l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem map_map {M : DMatrix m n α} {β : m → n → Type w} {γ : m → n → Type z}
    {f : ∀ ⦃i j⦄, α i j → β i j} {g : ∀ ⦃i j⦄, β i j → γ i j} :
    (M.map f).map g = M.map fun i j x => g (f x) := by ext; 

Proof state:
case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : Fintype o
α : m → n → Type v
M : DMatrix m n α
β : m → n → Type w
γ : m → n → Type z
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j → β i j
g : ⦃i : m⦄ → ⦃j : n⦄ → β i j → γ i j
i✝ : m
j✝ : n
⊢ map (map M f) g i✝ j✝ = map M (fun i j x => g (f x)) i✝ j✝

Next tactic:
simp
instance [∀ i j, Subsingleton (α i j)] : Subsingleton (DMatrix m n α) :=
  by 

Proof state:
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁴ : Fintype l
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : Fintype o
α : m → n → Type v
inst✝ : ∀ (i : m) (j : n), Subsingleton (α i j)
⊢ Subsingleton (DMatrix m n α)

Next tactic:
constructor
instance [∀ i j, Subsingleton (α i j)] : Subsingleton (DMatrix m n α) :=
  by constructor; 

Proof state:
case allEq
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁴ : Fintype l
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : Fintype o
α : m → n → Type v
inst✝ : ∀ (i : m) (j : n), Subsingleton (α i j)
⊢ ∀ (a b : DMatrix m n α), a = b

Next tactic:
simp only [DMatrix, eq_iff_true_of_subsingleton, implies_true]
@[simp]
theorem map_zero [∀ i j, Zero (α i j)] {β : m → n → Type w} [∀ i j, Zero (β i j)]
    {f : ∀ ⦃i j⦄, α i j → β i j} (h : ∀ i j, f (0 : α i j) = 0) : (0 : DMatrix m n α).map f = 0 :=
  by 

Proof state:
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁵ : Fintype l
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : Fintype o
α : m → n → Type v
inst✝¹ : (i : m) → (j : n) → Zero (α i j)
β : m → n → Type w
inst✝ : (i : m) → (j : n) → Zero (β i j)
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j → β i j
h : ∀ (i : m) (j : n), f 0 = 0
⊢ map 0 f = 0

Next tactic:
ext
@[simp]
theorem map_zero [∀ i j, Zero (α i j)] {β : m → n → Type w} [∀ i j, Zero (β i j)]
    {f : ∀ ⦃i j⦄, α i j → β i j} (h : ∀ i j, f (0 : α i j) = 0) : (0 : DMatrix m n α).map f = 0 :=
  by ext; 

Proof state:
case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁵ : Fintype l
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : Fintype o
α : m → n → Type v
inst✝¹ : (i : m) → (j : n) → Zero (α i j)
β : m → n → Type w
inst✝ : (i : m) → (j : n) → Zero (β i j)
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j → β i j
h : ∀ (i : m) (j : n), f 0 = 0
i✝ : m
j✝ : n
⊢ map 0 f i✝ j✝ = OfNat.ofNat 0 i✝ j✝

Next tactic:
simp [h]
theorem map_add [∀ i j, AddMonoid (α i j)] {β : m → n → Type w} [∀ i j, AddMonoid (β i j)]
    (f : ∀ ⦃i j⦄, α i j →+ β i j) (M N : DMatrix m n α) :
    ((M + N).map fun i j => @f i j) = (M.map fun i j => @f i j) + N.map fun i j => @f i j := by
  

Proof state:
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁵ : Fintype l
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : Fintype o
α : m → n → Type v
inst✝¹ : (i : m) → (j : n) → AddMonoid (α i j)
β : m → n → Type w
inst✝ : (i : m) → (j : n) → AddMonoid (β i j)
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j
M N : DMatrix m n α
⊢ (map (M + N) fun i j => ⇑f) = (map M fun i j => ⇑f) + map N fun i j => ⇑f

