state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← h1, sinh_add x (2 * x)] | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
have h1 : x + 2 * x = 3 * x := by ring
| Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ sinh x * cosh (2 * x) + cosh x * sinh (2 * x) = 4 * sinh x ^ 3 + 3 * sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [cosh_two_mul, sinh_two_mul] | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
| Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ sinh x * (cosh x ^ 2 + sinh x ^ 2) + cosh x * (2 * sinh x * cosh x) = 4 * sinh x ^ 3 + 3 * sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
| Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by | Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2
⊢ sinh x * (cosh x ^ 2 + sinh x ^ 2) + cosh x * (2 * sinh x * cosh x) = 4 * sinh x ^ 3 + 3 * sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [h2, cosh_sq] | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
| Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2
⊢ sinh x * (sinh x ^ 2 + 1 + sinh x ^ 2) + 2 * sinh x * (sinh x ^ 2 + 1) = 4 * sinh x ^ 3 + 3 * sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
rw [h2, cosh_sq]
| Mathlib.Data.Complex.Exponential.793_0.1bixbwhfBeJKySp | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin 0 = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin] | @[simp]
theorem sin_zero : sin 0 = 0 := by | Mathlib.Data.Complex.Exponential.802_0.1bixbwhfBeJKySp | @[simp]
theorem sin_zero : sin 0 = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin (-x) = -sin x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sin, sub_eq_add_neg, exp_neg, (neg_div _ _).symm, add_mul] | @[simp]
theorem sin_neg : sin (-x) = -sin x := by
| Mathlib.Data.Complex.Exponential.806_0.1bixbwhfBeJKySp | @[simp]
theorem sin_neg : sin (-x) = -sin x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh (x * I) = sin x * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, ← mul_assoc, two_sin, mul_assoc, I_mul_I,
mul_neg_one, neg_sub, neg_mul_eq_neg_mul] | theorem sinh_mul_I : sinh (x * I) = sin x * I := by
| Mathlib.Data.Complex.Exponential.819_0.1bixbwhfBeJKySp | theorem sinh_mul_I : sinh (x * I) = sin x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cosh (x * I) = cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, two_cos, neg_mul_eq_neg_mul] | theorem cosh_mul_I : cosh (x * I) = cos x := by
| Mathlib.Data.Complex.Exponential.825_0.1bixbwhfBeJKySp | theorem cosh_mul_I : cosh (x * I) = cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ tanh (x * I) = tan x * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [tanh_eq_sinh_div_cosh, cosh_mul_I, sinh_mul_I, mul_div_right_comm, tan] | theorem tanh_mul_I : tanh (x * I) = tan x * I := by
| Mathlib.Data.Complex.Exponential.830_0.1bixbwhfBeJKySp | theorem tanh_mul_I : tanh (x * I) = tan x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos (x * I) = cosh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← cosh_mul_I] | theorem cos_mul_I : cos (x * I) = cosh x := by | Mathlib.Data.Complex.Exponential.835_0.1bixbwhfBeJKySp | theorem cos_mul_I : cos (x * I) = cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cosh (x * I * I) = cosh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring_nf | theorem cos_mul_I : cos (x * I) = cosh x := by rw [← cosh_mul_I]; | Mathlib.Data.Complex.Exponential.835_0.1bixbwhfBeJKySp | theorem cos_mul_I : cos (x * I) = cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cosh (x * I ^ 2) = cosh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | theorem cos_mul_I : cos (x * I) = cosh x := by rw [← cosh_mul_I]; ring_nf; | Mathlib.Data.Complex.Exponential.835_0.1bixbwhfBeJKySp | theorem cos_mul_I : cos (x * I) = cosh x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin (x * I) = sinh x * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h : I * sin (x * I) = -sinh x := by
rw [mul_comm, ← sinh_mul_I]
ring_nf
simp | theorem sin_mul_I : sin (x * I) = sinh x * I := by
| Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ I * sin (x * I) = -sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [mul_comm, ← sinh_mul_I] | theorem sin_mul_I : sin (x * I) = sinh x * I := by
have h : I * sin (x * I) = -sinh x := by
| Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh (x * I * I) = -sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring_nf | theorem sin_mul_I : sin (x * I) = sinh x * I := by
have h : I * sin (x * I) = -sinh x := by
rw [mul_comm, ← sinh_mul_I]
| Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sinh (x * I ^ 2) = -sinh x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | theorem sin_mul_I : sin (x * I) = sinh x * I := by
have h : I * sin (x * I) = -sinh x := by
rw [mul_comm, ← sinh_mul_I]
ring_nf
| Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
