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 : ℝ
hx0 : 0 < x
hx : x ≤ 2
⊢ 0 < 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... | norm_num | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=
have : x / 2 ≤ 1 := (div_le_iff (by | Mathlib.Data.Complex.Exponential.1931_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 2
⊢ x ≤ 1 * 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... | simpa | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=
have : x / 2 ≤ 1 := (div_le_iff (by norm_num)).mpr (by | Mathlib.Data.Complex.Exponential.1931_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 2
this : x / 2 ≤ 1
⊢ 0 < 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... | norm_num | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=
have : x / 2 ≤ 1 := (div_le_iff (by norm_num)).mpr (by simpa)
calc
0 < 2 * sin (x / 2) * cos (x / 2) :=
mul_pos (mul_pos (by | Mathlib.Data.Complex.Exponential.1931_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 2
this : x / 2 ≤ 1
⊢ abs' (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... | rwa [_root_.abs_of_nonneg (le_of_lt (half_pos hx0))] | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=
have : x / 2 ≤ 1 := (div_le_iff (by norm_num)).mpr (by simpa)
calc
0 < 2 * sin (x / 2) * cos (x / 2) :=
mul_pos (mul_pos (by norm_num) (sin_pos_of_pos_of_le_one (half_pos hx0) this))
(cos_pos_of_le_one (by | Mathlib.Data.Complex.Exponential.1931_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 2
this : x / 2 ≤ 1
⊢ 2 * sin (x / 2) * cos (x / 2) = 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_two_mul, two_mul, add_halves] | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=
have : x / 2 ≤ 1 := (div_le_iff (by norm_num)).mpr (by simpa)
calc
0 < 2 * sin (x / 2) * cos (x / 2) :=
mul_pos (mul_pos (by norm_num) (sin_pos_of_pos_of_le_one (half_pos hx0) this))
(cos_pos_of_le_one (by rwa [_roo... | Mathlib.Data.Complex.Exponential.1931_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x | Mathlib_Data_Complex_Exponential |
⊢ abs' 1 ≤ 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 | theorem cos_one_le : cos 1 ≤ 2 / 3 :=
calc
cos 1 ≤ |(1 : ℝ)| ^ 4 * (5 / 96) + (1 - 1 ^ 2 / 2) :=
sub_le_iff_le_add.1 (abs_sub_le_iff.1 (cos_bound (by | Mathlib.Data.Complex.Exponential.1940_0.1bixbwhfBeJKySp | theorem cos_one_le : cos 1 ≤ 2 / 3 | Mathlib_Data_Complex_Exponential |
⊢ abs' 1 ^ 4 * (5 / 96) + (1 - 1 ^ 2 / 2) ≤ 2 / 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... | norm_num | theorem cos_one_le : cos 1 ≤ 2 / 3 :=
calc
cos 1 ≤ |(1 : ℝ)| ^ 4 * (5 / 96) + (1 - 1 ^ 2 / 2) :=
sub_le_iff_le_add.1 (abs_sub_le_iff.1 (cos_bound (by simp))).1
_ ≤ 2 / 3 := by | Mathlib.Data.Complex.Exponential.1940_0.1bixbwhfBeJKySp | theorem cos_one_le : cos 1 ≤ 2 / 3 | Mathlib_Data_Complex_Exponential |
⊢ 2 * cos 1 ^ 2 - 1 ≤ 2 * (2 / 3) ^ 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... | gcongr | theorem cos_two_neg : cos 2 < 0 :=
calc cos 2 = cos (2 * 1) := congr_arg cos (mul_one _).symm
_ = _ := (Real.cos_two_mul 1)
_ ≤ 2 * (2 / 3) ^ 2 - 1 := by
| Mathlib.Data.Complex.Exponential.1951_0.1bixbwhfBeJKySp | theorem cos_two_neg : cos 2 < 0 | Mathlib_Data_Complex_Exponential |
case h.h.ha
⊢ 0 ≤ cos 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... | exact cos_one_pos.le | theorem cos_two_neg : cos 2 < 0 :=
calc cos 2 = cos (2 * 1) := congr_arg cos (mul_one _).symm
_ = _ := (Real.cos_two_mul 1)
_ ≤ 2 * (2 / 3) ^ 2 - 1 := by
gcongr
· | Mathlib.Data.Complex.Exponential.1951_0.1bixbwhfBeJKySp | theorem cos_two_neg : cos 2 < 0 | Mathlib_Data_Complex_Exponential |
case h.h.hab
⊢ cos 1 ≤ 2 / 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... | apply cos_one_le | theorem cos_two_neg : cos 2 < 0 :=
calc cos 2 = cos (2 * 1) := congr_arg cos (mul_one _).symm
_ = _ := (Real.cos_two_mul 1)
_ ≤ 2 * (2 / 3) ^ 2 - 1 := by
gcongr
· exact cos_one_pos.le
· | Mathlib.Data.Complex.Exponential.1951_0.1bixbwhfBeJKySp | theorem cos_two_neg : cos 2 < 0 | Mathlib_Data_Complex_Exponential |
⊢ 2 * (2 / 3) ^ 2 - 1 < 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... | norm_num | theorem cos_two_neg : cos 2 < 0 :=
calc cos 2 = cos (2 * 1) := congr_arg cos (mul_one _).symm
_ = _ := (Real.cos_two_mul 1)
_ ≤ 2 * (2 / 3) ^ 2 - 1 := by
gcongr
· exact cos_one_pos.le
· apply cos_one_le
_ < 0 := by | Mathlib.Data.Complex.Exponential.1951_0.1bixbwhfBeJKySp | theorem cos_two_neg : cos 2 < 0 | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
⊢ rexp x < 1 / (1 - 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 H : 0 < 1 - (1 + x + x ^ 2) * (1 - x) | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
| Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
case H
x : ℝ
h1 : 0 < x
h2 : x < 1
⊢ 0 < 1 - (1 + x + x ^ 2) * (1 - 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... | calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
⊢ 0 < 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... | positivity | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
⊢ x ^ 3 = 1 - (1 + x + x ^ 2) * (1 - 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 exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ rexp x < 1 / (1 - 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... | calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤ 1 + x + x ^ 2 := by
-- Porting note: was `norm_num [Finset.sum] <;> nlinarith`
-- This proof should be restored after the norm_num plugin for big operators is ported.
