mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
g : MvPowerSeries σ R
⊢ HasSum f g ↔ ∀ (d : σ →₀ ℕ), HasSum (fun i ↦ (coeff d) (f i)) ((coeff d) g)

Tactic:
HasSum,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
g : MvPowerSeries σ R
⊢ Tendsto (fun s ↦ ∑ b ∈ s, f b) (SummationFilter.unconditional ι).filter (𝓝 g) ↔
    ∀ (d : σ →₀ ℕ),
      Tendsto (fun s ↦ ∑ x ∈ s, (coeff d) (f x)) (SummationFilter.unconditional ι).filter (𝓝 ((coeff d) g))
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
a : R
m : ℕ
⊢ ((trunc (n + 1)) (C a)).coeff m = (Polynomial.C a).coeff m

Tactic:
rw [coeff_trunc, coeff_C, Polynomial.coeff_C]

Goal state after:
R : Type u_1
inst✝ : Semiring R
n : ℕ
a : R
m : ℕ
⊢ (if m < n + 1 then if m = 0 then a else 0 else 0) = if m = 0 then a else 0
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
ih : n • φ.order ≤ (φ ^ n).order
⊢ (n + 1) • φ.order ≤ (φ ^ (n + 1)).order

Tactic:
grw [add_smul, one_smul, pow_succ, ih, le_order_mul]

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
f : R⟦X⟧
j : ℕ
⊢ (coeff j) f = (coeff j) ((mk fun i ↦ (coeff (i + n)) f) * X ^ n + ↑((trunc n) f))

Tactic:
rw [map_add, Polynomial.coeff_coe, coeff_mul_X_pow', coeff_trunc]

Goal state after:
R : Type u_1
inst✝ : Semiring R
n : ℕ
f : R⟦X⟧
j : ℕ
⊢ (coeff j) f = (if n ≤ j then (coeff (j - n)) (mk fun i ↦ (coeff (i + n)) f) else 0) + if j < n then (coeff j) f else 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ (v.adicAbv hb) ((algebraMap R K) r) ≤ 1

Tactic:
adicAbv_of_algebraMap

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ (v.intAdicAbv hb) r ≤ 1
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
n : ℕ
h : ↑n < ψ.order
⊢ (coeff n) (φ * ψ) = 0

Tactic:
suffices coeff n (φ * ψ) = ∑ p ∈ antidiagonal n, 0 by rw [this, Finset.sum_const_zero]

Goal state after:
R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
n : ℕ
h : ↑n < ψ.order
⊢ (coeff n) (φ * ψ) = ∑ p ∈ antidiagonal n, 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ (v.adicAbv hb) ((algebraMap R K) r) < 1 ↔ r ∈ v.asIdeal

Tactic:
adicAbv_of_algebraMap

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ (v.intAdicAbv hb) r < 1 ↔ r ∈ v.asIdeal
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ (v.adicAbv hb) ((algebraMap R K) r) = 1 ↔ r ∉ v.asIdeal

Tactic:
adicAbv_of_algebraMap

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ (v.intAdicAbv hb) r = 1 ↔ r ∉ v.asIdeal
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
g : R⟦X⟧
⊢ HasSum f g ↔ ∀ (d : ℕ), HasSum (fun i ↦ (coeff d) (f i)) ((coeff d) g)

Tactic:
HasSum,

Goal state after:
R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
g : R⟦X⟧
⊢ Tendsto (fun s ↦ ∑ b ∈ s, f b) (SummationFilter.unconditional ι).filter (𝓝 g) ↔
    ∀ (d : ℕ), Tendsto (fun s ↦ ∑ x ∈ s, (coeff d) (f x)) (SummationFilter.unconditional ι).filter (𝓝 ((coeff d) g))
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: case a
x : Cardinal.{u_1}
h₁ : 2 ≤ x
h₂ : x ≤ 𝔠
⊢ x ^ ℵ₀ ≤ 𝔠