h : I * sin (x * I) = -sinh x
⊢ sin (x * I) = sinh x * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← neg_neg (sinh x), ← h] | theorem sin_mul_I : sin (x * I) = sinh x * I := by
have h : I * sin (x * I) = -sinh x := by
rw [mul_comm, ← sinh_mul_I]
ring_nf
simp
| Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
h : I * sin (x * I) = -sinh x
⊢ sin (x * I) = -(I * sin (x * I)) * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | apply Complex.ext | theorem sin_mul_I : sin (x * I) = sinh x * I := by
have h : I * sin (x * I) = -sinh x := by
rw [mul_comm, ← sinh_mul_I]
ring_nf
simp
rw [← neg_neg (sinh x), ← h]
| Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
case a
x y : ℂ
h : I * sin (x * I) = -sinh x
⊢ (sin (x * I)).re = (-(I * sin (x * I)) * I).re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | theorem sin_mul_I : sin (x * I) = sinh x * I := by
have h : I * sin (x * I) = -sinh x := by
rw [mul_comm, ← sinh_mul_I]
ring_nf
simp
rw [← neg_neg (sinh x), ← h]
apply Complex.ext <;> | Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
case a
x y : ℂ
h : I * sin (x * I) = -sinh x
⊢ (sin (x * I)).im = (-(I * sin (x * I)) * I).im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | theorem sin_mul_I : sin (x * I) = sinh x * I := by
have h : I * sin (x * I) = -sinh x := by
rw [mul_comm, ← sinh_mul_I]
ring_nf
simp
rw [← neg_neg (sinh x), ← h]
apply Complex.ext <;> | Mathlib.Data.Complex.Exponential.839_0.1bixbwhfBeJKySp | theorem sin_mul_I : sin (x * I) = sinh x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ tan (x * I) = tanh x * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [tan, sin_mul_I, cos_mul_I, mul_div_right_comm, tanh_eq_sinh_div_cosh] | theorem tan_mul_I : tan (x * I) = tanh x * I := by
| Mathlib.Data.Complex.Exponential.849_0.1bixbwhfBeJKySp | theorem tan_mul_I : tan (x * I) = tanh x * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin (x + y) = sin x * cos y + cos x * sin y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I, add_mul, add_mul, mul_right_comm, ← sinh_mul_I,
mul_assoc, ← sinh_mul_I, ← cosh_mul_I, ← cosh_mul_I, sinh_add] | theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y := by
| Mathlib.Data.Complex.Exponential.854_0.1bixbwhfBeJKySp | theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos 0 = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos] | @[simp]
theorem cos_zero : cos 0 = 1 := by | Mathlib.Data.Complex.Exponential.859_0.1bixbwhfBeJKySp | @[simp]
theorem cos_zero : cos 0 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos (-x) = cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos, sub_eq_add_neg, exp_neg, add_comm] | @[simp]
theorem cos_neg : cos (-x) = cos x := by | Mathlib.Data.Complex.Exponential.863_0.1bixbwhfBeJKySp | @[simp]
theorem cos_neg : cos (-x) = cos x | Mathlib_Data_Complex_Exponential |
x y a b c d : ℂ
⊢ (a + b) * (c + d) - (b - a) * (d - c) * -1 = 2 * (a * c + b * d) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | private theorem cos_add_aux {a b c d : ℂ} :
(a + b) * (c + d) - (b - a) * (d - c) * -1 = 2 * (a * c + b * d) := by | Mathlib.Data.Complex.Exponential.867_0.1bixbwhfBeJKySp | private theorem cos_add_aux {a b c d : ℂ} :
(a + b) * (c + d) - (b - a) * (d - c) * -1 = 2 * (a * c + b * d) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos (x + y) = cos x * cos y - sin x * sin y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← cosh_mul_I, add_mul, cosh_add, cosh_mul_I, cosh_mul_I, sinh_mul_I, sinh_mul_I,
mul_mul_mul_comm, I_mul_I, mul_neg_one, sub_eq_add_neg] | theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y := by
| Mathlib.Data.Complex.Exponential.870_0.1bixbwhfBeJKySp | theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin (x - y) = sin x * cos y - cos x * sin y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg] | theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by
| Mathlib.Data.Complex.Exponential.875_0.1bixbwhfBeJKySp | theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos (x - y) = cos x * cos y + sin x * sin y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg] | theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y := by
| Mathlib.Data.Complex.Exponential.879_0.1bixbwhfBeJKySp | theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y | Mathlib_Data_Complex_Exponential |
x✝ y✝ x y : ℂ
⊢ sin (x + y * I) = sin x * cosh y + cos x * sinh y * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [sin_add, cos_mul_I, sin_mul_I, mul_assoc] | theorem sin_add_mul_I (x y : ℂ) : sin (x + y * I) = sin x * cosh y + cos x * sinh y * I := by
| Mathlib.Data.Complex.Exponential.883_0.1bixbwhfBeJKySp | theorem sin_add_mul_I (x y : ℂ) : sin (x + y * I) = sin x * cosh y + cos x * sinh y * I | Mathlib_Data_Complex_Exponential |
x y z : ℂ