-- (It may also need the positivity extensions in #3907.)
repe... | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
| Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ m in range 3, x ^ m / ↑(Nat.factorial m) + x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤ 1 + x + 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... | repeat erw [Finset.sum_range_succ] | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ m in range 3, x ^ m / ↑(Nat.factorial m) + x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤ 1 + x + 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... | erw [Finset.sum_range_succ] | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ x_1 in range 2, x ^ x_1 / ↑(Nat.factorial x_1) + x ^ 2 / ↑(Nat.factorial 2) +
x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤
1 + x + 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... | erw [Finset.sum_range_succ] | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ x_1 in range 1, x ^ x_1 / ↑(Nat.factorial x_1) + x ^ 1 / ↑(Nat.factorial 1) + x ^ 2 / ↑(Nat.factorial 2) +
x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤
1 + x + 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... | erw [Finset.sum_range_succ] | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ x_1 in range 0, x ^ x_1 / ↑(Nat.factorial x_1) + x ^ 0 / ↑(Nat.factorial 0) + x ^ 1 / ↑(Nat.factorial 1) +
x ^ 2 / ↑(Nat.factorial 2) +
x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤
1 + x + 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... | erw [Finset.sum_range_succ] | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ x_1 in range 0, x ^ x_1 / ↑(Nat.factorial x_1) + x ^ 0 / ↑(Nat.factorial 0) + x ^ 1 / ↑(Nat.factorial 1) +
x ^ 2 / ↑(Nat.factorial 2) +
x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤
1 + x + 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... | norm_num [Nat.factorial] | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ 1 + x + x ^ 2 / 2 + x ^ 3 * 4 / 18 ≤ 1 + x + 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... | nlinarith | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ 1 + x + x ^ 2 < 1 / (1 - 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 [lt_div_iff] | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ (1 + x + x ^ 2) * (1 - x) < 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... | nlinarith | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ 0 < 1 - 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... | nlinarith | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) := by
have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
· calc
0 < x ^ 3 := by positivity
_ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring
calc
exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three
_ ≤... | Mathlib.Data.Complex.Exponential.1961_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
Real.exp x < 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x < 1
⊢ rexp x ≤ 1 / (1 - 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... | rcases eq_or_lt_of_le h1 with (rfl | h1) | theorem exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :
Real.exp x ≤ 1 / (1 - x) := by
| Mathlib.Data.Complex.Exponential.1979_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :
Real.exp x ≤ 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
case inl
h1 : 0 ≤ 0
h2 : 0 < 1
⊢ rexp 0 ≤ 1 / (1 - 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 | theorem exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :
Real.exp x ≤ 1 / (1 - x) := by
rcases eq_or_lt_of_le h1 with (rfl | h1)
· | Mathlib.Data.Complex.Exponential.1979_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :
Real.exp x ≤ 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
case inr
x : ℝ
h1✝ : 0 ≤ x
h2 : x < 1
h1 : 0 < x
⊢ rexp x ≤ 1 / (1 - 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... | exact (exp_bound_div_one_sub_of_interval' h1 h2).le | theorem exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :
Real.exp x ≤ 1 / (1 - x) := by
rcases eq_or_lt_of_le h1 with (rfl | h1)
· simp
· | Mathlib.Data.Complex.Exponential.1979_0.1bixbwhfBeJKySp | theorem exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :
Real.exp x ≤ 1 / (1 - x) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : x ≠ 0
⊢ x + 1 < rexp 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... | obtain hx | hx := hx.symm.lt_or_lt | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
| Mathlib.Data.Complex.Exponential.1986_0.1bixbwhfBeJKySp | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x | Mathlib_Data_Complex_Exponential |
case inl
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
⊢ x + 1 < rexp 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... | exact add_one_lt_exp_of_pos hx | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
obtain hx | hx := hx.symm.lt_or_lt
· | Mathlib.Data.Complex.Exponential.1986_0.1bixbwhfBeJKySp | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x | Mathlib_Data_Complex_Exponential |
case inr
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ x + 1 < rexp 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... | obtain h' | h' := le_or_lt 1 (-x) | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
obtain hx | hx := hx.symm.lt_or_lt
· exact add_one_lt_exp_of_pos hx
| Mathlib.Data.Complex.Exponential.1986_0.1bixbwhfBeJKySp | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x | Mathlib_Data_Complex_Exponential |
case inr.inl
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
h' : 1 ≤ -x
⊢ x + 1 < rexp 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... | linarith [x.exp_pos] | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
obtain hx | hx := hx.symm.lt_or_lt
· exact add_one_lt_exp_of_pos hx
obtain h' | h' := le_or_lt 1 (-x)
