l3lab/ntp-mathlib huggingface dataset:
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  change Icc a b ⊆ { x | f x ≤ B x }
  set s := { x | f x ≤ B x } ∩ Icc a b
  have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
  have : IsClosed s := by
    simp only [inter_comm]
    exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
  apply this.Icc_subset_of_forall_exists_gt ha
  rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
  cases' hxB.lt_or_eq with hxB hxB
  · -- If `f x < B x`, then all we need is continuity of both sides
    refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
    have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
      A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
    have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
    exact this.mono fun y => le_of_lt
  · 

Proof state:
case intro.inr
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 f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x => (f x, B x)) (Icc a b)
this : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x = B x
⊢ Set.Nonempty ({x | f x ≤ B x} ∩ Ioc x y)

Next tactic:
rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  change Icc a b ⊆ { x | f x ≤ B x }
  set s := { x | f x ≤ B x } ∩ Icc a b
  have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
  have : IsClosed s := by
    simp only [inter_comm]
    exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
  apply this.Icc_subset_of_forall_exists_gt ha
  rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
  cases' hxB.lt_or_eq with hxB hxB
  · -- If `f x < B x`, then all we need is continuity of both sides
    refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
    have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
      A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
    have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
    exact this.mono fun y => le_of_lt
  · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
    

Proof state:
case intro.inr.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 f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x => (f x, B x)) (Icc a b)
this : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x = B x
r : ℝ
hfr : f' x < r
hrB : r < B' x
⊢ Set.Nonempty ({x | f x ≤ B x} ∩ Ioc x y)

Next tactic:
specialize hf' x xab r hfr
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  change Icc a b ⊆ { x | f x ≤ B x }
  set s := { x | f x ≤ B x } ∩ Icc a b
  have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
  have : IsClosed s := by
    simp only [inter_comm]
    exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
  apply this.Icc_subset_of_forall_exists_gt ha
  rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
  cases' hxB.lt_or_eq with hxB hxB
  · -- If `f x < B x`, then all we need is continuity of both sides
    refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
    have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
      A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
    have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
    exact this.mono fun y => le_of_lt
  · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
    specialize hf' x xab r hfr
    

Proof state:
case intro.inr.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 f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x => (f x, B x)) (Icc a b)
this : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x = B x
r : ℝ
hfr : f' x < r
hrB : r < B' x
hf' : ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
⊢ Set.Nonempty ({x | f x ≤ B x} ∩ Ioc x y)

Next tactic:
have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
      (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
        (Ioi_mem_nhds hrB)
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  change Icc a b ⊆ { x | f x ≤ B x }
  set s := { x | f x ≤ B x } ∩ Icc a b
  have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
  have : IsClosed s := by
    simp only [inter_comm]
    exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
  apply this.Icc_subset_of_forall_exists_gt ha
  rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
  cases' hxB.lt_or_eq with hxB hxB
  · -- If `f x < B x`, then all we need is continuity of both sides
    refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
    have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
      A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
    have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
    exact this.mono fun y => le_of_lt
  · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
    specialize hf' x xab r hfr
    have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
      (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
        (Ioi_mem_nhds hrB)
    

Proof state:
case intro.inr.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 f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x => (f x, B x)) (Icc a b)
this : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x = B x
r : ℝ
hfr : f' x < r
hrB : r < B' x
hf' : ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
HB : ∀ᶠ (z : ℝ) in 𝓝[>] x, r < slope B x z
⊢ Set.Nonempty ({x | f x ≤ B x} ∩ Ioc x y)

Next tactic:
obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  change Icc a b ⊆ { x | f x ≤ B x }
  set s := { x | f x ≤ B x } ∩ Icc a b
  have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
  have : IsClosed s := by
    simp only [inter_comm]
    exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
  apply this.Icc_subset_of_forall_exists_gt ha
  rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
  cases' hxB.lt_or_eq with hxB hxB
  · -- If `f x < B x`, then all we need is continuity of both sides
    refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
    have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
      A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
    have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
    exact this.mono fun y => le_of_lt
  · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
    specialize hf' x xab r hfr
    have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
      (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
        (Ioi_mem_nhds hrB)
    obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y
    exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists
    

Proof state:
case intro.inr.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
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x => (f x, B x)) (Icc a b)
this : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x = B x
r : ℝ
hfr : f' x < r
hrB : r < B' x
hf' : ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
HB : ∀ᶠ (z : ℝ) in 𝓝[>] x, r < slope B x z
z : ℝ
hfz : slope f x z < r
hzB : r < slope B x z
hz : z ∈ Ioc x y
⊢ Set.Nonempty ({x | f x ≤ B x} ∩ Ioc x y)

Next tactic:
refine' ⟨z, _, hz⟩
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by a function `f'`;
* we have `f' x < B' x` whenever `f x = B x`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
    {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  change Icc a b ⊆ { x | f x ≤ B x }
  set s := { x | f x ≤ B x } ∩ Icc a b
  have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
  have : IsClosed s := by
    simp only [inter_comm]
    exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
  apply this.Icc_subset_of_forall_exists_gt ha
  rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
  cases' hxB.lt_or_eq with hxB hxB
  · -- If `f x < B x`, then all we need is continuity of both sides
    refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
    have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
      A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
    have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
    exact this.mono fun y => le_of_lt
  · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
    specialize hf' x xab r hfr
    have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
      (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
        (Ioi_mem_nhds hrB)
    obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y
    exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists
    refine' ⟨z, _, hz⟩
    

Proof state:
case intro.inr.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
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x
s : Set ℝ := {x | f x ≤ B x} ∩ Icc a b
A : ContinuousOn (fun x => (f x, B x)) (Icc a b)
this : IsClosed s
x : ℝ
hxB✝ : f x ≤ B x
xab : x ∈ Ico a b
y : ℝ
hy : y ∈ Ioi x
hxB : f x = B x
r : ℝ
hfr : f' x < r
hrB : r < B' x
hf' : ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
HB : ∀ᶠ (z : ℝ) in 𝓝[>] x, r < slope B x z
z : ℝ
hfz : slope f x z < r
hzB : r < slope B x z
hz : z ∈ Ioc x y
⊢ z ∈ {x | f x ≤ B x}

Next tactic:
have := (hfz.trans hzB).le
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by `B'`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
    (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) :
    ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B 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
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
⊢ ∀ ⦃x : ℝ⦄, x ∈ Icc a b → f x ≤ B x

Next tactic:
have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by
    apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
    · rwa [sub_self, mul_zero, add_zero]
    · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))
    · intro x hx
      exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r)
    · intro x _ _
      rw [mul_one]
      exact (lt_add_iff_pos_right _).2 hr
    exact hx
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by `B'`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
    (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) :
    ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => 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
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x : ℝ
hx : x ∈ Icc a b
r : ℝ
hr : r > 0
⊢ f x ≤ B x + r * (x - a)

Next tactic:
apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by `B'`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
    (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) :
    ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by
    apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
    · 

Proof state:
case ha
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 : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x : ℝ
hx : x ∈ Icc a b
r : ℝ
hr : r > 0
⊢ f a ≤ B a + r * (a - a)

Next tactic:
rwa [sub_self, mul_zero, add_zero]
/-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that

* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
  is bounded above by `B'`.

Then `f x ≤ B x` everywhere on `[a, b]`. -/
theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
    (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
    -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
    (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) :
    ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
  have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by
    apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
    · rwa [sub_self, mul_zero, add_zero]
    · 

Proof state:
case hB
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 : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x : ℝ
hx : x ∈ Icc a b
r : ℝ
hr : r > 0
⊢ ContinuousOn (fun x => B x + r * (x - a)) (Icc a b)

Next tactic:
exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))