Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

[Merged by Bors] - feat(Trigonometric): add lemmas about cos x = -1 ↔ _ etc #9878

Closed
wants to merge 1 commit into from
+27 −2
Closed

[Merged by Bors] - feat(Trigonometric): add lemmas about cos x = -1 ↔ _ etc #9878

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
29 changes: 27 additions & 2 deletions Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -98,6 +98,22 @@ theorem sin_eq_sin_iff {x y : ℂ} :
refine' exists_congr fun k => or_congr _ _ <;> refine' Eq.congr rfl _ <;> field_simp <;> ring
#align complex.sin_eq_sin_iff Complex.sin_eq_sin_iff

theorem cos_eq_one_iff {x : ℂ} : cos x = 1 ↔ ∃ k : ℤ, k * (2 * π) = x := by
Copy link
Contributor

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Why don't you write it as

Suggested change
theorem cos_eq_one_iff {x : ℂ} : cos x = 1 ↔ ∃ k : ℤ, k * (2 * π) = x := by
theorem cos_eq_one_iff {x : ℂ} : cos x = 1 ↔ ∃ k : ℤ, x = k * (2 * π) := by

with the x on the left, like in the other statements of the file? Same thing below.

Copy link
Member Author

@urkud urkud Jan 23, 2024 •
edited
Loading

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

It aligns with Real.cos_eq_one_iff. Probably, we should have a convention about this. I asked on Zulip.

Copy link
Collaborator

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

I agree with the current spelling that Yury uses, but an even better one would use AddCommGroup.ModEq

rw [← cos_zero, eq_comm, cos_eq_cos_iff]
simp [mul_assoc, mul_left_comm, eq_comm]

theorem cos_eq_neg_one_iff {x : ℂ} : cos x = -1 ↔ ∃ k : ℤ, π + k * (2 * π) = x := by
rw [← neg_eq_iff_eq_neg, ← cos_sub_pi, cos_eq_one_iff]
simp only [eq_sub_iff_add_eq']

theorem sin_eq_one_iff {x : ℂ} : sin x = 1 ↔ ∃ k : ℤ, π / 2 + k * (2 * π) = x := by
rw [← cos_sub_pi_div_two, cos_eq_one_iff]
simp only [eq_sub_iff_add_eq']

theorem sin_eq_neg_one_iff {x : ℂ} : sin x = -1 ↔ ∃ k : ℤ, -(π / 2) + k * (2 * π) = x := by
rw [← neg_eq_iff_eq_neg, ← cos_add_pi_div_two, cos_eq_one_iff]
simp only [← sub_eq_neg_add, sub_eq_iff_eq_add]

theorem tan_add {x y : ℂ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) ∨
(∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2) :
Expand Down Expand Up @@ -208,8 +224,8 @@ theorem cos_eq_zero_iff {θ : ℝ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1
mod_cast @Complex.cos_eq_zero_iff θ
#align real.cos_eq_zero_iff Real.cos_eq_zero_iff

theorem cos_ne_zero_iff {θ : ℝ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 := by
rw [← not_exists, not_iff_not, cos_eq_zero_iff]
theorem cos_ne_zero_iff {θ : ℝ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 :=
mod_cast @Complex.cos_ne_zero_iff θ
#align real.cos_ne_zero_iff Real.cos_ne_zero_iff

theorem cos_eq_cos_iff {x y : ℝ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x :=
Expand All @@ -221,6 +237,15 @@ theorem sin_eq_sin_iff {x y : ℝ} :
mod_cast @Complex.sin_eq_sin_iff x y
#align real.sin_eq_sin_iff Real.sin_eq_sin_iff

theorem cos_eq_neg_one_iff {x : ℝ} : cos x = -1 ↔ ∃ k : ℤ, π + k * (2 * π) = x :=
mod_cast @Complex.cos_eq_neg_one_iff x

theorem sin_eq_one_iff {x : ℝ} : sin x = 1 ↔ ∃ k : ℤ, π / 2 + k * (2 * π) = x :=
mod_cast @Complex.sin_eq_one_iff x

theorem sin_eq_neg_one_iff {x : ℝ} : sin x = -1 ↔ ∃ k : ℤ, -(π / 2) + k * (2 * π) = x :=
mod_cast @Complex.sin_eq_neg_one_iff x

theorem lt_sin_mul {x : ℝ} (hx : 0 < x) (hx' : x < 1) : x < sin (π / 2 * x) := by
simpa [mul_comm x] using
strictConcaveOn_sin_Icc.2 ⟨le_rfl, pi_pos.le⟩ ⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩
Expand Down
Loading