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

Goal state before:
pp: c : ℝ
f : ℕ → Bool
n : ℕ
h : 0 ≤ c
⊢ 0 ≤ cantorFunctionAux c f n

Tactic:
cases h' : f n

Goal state after:
case false
c : ℝ
f : ℕ → Bool
n : ℕ
h : 0 ≤ c
h' : f n = false
⊢ 0 ≤ cantorFunctionAux c f n

case true
c : ℝ
f : ℕ → Bool
n : ℕ
h : 0 ≤ c
h' : f n = true
⊢ 0 ≤ cantorFunctionAux c f n
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: 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
n : ℕ
⊢ f n = true → f_max n = true

Tactic:
hn

Goal state after:
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
n : ℕ
hn : f n = true
⊢ f_max n = true
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: 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
n : ℕ
⊢ g_min n = true → g n = true

Tactic:
hn

Goal state after:
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
n : ℕ
hn : g_min n = true
⊢ g n = true
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: 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
⊢ c + c < 1

pp: 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
⊢ 0 < 1 - c

Tactic:
· convert! _root_.add_lt_add h2 h2
        norm_num

Goal state after:
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
⊢ 0 < 1 - c
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
n : ℕ
⊢ n ≠ 0 → cantorFunctionAux c g_min n = 0

Tactic:
hn

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
n : ℕ
hn : n ≠ 0
⊢ cantorFunctionAux c g_min n = 0
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f g : ℕ → ℂ
s : ℂ
hf : LSeriesSummable f s
S : Set ℕ
hS : S ∈ cofinite
hS' : Set.EqOn f g S
n : ℕ
hn : n ∈ S ∧ ¬n = 0
⊢ term f s n = term g s n

Tactic:
simp [hn.2, hS' hn.1]

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

Goal state before:
pp: τ : Type u_3
S : Type u_4
inst✝² : CommRing S
inst✝¹ : TopologicalSpace S
inst✝ : DiscreteTopology S
a : MvPowerSeries τ S
⊢ MvPowerSeries.HasEval (Function.const Unit a) ↔ MvPowerSeries.HasEval fun x ↦ a

Tactic:
Function.const_def

Goal state after:
τ : Type u_3
S : Type u_4
inst✝² : CommRing S
inst✝¹ : TopologicalSpace S
inst✝ : DiscreteTopology S
a : MvPowerSeries τ S
⊢ MvPowerSeries.HasEval (Function.const Unit a) ↔ MvPowerSeries.HasEval (Function.const Unit a)
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: case inr
f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)
hC₀ : 0 ≤ C
hsum : Summable fun n ↦ ‖↑C / ↑n ^ (s + (1 - ↑x))‖
n : ℕ
hn : n > 0
⊢ ‖term f s n‖ ≤ ‖↑C / ↑n ^ (s + (1 - ↑x))‖

Tactic:
have hn' : 0 < (n : ℝ) ^ s.re := Real.rpow_pos_of_pos (Nat.cast_pos.mpr hn) _

Goal state after:
case inr
f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)
hC₀ : 0 ≤ C
hsum : Summable fun n ↦ ‖↑C / ↑n ^ (s + (1 - ↑x))‖
n : ℕ
hn : n > 0
hn' : 0 < ↑n ^ s.re
⊢ ‖term f s n‖ ≤ ‖↑C / ↑n ^ (s + (1 - ↑x))‖
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
τ : Type u_3
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : MvPowerSeries τ S
ha : HasSubst a
f : R⟦X⟧
⊢ MvPowerSeries.constantCoeff (subst a f) = ∑ᶠ (d : ℕ), (coeff d) f • MvPowerSeries.constantCoeff (a ^ d)

Tactic:
simp only [← MvPowerSeries.coeff_zero_eq_constantCoeff_apply, coeff_subst ha f 0]

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), (f i).IsMultiplicative
hf' : ¬Multipliable f
⊢ IsMultiplicative 1

Tactic:
exact isMultiplicative_one

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
b : (Fin 2 →₀ ℕ) × (Fin 2 →₀ ℕ)
hb : b ∈ antidiagonal e
hb' : b ≠ (single 0 (e 0), single 1 (e 1))
hmul_ne_zero : ¬(MvPowerSeries.coeff b.1) (subst X₀ f) * (MvPowerSeries.coeff b.2) (subst X₁ f) = 0
h0 : b.1 = single 0 (b.1 0) ∧ ¬(coeff (b.1 0)) f = 0
h1 : b.2 = single 1 (b.2 1) ∧ ¬(coeff (b.2 1)) f = 0
⊢ b = (single 0 (e 0), single 1 (e 1))

Tactic:
rw [Prod.ext_iff, ← mem_antidiagonal.mp hb, h0.1, h1.1]

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
b : (Fin 2 →₀ ℕ) × (Fin 2 →₀ ℕ)
hb : b ∈ antidiagonal e
hb' : b ≠ (single 0 (e 0), single 1 (e 1))
hmul_ne_zero : ¬(MvPowerSeries.coeff b.1) (subst X₀ f) * (MvPowerSeries.coeff b.2) (subst X₁ f) = 0
h0 : b.1 = single 0 (b.1 0) ∧ ¬(coeff (b.1 0)) f = 0
h1 : b.2 = single 1 (b.2 1) ∧ ¬(coeff (b.2 1)) f = 0
⊢ single 0 (b.1 0) =
      (single 0 ((single 0 (b.1 0) + single 1 (b.2 1)) 0), single 1 ((single 0 (b.1 0) + single 1 (b.2 1)) 1)).1 ∧
    single 1 (b.2 1) =
      (single 0 ((single 0 (b.1 0) + single 1 (b.2 1)) 0), single 1 ((single 0 (b.1 0) + single 1 (b.2 1)) 1)).2
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
H : ¬DifferentiableWithinAt ℝ f (Ioi x) x
⊢ derivWithin f (Ioi x) x = derivWithin f (Ici x) x

Tactic:
rw [derivWithin_zero_of_not_differentiableWithinAt H,
      derivWithin_zero_of_not_differentiableWithinAt]

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
H : ¬DifferentiableWithinAt ℝ f (Ioi x) x
⊢ ¬DifferentiableWithinAt ℝ f (Ici x) x
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
⊢ HasFDerivAt f f' x ↔ (fun h ↦ f (x + h) - f x - f' h) =o[𝓝 0] fun h ↦ h

Tactic:
rw [hasFDerivAt_iff_isLittleO, ← map_add_left_nhds_zero x, isLittleO_map]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
⊢ ((fun x' ↦ f x' - f x - f' (x' - x)) ∘ fun x_1 ↦ x + x_1) =o[𝓝 0] ((fun x' ↦ x' - x) ∘ fun x_1 ↦ x + x_1) ↔
    (fun h ↦ f (x + h) - f x - f' h) =o[𝓝 0] fun h ↦ h
Module: Mathlib.Analysis.Calculus.FormalMultilinearSeries

Goal state before:
pp: 𝕜 : Type u
E : Type v
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
n : ℕ
y : Fin n → 𝕜
⊢ (p n) y = (∏ i, y i) • p.coeff n

Tactic:
by
  convert! (p n).toMultilinearMap.map_smul_univ y 1
  simp only [Pi.one_apply, smul_eq_mul, mul_one]

Goal state after:
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
n : ℕ
⊢ ‖p.compContinuousLinearMap u n‖ₑ ≤ ‖p n‖ₑ * ‖u‖ₑ ^ n

Tactic:
← ofReal_norm,

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
n : ℕ
⊢ ENNReal.ofReal ‖p.compContinuousLinearMap u n‖ ≤ ‖p n‖ₑ * ‖u‖ₑ ^ n
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
n : ℕ
⊢ ENNReal.ofReal ‖p.compContinuousLinearMap u n‖ ≤ ‖p n‖ₑ * ‖u‖ₑ ^ n

Tactic:
← ofReal_norm,

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
n : ℕ
⊢ ENNReal.ofReal ‖p.compContinuousLinearMap u n‖ ≤ ENNReal.ofReal ‖p n‖ * ‖u‖ₑ ^ n
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
n : ℕ
⊢ ENNReal.ofReal ‖p.compContinuousLinearMap u n‖ ≤ ENNReal.ofReal ‖p n‖ * ‖u‖ₑ ^ n

