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 |
|---|---|---|---|---|---|---|
b x y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
hx : x < 0
⊢ b ^ logb b x = -x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [rpow_logb_eq_abs b_pos b_ne_one (ne_of_lt hx)] | theorem rpow_logb_of_neg (hx : x < 0) : b ^ logb b x = -x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.146_0.egNyp4fdqSCAE7f | theorem rpow_logb_of_neg (hx : x < 0) : b ^ logb b x = -x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
hx : x < 0
⊢ |x| = -x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact abs_of_neg hx | theorem rpow_logb_of_neg (hx : x < 0) : b ^ logb b x = -x := by
rw [rpow_logb_eq_abs b_pos b_ne_one (ne_of_lt hx)]
| Mathlib.Analysis.SpecialFunctions.Log.Base.146_0.egNyp4fdqSCAE7f | theorem rpow_logb_of_neg (hx : x < 0) : b ^ logb b x = -x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
⊢ SurjOn (logb b) (Iio 0) univ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | intro x _ | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.163_0.egNyp4fdqSCAE7f | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x✝ y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
x : ℝ
a✝ : x ∈ univ
⊢ x ∈ logb b '' Iio 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | use -b ^ x | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ := by
intro x _
| Mathlib.Analysis.SpecialFunctions.Log.Base.163_0.egNyp4fdqSCAE7f | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h
b x✝ y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
x : ℝ
a✝ : x ∈ univ
⊢ -b ^ x ∈ Iio 0 ∧ logb b (-b ^ x) = x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | constructor | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ := by
intro x _
use -b ^ x
| Mathlib.Analysis.SpecialFunctions.Log.Base.163_0.egNyp4fdqSCAE7f | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h.left
b x✝ y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
x : ℝ
a✝ : x ∈ univ
⊢ -b ^ x ∈ Iio 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp only [Right.neg_neg_iff, Set.mem_Iio] | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ := by
intro x _
use -b ^ x
constructor
· | Mathlib.Analysis.SpecialFunctions.Log.Base.163_0.egNyp4fdqSCAE7f | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h.left
b x✝ y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
x : ℝ
a✝ : x ∈ univ
⊢ 0 < b ^ x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | apply rpow_pos_of_pos b_pos | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ := by
intro x _
use -b ^ x
constructor
· simp only [Right.neg_neg_iff, Set.mem_Iio]
| Mathlib.Analysis.SpecialFunctions.Log.Base.163_0.egNyp4fdqSCAE7f | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h.right
b x✝ y : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
x : ℝ
a✝ : x ∈ univ
⊢ logb b (-b ^ x) = x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_neg_eq_logb, logb_rpow b_pos b_ne_one] | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ := by
intro x _
use -b ^ x
constructor
· simp only [Right.neg_neg_iff, Set.mem_Iio]
apply rpow_pos_of_pos b_pos
· | Mathlib.Analysis.SpecialFunctions.Log.Base.163_0.egNyp4fdqSCAE7f | theorem surjOn_logb' : SurjOn (logb b) (Iio 0) univ | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
⊢ 0 < b | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | linarith | private theorem b_pos : 0 < b := by | Mathlib.Analysis.SpecialFunctions.Log.Base.178_0.egNyp4fdqSCAE7f | private theorem b_pos : 0 < b | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
⊢ b ≠ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | linarith | private theorem b_ne_one' : b ≠ 1 := by | Mathlib.Analysis.SpecialFunctions.Log.Base.181_0.egNyp4fdqSCAE7f | private theorem b_ne_one' : b ≠ 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
h : 0 < x
h₁ : 0 < y
⊢ logb b x ≤ logb b y ↔ x ≤ y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb, logb, div_le_div_right (log_pos hb), log_le_log_iff h h₁] | @[simp]
theorem logb_le_logb (h : 0 < x) (h₁ : 0 < y) : logb b x ≤ logb b y ↔ x ≤ y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.183_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_le_logb (h : 0 < x) (h₁ : 0 < y) : logb b x ≤ logb b y ↔ x ≤ y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
h : 0 < x
hxy : x ≤ y
⊢ 0 < y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | linarith | @[gcongr]
theorem logb_le_logb_of_le (h : 0 < x) (hxy : x ≤ y) : logb b x ≤ logb b y :=
(logb_le_logb hb h (by | Mathlib.Analysis.SpecialFunctions.Log.Base.188_0.egNyp4fdqSCAE7f | @[gcongr]
theorem logb_le_logb_of_le (h : 0 < x) (hxy : x ≤ y) : logb b x ≤ logb b y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
hxy : x < y
⊢ logb b x < logb b y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb, logb, div_lt_div_right (log_pos hb)] | @[gcongr]
theorem logb_lt_logb (hx : 0 < x) (hxy : x < y) : logb b x < logb b y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.192_0.egNyp4fdqSCAE7f | @[gcongr]
theorem logb_lt_logb (hx : 0 < x) (hxy : x < y) : logb b x < logb b y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
hxy : x < y
⊢ log x < log y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact log_lt_log hx hxy | @[gcongr]
theorem logb_lt_logb (hx : 0 < x) (hxy : x < y) : logb b x < logb b y := by
rw [logb, logb, div_lt_div_right (log_pos hb)]
| Mathlib.Analysis.SpecialFunctions.Log.Base.192_0.egNyp4fdqSCAE7f | @[gcongr]
theorem logb_lt_logb (hx : 0 < x) (hxy : x < y) : logb b x < logb b y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
hy : 0 < y
⊢ logb b x < logb b y ↔ x < y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb, logb, div_lt_div_right (log_pos hb)] | @[simp]
theorem logb_lt_logb_iff (hx : 0 < x) (hy : 0 < y) : logb b x < logb b y ↔ x < y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.198_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_lt_logb_iff (hx : 0 < x) (hy : 0 < y) : logb b x < logb b y ↔ x < y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
hy : 0 < y
⊢ log x < log y ↔ x < y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact log_lt_log_iff hx hy | @[simp]
theorem logb_lt_logb_iff (hx : 0 < x) (hy : 0 < y) : logb b x < logb b y ↔ x < y := by
rw [logb, logb, div_lt_div_right (log_pos hb)]
| Mathlib.Analysis.SpecialFunctions.Log.Base.198_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_lt_logb_iff (hx : 0 < x) (hy : 0 < y) : logb b x < logb b y ↔ x < y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
⊢ logb b x ≤ y ↔ x ≤ b ^ y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hx] | theorem logb_le_iff_le_rpow (hx : 0 < x) : logb b x ≤ y ↔ x ≤ b ^ y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.204_0.egNyp4fdqSCAE7f | theorem logb_le_iff_le_rpow (hx : 0 < x) : logb b x ≤ y ↔ x ≤ b ^ y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
⊢ logb b x < y ↔ x < b ^ y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_lt_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hx] | theorem logb_lt_iff_lt_rpow (hx : 0 < x) : logb b x < y ↔ x < b ^ y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.208_0.egNyp4fdqSCAE7f | theorem logb_lt_iff_lt_rpow (hx : 0 < x) : logb b x < y ↔ x < b ^ y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hy : 0 < y
⊢ x ≤ logb b y ↔ b ^ x ≤ y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hy] | theorem le_logb_iff_rpow_le (hy : 0 < y) : x ≤ logb b y ↔ b ^ x ≤ y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.212_0.egNyp4fdqSCAE7f | theorem le_logb_iff_rpow_le (hy : 0 < y) : x ≤ logb b y ↔ b ^ x ≤ y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hy : 0 < y
⊢ x < logb b y ↔ b ^ x < y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_lt_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hy] | theorem lt_logb_iff_rpow_lt (hy : 0 < y) : x < logb b y ↔ b ^ x < y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.216_0.egNyp4fdqSCAE7f | theorem lt_logb_iff_rpow_lt (hy : 0 < y) : x < logb b y ↔ b ^ x < y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
⊢ 0 < logb b x ↔ 1 < x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← @logb_one b] | theorem logb_pos_iff (hx : 0 < x) : 0 < logb b x ↔ 1 < x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.220_0.egNyp4fdqSCAE7f | theorem logb_pos_iff (hx : 0 < x) : 0 < logb b x ↔ 1 < x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
⊢ logb b 1 < logb b x ↔ 1 < x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_lt_logb_iff hb zero_lt_one hx] | theorem logb_pos_iff (hx : 0 < x) : 0 < logb b x ↔ 1 < x := by
rw [← @logb_one b]
| Mathlib.Analysis.SpecialFunctions.Log.Base.220_0.egNyp4fdqSCAE7f | theorem logb_pos_iff (hx : 0 < x) : 0 < logb b x ↔ 1 < x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 1 < x
⊢ 0 < logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_pos_iff hb (lt_trans zero_lt_one hx)] | theorem logb_pos (hx : 1 < x) : 0 < logb b x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.225_0.egNyp4fdqSCAE7f | theorem logb_pos (hx : 1 < x) : 0 < logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 1 < x
