| import OAI.Geometry.EinsteinFour.FiniteExceptionalMeanValue |
| import Mathlib.Topology.Order.IntermediateValue |
|
|
| namespace OAI |
|
|
| /-! Angular distance starting at zero needs a derivative bound only between its extreme values. -/ |
| namespace PositiveEinsteinFour.Geometry |
| open Set |
|
|
| theorem bounded_angle_le_of_deriv_le (f : ℝ → ℝ) (a b B C : ℝ) (E : Finset ℝ) |
| (hab : a ≤ b) (hC : 0 ≤ C) (hf : ContinuousOn f (Icc a b)) (hfa : f a = 0) |
| (hpos : ∀ t ∈ Icc a b, 0 ≤ f t) (hB : f b ≤ B) |
| (hd : ∀ t ∈ Ioo a b, t ∉ E → 0 < f t → f t < B → DifferentiableAt ℝ f t) |
| (hb : ∀ t ∈ Ioo a b, t ∉ E → 0 < f t → f t < B → deriv f t ≤ C) : |
| f b ≤ C*(b-a) := by |
| by_contra h |
| have hstrict : C*(b-a) < f b := lt_of_not_ge h |
| obtain ⟨y,hylo,hyhi⟩ := exists_between hstrict |
| have hy0 : 0 < y := lt_of_le_of_lt (mul_nonneg hC (sub_nonneg.mpr hab)) hylo |
| have hyB : y < B := hyhi.trans_le hB |
| obtain ⟨t,ht,hty⟩ := intermediate_value_Icc hab hf ⟨by rw [hfa]; exact hy0.le,hyhi.le⟩ |
| let L : Set ℝ := Icc a b ∩ f ⁻¹' {y} |
| have hLclosed : IsClosed L := hf.preimage_isClosed_of_isClosed isClosed_Icc isClosed_singleton |
| have hLcompact : IsCompact L := isCompact_Icc.of_isClosed_subset hLclosed inter_subset_left |
| obtain ⟨z,hz,hmin⟩ := hLcompact.exists_isMinOn ⟨t,ht,hty⟩ continuousOn_id |
| have haz : a ≤ z := hz.1.1 |
| have hzb : z ≤ b := hz.1.2 |
| have hfz : f z = y := hz.2 |
| have hclosed : IsClosed (Icc a z ∩ f ⁻¹' {0}) := |
| (hf.mono (Icc_subset_Icc_right hzb)).preimage_isClosed_of_isClosed isClosed_Icc isClosed_singleton |
| have hcompact : IsCompact (Icc a z ∩ f ⁻¹' {0}) := |
| isCompact_Icc.of_isClosed_subset hclosed inter_subset_left |
| obtain ⟨x,hx,hmax⟩ := hcompact.exists_isMaxOn ⟨a,⟨le_rfl,haz⟩,hfa⟩ continuousOn_id |
| have hax : a ≤ x := hx.1.1 |
| have hxz : x ≤ z := hx.1.2 |
| have hfx : f x = 0 := hx.2 |
| have hreg (r : ℝ) (hr : r ∈ Ioo x z) : r ∈ Ioo a b ∧ 0 < f r ∧ f r < B := by |
| have hra : a < r := hax.trans_lt hr.1 |
| have hrb : r < b := hr.2.trans_le hzb |
| have hfr : 0 < f r := by |
| apply lt_of_le_of_ne (hpos r ⟨hra.le,hrb.le⟩) |
| intro heq |
| have hh : r ≤ x := hmax ⟨⟨hra.le,hr.2.le⟩,heq.symm⟩ |
| exact (not_le_of_gt hr.1) hh |
| have hfry : f r < y := by |
| by_contra hn |
| obtain ⟨k,hk,hky⟩ := intermediate_value_Icc hra.le |
| (hf.mono (Icc_subset_Icc_right hrb.le)) ⟨by rw [hfa]; exact hy0.le,le_of_not_gt hn⟩ |
| have hzk : z ≤ k := hmin ⟨⟨hk.1,hk.2.trans hrb.le⟩,hky⟩ |
| exact (not_le_of_gt hr.2) (hzk.trans hk.2) |
| exact ⟨⟨hra,hrb⟩,hfr,hfry.trans hyB⟩ |
| have hbound := image_sub_le_mul_sub_of_deriv_le_off_finite E hxz |
| (hf.mono (Icc_subset_Icc hax hzb)) |
| (fun r hr he => hd r (hreg r hr).1 he (hreg r hr).2.1 (hreg r hr).2.2) |
| (fun r hr he => hb r (hreg r hr).1 he (hreg r hr).2.1 (hreg r hr).2.2) |
| rw [hfz,hfx,sub_zero] at hbound |
| have hlength : C*(z-x) ≤ C*(b-a) := mul_le_mul_of_nonneg_left (by linarith) hC |
| exact (not_le_of_gt hylo) (hbound.trans hlength) |
|
|
| end PositiveEinsteinFour.Geometry |
|
|
| end OAI |
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