mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_3
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ ((bif f_max 0 then 1 else 0) + c * cantorFunction c fun n ↦ f_max (n + 1)) = c * ∑' (n : ℕ), c ^ n

Tactic:
]

Goal state after:
case e'_3
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ ((bif f_max 0 then 1 else 0) + c * cantorFunction c fun n ↦ f_max (n + 1)) = c * ∑' (n : ℕ), c ^ n
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
⊢ 0 = 0

pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
⊢ ∀ (x : σ →₀ ℕ), (coeff x) f • constantCoeff (x.prod fun s e ↦ a s ^ e) = 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
⊢ ∀ (x : σ →₀ ℕ), (coeff x) f • constantCoeff (x.prod fun s e ↦ a s ^ e) = 0
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case succ
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
n : ℕ
ih : ∀ {f g : ℕ → Bool}, (∀ k < n, f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g
f g : ℕ → Bool
hn : ∀ k < n + 1, f k = g k
fn : f (n + 1) = false
gn : g (n + 1) = true
⊢ ((bif f 0 then 1 else 0) + c * cantorFunction c fun n ↦ f (n + 1)) <
    (bif g 0 then 1 else 0) + c * cantorFunction c fun n ↦ g (n + 1)

Tactic:
]

Goal state after:
case succ
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
n : ℕ
ih : ∀ {f g : ℕ → Bool}, (∀ k < n, f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g
f g : ℕ → Bool
hn : ∀ k < n + 1, f k = g k
fn : f (n + 1) = false
gn : g (n + 1) = true
⊢ ((bif f 0 then 1 else 0) + c * cantorFunction c fun n ↦ f (n + 1)) <
    (bif g 0 then 1 else 0) + c * cantorFunction c fun n ↦ g (n + 1)
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case succ
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
n : ℕ
ih : ∀ {f g : ℕ → Bool}, (∀ k < n, f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g
f g : ℕ → Bool
hn : ∀ k < n + 1, f k = g k
fn : f (n + 1) = false
gn : g (n + 1) = true
⊢ ((bif g 0 then 1 else 0) + c * cantorFunction c fun n ↦ f (n + 1)) <
    (bif g 0 then 1 else 0) + c * cantorFunction c fun n ↦ g (n + 1)

Tactic:
]

Goal state after:
case succ
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
n : ℕ
ih : ∀ {f g : ℕ → Bool}, (∀ k < n, f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g
f g : ℕ → Bool
hn : ∀ k < n + 1, f k = g k
fn : f (n + 1) = false
gn : g (n + 1) = true
⊢ ((bif g 0 then 1 else 0) + c * cantorFunction c fun n ↦ f (n + 1)) <
    (bif g 0 then 1 else 0) + c * cantorFunction c fun n ↦ g (n + 1)
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
⊢ (PowerSeries.coeff k) (f * g) =
    ∑ x ∈ (q ^ k).divisorsAntidiagonal,
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
        Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
⊢ (PowerSeries.coeff k) (f * g) =
    ∑ x ∈ (q ^ k).divisorsAntidiagonal,
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
        Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
d : σ →₀ ℕ
hd : ¬d = 0
this : constantCoeff (d.prod fun s e ↦ a s ^ e) = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case false
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
f g : ℕ → Bool
h : ¬f = g
this : ∃ n, f n ≠ g n
n : ℕ := Nat.find this
hn : ∀ k < n, f k = g k
fn : f n = false
⊢ ¬g n = f n

Tactic:
]

Goal state after:
case false
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
f g : ℕ → Bool
h : ¬f = g
this : ∃ n, f n ≠ g n
n : ℕ := Nat.find this
hn : ∀ k < n, f k = g k
fn : f n = false
⊢ ¬g n = f n
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
⊢ ∑ p ∈ Finset.antidiagonal k, (PowerSeries.coeff p.1) f * (PowerSeries.coeff p.2) g =
    ∑
      x ∈
        Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
          (Finset.antidiagonal k),
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
        Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2

pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
⊢ ∀ x ∈ (q ^ k).divisorsAntidiagonal,
    x ∉
        Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
          (Finset.antidiagonal k) →
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
          Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2 =
        0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
⊢ ∑ p ∈ Finset.antidiagonal k, (PowerSeries.coeff p.1) f * (PowerSeries.coeff p.2) g =
    ∑
      x ∈
        Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
          (Finset.antidiagonal k),
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
        Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2

case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
⊢ ∀ x ∈ (q ^ k).divisorsAntidiagonal,
    x ∉
        Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
          (Finset.antidiagonal k) →
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
          Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2 =
        0
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case true
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
f g : ℕ → Bool
h : ¬f = g
this : ∃ n, f n ≠ g n
n : ℕ := Nat.find this
hn : ∀ k < n, f k = g k
fn : f n = true
⊢ ¬g n = f n