⊢ 1 < x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact hx | theorem logb_pos (hx : 1 < x) : 0 < logb b x := by
rw [logb_pos_iff hb (lt_trans zero_lt_one hx)]
| Mathlib.Analysis.SpecialFunctions.Log.Base.225_0.egNyp4fdqSCAE7f | theorem logb_pos (hx : 1 < x) : 0 < logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
h : 0 < x
⊢ logb b x < 0 ↔ x < 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← logb_one] | theorem logb_neg_iff (h : 0 < x) : logb b x < 0 ↔ x < 1 := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.230_0.egNyp4fdqSCAE7f | theorem logb_neg_iff (h : 0 < x) : logb b x < 0 ↔ x < 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
h : 0 < x
⊢ logb b x < logb ?m.52635 1 ↔ x < 1
b x y : ℝ hb : 1 < b h : 0 < x ⊢ ℝ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact logb_lt_logb_iff hb h zero_lt_one | theorem logb_neg_iff (h : 0 < x) : logb b x < 0 ↔ x < 1 := by
rw [← logb_one]
| Mathlib.Analysis.SpecialFunctions.Log.Base.230_0.egNyp4fdqSCAE7f | theorem logb_neg_iff (h : 0 < x) : logb b x < 0 ↔ x < 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
⊢ 0 ≤ logb b x ↔ 1 ≤ x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← not_lt, logb_neg_iff hb hx, not_lt] | theorem logb_nonneg_iff (hx : 0 < x) : 0 ≤ logb b x ↔ 1 ≤ x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.239_0.egNyp4fdqSCAE7f | theorem logb_nonneg_iff (hx : 0 < x) : 0 ≤ logb b x ↔ 1 ≤ x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 < x
⊢ logb b x ≤ 0 ↔ x ≤ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← not_lt, logb_pos_iff hb hx, not_lt] | theorem logb_nonpos_iff (hx : 0 < x) : logb b x ≤ 0 ↔ x ≤ 1 := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.247_0.egNyp4fdqSCAE7f | theorem logb_nonpos_iff (hx : 0 < x) : logb b x ≤ 0 ↔ x ≤ 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
hx : 0 ≤ x
⊢ logb b x ≤ 0 ↔ x ≤ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rcases hx.eq_or_lt with (rfl | hx) | theorem logb_nonpos_iff' (hx : 0 ≤ x) : logb b x ≤ 0 ↔ x ≤ 1 := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.251_0.egNyp4fdqSCAE7f | theorem logb_nonpos_iff' (hx : 0 ≤ x) : logb b x ≤ 0 ↔ x ≤ 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inl
b y : ℝ
hb : 1 < b
hx : 0 ≤ 0
⊢ logb b 0 ≤ 0 ↔ 0 ≤ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp [le_refl, zero_le_one] | theorem logb_nonpos_iff' (hx : 0 ≤ x) : logb b x ≤ 0 ↔ x ≤ 1 := by
rcases hx.eq_or_lt with (rfl | hx)
· | Mathlib.Analysis.SpecialFunctions.Log.Base.251_0.egNyp4fdqSCAE7f | theorem logb_nonpos_iff' (hx : 0 ≤ x) : logb b x ≤ 0 ↔ x ≤ 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr
b x y : ℝ
hb : 1 < b
hx✝ : 0 ≤ x
hx : 0 < x
⊢ logb b x ≤ 0 ↔ x ≤ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact logb_nonpos_iff hb hx | theorem logb_nonpos_iff' (hx : 0 ≤ x) : logb b x ≤ 0 ↔ x ≤ 1 := by
rcases hx.eq_or_lt with (rfl | hx)
· simp [le_refl, zero_le_one]
| Mathlib.Analysis.SpecialFunctions.Log.Base.251_0.egNyp4fdqSCAE7f | theorem logb_nonpos_iff' (hx : 0 ≤ x) : logb b x ≤ 0 ↔ x ≤ 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
hb : 1 < b
⊢ StrictAntiOn (logb b) (Iio 0) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rintro x (hx : x < 0) y (hy : y < 0) hxy | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.265_0.egNyp4fdqSCAE7f | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x✝ y✝ : ℝ
hb : 1 < b
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ logb b y < logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← logb_abs y, ← logb_abs x] | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) := by
rintro x (hx : x < 0) y (hy : y < 0) hxy
| Mathlib.Analysis.SpecialFunctions.Log.Base.265_0.egNyp4fdqSCAE7f | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x✝ y✝ : ℝ
hb : 1 < b
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ logb b |y| < logb b |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | refine' logb_lt_logb hb (abs_pos.2 hy.ne) _ | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) := by
rintro x (hx : x < 0) y (hy : y < 0) hxy
rw [← logb_abs y, ← logb_abs x]
| Mathlib.Analysis.SpecialFunctions.Log.Base.265_0.egNyp4fdqSCAE7f | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x✝ y✝ : ℝ
hb : 1 < b
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ |y| < |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff] | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) := by