· | Mathlib.Data.Complex.Exponential.1986_0.1bixbwhfBeJKySp | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x | Mathlib_Data_Complex_Exponential |
case inr.inr
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
h' : -x < 1
⊢ x + 1 < rexp 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 hx' : 0 < x + 1 := by linarith | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
obtain hx | hx := hx.symm.lt_or_lt
· exact add_one_lt_exp_of_pos hx
obtain h' | h' := le_or_lt 1 (-x)
· linarith [x.exp_pos]
| Mathlib.Data.Complex.Exponential.1986_0.1bixbwhfBeJKySp | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
h' : -x < 1
⊢ 0 < x + 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... | linarith | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
obtain hx | hx := hx.symm.lt_or_lt
· exact add_one_lt_exp_of_pos hx
obtain h' | h' := le_or_lt 1 (-x)
· linarith [x.exp_pos]
have hx' : 0 < x + 1 := by | Mathlib.Data.Complex.Exponential.1986_0.1bixbwhfBeJKySp | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x | Mathlib_Data_Complex_Exponential |
case inr.inr
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
h' : -x < 1
hx' : 0 < x + 1
⊢ x + 1 < rexp 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... | simpa [add_comm, exp_neg, inv_lt_inv (exp_pos _) hx']
using exp_bound_div_one_sub_of_interval' (neg_pos.2 hx) h' | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
obtain hx | hx := hx.symm.lt_or_lt
· exact add_one_lt_exp_of_pos hx
obtain h' | h' := le_or_lt 1 (-x)
· linarith [x.exp_pos]
have hx' : 0 < x + 1 := by linarith
| Mathlib.Data.Complex.Exponential.1986_0.1bixbwhfBeJKySp | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x | Mathlib_Data_Complex_Exponential |
x : ℝ
⊢ x + 1 ≤ rexp 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... | obtain rfl | hx := eq_or_ne x 0 | theorem add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x := by
| Mathlib.Data.Complex.Exponential.1997_0.1bixbwhfBeJKySp | theorem add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x | Mathlib_Data_Complex_Exponential |
case inl
⊢ 0 + 1 ≤ rexp 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 | theorem add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x := by
obtain rfl | hx := eq_or_ne x 0
· | Mathlib.Data.Complex.Exponential.1997_0.1bixbwhfBeJKySp | theorem add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x | Mathlib_Data_Complex_Exponential |
case inr
x : ℝ
hx : x ≠ 0
⊢ x + 1 ≤ rexp 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... | exact (add_one_lt_exp hx).le | theorem add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x := by
obtain rfl | hx := eq_or_ne x 0
· simp
· | Mathlib.Data.Complex.Exponential.1997_0.1bixbwhfBeJKySp | theorem add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x | Mathlib_Data_Complex_Exponential |
n : ℕ
t : ℝ
ht' : t ≤ ↑n
⊢ (1 - t / ↑n) ^ n ≤ rexp (-t) | /-
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... | rcases eq_or_ne n 0 with (rfl | hn) | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
| Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case inl
t : ℝ
ht' : t ≤ ↑0
⊢ (1 - t / ↑0) ^ 0 ≤ rexp (-t) | /-
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 one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· | Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case inl
t : ℝ
ht' : t ≤ ↑0
⊢ t ≤ 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... | rwa [Nat.cast_zero] at ht' | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
| Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case inr
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ (1 - t / ↑n) ^ n ≤ rexp (-t) | /-
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 pow_le_pow_left ?_ (one_sub_le_exp_neg (t / n)) n using 2 | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rwa [Nat.cast_zero] at ht'
| Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case h.e'_4
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ rexp (-t) = rexp (-(t / ↑n)) ^ n | /-
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 [← Real.exp_nat_mul] | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rwa [Nat.cast_zero] at ht'
convert pow_le_pow_left ?_ (one_sub_le_exp_neg (t / n)) n using 2
· | Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case h.e'_4
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ rexp (-t) = rexp (↑n * -(t / ↑n)) | /-
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 1 | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rwa [Nat.cast_zero] at ht'
convert pow_le_pow_left ?_ (one_sub_le_exp_neg (t / n)) n using 2
· rw [← Real.exp_nat_mul]
| Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case h.e'_4.e_x
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ -t = ↑n * -(t / ↑n) | /-
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 one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rwa [Nat.cast_zero] at ht'
convert pow_le_pow_left ?_ (one_sub_le_exp_neg (t / n)) n using 2
· rw [← Real.exp_nat_mul]
congr 1
| Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case h.e'_4.e_x
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ t * ↑n = ↑n * t | /-
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 one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rwa [Nat.cast_zero] at ht'
convert pow_le_pow_left ?_ (one_sub_le_exp_neg (t / n)) n using 2
· rw [← Real.exp_nat_mul]
congr 1
field_simp
| Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case inr
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ 0 ≤ 1 - t / ↑n | /-
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... | rwa [sub_nonneg, div_le_one] | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rwa [Nat.cast_zero] at ht'
convert pow_le_pow_left ?_ (one_sub_le_exp_neg (t / n)) n using 2
· rw [← Real.exp_nat_mul]
congr 1
field_simp
ring_nf
· | Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
case inr