Next tactic:
ext
theorem map_add [∀ i j, AddMonoid (α i j)] {β : m → n → Type w} [∀ i j, AddMonoid (β i j)]
    (f : ∀ ⦃i j⦄, α i j →+ β i j) (M N : DMatrix m n α) :
    ((M + N).map fun i j => @f i j) = (M.map fun i j => @f i j) + N.map fun i j => @f i j := by
  ext; 

Proof state:
case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁵ : Fintype l
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : Fintype o
α : m → n → Type v
inst✝¹ : (i : m) → (j : n) → AddMonoid (α i j)
β : m → n → Type w
inst✝ : (i : m) → (j : n) → AddMonoid (β i j)
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j
M N : DMatrix m n α
i✝ : m
j✝ : n
⊢ map (M + N) (fun i j => ⇑f) i✝ j✝ = ((map M fun i j => ⇑f) + map N fun i j => ⇑f) i✝ j✝

Next tactic:
simp
theorem map_sub [∀ i j, AddGroup (α i j)] {β : m → n → Type w} [∀ i j, AddGroup (β i j)]
    (f : ∀ ⦃i j⦄, α i j →+ β i j) (M N : DMatrix m n α) :
    ((M - N).map fun i j => @f i j) = (M.map fun i j => @f i j) - N.map fun i j => @f i j := by
  

Proof state:
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁵ : Fintype l
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : Fintype o
α : m → n → Type v
inst✝¹ : (i : m) → (j : n) → AddGroup (α i j)
β : m → n → Type w
inst✝ : (i : m) → (j : n) → AddGroup (β i j)
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j
M N : DMatrix m n α
⊢ (map (M - N) fun i j => ⇑f) = (map M fun i j => ⇑f) - map N fun i j => ⇑f

Next tactic:
ext
theorem map_sub [∀ i j, AddGroup (α i j)] {β : m → n → Type w} [∀ i j, AddGroup (β i j)]
    (f : ∀ ⦃i j⦄, α i j →+ β i j) (M N : DMatrix m n α) :
    ((M - N).map fun i j => @f i j) = (M.map fun i j => @f i j) - N.map fun i j => @f i j := by
  ext; 

Proof state:
case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁵ : Fintype l
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : Fintype o
α : m → n → Type v
inst✝¹ : (i : m) → (j : n) → AddGroup (α i j)
β : m → n → Type w
inst✝ : (i : m) → (j : n) → AddGroup (β i j)
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j
M N : DMatrix m n α
i✝ : m
j✝ : n
⊢ map (M - N) (fun i j => ⇑f) i✝ j✝ = ((map M fun i j => ⇑f) - map N fun i j => ⇑f) i✝ j✝

Next tactic:
simp
instance subsingleton_of_empty_left [IsEmpty m] : Subsingleton (DMatrix m n α) :=
  ⟨fun M N => by
    

Proof state:
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁴ : Fintype l
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : Fintype o
α : m → n → Type v
inst✝ : IsEmpty m
M N : DMatrix m n α
⊢ M = N

Next tactic:
ext i
instance subsingleton_of_empty_left [IsEmpty m] : Subsingleton (DMatrix m n α) :=
  ⟨fun M N => by
    ext i
    

Proof state:
case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁴ : Fintype l
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : Fintype o
α : m → n → Type v
inst✝ : IsEmpty m
M N : DMatrix m n α
i : m
j✝ : n
⊢ M i j✝ = N i j✝

Next tactic:
exact isEmptyElim i
instance subsingleton_of_empty_right [IsEmpty n] : Subsingleton (DMatrix m n α) :=
  ⟨fun M N => by 

Proof state:
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁴ : Fintype l
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : Fintype o
α : m → n → Type v
inst✝ : IsEmpty n
M N : DMatrix m n α
⊢ M = N

Next tactic:
ext i j
instance subsingleton_of_empty_right [IsEmpty n] : Subsingleton (DMatrix m n α) :=
  ⟨fun M N => by ext i j; 

Proof state:
case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁴ : Fintype l
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : Fintype o
α : m → n → Type v
inst✝ : IsEmpty n
M N : DMatrix m n α
i : m
j : n
⊢ M i j = N i j

Next tactic:
exact isEmptyElim j
/-- The `AddMonoidHom` between spaces of dependently typed matrices
induced by an `AddMonoidHom` between their coefficients. -/
def AddMonoidHom.mapDMatrix [∀ i j, AddMonoid (α i j)] {β : m → n → Type w}
    [∀ i j, AddMonoid (β i j)] (f : ∀ ⦃i j⦄, α i j →+ β i j) : DMatrix m n α →+ DMatrix m n β
    where
  toFun M := M.map fun i j => @f i j
  map_zero' := by 

Proof state:
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
inst✝⁵ : Fintype l
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : Fintype o
α : m → n → Type v
inst✝¹ : (i : m) → (j : n) → AddMonoid (α i j)
β : m → n → Type w
inst✝ : (i : m) → (j : n) → AddMonoid (β i j)
f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j
⊢ (fun M => DMatrix.map M fun i j => ⇑f) 0 = 0

Next tactic:
simp
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  

Proof state:
p : ℝ[X]
⊢ Finset.card (Multiset.toFinset (roots p)) ≤
    Finset.card (Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p)) + 1

Next tactic:
rcases eq_or_ne (derivative p) 0 with hp' | hp'
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · 

Proof state:
case inl
p : ℝ[X]
hp' : derivative p = 0
⊢ Finset.card (Multiset.toFinset (roots p)) ≤
    Finset.card (Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p)) + 1

Next tactic:
rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    

Proof state:
case inl
p : ℝ[X]
hp' : derivative p = 0
⊢ 0 ≤ Finset.card (Multiset.toFinset (roots (derivative (C (coeff p 0)))) \ ∅) + 1

Next tactic:
exact zero_le _
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    exact zero_le _
  

Proof state:
case inr
p : ℝ[X]
hp' : derivative p ≠ 0
⊢ Finset.card (Multiset.toFinset (roots p)) ≤
    Finset.card (Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p)) + 1

Next tactic:
have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    exact zero_le _
  have hp : p ≠ 0 := ne_of_apply_ne derivative (by 

Proof state:
p : ℝ[X]
hp' : derivative p ≠ 0
⊢ derivative p ≠ derivative 0

Next tactic:
rwa [derivative_zero]
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    exact zero_le _
  have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])
  

Proof state:
case inr
p : ℝ[X]
hp' : derivative p ≠ 0
hp : p ≠ 0
⊢ Finset.card (Multiset.toFinset (roots p)) ≤
    Finset.card (Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p)) + 1

Next tactic:
refine' Finset.card_le_diff_of_interleaved fun x hx y hy hxy hxy' => _
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    exact zero_le _
  have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])
  refine' Finset.card_le_diff_of_interleaved fun x hx y hy hxy hxy' => _
  

Proof state:
case inr
p : ℝ[X]
hp' : derivative p ≠ 0
hp : p ≠ 0
x : ℝ
hx : x ∈ Multiset.toFinset (roots p)
y : ℝ
hy : y ∈ Multiset.toFinset (roots p)
hxy : x < y
hxy' : ∀ z ∈ Multiset.toFinset (roots p), z ∉ Set.Ioo x y
⊢ ∃ z ∈ Multiset.toFinset (roots (derivative p)), x < z ∧ z < y

Next tactic:
rw [Multiset.mem_toFinset, mem_roots hp] at hx hy
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    exact zero_le _
  have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])
  refine' Finset.card_le_diff_of_interleaved fun x hx y hy hxy hxy' => _
  rw [Multiset.mem_toFinset, mem_roots hp] at hx hy
  