Tactic:
← ofReal_norm,

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 F G
u : E →L[𝕜] F
n : ℕ
⊢ ENNReal.ofReal ‖p.compContinuousLinearMap u n‖ ≤ ENNReal.ofReal ‖p n‖ * ENNReal.ofReal ‖u‖ ^ n
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: case h.inl
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
hr : r = 0
⊢ ‖p (n + 1)‖ * ↑0 ^ n ≤ ‖p 1‖ ∨ ‖p (n + 1)‖ * ↑0 ^ n ≤ C / ↑0

Tactic:
cases n <;> simp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
G : Type wG
G' : Type wG'
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : G [×0]→L[𝕜] G'
x : Fin 0 → G
⊢ ‖f x‖ₑ = ‖f‖ₑ

Tactic:
← ofReal_norm,

Goal state after:
𝕜 : Type u
G : Type wG
G' : Type wG'
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : G [×0]→L[𝕜] G'
x : Fin 0 → G
⊢ ENNReal.ofReal ‖f x‖ = ENNReal.ofReal ‖f‖
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
s : Set E
g : F →L[𝕜] G
h : CPolynomialOn 𝕜 f s
⊢ CPolynomialOn 𝕜 (⇑g ∘ f) s

Tactic:
by
  rintro x hx
  rcases h x hx with ⟨p, n, r, hp⟩
  exact ⟨g.compFormalMultilinearSeries p, n, r, g.comp_hasFiniteFPowerSeriesOnBall hp⟩

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
n : ℕ
⊢ ((c n • ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E) fun x ↦ 0) = Pi.single 0 (c 0 • 1) n

Tactic:
cases n <;> simp

Goal state after:
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : r' * r < 1
hrz : ¬r' = 0
n : ℕ
⊢ n ≠ 0 → ‖‖ofScalars E c n‖ * ↑r' ^ n‖ ≤ ‖‖c n‖ * ↑r' ^ n‖

Tactic:
hn

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : r' * r < 1
hrz : ¬r' = 0
n : ℕ
hn : n ≠ 0
⊢ ‖‖ofScalars E c n‖ * ↑r' ^ n‖ ≤ ‖‖c n‖ * ↑r' ^ n‖
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
n : ℕ
⊢ n ≠ 0 → ‖‖ofScalars E c n‖ * ↑r' ^ n‖ ≤ ‖‖c n‖ * ↑r' ^ n‖

Tactic:
hn

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
n : ℕ
hn : n ≠ 0
⊢ ‖‖ofScalars E c n‖ * ↑r' ^ n‖ ≤ ‖‖c n‖ * ↑r' ^ n‖
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
hc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖
⊢ n ≠ 0 → 2 * ‖‖ofScalars E c n‖ * ↑r ^ n‖ ≤ ‖‖ofScalars E c (n + 1)‖ * ↑r ^ (n + 1)‖

Tactic:
hn

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
hc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖
hn : n ≠ 0
⊢ 2 * ‖‖ofScalars E c n‖ * ↑r ^ n‖ ≤ ‖‖ofScalars E c (n + 1)‖ * ↑r ^ (n + 1)‖
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hr : r = ∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 ∞)
n : ℕ
⊢ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖ ≠ 0 →
    ENNReal.ofReal ‖c (n + 1).succ‖ / ENNReal.ofReal ‖c (n + 1)‖ = ENNReal.ofReal (‖c (n + 1).succ‖ / ‖c (n + 1)‖)

Tactic:
hn

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hr : r = ∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 ∞)
n : ℕ
hn : ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖ ≠ 0
⊢ ENNReal.ofReal ‖c (n + 1).succ‖ / ENNReal.ofReal ‖c (n + 1)‖ = ENNReal.ofReal (‖c (n + 1).succ‖ / ‖c (n + 1)‖)
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : ‖y - (x, x)‖₊ < r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := fun n ↦ ((p n) fun x_1 ↦ y.1 - x) - (p n) fun x_1 ↦ y.2 - x
hA : HasSum (fun n ↦ A (n + 2)) (f y.1 - f y.2 - (p 1) fun x ↦ y.1 - y.2)
B : ℕ → ℝ := fun n ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((↑n + 2) * a ^ n)
n : ℕ
⊢ ‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖

Tactic:
by
          simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
            Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using!
            (p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x