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ 0 < ↑n | /-
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... | positivity | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) := by
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rwa [Nat.cast_zero] at ht'
convert pow_le_pow_left ?_ (one_sub_le_exp_neg (t / n)) n using 2
· rw [← Real.exp_nat_mul]
congr 1
field_simp
ring_nf
· rwa... | Mathlib.Data.Complex.Exponential.2012_0.1bixbwhfBeJKySp | theorem one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) | Mathlib_Data_Complex_Exponential |
x : ℝ
⊢ abs (cos ↑x + sin ↑x * I) = 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... | have := Real.sin_sq_add_cos_sq x | @[simp]
theorem abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 := by
| Mathlib.Data.Complex.Exponential.2040_0.1bixbwhfBeJKySp | @[simp]
theorem abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 | Mathlib_Data_Complex_Exponential |
x : ℝ
this : Real.sin x ^ 2 + Real.cos x ^ 2 = 1
⊢ abs (cos ↑x + sin ↑x * I) = 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_all [add_comm, abs, normSq, sq, sin_ofReal_re, cos_ofReal_re, mul_re] | @[simp]
theorem abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 := by
have := Real.sin_sq_add_cos_sq x
| Mathlib.Data.Complex.Exponential.2040_0.1bixbwhfBeJKySp | @[simp]
theorem abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 | Mathlib_Data_Complex_Exponential |
x : ℝ
⊢ abs (cexp ↑x) = Real.exp 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 [← ofReal_exp] | @[simp]
theorem abs_exp_ofReal (x : ℝ) : abs (exp x) = Real.exp x := by
| Mathlib.Data.Complex.Exponential.2047_0.1bixbwhfBeJKySp | @[simp]
theorem abs_exp_ofReal (x : ℝ) : abs (exp x) = Real.exp x | Mathlib_Data_Complex_Exponential |
x : ℝ
⊢ abs ↑(Real.exp x) = Real.exp 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... | exact abs_of_nonneg (le_of_lt (Real.exp_pos _)) | @[simp]
theorem abs_exp_ofReal (x : ℝ) : abs (exp x) = Real.exp x := by
rw [← ofReal_exp]
| Mathlib.Data.Complex.Exponential.2047_0.1bixbwhfBeJKySp | @[simp]
theorem abs_exp_ofReal (x : ℝ) : abs (exp x) = Real.exp x | Mathlib_Data_Complex_Exponential |
x : ℝ
⊢ abs (cexp (↑x * I)) = 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 [exp_mul_I, abs_cos_add_sin_mul_I] | @[simp]
theorem abs_exp_ofReal_mul_I (x : ℝ) : abs (exp (x * I)) = 1 := by
| Mathlib.Data.Complex.Exponential.2053_0.1bixbwhfBeJKySp | @[simp]
theorem abs_exp_ofReal_mul_I (x : ℝ) : abs (exp (x * I)) = 1 | Mathlib_Data_Complex_Exponential |
z : ℂ
⊢ abs (cexp z) = Real.exp z.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... | rw [exp_eq_exp_re_mul_sin_add_cos, map_mul, abs_exp_ofReal, abs_cos_add_sin_mul_I, mul_one] | theorem abs_exp (z : ℂ) : abs (exp z) = Real.exp z.re := by
| Mathlib.Data.Complex.Exponential.2059_0.1bixbwhfBeJKySp | theorem abs_exp (z : ℂ) : abs (exp z) = Real.exp z.re | Mathlib_Data_Complex_Exponential |
x y : ℂ
⊢ abs (cexp x) = abs (cexp y) ↔ x.re = y.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... | rw [abs_exp, abs_exp, Real.exp_eq_exp] | theorem abs_exp_eq_iff_re_eq {x y : ℂ} : abs (exp x) = abs (exp y) ↔ x.re = y.re := by
| Mathlib.Data.Complex.Exponential.2063_0.1bixbwhfBeJKySp | theorem abs_exp_eq_iff_re_eq {x y : ℂ} : abs (exp x) = abs (exp y) ↔ x.re = y.re | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : DivisionRing α
σα : OfScientific α
x✝¹ x✝ : ℕ
num✝ : ℤ
denom✝ : ℕ
σh : OfScientific.ofScientific = fun m s e => ↑(Rat.ofScientific m s e)
inv✝ : Invertible ↑denom✝
eq : ↑(mkRat (↑x✝¹) (10 ^ x✝)) = ↑num✝ * ⅟↑denom✝
⊢ OfScientific.ofScientific x✝¹ true x✝ = ↑num✝ * ⅟↑denom✝ | /-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.Defs
/-!
## `norm_num` plugin for scientific notation.
-/
set_option autoImplicit ... | simp only [σh, Rat.ofScientific_true_def] | theorem isRat_ofScientific_of_true [DivisionRing α] (σα : OfScientific α) :
{m e : ℕ} → {n : ℤ} → {d : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsRat (mkRat m (10 ^ e) : α) n d → IsRat (@OfScientific.ofScientific α σα m true e) n d
| _, _, _, _, σh, ⟨_, eq⟩ => ⟨_... | Mathlib.Tactic.NormNum.OfScientific.28_0.6nbfMwPMGvwsEsq | theorem isRat_ofScientific_of_true [DivisionRing α] (σα : OfScientific α) :
{m e : ℕ} → {n : ℤ} → {d : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsRat (mkRat m (10 ^ e) : α) n d → IsRat (@OfScientific.ofScientific α σα m true e) n d
| _, _, _, _, σh, ⟨_, eq⟩ => ⟨_... | Mathlib_Tactic_NormNum_OfScientific |
α : Type u_1
inst✝ : DivisionRing α
σα : OfScientific α
x✝¹ x✝ : ℕ
num✝ : ℤ
denom✝ : ℕ
σh : OfScientific.ofScientific = fun m s e => ↑(Rat.ofScientific m s e)
inv✝ : Invertible ↑denom✝
eq : ↑(mkRat (↑x✝¹) (10 ^ x✝)) = ↑num✝ * ⅟↑denom✝
⊢ ↑(mkRat (↑x✝¹) (10 ^ x✝)) = ↑num✝ * ⅟↑denom✝ | /-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.Defs
/-!
## `norm_num` plugin for scientific notation.
-/
set_option autoImplicit ... | exact eq | theorem isRat_ofScientific_of_true [DivisionRing α] (σα : OfScientific α) :
{m e : ℕ} → {n : ℤ} → {d : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsRat (mkRat m (10 ^ e) : α) n d → IsRat (@OfScientific.ofScientific α σα m true e) n d
| _, _, _, _, σh, ⟨_, eq⟩ => ⟨_... | Mathlib.Tactic.NormNum.OfScientific.28_0.6nbfMwPMGvwsEsq | theorem isRat_ofScientific_of_true [DivisionRing α] (σα : OfScientific α) :
{m e : ℕ} → {n : ℤ} → {d : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsRat (mkRat m (10 ^ e) : α) n d → IsRat (@OfScientific.ofScientific α σα m true e) n d
| _, _, _, _, σh, ⟨_, eq⟩ => ⟨_... | Mathlib_Tactic_NormNum_OfScientific |
α : Type u_1
inst✝ : DivisionRing α
σα : OfScientific α
n✝¹ n✝ x✝ : ℕ
σh : OfScientific.ofScientific = fun m s e => ↑(Rat.ofScientific m s e)
h : x✝ = Nat.mul n✝¹ (10 ^ n✝)
⊢ OfScientific.ofScientific (↑n✝¹) false ↑n✝ = ↑x✝ | /-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.Defs
/-!
