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 |
|---|---|---|---|---|---|---|
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
z : ℝ
hz : z ∈ Ioo x w
⊢ deriv f... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | apply ne_of_lt | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case h
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
z : ℝ
hz : z ∈ Ioo x w
⊢ ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2 | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case neg.intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, deriv f b < (f y - f w) / (y - w) := by
apply StrictMonoOn.exists_deriv_lt_slope_aux _ hwy _ _
· refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
· exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
· intro z hz
rw [← hw]
apply ne_of_gt
... | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | apply StrictMonoOn.exists_deriv_lt_slope_aux _ hwy _ _ | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' hf.mono (Icc_subset_Icc hxw.le le_rfl) | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl) | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | intro z hz | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rw [← hw] | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | apply ne_of_gt | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case h
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1 | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' ⟨a, ⟨hxa, haw.trans hwy⟩, _⟩ | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | simp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢ | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have : deriv f a * (y - w) < deriv f b * (y - w) := by
apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _
· exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
· rw [← hw]
exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _ | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb) | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rw [← hw] | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | linarith | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
· apply StrictMonoOn.exists_deriv_lt_slope_aux hf hx... | Mathlib.Analysis.Calculus.MeanValue.1082_0.ReDurB0qNQAwk9I | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
⊢ (f y - f x) / (y - x) < (f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hxzD : Icc x z ⊆ D := hD.ordConnected.out hx hz | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
⊢ (f y - f... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hxyD : Icc x y ⊆ D := (Icc_subset_Icc_right hyz.le).trans hxzD | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hxyD' : Ioo x y ⊆ interior D :=
subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩ | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hyzD : Icc y z ⊆ D := (Icc_subset_Icc_left hxy.le).trans hxzD | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hyzD' : Ioo y z ⊆ interior D :=
subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD⟩ | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact StrictMonoOn.exists_slope_lt_deriv (hf.mono hxyD) hxy (hf'.mono hxyD') | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD :... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b < (f z - f y) / (z - y) | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact StrictMonoOn.exists_deriv_lt_slope (hf.mono hyzD) hyz (hf'.mono hyzD') | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | apply ha.trans (lt_trans _ hb) | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact hf' (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb) | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib.Analysis.Calculus.MeanValue.1115_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. -/
theorem StrictMonoOn.strictConvexOn_of_... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
h_anti : StrictAntiOn (deriv f) (interior D)
⊢ StrictMonoOn (deriv (-f)) (interior D) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | simpa only [← deriv.neg] using h_anti.neg | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f'` is strictly antitone on the
interior, then `f` is strictly concave on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict antitonicity of `f'`. -/
theorem StrictAntiOn.strictConcaveOn_of... | Mathlib.Analysis.Calculus.MeanValue.1140_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f'` is strictly antitone on the
interior, then `f` is strictly concave on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict antitonicity of `f'`. -/
theorem StrictAntiOn.strictConcaveOn_of... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonneg : ∀ x ∈ interior D, 0 ≤ de... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [interior_interior] | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)) ... | Mathlib.Analysis.Calculus.MeanValue.1181_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)) ... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonneg : ∀ x ∈ interior D, 0 ≤ de... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [interior_interior] | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)) ... | Mathlib.Analysis.Calculus.MeanValue.1181_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. -/
theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)) ... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [interior_interior] | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)... | Mathlib.Analysis.Calculus.MeanValue.1191_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [interior_interior] | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)... | Mathlib.Analysis.Calculus.MeanValue.1191_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. -/
theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
(hf' : DifferentiableOn ℝ f (interior D)... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, 0 < deriv^[2] f x
⊢ ∀ x ∈ interior (interior D), 0 < deriv (deriv f) x | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [interior_interior] | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on the
interior, then `f` is strictly convex on `D`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly positive, except at at most one point. -/
theorem... | Mathlib.Analysis.Calculus.MeanValue.1201_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on the
interior, then `f` is strictly convex on `D`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly positive, except at at most one point. -/
theorem... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0
⊢ ∀ x ∈ interior (interior D), deriv (deriv f) x < 0 | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [interior_interior] | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on the
interior, then `f` is strictly concave on `D`.
