l3lab/ntp-mathlib huggingface dataset:
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · 

Proof state:
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
⊢ ∀ w_1 ∈ Ioo x w, deriv (fun w => f w) w_1 ≠ 0

Next tactic:
intro z hz
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        

Proof state:
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 (fun w => f w) z ≠ 0

Next tactic:
rw [← hw]
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        

Proof state:
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 (fun w => f w) z ≠ deriv f w

Next tactic:
apply ne_of_lt
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        

Proof state:
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
⊢ deriv (fun w => f w) z < deriv f w

Next tactic:
exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    

Proof state:
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
hwy : w < y
a : ℝ
ha : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a

Next tactic:
obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      

Proof state:
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 : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
⊢ ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b

Next tactic:
apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
      · 

Proof state:
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 : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
⊢ ContinuousOn (fun {y} => f y) (Icc w y)

Next tactic:
refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
      · refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
      · 

Proof state:
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 : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
⊢ StrictMonoOn (deriv fun {y} => f y) (Ioo w y)

Next tactic:
exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
      · refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)
      · exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
      · 

Proof state:
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 : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
⊢ ∀ w_1 ∈ Ioo w y, deriv (fun {y} => f y) w_1 ≠ 0

Next tactic:
intro z hz
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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
        

Proof state:
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 : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
z : ℝ
hz : z ∈ Ioo w y
⊢ deriv (fun {y} => f y) z ≠ 0

Next tactic:
rw [← hw]
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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]
        

Proof state:
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 : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
z : ℝ
hz : z ∈ Ioo w y
⊢ deriv (fun {y} => f y) z ≠ deriv f w

Next tactic:
apply ne_of_gt
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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
        

Proof state:
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 : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
z : ℝ
hz : z ∈ Ioo w y
⊢ deriv f w < deriv (fun {y} => f y) z

Next tactic:
exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
    

Proof state:
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 w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
b : ℝ
hb : (f y - f w) / (y - w) < deriv f b
hwb : w < b
hby : b < y
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a

Next tactic:
refine' ⟨b, ⟨hxw.trans hwb, hby⟩, _⟩
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
    refine' ⟨b, ⟨hxw.trans hwb, hby⟩, _⟩
    

Proof state:
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 w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
b : ℝ
hb : (f y - f w) / (y - w) < deriv f b
hwb : w < b
hby : b < y
⊢ (f y - f x) / (y - x) < deriv f b

Next tactic:
simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
theorem StrictMonoOn.exists_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
    refine' ⟨b, ⟨hxw.trans hwb, hby⟩, _⟩
    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
    

Proof state:
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 w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : f w - f x < deriv f a * (w - x)
hb : f y - f w < deriv f b * (y - w)
⊢ f y - f x < deriv f b * (y - x)

Next tactic:
have : deriv f a * (w - x) < deriv f b * (w - x) := by
      apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _
      · 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_slope_lt_deriv {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, (f y - f x) / (y - x) < deriv f a := by
  by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h
  · push_neg at h
    rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_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
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
    refine' ⟨b, ⟨hxw.trans hwb, hby⟩, _⟩
    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
    have : deriv f a * (w - x) < deriv f b * (w - x) := by
      

Proof state:
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 : ℝ
hwb : w < b
hby : b < y
ha : f w - f x < deriv f a * (w - x)
hb : f y - f w < deriv f b * (y - w)
⊢ deriv f a * (w - x) < deriv f b * (w - x)

Next tactic:
apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _