90 lines
2.8 KiB
Text
90 lines
2.8 KiB
Text
-- actually it's analytical trigonometry
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import Mathlib.Data.Real.Basic
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import Mathlib.Tactic.LibrarySearch
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axiom sin : Real → Real
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axiom csc : Real → Real
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axiom sin_eq_reciprocal_of_csc : ∀x, sin x = 1 / (csc x)
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axiom csc_eq_reciprocal_of_sin : ∀x, csc x = 1 / (sin x)
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axiom cos : Real → Real
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axiom sec : Real → Real
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axiom cos_eq_reciprocal_of_sec : ∀x, cos x = (sec x)⁻¹
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axiom sec_eq_reciprocal_of_cos : ∀x, sec x = (cos x)⁻¹
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@[simp]
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noncomputable def tan (x : Real) : Real := sin x / cos x
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theorem tan_def : ∀x, tan x = sin x / cos x := by simp
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@[simp]
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noncomputable def cot (x : Real) : Real := cos x / sin x
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theorem tan_eq_sin_div_cos : ∀x, tan x = sin x / cos x := by simp
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theorem cot_eq_cos_div_sin : ∀x, cot x = cos x / sin x := by simp
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theorem tan_eq_reciprocal_of_cot : ∀x, tan x = 1 / (cot x) := by simp
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theorem cot_eq_reciprocal_of_tan : ∀x, cot x = 1 / (tan x) := by simp
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noncomputable def sin2 (x : Real) : Real := sin x * sin x
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noncomputable def cos2 (x : Real) : Real := cos x * cos x
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noncomputable def tan2 (x : Real) : Real := tan x * tan x
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theorem tan2_def : ∀x, tan2 x = sin2 x / cos2 x := by
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intro x
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simp [tan2, sin2, cos2]
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exact div_mul_div_comm (sin x) (cos x) (sin x) (cos x)
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noncomputable def csc2 (x : Real) : Real := csc x * csc x
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noncomputable def sec2 (x : Real) : Real := sec x * sec x
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theorem sec2_eq_reciprocal_of_cos2 : ∀x, sec2 x = (cos2 x)⁻¹ := by
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intro x
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rw [sec2, cos2]
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simp
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rw [sec_eq_reciprocal_of_cos]
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noncomputable def cot2 (x : Real) : Real := cot x * cot x
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noncomputable def sin3 (x : Real) : Real := sin2 x * sin x
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noncomputable def cos3 (x : Real) : Real := cos2 x * cos x
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noncomputable def tan3 (x : Real) : Real := tan2 x * tan x
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noncomputable def csc3 (x : Real) : Real := csc2 x * csc x
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noncomputable def sec3 (x : Real) : Real := sec2 x * sec x
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noncomputable def cot3 (x : Real) : Real := cot2 x * cot x
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@[simp]
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axiom pyth_id : ∀x, sin2 x + cos2 x = 1
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theorem sin_squared_eq {x : Real} : sin2 x = 1 - cos2 x := by
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rw [← add_left_inj (cos2 x)]
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rw [sub_add_cancel]
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exact pyth_id x
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theorem cos_squared_eq {x : Real} : cos2 x = 1 - sin2 x := by
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rw [← add_left_inj (sin2 x)]
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rw [sub_add_cancel]
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rw [add_comm]
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exact pyth_id x
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theorem bleh {a b c : Real} (d : Real) : a + b = c → a / d + b / d = c / d := by
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intro h
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rw [← add_div, h]
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def tan_pyth_id : ∀x, cos2 x ≠ 0 → tan2 x + 1 = sec2 x := by
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intro x
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intro hnz
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have h := pyth_id x
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have h : sin2 x / cos2 x + cos2 x / cos2 x = 1 / cos2 x :=
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bleh (cos2 x) h
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rw [← tan2_def] at h
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simp at h
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rw [← sec2_eq_reciprocal_of_cos2, div_self] at h
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exact h
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exact hnz
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example (x : Real) : 2 * (1 / (csc x)) = 2 * (sin x) := by
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rw [sin_eq_reciprocal_of_csc]
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--todo evens and odds and the sin/cos (a ± b) ones
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