⊢ sin z = sin ↑z.re * cosh ↑z.im + cos ↑z.re * sinh ↑z.im * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | convert sin_add_mul_I z.re z.im | theorem sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I := by
| Mathlib.Data.Complex.Exponential.888_0.1bixbwhfBeJKySp | theorem sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I | Mathlib_Data_Complex_Exponential |
case h.e'_2.h.e'_1
x y z : ℂ
⊢ z = ↑z.re + ↑z.im * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact (re_add_im z).symm | theorem sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I := by
convert sin_add_mul_I z.re z.im; | Mathlib.Data.Complex.Exponential.888_0.1bixbwhfBeJKySp | theorem sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I | Mathlib_Data_Complex_Exponential |
x✝ y✝ x y : ℂ
⊢ cos (x + y * I) = cos x * cosh y - sin x * sinh y * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [cos_add, cos_mul_I, sin_mul_I, mul_assoc] | theorem cos_add_mul_I (x y : ℂ) : cos (x + y * I) = cos x * cosh y - sin x * sinh y * I := by
| Mathlib.Data.Complex.Exponential.892_0.1bixbwhfBeJKySp | theorem cos_add_mul_I (x y : ℂ) : cos (x + y * I) = cos x * cosh y - sin x * sinh y * I | Mathlib_Data_Complex_Exponential |
x y z : ℂ
⊢ cos z = cos ↑z.re * cosh ↑z.im - sin ↑z.re * sinh ↑z.im * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | convert cos_add_mul_I z.re z.im | theorem cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I := by
| Mathlib.Data.Complex.Exponential.897_0.1bixbwhfBeJKySp | theorem cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I | Mathlib_Data_Complex_Exponential |
case h.e'_2.h.e'_1
x y z : ℂ
⊢ z = ↑z.re + ↑z.im * I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact (re_add_im z).symm | theorem cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I := by
convert cos_add_mul_I z.re z.im; | Mathlib.Data.Complex.Exponential.897_0.1bixbwhfBeJKySp | theorem cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have s1 := sin_add ((x + y) / 2) ((x - y) / 2) | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
| Mathlib.Data.Complex.Exponential.901_0.1bixbwhfBeJKySp | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : sin ((x + y) / 2 + (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)
⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have s2 := sin_sub ((x + y) / 2) ((x - y) / 2) | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
have s1 := sin_add ((x + y) / 2) ((x - y) / 2)
| Mathlib.Data.Complex.Exponential.901_0.1bixbwhfBeJKySp | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : sin ((x + y) / 2 + (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)
s2 : sin ((x + y) / 2 - (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)
⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1 | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
have s1 := sin_add ((x + y) / 2) ((x - y) / 2)
have s2 := sin_sub ((x + y) / 2) ((x - y) / 2)
| Mathlib.Data.Complex.Exponential.901_0.1bixbwhfBeJKySp | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)
s2 : sin ((x + y) / 2 - (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)
⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [div_sub_div_same, ← sub_add, add_sub_cancel', half_add_self] at s2 | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
have s1 := sin_add ((x + y) / 2) ((x - y) / 2)
have s2 := sin_sub ((x + y) / 2) ((x - y) / 2)
rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1
| Mathlib.Data.Complex.Exponential.901_0.1bixbwhfBeJKySp | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)
s2 : sin y = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)
⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [s1, s2] | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
have s1 := sin_add ((x + y) / 2) ((x - y) / 2)
have s2 := sin_sub ((x + y) / 2) ((x - y) / 2)
rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1
rw [div_sub_div_same, ← sub_add, add_sub_cancel'... | Mathlib.Data.Complex.Exponential.901_0.1bixbwhfBeJKySp | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)
s2 : sin y = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)
⊢ sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2) -
(sin ((x + y) / 2) * cos ((x - y) / 2) - c... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
have s1 := sin_add ((x + y) / 2) ((x - y) / 2)
have s2 := sin_sub ((x + y) / 2) ((x - y) / 2)
rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1
rw [div_sub_div_same, ← sub_add, add_sub_cancel'... | Mathlib.Data.Complex.Exponential.901_0.1bixbwhfBeJKySp | theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have s1 := cos_add ((x + y) / 2) ((x - y) / 2) | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
| Mathlib.Data.Complex.Exponential.910_0.1bixbwhfBeJKySp | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : cos ((x + y) / 2 + (x - y) / 2) = cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2)