Note that we don't require twice differentiability explicitly as it already implied by the second
derivative being strictly negative, except at at most one point. -/
theorem s... | Mathlib.Analysis.Calculus.MeanValue.1215_0.ReDurB0qNQAwk9I | /-- If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on the
interior, then `f` is strictly concave on `D`.
Note that we don't require twice differentiability explicitly as it already implied by the second
derivative being strictly negative, except at at most one point. -/
theorem s... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
⊢ ∃ z ∈ segment ℝ x y, f y - f x = (f' z) (y - x) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | set g : ℝ → E := fun t => AffineMap.lineMap x y t | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
⊢ ∃ z ∈... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | set I := Icc (0 : ℝ) 1 | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
I : Set... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hsub : Ioo (0 : ℝ) 1 ⊆ I := Ioo_subset_Icc_self | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
I : Set... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hmaps : MapsTo g I s := hs.mapsTo_lineMap xs ys | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
I : Set... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hfg : ∀ t ∈ I, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) I t := fun t ht =>
(hf _ (hmaps ht)).comp_hasDerivWithinAt t AffineMap.hasDerivWithinAt_lineMap hmaps | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
I : Set... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hMVT : ∃ t ∈ Ioo (0 : ℝ) 1, f' (g t) (y - x) = (f (g 1) - f (g 0)) / (1 - 0) := by
refine' exists_hasDerivAt_eq_slope (f ∘ g) _ (by norm_num) _ _
· exact fun t Ht => (hfg t Ht).continuousWithinAt
· exact fun t Ht => (hfg t <| hsub Ht).hasDerivAt (Icc_mem_nhds Ht.1 Ht.2) | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
I : Set... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' exists_hasDerivAt_eq_slope (f ∘ g) _ (by norm_num) _ _ | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
I : Set... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | norm_num | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case refine'_1
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact fun t Ht => (hfg t Ht).continuousWithinAt | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case refine'_2
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact fun t Ht => (hfg t <| hsub Ht).hasDerivAt (Icc_mem_nhds Ht.1 Ht.2) | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineMap x y) t
I : Set... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rcases hMVT with ⟨t, Ht, hMVT'⟩ | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineM... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rw [segment_eq_image_lineMap, bex_image_iff] | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineM... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine ⟨t, hsub Ht, ?_⟩ | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → ℝ
s : Set E
x y : E
f' : E → E →L[ℝ] ℝ
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t => (AffineMap.lineM... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | simpa using hMVT'.symm | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) := by
-- Use `g = AffineMap.... | Mathlib.Analysis.Calculus.MeanValue.1304_0.ReDurB0qNQAwk9I | /-- Lagrange's **Mean Value Theorem**, applied to convex domains. -/
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' :... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' isLittleO_iff.mpr fun c hc => Metric.eventually_nhds_iff_ball.mpr _ | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' :... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rcases Metric.mem_nhds_iff.mp (inter_mem hder (hcont <| ball_mem_nhds _ hc)) with ⟨ε, ε0, hε⟩ | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' ⟨ε, ε0, _⟩ | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rintro ⟨a, b⟩ h | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro.mk
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rw [← ball_prod_same, prod_mk_mem_set_prod_eq] at h | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro.mk
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have hf' : ∀ x' ∈ ball x ε, ‖f' x' - f' x‖ ≤ c := fun x' H' => by
rw [← dist_eq_norm]
exact le_of_lt (hε H').2 | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' :... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rw [← dist_eq_norm] | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' :... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact le_of_lt (hε H').2 | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro.mk
