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
⊢ cantorFunction c f_max = c / (1 - c)

pp: case e'_4
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
⊢ cantorFunction c g_min = 1

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
⊢ cantorFunction c f_max = c / (1 - c)
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_4
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
⊢ cantorFunction c g_min = 1

Tactic:
·

Goal state after:
case e'_4
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
⊢ cantorFunction c g_min = 1
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : 1 < q
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) 1) 0, map_zero' := ⋯ } n = 1 n

pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : ¬1 < q
⊢ ((algebraMap R (ArithmeticFunction R)) (PowerSeries.constantCoeff 1)) n = 1 n

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : 1 < q
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) 1) 0, map_zero' := ⋯ } n = 1 n
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_4.refine_1
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
⊢ ∀ (b' : ℕ), b' ≠ 0 → cantorFunctionAux c g_min b' = 0

pp: case e'_4.refine_2
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
⊢ cantorFunctionAux c g_min 0 = 1

Tactic:
·

Goal state after:
case e'_4.refine_1
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
⊢ ∀ (b' : ℕ), b' ≠ 0 → cantorFunctionAux c g_min b' = 0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : 1 < q
hn : ∃ k, q ^ k = n
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) 1) 0, map_zero' := ⋯ } n = 1 n

pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : 1 < q
hn : ¬∃ k, q ^ k = n
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) 1) 0, map_zero' := ⋯ } n = 1 n

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : 1 < q
hn : ∃ k, q ^ k = n
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) 1) 0, map_zero' := ⋯ } n = 1 n
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : 1 < q
hn : ¬∃ k, q ^ k = n
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) 1) 0, map_zero' := ⋯ } n = 1 n

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : 1 < q
hn : ¬∃ k, q ^ k = n
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) 1) 0, map_zero' := ⋯ } n = 1 n
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_4.refine_1.zero
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
hn : 0 ≠ 0
⊢ cantorFunctionAux c g_min 0 = 0

pp: case e'_4.refine_1.succ
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
n✝ : ℕ
hn : n✝ + 1 ≠ 0
⊢ cantorFunctionAux c g_min (n✝ + 1) = 0

Tactic:
·

Goal state after:
case e'_4.refine_1.zero
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
hn : 0 ≠ 0
⊢ cantorFunctionAux c g_min 0 = 0
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
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
d : σ →₀ ℕ
hd : d = 0
⊢ (coeff d) f • constantCoeff (d.prod fun s e ↦ a s ^ e) = 0

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
⊢ (coeff d) f • constantCoeff (d.prod fun s e ↦ a s ^ e) = 0

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
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
d : σ →₀ ℕ
hd : d = 0
⊢ (coeff d) f • constantCoeff (d.prod fun s e ↦ a s ^ e) = 0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : ¬1 < q
⊢ ((algebraMap R (ArithmeticFunction R)) (PowerSeries.constantCoeff 1)) n = 1 n

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommSemiring R
q n : ℕ
hq : ¬1 < q
⊢ ((algebraMap R (ArithmeticFunction R)) (PowerSeries.constantCoeff 1)) n = 1 n
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_4.refine_2
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
⊢ cantorFunctionAux c g_min 0 = 1

Tactic:
·

Goal state after:
case e'_4.refine_2
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
⊢ cantorFunctionAux c g_min 0 = 1
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
⊢ (coeff d) f • constantCoeff (d.prod fun s e ↦ a s ^ e) = 0

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
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
d : σ →₀ ℕ
hd : ¬d = 0
⊢ (coeff d) f • constantCoeff (d.prod fun s e ↦ a 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
n : ℕ
hq : 1 < q
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (f * g)) 0, map_zero' := ⋯ } n =
    ({ toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0, map_zero' := ⋯ } *
        { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0, map_zero' := ⋯ })
      n

pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : ¬1 < q
⊢ ((algebraMap R (ArithmeticFunction R)) (PowerSeries.constantCoeff (f * g))) n =
    ((algebraMap R (ArithmeticFunction R)) (PowerSeries.constantCoeff f) *
        (algebraMap R (ArithmeticFunction R)) (PowerSeries.constantCoeff g))
      n

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (f * g)) 0, map_zero' := ⋯ } n =
    ({ toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0, map_zero' := ⋯ } *
        { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0, map_zero' := ⋯ })
      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
n : ℕ
hq : 1 < q
hn : ∃ a, q ^ a = n
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (f * g)) 0 n =
    ∑ x ∈ n.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

pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (f * g)) 0 n =
    ∑ x ∈ n.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
n : ℕ
hq : 1 < q
hn : ∃ a, q ^ a = n
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (f * g)) 0 n =
    ∑ x ∈ n.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.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
⊢ ¬cantorFunction c f = cantorFunction c g

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
⊢ ¬cantorFunction c f = cantorFunction c g

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
⊢ ¬cantorFunction c f = cantorFunction c g
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
⊢ ¬cantorFunction c f = cantorFunction c g

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
⊢ ¬cantorFunction c f = cantorFunction c g
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
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case a
⊢ #ℝ ≤ 𝔠

pp: case a
⊢ 𝔠 ≤ #ℝ

Tactic:
·

Goal state after:
case a
⊢ #ℝ ≤ 𝔠
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case a
⊢ 𝔠 ≤ #ℝ

Tactic:
·

Goal state after:
case a
⊢ 𝔠 ≤ #ℝ
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case e'_3
⊢ 𝔠 = #(ℕ → Bool)

pp: case a.convert_1
⊢ ℝ

pp: case a.convert_2
⊢ 0 < ?a.convert_1✝

pp: case a.convert_3
⊢ ?a.convert_1✝ < 1 / 2

Tactic:
·

Goal state after:
case e'_3
⊢ 𝔠 = #(ℕ → Bool)
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case a.convert_1
⊢ ℝ

pp: case a.convert_2
⊢ 0 < ?a.convert_1✝

pp: case a.convert_3
⊢ ?a.convert_1✝ < 1 / 2

Tactic:
·

Goal state after:
case a.convert_1
⊢ ℝ
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case a.convert_2
⊢ 0 < 1 / 3

pp: case a.convert_3
⊢ 1 / 3 < 1 / 2

Tactic:
·

Goal state after:
case a.convert_2
⊢ 0 < 1 / 3
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case a.convert_3
⊢ 1 / 3 < 1 / 2

Tactic:
·

Goal state after:
case a.convert_3
⊢ 1 / 3 < 1 / 2
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
⊢ ∀ 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
⊢ ∀ 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.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)
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
⊢ ∀ (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
⊢ ∀ (x : σ →₀ ℕ), x ≠ e → (coeff x) f • (coeff e) (x.prod fun s e ↦ X 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
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
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0

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
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).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
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
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    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
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
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0

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
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).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
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
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0