⊢ cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have s2 := cos_sub ((x + y) / 2) ((x - y) / 2) | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
have s1 := cos_add ((x + y) / 2) ((x - y) / 2)
| Mathlib.Data.Complex.Exponential.910_0.1bixbwhfBeJKySp | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : cos ((x + y) / 2 + (x - y) / 2) = cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2)
s2 : cos ((x + y) / 2 - (x - y) / 2) = cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)
⊢ cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1 | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
have s1 := cos_add ((x + y) / 2) ((x - y) / 2)
have s2 := cos_sub ((x + y) / 2) ((x - y) / 2)
| Mathlib.Data.Complex.Exponential.910_0.1bixbwhfBeJKySp | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : cos x = cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2)
s2 : cos ((x + y) / 2 - (x - y) / 2) = cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)
⊢ cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [div_sub_div_same, ← sub_add, add_sub_cancel', half_add_self] at s2 | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
have s1 := cos_add ((x + y) / 2) ((x - y) / 2)
have s2 := cos_sub ((x + y) / 2) ((x - y) / 2)
rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1
| Mathlib.Data.Complex.Exponential.910_0.1bixbwhfBeJKySp | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : cos x = cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2)
s2 : cos y = cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)
⊢ cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [s1, s2] | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
have s1 := cos_add ((x + y) / 2) ((x - y) / 2)
have s2 := cos_sub ((x + y) / 2) ((x - y) / 2)
rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1
rw [div_sub_div_same, ← sub_add, add_sub_cancel... | Mathlib.Data.Complex.Exponential.910_0.1bixbwhfBeJKySp | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
s1 : cos x = cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2)
s2 : cos y = cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)
⊢ cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) -
(cos ((x + y) / 2) * cos ((x - y) / 2) + s... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
have s1 := cos_add ((x + y) / 2) ((x - y) / 2)
have s2 := cos_sub ((x + y) / 2) ((x - y) / 2)
rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1
rw [div_sub_div_same, ← sub_add, add_sub_cancel... | Mathlib.Data.Complex.Exponential.910_0.1bixbwhfBeJKySp | theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | calc
cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
_ =
cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) :=
?_
_ = 2 * cos ((x + y) ... | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
| Mathlib.Data.Complex.Exponential.919_0.1bixbwhfBeJKySp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
case calc_1
x y : ℂ
⊢ cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | congr | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
calc
cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
_ =
cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x ... | Mathlib.Data.Complex.Exponential.919_0.1bixbwhfBeJKySp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
case calc_1.e_a.e_z
x y : ℂ
⊢ x = (x + y) / 2 + (x - y) / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | field_simp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
calc
cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
_ =
cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x ... | Mathlib.Data.Complex.Exponential.919_0.1bixbwhfBeJKySp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
case calc_1.e_a.e_z
x y : ℂ
⊢ y = (x + y) / 2 - (x - y) / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | field_simp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
calc
cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
_ =
cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x ... | Mathlib.Data.Complex.Exponential.919_0.1bixbwhfBeJKySp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
case calc_2
x y : ℂ
⊢ cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) =
cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [cos_add, cos_sub] | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
calc
cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
_ =
cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x ... | Mathlib.Data.Complex.Exponential.919_0.1bixbwhfBeJKySp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
case calc_3
x y : ℂ
⊢ cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) =
2 * cos ((x + y) / 2) * cos ((x - y) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
calc
cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