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro.mk
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' (convex_ball _ _).norm_image_sub_le_of_norm_hasFDerivWithin_le' _ hf' h.2 h.1 | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
case intro.intro.mk
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact fun y hy => (hε hy).1.hasFDerivWithinAt | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x := by
-- turn little-o definition of stri... | Mathlib.Analysis.Calculus.MeanValue.1343_0.ReDurB0qNQAwk9I | /-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | Mathlib_Analysis_Calculus_MeanValue |
S : Type u
inst✝ : Semiring S
⊢ ascPochhammer S 1 = X | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [ascPochhammer] | @[simp]
theorem ascPochhammer_one : ascPochhammer S 1 = X := by | Mathlib.RingTheory.Polynomial.Pochhammer.60_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_one : ascPochhammer S 1 = X | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ ascPochhammer S (n + 1) = X * comp (ascPochhammer S n) (X + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer] | theorem ascPochhammer_succ_left (n : ℕ) :
ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1) :=
by | Mathlib.RingTheory.Polynomial.Pochhammer.64_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_left (n : ℕ) :
ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝² : Semiring S
n : ℕ
inst✝¹ : Nontrivial S
inst✝ : NoZeroDivisors S
⊢ Monic (ascPochhammer S n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n hn | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n := by
| Mathlib.RingTheory.Polynomial.Pochhammer.69_0.yf6mY7NVFIgfXWQ | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u
inst✝² : Semiring S
inst✝¹ : Nontrivial S
inst✝ : NoZeroDivisors S
⊢ Monic (ascPochhammer S Nat.zero) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n := by
induction' n with n hn
· | Mathlib.RingTheory.Polynomial.Pochhammer.69_0.yf6mY7NVFIgfXWQ | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝² : Semiring S
inst✝¹ : Nontrivial S
inst✝ : NoZeroDivisors S
n : ℕ
hn : Monic (ascPochhammer S n)
⊢ Monic (ascPochhammer S (Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | have : leadingCoeff (X + 1 : S[X]) = 1 := leadingCoeff_X_add_C 1 | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n := by
induction' n with n hn
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.69_0.yf6mY7NVFIgfXWQ | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝² : Semiring S
inst✝¹ : Nontrivial S
inst✝ : NoZeroDivisors S
n : ℕ
hn : Monic (ascPochhammer S n)
this : leadingCoeff (X + 1) = 1
⊢ Monic (ascPochhammer S (Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_left, Monic.def, leadingCoeff_mul,
leadingCoeff_comp (ne_zero_of_eq_one <| natDegree_X_add_C 1 : natDegree (X + 1) ≠ 0), hn,
monic_X, one_mul, one_mul, this, one_pow] | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n := by
induction' n with n hn
· simp
· have : leadingCoeff (X + 1 : S[X]) = 1 := leadingCoeff_X_add_C 1
| Mathlib.RingTheory.Polynomial.Pochhammer.69_0.yf6mY7NVFIgfXWQ | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝¹ : Semiring S
T : Type v
inst✝ : Semiring T
f : S →+* T
n : ℕ
⊢ map f (ascPochhammer S n) = ascPochhammer T n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n ih | @[simp]
theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n := by
| Mathlib.RingTheory.Polynomial.Pochhammer.82_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u
inst✝¹ : Semiring S
T : Type v
inst✝ : Semiring T
f : S →+* T
⊢ map f (ascPochhammer S Nat.zero) = ascPochhammer T Nat.zero | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | @[simp]
theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n := by
induction' n with n ih
· | Mathlib.RingTheory.Polynomial.Pochhammer.82_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝¹ : Semiring S
T : Type v
inst✝ : Semiring T
f : S →+* T
n : ℕ
ih : map f (ascPochhammer S n) = ascPochhammer T n
⊢ map f (ascPochhammer S (Nat.succ n)) = ascPochhammer T (Nat.succ n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [ih, ascPochhammer_succ_left, map_comp] | @[simp]
theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n := by
induction' n with n ih
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.82_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n k : ℕ