Proof state:
case inr
p : ℝ[X]
hp' : derivative p ≠ 0
hp : p ≠ 0
x : ℝ
hx : IsRoot p x
y : ℝ
hy : IsRoot p y
hxy : x < y
hxy' : ∀ z ∈ Multiset.toFinset (roots p), z ∉ Set.Ioo x y
⊢ ∃ z ∈ Multiset.toFinset (roots (derivative p)), x < z ∧ z < y

Next tactic:
obtain ⟨z, hz1, hz2⟩ := exists_deriv_eq_zero hxy p.continuousOn (hx.trans hy.symm)
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    exact zero_le _
  have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])
  refine' Finset.card_le_diff_of_interleaved fun x hx y hy hxy hxy' => _
  rw [Multiset.mem_toFinset, mem_roots hp] at hx hy
  obtain ⟨z, hz1, hz2⟩ := exists_deriv_eq_zero hxy p.continuousOn (hx.trans hy.symm)
  

Proof state:
case inr.intro.intro
p : ℝ[X]
hp' : derivative p ≠ 0
hp : p ≠ 0
x : ℝ
hx : IsRoot p x
y : ℝ
hy : IsRoot p y
hxy : x < y
hxy' : ∀ z ∈ Multiset.toFinset (roots p), z ∉ Set.Ioo x y
z : ℝ
hz1 : z ∈ Set.Ioo x y
hz2 : deriv (fun x => eval x p) z = 0
⊢ ∃ z ∈ Multiset.toFinset (roots (derivative p)), x < z ∧ z < y

Next tactic:
refine' ⟨z, _, hz1⟩
/-- The number of roots of a real polynomial `p` is at most the number of roots of its derivative
that are not roots of `p` plus one. -/
theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
    p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
    exact zero_le _
  have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])
  refine' Finset.card_le_diff_of_interleaved fun x hx y hy hxy hxy' => _
  rw [Multiset.mem_toFinset, mem_roots hp] at hx hy
  obtain ⟨z, hz1, hz2⟩ := exists_deriv_eq_zero hxy p.continuousOn (hx.trans hy.symm)
  refine' ⟨z, _, hz1⟩
  

Proof state:
case inr.intro.intro
p : ℝ[X]
hp' : derivative p ≠ 0
hp : p ≠ 0
x : ℝ
hx : IsRoot p x
y : ℝ
hy : IsRoot p y
hxy : x < y
hxy' : ∀ z ∈ Multiset.toFinset (roots p), z ∉ Set.Ioo x y
z : ℝ
hz1 : z ∈ Set.Ioo x y
hz2 : deriv (fun x => eval x p) z = 0
⊢ z ∈ Multiset.toFinset (roots (derivative p))

Next tactic:
rwa [Multiset.mem_toFinset, mem_roots hp', IsRoot, ← p.deriv]
/-- The number of roots of a real polynomial (counted with multiplicities) is at most the number of
roots of its derivative (counted with multiplicities) plus one. -/
theorem card_roots_le_derivative (p : ℝ[X]) :
    Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 :=
  calc
    Multiset.card p.roots = ∑ x in p.roots.toFinset, p.roots.count x :=
      (Multiset.toFinset_sum_count_eq _).symm
    _ = ∑ x in p.roots.toFinset, (p.roots.count x - 1 + 1) :=
      (Eq.symm <| Finset.sum_congr rfl fun x hx => tsub_add_cancel_of_le <|
        Nat.succ_le_iff.2 <| Multiset.count_pos.2 <| Multiset.mem_toFinset.1 hx)
    _ = (∑ x in p.roots.toFinset, (p.rootMultiplicity x - 1)) + p.roots.toFinset.card := by
      

Proof state:
p : ℝ[X]
⊢ ∑ x in Multiset.toFinset (roots p), (Multiset.count x (roots p) - 1 + 1) =
    ∑ x in Multiset.toFinset (roots p), (rootMultiplicity x p - 1) + Finset.card (Multiset.toFinset (roots p))

Next tactic:
simp only [Finset.sum_add_distrib, Finset.card_eq_sum_ones, count_roots]
/-- The number of roots of a real polynomial (counted with multiplicities) is at most the number of
roots of its derivative (counted with multiplicities) plus one. -/
theorem card_roots_le_derivative (p : ℝ[X]) :
    Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 :=
  calc
    Multiset.card p.roots = ∑ x in p.roots.toFinset, p.roots.count x :=
      (Multiset.toFinset_sum_count_eq _).symm
    _ = ∑ x in p.roots.toFinset, (p.roots.count x - 1 + 1) :=
      (Eq.symm <| Finset.sum_congr rfl fun x hx => tsub_add_cancel_of_le <|
        Nat.succ_le_iff.2 <| Multiset.count_pos.2 <| Multiset.mem_toFinset.1 hx)
    _ = (∑ x in p.roots.toFinset, (p.rootMultiplicity x - 1)) + p.roots.toFinset.card := by
      simp only [Finset.sum_add_distrib, Finset.card_eq_sum_ones, count_roots]
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.rootMultiplicity x) +
          ((p.derivative.roots.toFinset \ p.roots.toFinset).card + 1) :=
      (add_le_add
        (Finset.sum_le_sum fun x _ => rootMultiplicity_sub_one_le_derivative_rootMultiplicity _ _)
        p.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ)
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.roots.count x) +
          ((∑ x in p.derivative.roots.toFinset \ p.roots.toFinset,
            p.derivative.roots.count x) + 1) := by
      

Proof state:
p : ℝ[X]
⊢ ∑ x in Multiset.toFinset (roots p), rootMultiplicity x (derivative p) +
      (Finset.card (Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p)) + 1) ≤
    ∑ x in Multiset.toFinset (roots p), Multiset.count x (roots (derivative p)) +
      (∑ x in Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p),
          Multiset.count x (roots (derivative p)) +
        1)

Next tactic:
simp only [← count_roots]
/-- The number of roots of a real polynomial (counted with multiplicities) is at most the number of
roots of its derivative (counted with multiplicities) plus one. -/
theorem card_roots_le_derivative (p : ℝ[X]) :
    Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 :=
  calc
    Multiset.card p.roots = ∑ x in p.roots.toFinset, p.roots.count x :=
      (Multiset.toFinset_sum_count_eq _).symm
    _ = ∑ x in p.roots.toFinset, (p.roots.count x - 1 + 1) :=
      (Eq.symm <| Finset.sum_congr rfl fun x hx => tsub_add_cancel_of_le <|
        Nat.succ_le_iff.2 <| Multiset.count_pos.2 <| Multiset.mem_toFinset.1 hx)
    _ = (∑ x in p.roots.toFinset, (p.rootMultiplicity x - 1)) + p.roots.toFinset.card := by
      simp only [Finset.sum_add_distrib, Finset.card_eq_sum_ones, count_roots]
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.rootMultiplicity x) +
          ((p.derivative.roots.toFinset \ p.roots.toFinset).card + 1) :=
      (add_le_add
        (Finset.sum_le_sum fun x _ => rootMultiplicity_sub_one_le_derivative_rootMultiplicity _ _)
        p.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ)
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.roots.count x) +
          ((∑ x in p.derivative.roots.toFinset \ p.roots.toFinset,
            p.derivative.roots.count x) + 1) := by
      simp only [← count_roots]
      

Proof state:
p : ℝ[X]
⊢ ∑ x in Multiset.toFinset (roots p), Multiset.count x (roots (derivative p)) +
      (Finset.card (Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p)) + 1) ≤
    ∑ x in Multiset.toFinset (roots p), Multiset.count x (roots (derivative p)) +
      (∑ x in Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p),
          Multiset.count x (roots (derivative p)) +
        1)