## `norm_num` plugin for scientific notation.
-/
set_option autoImplicit ... | simp [σh, Rat.ofScientific_false_def, h] | theorem isNat_ofScientific_of_false [DivisionRing α] (σα : OfScientific α) : {m e nm ne n : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsNat m nm → IsNat e ne → n = Nat.mul nm ((10 : ℕ) ^ ne) →
IsNat (@OfScientific.ofScientific α σα m false e : α) n
| _, _, _, _, _... | Mathlib.Tactic.NormNum.OfScientific.35_0.6nbfMwPMGvwsEsq | theorem isNat_ofScientific_of_false [DivisionRing α] (σα : OfScientific α) : {m e nm ne n : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsNat m nm → IsNat e ne → n = Nat.mul nm ((10 : ℕ) ^ ne) →
IsNat (@OfScientific.ofScientific α σα m false e : α) n
| _, _, _, _, _... | Mathlib_Tactic_NormNum_OfScientific |
α : Type u_1
inst✝ : DivisionRing α
σα : OfScientific α
n✝¹ n✝ x✝ : ℕ
σh : OfScientific.ofScientific = fun m s e => ↑(Rat.ofScientific m s e)
h : x✝ = Nat.mul n✝¹ (10 ^ n✝)
⊢ ↑(↑n✝¹ * ↑(10 ^ n✝)) = ↑n✝¹ * ↑(10 ^ n✝) | /-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.Defs
/-!
## `norm_num` plugin for scientific notation.
-/
set_option autoImplicit ... | norm_cast | theorem isNat_ofScientific_of_false [DivisionRing α] (σα : OfScientific α) : {m e nm ne n : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsNat m nm → IsNat e ne → n = Nat.mul nm ((10 : ℕ) ^ ne) →
IsNat (@OfScientific.ofScientific α σα m false e : α) n
| _, _, _, _, _... | Mathlib.Tactic.NormNum.OfScientific.35_0.6nbfMwPMGvwsEsq | theorem isNat_ofScientific_of_false [DivisionRing α] (σα : OfScientific α) : {m e nm ne n : ℕ} →
@OfScientific.ofScientific α σα = (fun m s e ↦ (Rat.ofScientific m s e : α)) →
IsNat m nm → IsNat e ne → n = Nat.mul nm ((10 : ℕ) ^ ne) →
IsNat (@OfScientific.ofScientific α σα m false e : α) n
| _, _, _, _, _... | Mathlib_Tactic_NormNum_OfScientific |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
⊢ Category.{?u.1815, u + 1} (Bundled c) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | refine' { Hom := fun X Y => @hom X Y X.str Y.str
id := fun X => @BundledHom.id c hom 𝒞 X X.str
comp := @fun X Y Z f g => @BundledHom.comp c hom 𝒞 X Y Z X.str Y.str Z.str g f
comp_id := _
id_comp := _
assoc := _ } | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_1
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
⊢ ∀ {X Y : Bundled c} (f : X ⟶ Y), 𝟙 X ≫ f = f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | intros | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_2
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
⊢ ∀ {X Y : Bundled c} (f : X ⟶ Y), f ≫ 𝟙 Y = f | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | intros | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_3
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
⊢ ∀ {W X Y Z : Bundled c} (f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z), (f ≫ g) ≫ h = f ≫ g ≫ h | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | intros | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_1
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ : Bundled c
f✝ : X✝ ⟶ Y✝
⊢ 𝟙 X✝ ≫ f✝ = f✝ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | apply 𝒞.hom_ext | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_2
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ : Bundled c
f✝ : X✝ ⟶ Y✝
⊢ f✝ ≫ 𝟙 Y✝ = f✝ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | apply 𝒞.hom_ext | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_3
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
W✝ X✝ Y✝ Z✝ : Bundled c
f✝ : W✝ ⟶ X✝
g✝ : X✝ ⟶ Y✝
h✝ : Y✝ ⟶ Z✝
⊢ (f✝ ≫ g✝) ≫ h✝ = f✝ ≫ g✝ ≫ h✝ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | apply 𝒞.hom_ext | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_1.a
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ : Bundled c
f✝ : X✝ ⟶ Y✝
⊢ toFun 𝒞 X✝.str Y✝.str (𝟙 X✝ ≫ f✝) = toFun 𝒞 X✝.str Y✝.str f✝ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | aesop_cat | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_2.a
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ : Bundled c
f✝ : X✝ ⟶ Y✝
⊢ toFun 𝒞 X✝.str Y✝.str (f✝ ≫ 𝟙 Y✝) = toFun 𝒞 X✝.str Y✝.str f✝ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | aesop_cat | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
case refine'_3.a
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
W✝ X✝ Y✝ Z✝ : Bundled c
f✝ : W✝ ⟶ X✝
g✝ : X✝ ⟶ Y✝
h✝ : Y✝ ⟶ Z✝
⊢ toFun 𝒞 W✝.str Z✝.str ((f✝ ≫ g✝) ≫ h✝) = toFun 𝒞 W✝.str Z✝.str (f✝ ≫ g✝ ≫ h✝) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | aesop_cat | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) :=... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.60_0.77N55zFlC0ujKHE | set_option synthInstance.checkSynthOrder false in
/-- Every `@BundledHom c _` defines a category with objects in `Bundled c`.
This instance generates the type-class problem `BundledHom ?m`.
Currently that is not a problem, as there are almost no instances of `BundledHom`.