⊢ ↑(eval k (ascPochhammer ℕ n)) = eval (↑k) (ascPochhammer S n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [← ascPochhammer_map (algebraMap ℕ S), eval_map, ← eq_natCast (algebraMap ℕ S),
eval₂_at_nat_cast,Nat.cast_id, eq_natCast] | @[simp, norm_cast]
theorem ascPochhammer_eval_cast (n k : ℕ) :
(((ascPochhammer ℕ n).eval k : ℕ) : S) = ((ascPochhammer S n).eval k : S) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.92_0.yf6mY7NVFIgfXWQ | @[simp, norm_cast]
theorem ascPochhammer_eval_cast (n k : ℕ) :
(((ascPochhammer ℕ n).eval k : ℕ) : S) = ((ascPochhammer S n).eval k : S) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ eval 0 (ascPochhammer S n) = if n = 0 then 1 else 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | cases n | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 := by
| Mathlib.RingTheory.Polynomial.Pochhammer.99_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u
inst✝ : Semiring S
⊢ eval 0 (ascPochhammer S Nat.zero) = if Nat.zero = 0 then 1 else 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 := by
cases n
· | Mathlib.RingTheory.Polynomial.Pochhammer.99_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝ : Semiring S
n✝ : ℕ
⊢ eval 0 (ascPochhammer S (Nat.succ n✝)) = if Nat.succ n✝ = 0 then 1 else 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [X_mul, Nat.succ_ne_zero, ascPochhammer_succ_left] | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 := by
cases n
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.99_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
⊢ eval 0 (ascPochhammer S 0) = 1 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem ascPochhammer_zero_eval_zero : (ascPochhammer S 0).eval 0 = 1 := by | Mathlib.RingTheory.Polynomial.Pochhammer.105_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_zero_eval_zero : (ascPochhammer S 0).eval 0 = 1 | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
h : n ≠ 0
⊢ eval 0 (ascPochhammer S n) = 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [ascPochhammer_eval_zero, h] | @[simp]
theorem ascPochhammer_ne_zero_eval_zero {n : ℕ} (h : n ≠ 0) : (ascPochhammer S n).eval 0 = 0 := by
| Mathlib.RingTheory.Polynomial.Pochhammer.108_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_ne_zero_eval_zero {n : ℕ} (h : n ≠ 0) : (ascPochhammer S n).eval 0 = 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X])) | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + ↑n)
⊢ ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | apply_fun Polynomial.map (algebraMap ℕ S) at h | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· | Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
h : map (algebraMap ℕ S) (ascPochhammer ℕ (n + 1)) = map (algebraMap ℕ S) (ascPochhammer ℕ n * (X + ↑n))
⊢ ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_nat_cast] using h | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· apply_fun Polynomial.map (algebraMap ℕ S) at h
| Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n ih | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, ... | Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h.zero
S : Type u
inst✝ : Semiring S
⊢ ascPochhammer ℕ (Nat.zero + 1) = ascPochhammer ℕ Nat.zero * (X + ↑Nat.zero) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, ... | Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h.succ
S : Type u
inst✝ : Semiring S
n : ℕ
ih : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + ↑n)
⊢ ascPochhammer ℕ (Nat.succ n + 1) = ascPochhammer ℕ (Nat.succ n) * (X + ↑(Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | conv_lhs =>
rw [ascPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← ascPochhammer_succ_left, add_comp,
X_comp, nat_cast_comp, add_assoc, add_comm (1 : ℕ[X]), ← Nat.cast_succ] | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, ... | Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
ih : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + ↑n)
| ascPochhammer ℕ (Nat.succ n + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← ascPochhammer_succ_left, add_comp,
X_comp, nat_cast_comp, add_assoc, add_comm (1 : ℕ[X]), ← Nat.cast_succ] | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, ... | Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
ih : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + ↑n)
| ascPochhammer ℕ (Nat.succ n + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← ascPochhammer_succ_left, add_comp,
X_comp, nat_cast_comp, add_assoc, add_comm (1 : ℕ[X]), ← Nat.cast_succ] | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, ... | Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
ih : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + ↑n)
| ascPochhammer ℕ (Nat.succ n + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← ascPochhammer_succ_left, add_comp,
X_comp, nat_cast_comp, add_assoc, add_comm (1 : ℕ[X]), ← Nat.cast_succ] | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X]))
· apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, ... | Mathlib.RingTheory.Polynomial.Pochhammer.113_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
S✝ : Type u
inst✝¹ : Semiring S✝
S : Type u_1
inst✝ : Semiring S
n : ℕ
k : S
⊢ eval k (ascPochhammer S (n + 1)) = eval k (ascPochhammer S n) * (k + ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_right, mul_add, eval_add, eval_mul_X, ← Nat.cast_comm, ← C_eq_nat_cast,
eval_C_mul, Nat.cast_comm, ← mul_add] | theorem ascPochhammer_succ_eval {S : Type*} [Semiring S] (n : ℕ) (k : S) :
(ascPochhammer S (n + 1)).eval k = (ascPochhammer S n).eval k * (k + n) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.126_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_eval {S : Type*} [Semiring S] (n : ℕ) (k : S) :
(ascPochhammer S (n + 1)).eval k = (ascPochhammer S n).eval k * (k + n) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ comp (ascPochhammer S (n + 1)) (X + 1) = ascPochhammer S (n + 1) + (n + 1) • comp (ascPochhammer S n) (X + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | suffices (ascPochhammer ℕ (n + 1)).comp (X + 1) =
ascPochhammer ℕ (n + 1) + (n + 1) * (ascPochhammer ℕ n).comp (X + 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Nat.castRingHom S)) this | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.132_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
this : comp (ascPochhammer ℕ (n + 1)) (X + 1) = ascPochhammer ℕ (n + 1) + (↑n + 1) * comp (ascPochhammer ℕ n) (X + 1)
⊢ comp (ascPochhammer S (n + 1)) (X + 1) = ascPochhammer S (n + 1) + (n + 1) • comp (ascPochhammer S n) (X + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simpa [map_comp] using congr_arg (Polynomial.map (Nat.castRingHom S)) this | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) := by
suffices (ascPochhammer ℕ (n + 1)).comp (X + 1) =
ascPochhammer ℕ (n + 1) + (n + 1) * (ascPochhammer ℕ n).comp (X + 1)
by | Mathlib.RingTheory.Polynomial.Pochhammer.132_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ comp (ascPochhammer ℕ (n + 1)) (X + 1) = ascPochhammer ℕ (n + 1) + (↑n + 1) * comp (ascPochhammer ℕ n) (X + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | nth_rw 2 [ascPochhammer_succ_left] | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) := by
suffices (ascPochhammer ℕ (n + 1)).comp (X + 1) =
ascPochhammer ℕ (n + 1) + (n + 1) * (ascPochhammer ℕ n).comp (X + 1)
by simpa [m... | Mathlib.RingTheory.Polynomial.Pochhammer.132_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ comp (ascPochhammer ℕ (n + 1)) (X + 1) =
X * comp (ascPochhammer ℕ n) (X + 1) + (↑n + 1) * comp (ascPochhammer ℕ n) (X + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [← add_mul, ascPochhammer_succ_right ℕ n, mul_comp, mul_comm, add_comp, X_comp, nat_cast_comp,
add_comm, ← add_assoc] | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) := by
suffices (ascPochhammer ℕ (n + 1)).comp (X + 1) =
ascPochhammer ℕ (n + 1) + (n + 1) * (ascPochhammer ℕ n).comp (X + 1)
by simpa [m... | Mathlib.RingTheory.Polynomial.Pochhammer.132_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ (↑n + X + 1) * comp (ascPochhammer ℕ n) (X + 1) = (X + (↑n + 1)) * comp (ascPochhammer ℕ n) (X + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | ring | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) := by
suffices (ascPochhammer ℕ (n + 1)).comp (X + 1) =
ascPochhammer ℕ (n + 1) + (n + 1) * (ascPochhammer ℕ n).comp (X + 1)
by simpa [m... | Mathlib.RingTheory.Polynomial.Pochhammer.132_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_succ_comp_X_add_one (n : ℕ) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n m : ℕ
⊢ ascPochhammer S n * comp (ascPochhammer S m) (X + ↑n) = ascPochhammer S (n + m) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' m with m ih | theorem ascPochhammer_mul (n m : ℕ) :
ascPochhammer S n * (ascPochhammer S m).comp (X + (n : S[X])) = ascPochhammer S (n + m) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.145_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_mul (n m : ℕ) :
ascPochhammer S n * (ascPochhammer S m).comp (X + (n : S[X])) = ascPochhammer S (n + m) | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ ascPochhammer S n * comp (ascPochhammer S Nat.zero) (X + ↑n) = ascPochhammer S (n + Nat.zero) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem ascPochhammer_mul (n m : ℕ) :