Tactic:
]

Goal state after:
case true
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
f g : ℕ → Bool
h : ¬f = g
this : ∃ n, f n ≠ g n
n : ℕ := Nat.find this
hn : ∀ k < n, f k = g k
fn : f n = true
⊢ ¬g n = f n
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case a
⊢ #(CauSeq.Completion.Cauchy abs) ≤ 𝔠

Tactic:
]

Goal state after:
case a
⊢ #(CauSeq.Completion.Cauchy abs) ≤ 𝔠
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case a
⊢ 𝔠 = 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_3
⊢ 𝔠 = 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: ⊢ 𝔠 = 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: c : ℝ
f g : ℕ → Bool
n : ℕ
⊢ ℵ₀ < 𝔠

Tactic:
]

Goal state after:
c : ℝ
f g : ℕ → Bool
n : ℕ
⊢ ℵ₀ < 𝔠
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (algebraMap R S) ((coeff e) f) = (coeff e) f • (coeff e) (e.prod fun s e ↦ X s ^ e)

pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ e → (coeff x) f • (coeff e) (x.prod fun s e ↦ X s ^ e) = 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (algebraMap R S) ((coeff e) f) = (coeff e) f • (coeff e) (e.prod fun s e ↦ X s ^ e)

σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ e → (coeff x) f • (coeff e) (x.prod fun s e ↦ X s ^ e) = 0
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a : ℝ
⊢ ¬#↑(Ioi a) < 𝔠

Tactic:
]

Goal state after:
a : ℝ
⊢ ¬#↑(Ioi a) < 𝔠
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑(Iio a ∪ {a} ∪ Ioi a) < 𝔠

Tactic:
]

Goal state after:
a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑(Iio a ∪ {a} ∪ Ioi a) < 𝔠
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (algebraMap R S) ((coeff e) f) = (algebraMap R S) ((coeff e) f)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑(Iio a) + #↑{a} + #↑(Ioi a) < 𝔠

Tactic:
]

Goal state after:
a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑(Iio a) + #↑{a} + #↑(Ioi a) < 𝔠
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e d : σ →₀ ℕ
hd : d ≠ e
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
h2 : (fun x ↦ a + a - x) '' Ioi a = Iio a
⊢ #↑((fun x ↦ a + a - x) '' Ioi a) + #↑{a} + #↑(Ioi a) < 𝔠

Tactic:
]

Goal state after:
a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
h2 : (fun x ↦ a + a - x) '' Ioi a = Iio a
⊢ #↑((fun x ↦ a + a - x) '' Ioi a) + #↑{a} + #↑(Ioi a) < 𝔠
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
f : MvPowerSeries σ R
n : τ →₀ ℕ
this : ∀ {r : R}, h r = h.toAddMonoidHom r
⊢ finsum ?m.132 = ∑ᶠ (d : σ →₀ ℕ), (coeff d) ((map h) f) • (coeff n) (d.prod fun s e ↦ (map h) (a s) ^ e)

pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
f : MvPowerSeries σ R
n : τ →₀ ℕ
this : ∀ {r : R}, h r = h.toAddMonoidHom r
⊢ ∀ (x : σ →₀ ℕ), h.toAddMonoidHom ((coeff x) f • (coeff n) (x.prod fun s e ↦ a s ^ e)) = ?m.132 x

pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
f : MvPowerSeries σ R
n : τ →₀ ℕ
this : ∀ {r : R}, h r = h.toAddMonoidHom r
⊢ (σ →₀ ℕ) → S

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
f : MvPowerSeries σ R
n : τ →₀ ℕ
this : ∀ {r : R}, h r = h.toAddMonoidHom r
⊢ ∀ (x : σ →₀ ℕ),
    h.toAddMonoidHom ((coeff x) f • (coeff n) (x.prod fun s e ↦ a s ^ e)) =
      (coeff x) ((map h) f) • (coeff n) (x.prod fun s e ↦ (map h) (a s) ^ e)
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
h2 : (fun x ↦ a + a - x) '' Ioi a = Iio a
⊢ 1 < 𝔠