_ =
cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
(cos ((x + y) / 2) * cos ((x ... | Mathlib.Data.Complex.Exponential.919_0.1bixbwhfBeJKySp | theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin ((starRingEnd ℂ) x) = (starRingEnd ℂ) (sin x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I, ← conj_neg_I, ← RingHom.map_mul, ← RingHom.map_mul,
sinh_conj, mul_neg, sinh_neg, sinh_mul_I, mul_neg] | theorem sin_conj : sin (conj x) = conj (sin x) := by
| Mathlib.Data.Complex.Exponential.933_0.1bixbwhfBeJKySp | theorem sin_conj : sin (conj x) = conj (sin x) | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (starRingEnd ℂ) (sin ↑x) = sin ↑x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sin_conj, conj_ofReal] | @[simp]
theorem ofReal_sin_ofReal_re (x : ℝ) : ((sin x).re : ℂ) = sin x :=
conj_eq_iff_re.1 <| by | Mathlib.Data.Complex.Exponential.938_0.1bixbwhfBeJKySp | @[simp]
theorem ofReal_sin_ofReal_re (x : ℝ) : ((sin x).re : ℂ) = sin x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (sin ↑x).im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_sin_ofReal_re, ofReal_im] | @[simp]
theorem sin_ofReal_im (x : ℝ) : (sin x).im = 0 := by | Mathlib.Data.Complex.Exponential.948_0.1bixbwhfBeJKySp | @[simp]
theorem sin_ofReal_im (x : ℝ) : (sin x).im = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos ((starRingEnd ℂ) x) = (starRingEnd ℂ) (cos x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← cosh_mul_I, ← conj_neg_I, ← RingHom.map_mul, ← cosh_mul_I, cosh_conj, mul_neg, cosh_neg] | theorem cos_conj : cos (conj x) = conj (cos x) := by
| Mathlib.Data.Complex.Exponential.956_0.1bixbwhfBeJKySp | theorem cos_conj : cos (conj x) = conj (cos x) | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (starRingEnd ℂ) (cos ↑x) = cos ↑x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← cos_conj, conj_ofReal] | @[simp]
theorem ofReal_cos_ofReal_re (x : ℝ) : ((cos x).re : ℂ) = cos x :=
conj_eq_iff_re.1 <| by | Mathlib.Data.Complex.Exponential.960_0.1bixbwhfBeJKySp | @[simp]
theorem ofReal_cos_ofReal_re (x : ℝ) : ((cos x).re : ℂ) = cos x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (cos ↑x).im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_cos_ofReal_re, ofReal_im] | @[simp]
theorem cos_ofReal_im (x : ℝ) : (cos x).im = 0 := by | Mathlib.Data.Complex.Exponential.970_0.1bixbwhfBeJKySp | @[simp]
theorem cos_ofReal_im (x : ℝ) : (cos x).im = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ tan 0 = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tan] | @[simp]
theorem tan_zero : tan 0 = 0 := by | Mathlib.Data.Complex.Exponential.978_0.1bixbwhfBeJKySp | @[simp]
theorem tan_zero : tan 0 = 0 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℂ
hx : cos x ≠ 0
⊢ tan x * cos x = sin x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [tan_eq_sin_div_cos, div_mul_cancel _ hx] | theorem tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x := by
| Mathlib.Data.Complex.Exponential.986_0.1bixbwhfBeJKySp | theorem tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ tan (-x) = -tan x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [tan, neg_div] | @[simp]
theorem tan_neg : tan (-x) = -tan x := by | Mathlib.Data.Complex.Exponential.990_0.1bixbwhfBeJKySp | @[simp]
theorem tan_neg : tan (-x) = -tan x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ tan ((starRingEnd ℂ) x) = (starRingEnd ℂ) (tan x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [tan, sin_conj, cos_conj, ← map_div₀, tan] | theorem tan_conj : tan (conj x) = conj (tan x) := by | Mathlib.Data.Complex.Exponential.994_0.1bixbwhfBeJKySp | theorem tan_conj : tan (conj x) = conj (tan x) | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (starRingEnd ℂ) (tan ↑x) = tan ↑x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← tan_conj, conj_ofReal] | @[simp]
theorem ofReal_tan_ofReal_re (x : ℝ) : ((tan x).re : ℂ) = tan x :=
conj_eq_iff_re.1 <| by | Mathlib.Data.Complex.Exponential.997_0.1bixbwhfBeJKySp | @[simp]
theorem ofReal_tan_ofReal_re (x : ℝ) : ((tan x).re : ℂ) = tan x | Mathlib_Data_Complex_Exponential |
x✝ y : ℂ
x : ℝ
⊢ (tan ↑x).im = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_tan_ofReal_re, ofReal_im] | @[simp]
theorem tan_ofReal_im (x : ℝ) : (tan x).im = 0 := by | Mathlib.Data.Complex.Exponential.1007_0.1bixbwhfBeJKySp | @[simp]
theorem tan_ofReal_im (x : ℝ) : (tan x).im = 0 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos x + sin x * I = cexp (x * I) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← cosh_add_sinh, sinh_mul_I, cosh_mul_I] | theorem cos_add_sin_I : cos x + sin x * I = exp (x * I) := by
| Mathlib.Data.Complex.Exponential.1015_0.1bixbwhfBeJKySp | theorem cos_add_sin_I : cos x + sin x * I = exp (x * I) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos x - sin x * I = cexp (-x * I) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [neg_mul, ← cosh_sub_sinh, sinh_mul_I, cosh_mul_I] | theorem cos_sub_sin_I : cos x - sin x * I = exp (-x * I) := by
| Mathlib.Data.Complex.Exponential.1020_0.1bixbwhfBeJKySp | theorem cos_sub_sin_I : cos x - sin x * I = exp (-x * I) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin x ^ 2 + cos x ^ 2 = cosh (x * I) ^ 2 - sinh (x * I) ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [cosh_mul_I, sinh_mul_I, mul_pow, I_sq, mul_neg_one, sub_neg_eq_add, add_comm] | @[simp]
theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=
Eq.trans (by | Mathlib.Data.Complex.Exponential.1025_0.1bixbwhfBeJKySp | @[simp]
theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos x ^ 2 + sin x ^ 2 = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [add_comm, sin_sq_add_cos_sq] | @[simp]
theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 := by | Mathlib.Data.Complex.Exponential.1031_0.1bixbwhfBeJKySp | @[simp]
theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos (2 * x) = cos x ^ 2 - sin x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [two_mul, cos_add, ← sq, ← sq] | theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 := by | Mathlib.Data.Complex.Exponential.1035_0.1bixbwhfBeJKySp | theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos (2 * x) = 2 * cos x ^ 2 - 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x), ← sub_add, sub_add_eq_add_sub,
two_mul] | theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 := by
| Mathlib.Data.Complex.Exponential.1038_0.1bixbwhfBeJKySp | theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin (2 * x) = 2 * sin x * cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [two_mul, sin_add, two_mul, add_mul, mul_comm] | theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x := by
| Mathlib.Data.Complex.Exponential.1043_0.1bixbwhfBeJKySp | theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [cos_two_mul, div_add_div_same, mul_div_cancel_left, two_ne_zero, -one_div] | theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 := by
| Mathlib.Data.Complex.Exponential.1047_0.1bixbwhfBeJKySp | theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos x ^ 2 = 1 - sin x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sin_sq_add_cos_sq x, add_sub_cancel'] | theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 := by | Mathlib.Data.Complex.Exponential.1051_0.1bixbwhfBeJKySp | theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin x ^ 2 = 1 - cos x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← sin_sq_add_cos_sq x, add_sub_cancel] | theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 := by | Mathlib.Data.Complex.Exponential.1054_0.1bixbwhfBeJKySp | theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℂ
hx : cos x ≠ 0
⊢ (1 + tan x ^ 2)⁻¹ = cos x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [tan_eq_sin_div_cos, div_pow] | theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 := by
| Mathlib.Data.Complex.Exponential.1057_0.1bixbwhfBeJKySp | theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℂ
hx : cos x ≠ 0
⊢ (1 + sin x ^ 2 / cos x ^ 2)⁻¹ = cos x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | field_simp | theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 := by
rw [tan_eq_sin_div_cos, div_pow]
| Mathlib.Data.Complex.Exponential.1057_0.1bixbwhfBeJKySp | theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 | Mathlib_Data_Complex_Exponential |
x✝ y x : ℂ
hx : cos x ≠ 0
⊢ tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul] | theorem tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) :
tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 := by
| Mathlib.Data.Complex.Exponential.1062_0.1bixbwhfBeJKySp | theorem tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) :
tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h1 : x + 2 * x = 3 * x := by ring | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
| Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ x + 2 * x = 3 * x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
have h1 : x + 2 * x = 3 * x := by | Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← h1, cos_add x (2 * x)] | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
have h1 : x + 2 * x = 3 * x := by ring
| Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos x * cos (2 * x) - sin x * sin (2 * x) = 4 * cos x ^ 3 - 3 * cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq] | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cos_add x (2 * x)]
| Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by ring | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cos_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]
| Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cos_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]
have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by | Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [h2, cos_sq'] | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cos_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]
have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by ring
| Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) =
2 * cos x * cos x * cos x + 2 * cos x * (1 - sin x ^ 2) - 3 * cos x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, cos_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]
have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by ring
rw [h2, cos_sq']
... | Mathlib.Data.Complex.Exponential.1067_0.1bixbwhfBeJKySp | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h1 : x + 2 * x = 3 * x := by ring | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
| Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ x + 2 * x = 3 * x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
have h1 : x + 2 * x = 3 * x := by | Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← h1, sin_add x (2 * x)] | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
have h1 : x + 2 * x = 3 * x := by ring
| Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ sin x * cos (2 * x) + cos x * sin (2 * x) = 3 * sin x - 4 * sin x ^ 3 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [cos_two_mul, sin_two_mul, cos_sq'] | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sin_add x (2 * x)]
| Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ sin x * (2 * (1 - sin x ^ 2) - 1) + cos x * (2 * sin x * cos x) = 3 * sin x - 4 * sin x ^ 3 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 := by ring | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sin_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, cos_sq']
| Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
⊢ cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sin_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, cos_sq']
have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 := by | Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2
⊢ sin x * (2 * (1 - sin x ^ 2) - 1) + cos x * (2 * sin x * cos x) = 3 * sin x - 4 * sin x ^ 3 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [h2, cos_sq'] | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sin_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, cos_sq']
have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 := by ring
| Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2
⊢ sin x * (2 * (1 - sin x ^ 2) - 1) + 2 * sin x * (1 - sin x ^ 2) = 3 * sin x - 4 * sin x ^ 3 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sin_add x (2 * x)]
simp only [cos_two_mul, sin_two_mul, cos_sq']
have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 := by ring
rw [h2, cos_sq']
| Mathlib.Data.Complex.Exponential.1076_0.1bixbwhfBeJKySp | theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp (x + y * I) = cexp x * (cos y + sin y * I) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [exp_add, exp_mul_I] | theorem exp_add_mul_I : exp (x + y * I) = exp x * (cos y + sin y * I) := by | Mathlib.Data.Complex.Exponential.1090_0.1bixbwhfBeJKySp | theorem exp_add_mul_I : exp (x + y * I) = exp x * (cos y + sin y * I) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ cexp x = cexp ↑x.re * (cos ↑x.im + sin ↑x.im * I) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← exp_add_mul_I, re_add_im] | theorem exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) := by
| Mathlib.Data.Complex.Exponential.1094_0.1bixbwhfBeJKySp | theorem exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (cexp x).re = Real.exp x.re * Real.cos x.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [exp_eq_exp_re_mul_sin_add_cos] | theorem exp_re : (exp x).re = Real.exp x.re * Real.cos x.im := by
| Mathlib.Data.Complex.Exponential.1098_0.1bixbwhfBeJKySp | theorem exp_re : (exp x).re = Real.exp x.re * Real.cos x.im | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (cexp ↑x.re * (cos ↑x.im + sin ↑x.im * I)).re = Real.exp x.re * Real.cos x.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [exp_ofReal_re, cos_ofReal_re] | theorem exp_re : (exp x).re = Real.exp x.re * Real.cos x.im := by
rw [exp_eq_exp_re_mul_sin_add_cos]
| Mathlib.Data.Complex.Exponential.1098_0.1bixbwhfBeJKySp | theorem exp_re : (exp x).re = Real.exp x.re * Real.cos x.im | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ (cexp x).im = Real.exp x.re * Real.sin x.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [exp_eq_exp_re_mul_sin_add_cos] | theorem exp_im : (exp x).im = Real.exp x.re * Real.sin x.im := by
| Mathlib.Data.Complex.Exponential.1103_0.1bixbwhfBeJKySp | theorem exp_im : (exp x).im = Real.exp x.re * Real.sin x.im | Mathlib_Data_Complex_Exponential |
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