ascPochhammer S n * (ascPochhammer S m).comp (X + (n : S[X])) = ascPochhammer S (n + m) := by
induction' m with m ih
· | Mathlib.RingTheory.Polynomial.Pochhammer.145_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_mul (n m : ℕ) :
ascPochhammer S n * (ascPochhammer S m).comp (X + (n : S[X])) = ascPochhammer S (n + m) | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝ : Semiring S
n m : ℕ
ih : ascPochhammer S n * comp (ascPochhammer S m) (X + ↑n) = ascPochhammer S (n + m)
⊢ ascPochhammer S n * comp (ascPochhammer S (Nat.succ m)) (X + ↑n) = ascPochhammer S (n + Nat.succ m) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_right, Polynomial.mul_X_add_nat_cast_comp, ← mul_assoc, ih,
Nat.succ_eq_add_one, ← add_assoc, ascPochhammer_succ_right, Nat.cast_add, add_assoc] | theorem ascPochhammer_mul (n m : ℕ) :
ascPochhammer S n * (ascPochhammer S m).comp (X + (n : S[X])) = ascPochhammer S (n + m) := by
induction' m with m ih
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.145_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_mul (n m : ℕ) :
ascPochhammer S n * (ascPochhammer S m).comp (X + (n : S[X])) = ascPochhammer S (n + m) | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n : ℕ
⊢ eval (n + 1) (ascPochhammer ℕ 0) = Nat.ascFactorial n 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero] | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by | Mathlib.RingTheory.Polynomial.Pochhammer.153_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero]
| t + 1 => by
rw [ascPochhammer_succ_right, eval_mul, ascPochhammer_nat_eq_ascFactorial n t]
simp only [eval_add, eval_X, eval_n... | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n t : ℕ
⊢ eval (n + 1) (ascPochhammer ℕ (t + 1)) = Nat.ascFactorial n (t + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_right, eval_mul, ascPochhammer_nat_eq_ascFactorial n t] | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero]
| t + 1 => by
| Mathlib.RingTheory.Polynomial.Pochhammer.153_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero]
| t + 1 => by
rw [ascPochhammer_succ_right, eval_mul, ascPochhammer_nat_eq_ascFactorial n t]
simp only [eval_add, eval_X, eval_n... | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n t : ℕ
⊢ Nat.ascFactorial n t * eval (n + 1) (X + ↑t) = Nat.ascFactorial n (t + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [eval_add, eval_X, eval_nat_cast, Nat.cast_id] | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero]
| t + 1 => by
rw [ascPochhammer_succ_right, eval_mul, ascPochhammer_nat_eq_ascFactorial n t]
| Mathlib.RingTheory.Polynomial.Pochhammer.153_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero]
| t + 1 => by
rw [ascPochhammer_succ_right, eval_mul, ascPochhammer_nat_eq_ascFactorial n t]
simp only [eval_add, eval_X, eval_n... | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
n t : ℕ
⊢ Nat.ascFactorial n t * (n + 1 + t) = Nat.ascFactorial n (t + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [Nat.ascFactorial_succ, add_right_comm, mul_comm] | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero]
| t + 1 => by
rw [ascPochhammer_succ_right, eval_mul, ascPochhammer_nat_eq_ascFactorial n t]
simp only [eval_add, eval_X, eval_n... | Mathlib.RingTheory.Polynomial.Pochhammer.153_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_ascFactorial (n : ℕ) :
∀ k, (ascPochhammer ℕ k).eval (n + 1) = n.ascFactorial k
| 0 => by rw [ascPochhammer_zero, eval_one, Nat.ascFactorial_zero]
| t + 1 => by
rw [ascPochhammer_succ_right, eval_mul, ascPochhammer_nat_eq_ascFactorial n t]
simp only [eval_add, eval_X, eval_n... | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝ : Semiring S
a b : ℕ
⊢ eval a (ascPochhammer ℕ b) = Nat.descFactorial (a + b - 1) b | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | cases' b with b | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b := by
| Mathlib.RingTheory.Polynomial.Pochhammer.162_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u
inst✝ : Semiring S
a : ℕ
⊢ eval a (ascPochhammer ℕ Nat.zero) = Nat.descFactorial (a + Nat.zero - 1) Nat.zero | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [Nat.descFactorial_zero, ascPochhammer_zero, Polynomial.eval_one] | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b := by
cases' b with b
· | Mathlib.RingTheory.Polynomial.Pochhammer.162_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b | Mathlib_RingTheory_Polynomial_Pochhammer |
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