Tactic:
]

Goal state after:
a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
h2 : (fun x ↦ a + a - x) '' Ioi a = Iio a
⊢ 1 < 𝔠
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ (coeff 0) f • (coeff n) (Finsupp.prod 0 fun s e ↦ 0 s ^ e) = (algebraMap R S) ((coeff n) (C (constantCoeff f)))

pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ 0 → (coeff x) f • (coeff n) (x.prod fun s e ↦ 0 s ^ e) = 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ (coeff 0) f • (coeff n) (Finsupp.prod 0 fun s e ↦ 0 s ^ e) = (algebraMap R S) ((coeff n) (C (constantCoeff f)))

σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ 0 → (coeff x) f • (coeff n) (x.prod fun s e ↦ 0 s ^ e) = 0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
i j : ℕ
hab : i + j = k ∧ q ^ k ≠ 0
h :
  (q ^ i, q ^ j) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (q ^ i, q ^ j).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (q ^ i, q ^ j).2 =
    0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
i j : ℕ
hab : i + j = k ∧ q ^ k ≠ 0
h :
  (q ^ i, q ^ j) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (q ^ i, q ^ j).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (q ^ i, q ^ j).2 =
    0
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a b : ℝ
h : a < b
h1 : #↑((fun x ↦ x - a) '' Ioo a b) ≤ #↑(Ioo a b)
⊢ 𝔠 ≤ #↑(Ioo 0 (b - a))

Tactic:
]

Goal state after:
a b : ℝ
h : a < b
h1 : #↑((fun x ↦ x - a) '' Ioo a b) ≤ #↑(Ioo a b)
⊢ 𝔠 ≤ #↑(Ioo 0 (b - a))
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a b : ℝ
h1 : #↑((fun x ↦ x - a) '' Ioo a b) ≤ #↑(Ioo a b)
h : 0 < b - a
h2 : #↑(Inv.inv '' Ioo 0 (b - a)) ≤ #↑(Ioo 0 (b - a))
⊢ 𝔠 ≤ 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹⁴ : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
a : σ → MvPowerSeries τ S
T : Type u_6
inst✝¹¹ : CommRing T
inst✝¹⁰ : UniformSpace T
inst✝⁹ : T2Space T
inst✝⁸ : CompleteSpace T
inst✝⁷ : IsUniformAddGroup T
inst✝⁶ : IsTopologicalRing T
inst✝⁵ : IsLinearTopology T T
inst✝⁴ : Algebra R T
ε : MvPowerSeries τ S →ₐ[R] T
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
ha : HasSubst a
hε : Continuous ⇑ε
⊢ ⇑ε ∘ subst a = ⇑ε ∘ subst a

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: x y : ℝ
h : x < y
⊢ ℵ₀ < 𝔠

Tactic:
]

Goal state after:
x y : ℝ
h : x < y
⊢ ℵ₀ < 𝔠
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
hf : IsNilpotent (constantCoeff f)
⊢ IsNilpotent (∑ᶠ (d : σ →₀ ℕ), (coeff d) f • constantCoeff (d.prod fun s e ↦ a s ^ e))

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
hf : IsNilpotent (constantCoeff f)
⊢ IsNilpotent (∑ᶠ (d : σ →₀ ℕ), (coeff d) f • constantCoeff (d.prod fun s e ↦ a s ^ e))
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
hf : IsNilpotent (constantCoeff f)
d : σ →₀ ℕ
hd : d = 0
⊢ IsNilpotent (constantCoeff (Finsupp.prod 0 fun s e ↦ a s ^ e) • (algebraMap R S) ((coeff 0) f))

Tactic:
]

Goal state after:
case pos
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
hf : IsNilpotent (constantCoeff f)
d : σ →₀ ℕ
hd : d = 0
⊢ IsNilpotent (constantCoeff (Finsupp.prod 0 fun s e ↦ a s ^ e) • (algebraMap R S) ((coeff 0) f))
Module: Mathlib.Analysis.Complex.Cardinality

Goal state before:
pp: ⊢ 𝔠 = 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Cardinality

Goal state before:
pp: ⊢ ℵ₀ < 𝔠

Tactic:
]

Goal state after:
⊢ ℵ₀ < 𝔠
Module: Mathlib.LinearAlgebra.Complex.FiniteDimensional

Goal state before:
pp: ⊢ 2 = 2

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.FiniteDimensional

Goal state before:
pp: ⊢ Cardinal.lift.{u, 0} (Module.rank ℝ ℂ) = 2

Tactic:
by
  rw [← finrank_eq_rank, finrank_real_complex, Cardinal.lift_natCast, Nat.cast_ofNat]

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.FiniteDimensional

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
⊢ 2 * Cardinal.lift.{0, u_1} (Module.rank ℂ E) = Cardinal.lift.{0, u_1} (2 * Module.rank ℂ E)

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
⊢ 2 * Cardinal.lift.{0, u_1} (Module.rank ℂ E) = Cardinal.lift.{0, u_1} (2 * Module.rank ℂ E)
Module: Mathlib.LinearAlgebra.Complex.FiniteDimensional

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
⊢ 2 * finrank ℂ E = 2 * finrank ℂ E

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.FiniteDimensional

Goal state before:
pp: ⊢ lift.{0, 0} #ℚ < lift.{0, 0} #ℝ

Tactic:
simpa [mk_real] using aleph0_lt_continuum

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
hf : IsNilpotent (constantCoeff f)
d : σ →₀ ℕ
hd : ¬d = 0
i : σ
hi : i ∈ d.support
⊢ IsNilpotent
    (((algebraMap R S) ((coeff d) f) * ∏ x ∈ d.support.erase i, constantCoeff (a x ^ d x)) * constantCoeff (a i) ^ d i)

Tactic:
]

Goal state after:
case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
hf : IsNilpotent (constantCoeff f)
d : σ →₀ ℕ
hd : ¬d = 0
i : σ
hi : i ∈ d.support
⊢ IsNilpotent
    (((algebraMap R S) ((coeff d) f) * ∏ x ∈ d.support.erase i, constantCoeff (a x ^ d x)) * constantCoeff (a i) ^ d i)
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ∃ i, q ^ i = a
hb : ¬∃ j, q ^ j = b
⊢ 0 = 0

pp: case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ∃ i, q ^ i = a
hb : ¬∃ j, q ^ j = b
⊢ ¬∃ a_1, q ^ a_1 = (a, b).2

Tactic:
]

Goal state after:
case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ∃ i, q ^ i = a
hb : ¬∃ j, q ^ j = b
⊢ ¬∃ a_1, q ^ a_1 = (a, b).2
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
a : σ → MvPowerSeries τ S
υ : Type u_7
T : Type u_8
inst✝¹ : CommRing T
inst✝ : Algebra S T
b : τ → MvPowerSeries υ T
ha : HasSubst a
hb : HasSubst b
this✝ : UniformSpace S := ⊥
this : UniformSpace T := ⊥
⊢ Filter.Tendsto (fun s ↦ (substAlgHom hb) (a s)) Filter.cofinite (nhds 0)

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
a : σ → MvPowerSeries τ S
υ : Type u_7
T : Type u_8
inst✝¹ : CommRing T
inst✝ : Algebra S T
b : τ → MvPowerSeries υ T
ha : HasSubst a
hb : HasSubst b
this✝ : UniformSpace S := ⊥
this : UniformSpace T := ⊥
⊢ Filter.Tendsto (fun s ↦ (substAlgHom hb) (a s)) Filter.cofinite (nhds 0)
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
hf : f 0 = 0
s : ℂ
n : ℕ
⊢ (if n = 0 then 0 else f n / ↑n ^ s) = f n * ↑n ^ (-s)

Tactic:
]

Goal state after:
f : ℕ → ℂ
hf : f 0 = 0
s : ℂ
n : ℕ
⊢ (if n = 0 then 0 else f n / ↑n ^ s) = f n * ↑n ^ (-s)
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: case inl
s : ℂ
hs : s ≠ 0
f : ℕ → ℂ
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: case inr
s : ℂ
hs : s ≠ 0
f : ℕ → ℂ
n : ℕ
hn : n ≠ 0
⊢ f n / ↑n ^ s = f n / ↑n ^ s

Tactic:
]

Goal state after: