : ℝ} (hx : x ≤ 7/2) : beta x = 1 := by |
| simp [beta,Real.smoothTransition.zero_of_nonpos (show 2*x-7 ≤ 0 by linarith)] |
| lemma beta_zero {x : ℝ} (hx : 4 ≤ x) : beta x = 0 := by |
| simp [beta,Real.smoothTransition.one_of_one_le (show 1 ≤ 2*x-7 by linarith)] |
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| lemma bendingV_smooth {R : ℝ} (hR : 0 < R) : |
| ContDiffOn ℝ ∞ (bendingV R) (Ioi 0) := by |
| unfold bendingV |
| have hc : ContDiffOn ℝ ∞ (fun r : ℝ => phi (r/R)) (Ioi 0) := |
| phi_smooth.comp (contDiffOn_id.div_const R) (fun r hr => div_pos hr hR) |
| exact (contDiffOn_id.mul hc).mul |
| ((xi_smooth.comp (contDiff_id.div_const (R^2))).contDiffOn) |
| lemma bendingZeta_smooth (R : ℝ) : ContDiff ℝ ∞ (bendingZeta R) := |
| zeta_smooth.comp (contDiff_id.div_const R) |
| lemma bendingBeta_smooth (R : ℝ) : ContDiff ℝ ∞ (bendingBeta R) := |
| beta_smooth.comp (contDiff_id.div_const R) |
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| lemma bendingV_nonneg {R r : ℝ} (hR : 0 < R) (hr : 0 < r) : 0 ≤ bendingV R r := |
| mul_nonneg (mul_nonneg hr.le (phi_pos (div_pos hr hR)).le) (xi_nonneg _) |
| lemma bendingV_le_radius {R r : ℝ} (hR : 0 < R) (hr : 0 < r) : bendingV R r ≤ r := by |
| calc |
| _ ≤ r * 1 * 1 := mul_le_mul (mul_le_mul_of_nonneg_left (phi_le_one (div_pos hr hR)) hr.le) |
| (xi_le_one _) (xi_nonneg _) (by positivity) |
| _ = r := by ring |
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| lemma bendingV_initial {R r : ℝ} (hR : 12 ≤ R) (hr : r ≤ 6*R) : bendingV R r = r := by |
| have hRp : 0 < R := by linarith |
| have hp : r/R ≤ 6 := (div_le_iff₀ hRp).mpr hr |
| have hx : r/R^2 ≤ 1 := (div_le_iff₀ (sq_pos_of_pos hRp)).mpr (by nlinarith) |
| simp [bendingV,phi_one hp,xi_one hx] |
| lemma bendingV_tail {R r : ℝ} (hR : 0 < R) (hr : 2*R^2 ≤ r) : bendingV R r = 0 := by |
| have hx : 2 ≤ r/R^2 := (le_div_iff₀ (sq_pos_of_pos hR)).mpr hr |
| simp [bendingV,xi_zero hx] |
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| lemma beta_support {R r : ℝ} (hR : 12 ≤ R) (hb : bendingBeta R r ≠ 0) : bendingV R r = r := by |
| apply bendingV_initial hR |
| have hRp : 0 < R := by linarith |
| have hh : r/R < 4 := by |
| by_contra h |
| exact hb (beta_zero (le_of_not_gt h)) |
| have hx := (div_lt_iff₀ hRp).mp hh |
| linarith |
| lemma zeta_support {R r : ℝ} (hR : 12 ≤ R) (hz : bendingZeta R r ≠ 1) : bendingV R r = r := by |
| apply bendingV_initial hR |
| have hRp : 0 < R := by linarith |
| have hh : r/R < 5 := by |
| by_contra h |
| exact hz (zeta_one (le_of_not_gt h)) |
| have hx := (div_lt_iff₀ hRp).mp hh |
| linarith |
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| end |
| end CKSBending |
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| end |
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| end OAI |
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