-/
instance category : Category (Bundled c) | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ Z✝ : Bundled c
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := fun X => ↑X, map := fun X Y f => toFun 𝒞 X.str Y.str f }.map (f ≫ g) =
{ obj := fun X => ↑X, map := fun X Y f => toFun 𝒞 X.str Y.str f }.map f ≫
{ obj := fun X => ↑X, map... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | dsimp | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget :=
{ obj := fun X => X
map := @fun X Y f => 𝒞.toFun X.str Y.str f
map_id := fun X => 𝒞.id_toFun X.str
map_comp := fun f g => by | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.76_0.77N55zFlC0ujKHE | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ Z✝ : Bundled c
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ toFun 𝒞 X✝.str Z✝.str (f ≫ g) = toFun 𝒞 X✝.str Y✝.str f ≫ toFun 𝒞 Y✝.str Z✝.str g | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | erw [𝒞.comp_toFun] | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget :=
{ obj := fun X => X
map := @fun X Y f => 𝒞.toFun X.str Y.str f
map_id := fun X => 𝒞.id_toFun X.str
map_comp := fun f g => by dsimp; | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.76_0.77N55zFlC0ujKHE | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ Z✝ : Bundled c
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ toFun 𝒞 Y✝.str Z✝.str g ∘ toFun 𝒞 X✝.str Y✝.str f = toFun 𝒞 X✝.str Y✝.str f ≫ toFun 𝒞 Y✝.str Z✝.str g | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | rfl | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget :=
{ obj := fun X => X
map := @fun X Y f => 𝒞.toFun X.str Y.str f
map_id := fun X => 𝒞.id_toFun X.str
map_comp := fun f g => by dsimp; erw [𝒞.comp_toFu... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.76_0.77N55zFlC0ujKHE | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
⊢ ∀ {X Y : Bundled c},
Function.Injective (Functor.mk { obj := fun X => ↑X, map := fun X Y f => toFun 𝒞 X.str Y.str f }).map | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | intros | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget :=
{ obj := fun X => X
map := @fun X Y f => 𝒞.toFun X.str Y.str f
map_id := fun X => 𝒞.id_toFun X.str
map_comp := fun f g => by dsimp; erw [𝒞.comp_toFu... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.76_0.77N55zFlC0ujKHE | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
X✝ Y✝ : Bundled c
⊢ Function.Injective (Functor.mk { obj := fun X => ↑X, map := fun X Y f => toFun 𝒞 X.str Y.str f }).map | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | apply 𝒞.hom_ext | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget :=
{ obj := fun X => X
map := @fun X Y f => 𝒞.toFun X.str Y.str f
map_id := fun X => 𝒞.id_toFun X.str
map_comp := fun f g => by dsimp; erw [𝒞.comp_toFu... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.76_0.77N55zFlC0ujKHE | /-- A category given by `BundledHom` is a concrete category. -/
instance concreteCategory : ConcreteCategory.{u} (Bundled c)
where
forget | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
d : Type u → Type u
hom_d : ⦃α β : Type u⦄ → d α → d β → Type u
inst✝ : BundledHom hom_d
obj : ⦃α : Type u⦄ → c α → d α
map : {X Y : Bundled c} → (X ⟶ Y) → (Bundled.map obj X ⟶ Bundled.map obj Y)
h_map : ∀ {X Y : Bundled c} (f : X ⟶ Y), ⇑... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | intros X Y f | /-- A version of `HasForget₂.mk'` for categories defined using `@BundledHom`. -/
def mkHasForget₂ {d : Type u → Type u} {hom_d : ∀ ⦃α β : Type u⦄ (_ : d α) (_ : d β), Type u}
[BundledHom hom_d] (obj : ∀ ⦃α⦄, c α → d α)
(map : ∀ {X Y : Bundled c}, (X ⟶ Y) → (Bundled.map @obj X ⟶ (Bundled.map @obj Y)))
(h_map... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.91_0.77N55zFlC0ujKHE | /-- A version of `HasForget₂.mk'` for categories defined using `@BundledHom`. -/
def mkHasForget₂ {d : Type u → Type u} {hom_d : ∀ ⦃α β : Type u⦄ (_ : d α) (_ : d β), Type u}
[BundledHom hom_d] (obj : ∀ ⦃α⦄, c α → d α)
(map : ∀ {X Y : Bundled c}, (X ⟶ Y) → (Bundled.map @obj X ⟶ (Bundled.map @obj Y)))
(h_map... | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
c : Type u → Type u
hom : ⦃α β : Type u⦄ → c α → c β → Type u
𝒞 : BundledHom hom
d : Type u → Type u
hom_d : ⦃α β : Type u⦄ → d α → d β → Type u
inst✝ : BundledHom hom_d
obj : ⦃α : Type u⦄ → c α → d α
map : {X Y : Bundled c} → (X ⟶ Y) → (Bundled.map obj X ⟶ Bundled.map obj Y)
h_map : ∀ {X Y : Bundled c} (f : X ⟶ Y), ⇑... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
#align_import category_theory.concrete_c... | rw [heq_eq_eq, forget_map_eq_coe, forget_map_eq_coe, h_map f] | /-- A version of `HasForget₂.mk'` for categories defined using `@BundledHom`. -/
def mkHasForget₂ {d : Type u → Type u} {hom_d : ∀ ⦃α β : Type u⦄ (_ : d α) (_ : d β), Type u}
[BundledHom hom_d] (obj : ∀ ⦃α⦄, c α → d α)
(map : ∀ {X Y : Bundled c}, (X ⟶ Y) → (Bundled.map @obj X ⟶ (Bundled.map @obj Y)))
(h_map... | Mathlib.CategoryTheory.ConcreteCategory.BundledHom.91_0.77N55zFlC0ujKHE | /-- A version of `HasForget₂.mk'` for categories defined using `@BundledHom`. -/
def mkHasForget₂ {d : Type u → Type u} {hom_d : ∀ ⦃α β : Type u⦄ (_ : d α) (_ : d β), Type u}
[BundledHom hom_d] (obj : ∀ ⦃α⦄, c α → d α)
(map : ∀ {X Y : Bundled c}, (X ⟶ Y) → (Bundled.map @obj X ⟶ (Bundled.map @obj Y)))
(h_map... | Mathlib_CategoryTheory_ConcreteCategory_BundledHom |
M : Type ?u.45390
N : Type ?u.45393
α : Type ?u.45396
β : Type ?u.45406
inst✝⁸ : MeasurableSpace M
inst✝⁷ : MeasurableSpace N
inst✝⁶ : MeasurableSpace α
μ : Measure α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : SMul M α
inst✝³ : MeasurableSMul M α
inst✝² : SMulInvariantMeasure M α μ
inst✝¹ : SMul N β
inst✝ : ContinuousConstSM... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Group.Action
import Mathlib.GroupTheory.GroupAction.DomAct.Basic
/-!
# Action of `DomMulA... | rintro _ _ ⟨_⟩ | instance [SMul N β] [ContinuousConstSMul N β] : SMulCommClass Mᵈᵐᵃ N (α →ₘ[μ] β) where
smul_comm := by | Mathlib.MeasureTheory.Function.AEEqFun.DomAct.56_0.VhVS4JnFyP2Nti0 | instance [SMul N β] [ContinuousConstSMul N β] : SMulCommClass Mᵈᵐᵃ N (α →ₘ[μ] β) where
smul_comm | Mathlib_MeasureTheory_Function_AEEqFun_DomAct |
case mk
M : Type ?u.45390
N : Type ?u.45393
α : Type ?u.45396
β : Type ?u.45406
inst✝⁸ : MeasurableSpace M
inst✝⁷ : MeasurableSpace N
inst✝⁶ : MeasurableSpace α
μ : Measure α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : SMul M α
inst✝³ : MeasurableSMul M α
inst✝² : SMulInvariantMeasure M α μ
inst✝¹ : SMul N β
inst✝ : Continuou... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Group.Action
import Mathlib.GroupTheory.GroupAction.DomAct.Basic
/-!
# Action of `DomMulA... | rfl | instance [SMul N β] [ContinuousConstSMul N β] : SMulCommClass Mᵈᵐᵃ N (α →ₘ[μ] β) where
smul_comm := by rintro _ _ ⟨_⟩; | Mathlib.MeasureTheory.Function.AEEqFun.DomAct.56_0.VhVS4JnFyP2Nti0 | instance [SMul N β] [ContinuousConstSMul N β] : SMulCommClass Mᵈᵐᵃ N (α →ₘ[μ] β) where
smul_comm | Mathlib_MeasureTheory_Function_AEEqFun_DomAct |
M : Type ?u.49886
N : Type ?u.49889
α : Type ?u.49892
β : Type ?u.49902
inst✝⁸ : MeasurableSpace M
inst✝⁷ : MeasurableSpace N
inst✝⁶ : MeasurableSpace α
μ : Measure α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : SMul M α
inst✝³ : MeasurableSMul M α
inst✝² : SMulInvariantMeasure M α μ
inst✝¹ : AddMonoid β
inst✝ : ContinuousAdd ... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Group.Action
import Mathlib.GroupTheory.GroupAction.DomAct.Basic
/-!
# Action of `DomMulA... | rintro _ ⟨⟩ ⟨⟩ | instance [AddMonoid β] [ContinuousAdd β] : DistribSMul Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_add := by | Mathlib.MeasureTheory.Function.AEEqFun.DomAct.72_0.VhVS4JnFyP2Nti0 | instance [AddMonoid β] [ContinuousAdd β] : DistribSMul Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_add | Mathlib_MeasureTheory_Function_AEEqFun_DomAct |
case mk.mk
M : Type ?u.49886
N : Type ?u.49889
α : Type ?u.49892
β : Type ?u.49902
inst✝⁸ : MeasurableSpace M
inst✝⁷ : MeasurableSpace N
inst✝⁶ : MeasurableSpace α
μ : Measure α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : SMul M α
inst✝³ : MeasurableSMul M α
inst✝² : SMulInvariantMeasure M α μ
inst✝¹ : AddMonoid β
inst✝ : Con... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Group.Action
import Mathlib.GroupTheory.GroupAction.DomAct.Basic
/-!
# Action of `DomMulA... | rfl | instance [AddMonoid β] [ContinuousAdd β] : DistribSMul Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_add := by rintro _ ⟨⟩ ⟨⟩; | Mathlib.MeasureTheory.Function.AEEqFun.DomAct.72_0.VhVS4JnFyP2Nti0 | instance [AddMonoid β] [ContinuousAdd β] : DistribSMul Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_add | Mathlib_MeasureTheory_Function_AEEqFun_DomAct |
M : Type ?u.61364
N : Type ?u.61367
α : Type ?u.61370
β : Type ?u.61380
inst✝⁹ : MeasurableSpace M
inst✝⁸ : MeasurableSpace N
inst✝⁷ : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : Monoid M
inst✝⁴ : MulAction M α
inst✝³ : MeasurableSMul M α
inst✝² : SMulInvariantMeasure M α μ
inst✝¹ : Monoid β
in... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Group.Action
import Mathlib.GroupTheory.GroupAction.DomAct.Basic
/-!
# Action of `DomMulA... | rintro _ ⟨⟩ ⟨⟩ | instance [Monoid β] [ContinuousMul β] : MulDistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_one _ := rfl
smul_mul := by | Mathlib.MeasureTheory.Function.AEEqFun.DomAct.88_0.VhVS4JnFyP2Nti0 | instance [Monoid β] [ContinuousMul β] : MulDistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_one _ | Mathlib_MeasureTheory_Function_AEEqFun_DomAct |
case mk.mk
M : Type ?u.61364
N : Type ?u.61367
α : Type ?u.61370
β : Type ?u.61380
inst✝⁹ : MeasurableSpace M
inst✝⁸ : MeasurableSpace N
inst✝⁷ : MeasurableSpace α
μ : Measure α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : Monoid M
inst✝⁴ : MulAction M α
inst✝³ : MeasurableSMul M α
inst✝² : SMulInvariantMeasure M α μ
inst✝¹ : ... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Group.Action
import Mathlib.GroupTheory.GroupAction.DomAct.Basic
/-!
# Action of `DomMulA... | rfl | instance [Monoid β] [ContinuousMul β] : MulDistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_one _ := rfl
smul_mul := by rintro _ ⟨⟩ ⟨⟩; | Mathlib.MeasureTheory.Function.AEEqFun.DomAct.88_0.VhVS4JnFyP2Nti0 | instance [Monoid β] [ContinuousMul β] : MulDistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β) where
smul_one _ | Mathlib_MeasureTheory_Function_AEEqFun_DomAct |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε : α
ε0 : 0 < ε
a₁ a₂ b₁ b₂ : β
h₁ : abv (a₁ - b₁) < ε / 2
h₂ : abv (a₂ - b₂) < ε / 2
⊢ abv (a₁ + a₂ - (b₁ + b₂)) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa [add_halves, sub_eq_add_neg, add_comm, add_left_comm, add_assoc] using
lt_of_le_of_lt (abv_add abv _ _) (add_lt_add h₁ h₂) | theorem rat_add_continuous_lemma {ε : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ →
abv (a₁ + a₂ - (b₁ + b₂)) < ε :=
⟨ε / 2, half_pos ε0, fun {a₁ a₂ b₁ b₂} h₁ h₂ => by
| Mathlib.Data.Real.CauSeq.52_0.b72JnnMmhSc5wVP | theorem rat_add_continuous_lemma {ε : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ →
abv (a₁ + a₂ - (b₁ + b₂)) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
⊢ ∃ δ > 0,
∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _) | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
| Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
K0 : 0 < max 1 (max K₁ K₂)
⊢ ∃ δ > 0,
∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have εK := div_pos (half_pos ε0) K0 | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
| Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
K0 : 0 < max 1 (max K₁ K₂)
εK : 0 < ε / 2 / max 1 (max K₁ K₂)
⊢ ∃ δ > 0,
∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ *... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' ⟨_, εK, fun {a₁ a₂ b₁ b₂} ha₁ hb₂ h₁ h₂ => _⟩ | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
K0 : 0 < max 1 (max K₁ K₂)
εK : 0 < ε / 2 / max 1 (max K₁ K₂)
a₁ a₂ b₁ b₂ : β
ha₁ : abv a₁ < K₁
hb₂ : abv b₂ < K₂
h₁ : abv (a₁ - b₁) < ε / 2 / max 1 (max K₁ K₂)
h₂ : abv (a₂ - b₂) < ε /... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | replace ha₁ := lt_of_lt_of_le ha₁ (le_trans (le_max_left _ K₂) (le_max_right 1 _)) | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
K0 : 0 < max 1 (max K₁ K₂)
εK : 0 < ε / 2 / max 1 (max K₁ K₂)
a₁ a₂ b₁ b₂ : β
hb₂ : abv b₂ < K₂
h₁ : abv (a₁ - b₁) < ε / 2 / max 1 (max K₁ K₂)
h₂ : abv (a₂ - b₂) < ε / 2 / max 1 (max K₁... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | replace hb₂ := lt_of_lt_of_le hb₂ (le_trans (le_max_right K₁ _) (le_max_right 1 _)) | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
K0 : 0 < max 1 (max K₁ K₂)
εK : 0 < ε / 2 / max 1 (max K₁ K₂)
a₁ a₂ b₁ b₂ : β
h₁ : abv (a₁ - b₁) < ε / 2 / max 1 (max K₁ K₂)
h₂ : abv (a₂ - b₂) < ε / 2 / max 1 (max K₁ K₂)
ha₁ : abv a₁ ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | set M := max 1 (max K₁ K₂) | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
a₁ a₂ b₁ b₂ : β
M : α := max 1 (max K₁ K₂)
K0 : 0 < M
εK : 0 < ε / 2 / M
h₁ : abv (a₁ - b₁) < ε / 2 / M
h₂ : abv (a₂ - b₂) < ε / 2 / M
ha₁ : abv a₁ < M
hb₂ : abv b₂ < M
⊢ abv (a₁ * a₂ -... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have : abv (a₁ - b₁) * abv b₂ + abv (a₂ - b₂) * abv a₁ < ε / 2 / M * M + ε / 2 / M * M := by
gcongr | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
a₁ a₂ b₁ b₂ : β
M : α := max 1 (max K₁ K₂)
K0 : 0 < M
εK : 0 < ε / 2 / M
h₁ : abv (a₁ - b₁) < ε / 2 / M
h₂ : abv (a₂ - b₂) < ε / 2 / M
ha₁ : abv a₁ < M
hb₂ : abv b₂ < M
⊢ abv (a₁ - b₁) ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | gcongr | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
a₁ a₂ b₁ b₂ : β
M : α := max 1 (max K₁ K₂)
K0 : 0 < M
εK : 0 < ε / 2 / M
h₁ : abv (a₁ - b₁) < ε / 2 / M
h₂ : abv (a₂ - b₂) < ε / 2 / M
ha₁ : abv a₁ < M
hb₂ : abv b₂ < M
this : abv (a₁ -... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [← abv_mul abv, mul_comm, div_mul_cancel _ (ne_of_gt K0), ← abv_mul abv, add_halves] at this | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K₁ K₂ : α
ε0 : 0 < ε
a₁ a₂ b₁ b₂ : β
M : α := max 1 (max K₁ K₂)
K0 : 0 < M
εK : 0 < ε / 2 / M
h₁ : abv (a₁ - b₁) < ε / 2 / M
h₂ : abv (a₂ - b₂) < ε / 2 / M
ha₁ : abv a₁ < M
hb₂ : abv b₂ < M
this : abv ((a₁ ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa [sub_eq_add_neg, mul_add, add_mul, add_left_comm] using
lt_of_le_of_lt (abv_add abv _ _) this | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by
have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
have εK := div_pos (half_p... | Mathlib.Data.Real.CauSeq.60_0.b72JnnMmhSc5wVP | theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
⊢ ∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩ | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
| Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
⊢ abv (a⁻¹ - b⁻¹) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have a0 := K0.trans_le ha | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩
| Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
⊢ abv (a⁻... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have b0 := K0.trans_le hb | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩
have a0 :=... | Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
b0 : 0 < ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv,
abv_inv abv, abv_sub abv] | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩
have a0 :=... | Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
b0 : 0 < ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right _ b0.le) a0.le | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩
have a0 :=... | Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
b0 : 0 < ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [mul_assoc, inv_mul_cancel_right₀ b0.ne', ← mul_assoc, mul_inv_cancel a0.ne', one_mul] | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩
have a0 :=... | Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
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