rintro x (hx : x < 0) y (hy : y < 0) hxy
rw [← logb_abs y, ← logb_abs x]
refine' logb_lt_logb hb (abs_pos.2 hy.ne) _
| Mathlib.Analysis.SpecialFunctions.Log.Base.265_0.egNyp4fdqSCAE7f | theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
⊢ b ≠ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | linarith | private theorem b_ne_one : b ≠ 1 := by | Mathlib.Analysis.SpecialFunctions.Log.Base.294_0.egNyp4fdqSCAE7f | private theorem b_ne_one : b ≠ 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
h : 0 < x
h₁ : 0 < y
⊢ logb b x ≤ logb b y ↔ y ≤ x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb, logb, div_le_div_right_of_neg (log_neg b_pos b_lt_one), log_le_log_iff h₁ h] | @[simp]
theorem logb_le_logb_of_base_lt_one (h : 0 < x) (h₁ : 0 < y) : logb b x ≤ logb b y ↔ y ≤ x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.296_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_le_logb_of_base_lt_one (h : 0 < x) (h₁ : 0 < y) : logb b x ≤ logb b y ↔ y ≤ x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hxy : x < y
⊢ logb b y < logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)] | theorem logb_lt_logb_of_base_lt_one (hx : 0 < x) (hxy : x < y) : logb b y < logb b x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.301_0.egNyp4fdqSCAE7f | theorem logb_lt_logb_of_base_lt_one (hx : 0 < x) (hxy : x < y) : logb b y < logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hxy : x < y
⊢ log x < log y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact log_lt_log hx hxy | theorem logb_lt_logb_of_base_lt_one (hx : 0 < x) (hxy : x < y) : logb b y < logb b x := by
rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)]
| Mathlib.Analysis.SpecialFunctions.Log.Base.301_0.egNyp4fdqSCAE7f | theorem logb_lt_logb_of_base_lt_one (hx : 0 < x) (hxy : x < y) : logb b y < logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hy : 0 < y
⊢ logb b x < logb b y ↔ y < x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)] | @[simp]
theorem logb_lt_logb_iff_of_base_lt_one (hx : 0 < x) (hy : 0 < y) :
logb b x < logb b y ↔ y < x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.306_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_lt_logb_iff_of_base_lt_one (hx : 0 < x) (hy : 0 < y) :
logb b x < logb b y ↔ y < x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hy : 0 < y
⊢ log y < log x ↔ y < x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact log_lt_log_iff hy hx | @[simp]
theorem logb_lt_logb_iff_of_base_lt_one (hx : 0 < x) (hy : 0 < y) :
logb b x < logb b y ↔ y < x := by
rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)]
| Mathlib.Analysis.SpecialFunctions.Log.Base.306_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_lt_logb_iff_of_base_lt_one (hx : 0 < x) (hy : 0 < y) :
logb b x < logb b y ↔ y < x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
⊢ logb b x ≤ y ↔ b ^ y ≤ x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_le_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hx] | theorem logb_le_iff_le_rpow_of_base_lt_one (hx : 0 < x) : logb b x ≤ y ↔ b ^ y ≤ x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.313_0.egNyp4fdqSCAE7f | theorem logb_le_iff_le_rpow_of_base_lt_one (hx : 0 < x) : logb b x ≤ y ↔ b ^ y ≤ x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
⊢ logb b x < y ↔ b ^ y < x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hx] | theorem logb_lt_iff_lt_rpow_of_base_lt_one (hx : 0 < x) : logb b x < y ↔ b ^ y < x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.317_0.egNyp4fdqSCAE7f | theorem logb_lt_iff_lt_rpow_of_base_lt_one (hx : 0 < x) : logb b x < y ↔ b ^ y < x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hy : 0 < y
⊢ x ≤ logb b y ↔ y ≤ b ^ x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_le_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hy] | theorem le_logb_iff_rpow_le_of_base_lt_one (hy : 0 < y) : x ≤ logb b y ↔ y ≤ b ^ x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.321_0.egNyp4fdqSCAE7f | theorem le_logb_iff_rpow_le_of_base_lt_one (hy : 0 < y) : x ≤ logb b y ↔ y ≤ b ^ x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hy : 0 < y
⊢ x < logb b y ↔ y < b ^ x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hy] | theorem lt_logb_iff_rpow_lt_of_base_lt_one (hy : 0 < y) : x < logb b y ↔ y < b ^ x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.325_0.egNyp4fdqSCAE7f | theorem lt_logb_iff_rpow_lt_of_base_lt_one (hy : 0 < y) : x < logb b y ↔ y < b ^ x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