Next tactic:
refine' add_le_add_left (add_le_add_right ((Finset.card_eq_sum_ones _).trans_le _) _) _
/-- The number of roots of a real polynomial (counted with multiplicities) is at most the number of
roots of its derivative (counted with multiplicities) plus one. -/
theorem card_roots_le_derivative (p : ℝ[X]) :
    Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 :=
  calc
    Multiset.card p.roots = ∑ x in p.roots.toFinset, p.roots.count x :=
      (Multiset.toFinset_sum_count_eq _).symm
    _ = ∑ x in p.roots.toFinset, (p.roots.count x - 1 + 1) :=
      (Eq.symm <| Finset.sum_congr rfl fun x hx => tsub_add_cancel_of_le <|
        Nat.succ_le_iff.2 <| Multiset.count_pos.2 <| Multiset.mem_toFinset.1 hx)
    _ = (∑ x in p.roots.toFinset, (p.rootMultiplicity x - 1)) + p.roots.toFinset.card := by
      simp only [Finset.sum_add_distrib, Finset.card_eq_sum_ones, count_roots]
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.rootMultiplicity x) +
          ((p.derivative.roots.toFinset \ p.roots.toFinset).card + 1) :=
      (add_le_add
        (Finset.sum_le_sum fun x _ => rootMultiplicity_sub_one_le_derivative_rootMultiplicity _ _)
        p.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ)
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.roots.count x) +
          ((∑ x in p.derivative.roots.toFinset \ p.roots.toFinset,
            p.derivative.roots.count x) + 1) := by
      simp only [← count_roots]
      refine' add_le_add_left (add_le_add_right ((Finset.card_eq_sum_ones _).trans_le _) _) _
      

Proof state:
p : ℝ[X]
⊢ ∑ x in Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p), 1 ≤
    ∑ x in Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p),
      Multiset.count x (roots (derivative p))

Next tactic:
refine' Finset.sum_le_sum fun x hx => Nat.succ_le_iff.2 <| _
/-- The number of roots of a real polynomial (counted with multiplicities) is at most the number of
roots of its derivative (counted with multiplicities) plus one. -/
theorem card_roots_le_derivative (p : ℝ[X]) :
    Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 :=
  calc
    Multiset.card p.roots = ∑ x in p.roots.toFinset, p.roots.count x :=
      (Multiset.toFinset_sum_count_eq _).symm
    _ = ∑ x in p.roots.toFinset, (p.roots.count x - 1 + 1) :=
      (Eq.symm <| Finset.sum_congr rfl fun x hx => tsub_add_cancel_of_le <|
        Nat.succ_le_iff.2 <| Multiset.count_pos.2 <| Multiset.mem_toFinset.1 hx)
    _ = (∑ x in p.roots.toFinset, (p.rootMultiplicity x - 1)) + p.roots.toFinset.card := by
      simp only [Finset.sum_add_distrib, Finset.card_eq_sum_ones, count_roots]
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.rootMultiplicity x) +
          ((p.derivative.roots.toFinset \ p.roots.toFinset).card + 1) :=
      (add_le_add
        (Finset.sum_le_sum fun x _ => rootMultiplicity_sub_one_le_derivative_rootMultiplicity _ _)
        p.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ)
    _ ≤ (∑ x in p.roots.toFinset, p.derivative.roots.count x) +
          ((∑ x in p.derivative.roots.toFinset \ p.roots.toFinset,
            p.derivative.roots.count x) + 1) := by
      simp only [← count_roots]
      refine' add_le_add_left (add_le_add_right ((Finset.card_eq_sum_ones _).trans_le _) _) _
      refine' Finset.sum_le_sum fun x hx => Nat.succ_le_iff.2 <| _
      

Proof state:
p : ℝ[X]
x : ℝ
hx : x ∈ Multiset.toFinset (roots (derivative p)) \ Multiset.toFinset (roots p)
⊢ 0 < Multiset.count x (roots (derivative p))

Next tactic:
rw [Multiset.count_pos, ← Multiset.mem_toFinset]