Goal state after:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
hq : 0 < q.radius
hp : 0 < p.radius
rq rp : ℝ≥0
hrp : rp < 1 ∧ ↑rp < p.radius
hrq : rq < 1 ∧ ↑rq < q.radius
rp_pos : 0 < rp
rq_pos : 0 < rq
Cq : ℝ≥0
_hCq0 : Cq > 0
hCq : ∀ (n : ℕ), ‖q n‖₊ * rq ^ n ≤ Cq
Cp : ℝ≥0
hCp1 : Cp ≥ 1
hCp : ∀ (n : ℕ), ‖p n‖₊ * rp ^ n ≤ Cp
r0 : ℝ≥0 := (4 * Cp)⁻¹
r0_pos : 0 < r0
r : ℝ≥0 := rp * rq * r0
r_pos : 0 < r
I : ∀ (i : (n : ℕ) × Composition n), ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst ≤ Cq / 4 ^ i.fst
⊢ Summable fun b ↦ (4 ^ b.fst)⁻¹

Tactic:
have : ∀ n : ℕ, HasSum (fun c : Composition n => (4 ^ n : ℝ≥0)⁻¹) (2 ^ (n - 1) / 4 ^ n) := by
    intro n
    convert! hasSum_fintype fun c : Composition n => (4 ^ n : ℝ≥0)⁻¹
    simp [Finset.card_univ, composition_card, div_eq_mul_inv]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
hq : 0 < q.radius
hp : 0 < p.radius
rq rp : ℝ≥0
hrp : rp < 1 ∧ ↑rp < p.radius
hrq : rq < 1 ∧ ↑rq < q.radius
rp_pos : 0 < rp
rq_pos : 0 < rq
Cq : ℝ≥0
_hCq0 : Cq > 0
hCq : ∀ (n : ℕ), ‖q n‖₊ * rq ^ n ≤ Cq
Cp : ℝ≥0
hCp1 : Cp ≥ 1
hCp : ∀ (n : ℕ), ‖p n‖₊ * rp ^ n ≤ Cp
r0 : ℝ≥0 := (4 * Cp)⁻¹
r0_pos : 0 < r0
r : ℝ≥0 := rp * rq * r0
r_pos : 0 < r
I : ∀ (i : (n : ℕ) × Composition n), ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst ≤ Cq / 4 ^ i.fst
this : ∀ (n : ℕ), HasSum (fun c ↦ (4 ^ n)⁻¹) (2 ^ (n - 1) / 4 ^ n)
⊢ Summable fun b ↦ (4 ^ b.fst)⁻¹
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
⊢ (∑ n ∈ Finset.range M, ∑ r ∈ Fintype.piFinset fun i ↦ Finset.Ico 1 N, (q n) fun i ↦ (p (r i)) fun _j ↦ z) =
    ∑ i ∈ compPartialSumTarget 0 M N, (q.compAlongComposition p i.snd) fun _j ↦ z

Tactic:
Finset.range_eq_Ico,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
⊢ (∑ n ∈ Finset.Ico 0 M, ∑ r ∈ Fintype.piFinset fun i ↦ Finset.Ico 1 N, (q n) fun i ↦ (p (r i)) fun _j ↦ z) =
    ∑ i ∈ compPartialSumTarget 0 M N, (q.compAlongComposition p i.snd) fun _j ↦ z
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
y : E
h'y : x + y ∈ insert x s
hy : y ∈ Metric.eball 0 (min rf' ↑r)
y_mem : y ∈ Metric.eball 0 rf
fy_mem : f (x + y) ∈ insert (f x) t ∩ Metric.eball (f x) rg
n : ℕ
hn : 1 ≤ n
⊢ (f x + ∑ a ∈ Finset.Ico 1 n, (p a) fun _b ↦ y) = ∑ a ∈ Finset.range n, (p a) fun _b ↦ y

Tactic:
Finset.range_eq_Ico,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
δ : ℝ≥0∞
δpos : 0 < δ
hδ : ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
rf' : ℝ≥0∞ := min rf δ
min_pos : 0 < min rf' ↑r
y : E
h'y : x + y ∈ insert x s
hy : y ∈ Metric.eball 0 (min rf' ↑r)
y_mem : y ∈ Metric.eball 0 rf
fy_mem : f (x + y) ∈ insert (f x) t ∩ Metric.eball (f x) rg
n : ℕ
hn : 1 ≤ n
⊢ (f x + ∑ a ∈ Finset.Ico 1 n, (p a) fun _b ↦ y) = ∑ a ∈ Finset.Ico 0 n, (p a) fun _b ↦ y