⊢ 0 < logb b x ↔ x < 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← @logb_one b, logb_lt_logb_iff_of_base_lt_one b_pos b_lt_one zero_lt_one hx] | theorem logb_pos_iff_of_base_lt_one (hx : 0 < x) : 0 < logb b x ↔ x < 1 := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.329_0.egNyp4fdqSCAE7f | theorem logb_pos_iff_of_base_lt_one (hx : 0 < x) : 0 < logb b x ↔ x < 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hx' : x < 1
⊢ 0 < logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_pos_iff_of_base_lt_one b_pos b_lt_one hx] | theorem logb_pos_of_base_lt_one (hx : 0 < x) (hx' : x < 1) : 0 < logb b x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.333_0.egNyp4fdqSCAE7f | theorem logb_pos_of_base_lt_one (hx : 0 < x) (hx' : x < 1) : 0 < logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hx' : x < 1
⊢ x < 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact hx' | theorem logb_pos_of_base_lt_one (hx : 0 < x) (hx' : x < 1) : 0 < logb b x := by
rw [logb_pos_iff_of_base_lt_one b_pos b_lt_one hx]
| Mathlib.Analysis.SpecialFunctions.Log.Base.333_0.egNyp4fdqSCAE7f | theorem logb_pos_of_base_lt_one (hx : 0 < x) (hx' : x < 1) : 0 < logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
h : 0 < x
⊢ logb b x < 0 ↔ 1 < x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← @logb_one b, logb_lt_logb_iff_of_base_lt_one b_pos b_lt_one h zero_lt_one] | theorem logb_neg_iff_of_base_lt_one (h : 0 < x) : logb b x < 0 ↔ 1 < x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.338_0.egNyp4fdqSCAE7f | theorem logb_neg_iff_of_base_lt_one (h : 0 < x) : logb b x < 0 ↔ 1 < x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
⊢ 0 ≤ logb b x ↔ x ≤ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← not_lt, logb_neg_iff_of_base_lt_one b_pos b_lt_one hx, not_lt] | theorem logb_nonneg_iff_of_base_lt_one (hx : 0 < x) : 0 ≤ logb b x ↔ x ≤ 1 := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.346_0.egNyp4fdqSCAE7f | theorem logb_nonneg_iff_of_base_lt_one (hx : 0 < x) : 0 ≤ logb b x ↔ x ≤ 1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hx' : x ≤ 1
⊢ 0 ≤ logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_nonneg_iff_of_base_lt_one b_pos b_lt_one hx] | theorem logb_nonneg_of_base_lt_one (hx : 0 < x) (hx' : x ≤ 1) : 0 ≤ logb b x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.350_0.egNyp4fdqSCAE7f | theorem logb_nonneg_of_base_lt_one (hx : 0 < x) (hx' : x ≤ 1) : 0 ≤ logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
hx' : x ≤ 1
⊢ x ≤ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact hx' | theorem logb_nonneg_of_base_lt_one (hx : 0 < x) (hx' : x ≤ 1) : 0 ≤ logb b x := by
rw [logb_nonneg_iff_of_base_lt_one b_pos b_lt_one hx]
| Mathlib.Analysis.SpecialFunctions.Log.Base.350_0.egNyp4fdqSCAE7f | theorem logb_nonneg_of_base_lt_one (hx : 0 < x) (hx' : x ≤ 1) : 0 ≤ logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
hx : 0 < x
⊢ logb b x ≤ 0 ↔ 1 ≤ x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← not_lt, logb_pos_iff_of_base_lt_one b_pos b_lt_one hx, not_lt] | theorem logb_nonpos_iff_of_base_lt_one (hx : 0 < x) : logb b x ≤ 0 ↔ 1 ≤ x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.355_0.egNyp4fdqSCAE7f | theorem logb_nonpos_iff_of_base_lt_one (hx : 0 < x) : logb b x ≤ 0 ↔ 1 ≤ x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
⊢ StrictMonoOn (logb b) (Iio 0) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rintro x (hx : x < 0) y (hy : y < 0) hxy | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.363_0.egNyp4fdqSCAE7f | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x✝ y✝ : ℝ
b_pos : 0 < b
b_lt_one : b < 1
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ logb b x < logb b y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← logb_abs y, ← logb_abs x] | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) := by
rintro x (hx : x < 0) y (hy : y < 0) hxy
| Mathlib.Analysis.SpecialFunctions.Log.Base.363_0.egNyp4fdqSCAE7f | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x✝ y✝ : ℝ
b_pos : 0 < b
b_lt_one : b < 1
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ logb b |x| < logb b |y| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | refine' logb_lt_logb_of_base_lt_one b_pos b_lt_one (abs_pos.2 hy.ne) _ | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) := by
rintro x (hx : x < 0) y (hy : y < 0) hxy
rw [← logb_abs y, ← logb_abs x]
| Mathlib.Analysis.SpecialFunctions.Log.Base.363_0.egNyp4fdqSCAE7f | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x✝ y✝ : ℝ
b_pos : 0 < b
b_lt_one : b < 1
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ |y| < |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff] | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) := by
rintro x (hx : x < 0) y (hy : y < 0) hxy
rw [← logb_abs y, ← logb_abs x]
refine' logb_lt_logb_of_base_lt_one b_pos b_lt_one (abs_pos.2 hy.ne) _
| Mathlib.Analysis.SpecialFunctions.Log.Base.363_0.egNyp4fdqSCAE7f | theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
⊢ Tendsto (logb b) atTop atBot | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [tendsto_atTop_atBot] | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
⊢ ∀ (b_1 : ℝ), ∃ i, ∀ (a : ℝ), i ≤ a → logb b a ≤ b_1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | intro e | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e : ℝ
⊢ ∃ i, ∀ (a : ℝ), i ≤ a → logb b a ≤ e | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | use 1 ⊔ b ^ e | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
intro e
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e : ℝ
⊢ ∀ (a : ℝ), 1 ⊔ b ^ e ≤ a → logb b a ≤ e | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | intro a | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
intro e
use 1 ⊔ b ^ e
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e a : ℝ
⊢ 1 ⊔ b ^ e ≤ a → logb b a ≤ e | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp only [and_imp, sup_le_iff] | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
intro e
use 1 ⊔ b ^ e
intro a
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e a : ℝ
⊢ 1 ≤ a → b ^ e ≤ a → logb b a ≤ e | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | intro ha | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
intro e
use 1 ⊔ b ^ e
intro a
simp only [and_imp, sup_le_iff]
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e a : ℝ
ha : 1 ≤ a
⊢ b ^ e ≤ a → logb b a ≤ e | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_le_iff_le_rpow_of_base_lt_one b_pos b_lt_one] | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
intro e
use 1 ⊔ b ^ e
intro a
simp only [and_imp, sup_le_iff]
intro ha
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e a : ℝ
ha : 1 ≤ a
⊢ b ^ e ≤ a → b ^ e ≤ a
case h b x y : ℝ b_pos : 0 < b b_lt_one : b < 1 e a : ℝ ha : 1 ≤ a ⊢ 0 < a | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | tauto | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
intro e
use 1 ⊔ b ^ e
intro a
simp only [and_imp, sup_le_iff]
intro ha
rw [logb_le_iff_le_rpow_of_base_lt_one b_pos b_lt_one]
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
case h
b x y : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e a : ℝ
ha : 1 ≤ a
⊢ 0 < a | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact lt_of_lt_of_le zero_lt_one ha | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
rw [tendsto_atTop_atBot]
intro e
use 1 ⊔ b ^ e
intro a
simp only [and_imp, sup_le_iff]
intro ha
rw [logb_le_iff_le_rpow_of_base_lt_one b_pos b_lt_one]
tauto
| Mathlib.Analysis.SpecialFunctions.Log.Base.383_0.egNyp4fdqSCAE7f | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | Mathlib_Analysis_SpecialFunctions_Log_Base |
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr : 0 ≤ r
⊢ ⌊logb (↑b) r⌋ = Int.log b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | obtain rfl | hr := hr.eq_or_lt | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inl
b✝ x y : ℝ
b : ℕ
hb : 1 < b
hr : 0 ≤ 0
⊢ ⌊logb (↑b) 0⌋ = Int.log b 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_zero, Int.log_zero_right, Int.floor_zero] | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· | Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
⊢ ⌊logb (↑b) r⌋ = Int.log b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
| Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ⌊logb (↑b) r⌋ = Int.log b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | apply le_antisymm | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
| Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ⌊logb (↑b) r⌋ ≤ Int.log b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_int_cast b] | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· | Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ↑b ^ ↑⌊logb (↑b) r⌋ ≤ r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | refine' le_of_le_of_eq _ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr) | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_int_ca... | Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ↑b ^ ↑⌊logb (↑b) r⌋ ≤ ↑b ^ logb (↑b) r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact rpow_le_rpow_of_exponent_le hb1'.le (Int.floor_le _) | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_int_ca... | Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ Int.log b r ≤ ⌊logb (↑b) r⌋ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_int_cast] | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_int_ca... | Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ↑b ^ Int.log b r ≤ r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact Int.zpow_log_le_self hb hr | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_int_ca... | Mathlib.Analysis.SpecialFunctions.Log.Base.397_0.egNyp4fdqSCAE7f | theorem floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌊logb b r⌋ = Int.log b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr : 0 ≤ r
⊢ ⌈logb (↑b) r⌉ = Int.clog b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | obtain rfl | hr := hr.eq_or_lt | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inl
b✝ x y : ℝ
b : ℕ
hb : 1 < b
hr : 0 ≤ 0
⊢ ⌈logb (↑b) 0⌉ = Int.clog b 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [logb_zero, Int.clog_zero_right, Int.ceil_zero] | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· | Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
⊢ ⌈logb (↑b) r⌉ = Int.clog b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
| Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ⌈logb (↑b) r⌉ = Int.clog b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | apply le_antisymm | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
| Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ⌈logb (↑b) r⌉ ≤ Int.clog b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpow_int_cast] | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· | Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ r ≤ ↑b ^ Int.clog b r | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | refine' Int.self_le_zpow_clog hb r | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpo... | Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ Int.clog b r ≤ ⌈logb (↑b) r⌉ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [← Int.le_zpow_iff_clog_le hb hr, ← rpow_int_cast b] | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpo... | Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ r ≤ ↑b ^ ↑⌈logb (↑b) r⌉ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | refine' (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr).symm.trans_le _ | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpo... | Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
case inr.a
b✝ x y : ℝ
b : ℕ
r : ℝ
hb : 1 < b
hr✝ : 0 ≤ r
hr : 0 < r
hb1' : 1 < ↑b
⊢ ↑b ^ logb (↑b) r ≤ ↑b ^ ↑⌈logb (↑b) r⌉ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact rpow_le_rpow_of_exponent_le hb1'.le (Int.le_ceil _) | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpo... | Mathlib.Analysis.SpecialFunctions.Log.Base.410_0.egNyp4fdqSCAE7f | theorem ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
⊢ logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp_rw [logb, div_eq_zero_iff, log_eq_zero] | @[simp]
theorem logb_eq_zero : logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.423_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_eq_zero : logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
⊢ (x = 0 ∨ x = 1 ∨ x = -1) ∨ b = 0 ∨ b = 1 ∨ b = -1 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | tauto | @[simp]
theorem logb_eq_zero : logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 := by
simp_rw [logb, div_eq_zero_iff, log_eq_zero]
| Mathlib.Analysis.SpecialFunctions.Log.Base.423_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_eq_zero : logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
α : Type u_1
s : Finset α
f : α → ℝ
hf : ∀ x ∈ s, f x ≠ 0
⊢ logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | classical
induction' s using Finset.induction_on with a s ha ih
· simp
simp only [Finset.mem_insert, forall_eq_or_imp] at hf
simp [ha, ih hf.2, logb_mul hf.1 (Finset.prod_ne_zero_iff.2 hf.2)] | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.432_0.egNyp4fdqSCAE7f | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : ℝ
α : Type u_1
s : Finset α
f : α → ℝ
hf : ∀ x ∈ s, f x ≠ 0
⊢ logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | induction' s using Finset.induction_on with a s ha ih | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) := by
classical
| Mathlib.Analysis.SpecialFunctions.Log.Base.432_0.egNyp4fdqSCAE7f | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | Mathlib_Analysis_SpecialFunctions_Log_Base |
case empty
b x y : ℝ
α : Type u_1
f : α → ℝ
hf : ∀ x ∈ ∅, f x ≠ 0
⊢ logb b (∏ i in ∅, f i) = ∑ i in ∅, logb b (f i) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) := by
classical
induction' s using Finset.induction_on with a s ha ih
· | Mathlib.Analysis.SpecialFunctions.Log.Base.432_0.egNyp4fdqSCAE7f | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | Mathlib_Analysis_SpecialFunctions_Log_Base |
case insert
b x y : ℝ
α : Type u_1
f : α → ℝ
a : α
s : Finset α
ha : a ∉ s
ih : (∀ x ∈ s, f x ≠ 0) → logb b (∏ i in s, f i) = ∑ i in s, logb b (f i)
hf : ∀ x ∈ insert a s, f x ≠ 0
⊢ logb b (∏ i in insert a s, f i) = ∑ i in insert a s, logb b (f i) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp only [Finset.mem_insert, forall_eq_or_imp] at hf | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) := by
classical
induction' s using Finset.induction_on with a s ha ih
· simp
| Mathlib.Analysis.SpecialFunctions.Log.Base.432_0.egNyp4fdqSCAE7f | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | Mathlib_Analysis_SpecialFunctions_Log_Base |
case insert
b x y : ℝ
α : Type u_1
f : α → ℝ
a : α
s : Finset α
ha : a ∉ s
ih : (∀ x ∈ s, f x ≠ 0) → logb b (∏ i in s, f i) = ∑ i in s, logb b (f i)
hf : f a ≠ 0 ∧ ∀ a ∈ s, f a ≠ 0
⊢ logb b (∏ i in insert a s, f i) = ∑ i in insert a s, logb b (f i) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp [ha, ih hf.2, logb_mul hf.1 (Finset.prod_ne_zero_iff.2 hf.2)] | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) := by
classical
induction' s using Finset.induction_on with a s ha ih
· simp
simp only [Finset.mem_insert, forall_eq_or_imp] at hf
| Mathlib.Analysis.SpecialFunctions.Log.Base.432_0.egNyp4fdqSCAE7f | theorem logb_prod {α : Type*} (s : Finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0) :
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
⊢ ∀ x ≥ x₀, P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | suffices ∀ n : ℕ, ∀ x ∈ Set.Ico x₀ (r ^ (n + 1) * x₀), P x by
intro x hx
have hx' : 0 < x / x₀ := div_pos (hx₀.trans_le hx) hx₀
refine this ⌊logb r (x / x₀)⌋₊ x ?_
rw [mem_Ico, ← div_lt_iff hx₀, ← rpow_nat_cast, ← logb_lt_iff_lt_rpow hr hx', Nat.cast_add,
Nat.cast_one]
exact ⟨hx, Nat.lt_floor_... | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
this : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
⊢ ∀ x ≥ x₀, P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | intro x hx | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
this : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
x : ℝ
hx : x ≥ x₀
⊢ P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | have hx' : 0 < x / x₀ := div_pos (hx₀.trans_le hx) hx₀ | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
this : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
x : ℝ
hx : x ≥ x₀
hx' : 0 < x / x₀
⊢ P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | refine this ⌊logb r (x / x₀)⌋₊ x ?_ | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
this : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
x : ℝ
hx : x ≥ x₀
hx' : 0 < x / x₀
⊢ x ∈ Ico x₀ (r ^ (⌊logb r (x / x₀)⌋₊ + 1) * x₀) | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | rw [mem_Ico, ← div_lt_iff hx₀, ← rpow_nat_cast, ← logb_lt_iff_lt_rpow hr hx', Nat.cast_add,
Nat.cast_one] | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
this : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
x : ℝ
hx : x ≥ x₀
hx' : 0 < x / x₀
⊢ x₀ ≤ x ∧ logb r (x / x₀) < ↑⌊logb r (x / x₀)⌋₊ + 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact ⟨hx, Nat.lt_floor_add_one _⟩ | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
⊢ ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | intro n | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
n : ℕ
⊢ ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | induction n with
| zero => simpa using base
| succ n ih =>
exact fun x hx => (Ico_subset_Ico_union_Ico hx).elim (ih x) (step (n + 1) (by simp) ih _) | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
n : ℕ
⊢ ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | induction n with
| zero => simpa using base
| succ n ih =>
exact fun x hx => (Ico_subset_Ico_union_Ico hx).elim (ih x) (step (n + 1) (by simp) ih _) | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
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