Tactic:
rw [← continuum_power_aleph0]
    exact power_le_power_right h₂

Goal state after:
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)

Tactic:
rw [cantorFunction_succ _ (le_of_lt h1) h3, div_eq_mul_inv, ←
        tsum_geometric_of_lt_one (le_of_lt h1) h3]
      apply zero_add

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

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

Tactic:
Complex.finrank_real_complex

Goal state after:
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
⊢ 2 * finrank ℂ E = 2 * finrank ℂ E
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
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
⊢ ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d ≤ weightedOrder w (subst a f)

Tactic:
apply MvPowerSeries.le_weightedOrder
  intro d hd
  rw [coeff_subst ha, finsum_eq_zero_of_forall_eq_zero]
  intro x
  by_cases hfx : f.coeff x = 0
  · simp [hfx]
  rw [coeff_eq_zero_of_lt_weightedOrder w, smul_zero]
  refine hd.trans_le (((biInf_le _ hfx).trans ?_).trans (le_weightedOrder_prod ..))
  simp only [Finsupp.weight_apply, Finsupp.sum, Function.comp_apply]
  exact Finset.sum_le_sum fun i hi ↦ .trans (by simp) (le_weightedOrder_pow ..)

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

Goal state before:
pp: f : ℕ → ℂ
s : ℂ
h : LSeriesSummable f s
⊢ ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ s.re

Tactic:
by
  replace h := h.norm
  use tsum fun n ↦ ‖term f s n‖
  by_contra! ⟨n, hn₀, hn⟩
  have := h.le_tsum n fun _ _ ↦ norm_nonneg _
  rw [norm_term_eq, if_neg hn₀,
    div_le_iff₀ <| Real.rpow_pos_of_pos (Nat.cast_pos.mpr <| Nat.pos_of_ne_zero hn₀) _] at this
  exact (this.trans_lt hn).false.elim

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

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
x : R
n : ℕ
hq : 1 < q
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x_1 ↦ (PowerSeries.coeff x_1) ((algebraMap R (PowerSeries R)) x)) 0,
        map_zero' := ⋯ }
      n =
    ((algebraMap R (ArithmeticFunction R)) x) n

Tactic:
simp only [Algebra.algebraMap_eq_smul_one, coe_mk]

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
x : R
n : ℕ
hq : 1 < q
⊢ Function.extend (fun x ↦ q ^ x) (fun x_1 ↦ (PowerSeries.coeff x_1) (x • 1)) 0 n = (x • 1) n
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg.hb
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
n : ℕ
hn : ¬∃ i, q ^ i = n
⊢ ¬∃ a, (q ^ k) ^ a = n

Tactic:
contrapose! hn

Goal state after:
case neg.hb
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
n : ℕ
hn : ∃ a, (q ^ k) ^ a = n
⊢ ∃ i, q ^ i = n
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg.hb
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i : ℕ
hmn : (p ^ i).Coprime n
hn : ¬∃ j, p ^ j = n
⊢ ¬∃ a, p ^ a = p ^ i * n

Tactic:
contrapose! hn

Goal state after:
case neg.hb
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i : ℕ
hmn : (p ^ i).Coprime n
hn : ∃ a, p ^ a = p ^ i * n
⊢ ∃ j, p ^ j = n
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
w : τ → ℕ
ha : HasSubst a
f : R⟦X⟧
i : Unit →₀ ℕ
hi : ¬(MvPowerSeries.coeff i) f = 0
⊢ f.order * MvPowerSeries.weightedOrder w a ≤ ↑(i ()) * MvPowerSeries.weightedOrder w a

Tactic:
exact mul_le_mul_left (f.order_le (i ()) (by delta PowerSeries.coeff; convert! hi; aesop)) _

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NNNorm 𝕜
E : Type u_2
inst✝ : SMul 𝕜 E
s t : Set E
x : E
⊢ ⨅ c, ⨅ (_ : x ∈ c • (s ∪ t)), ‖c‖ₑ = min (⨅ c, ⨅ (_ : x ∈ c • s), ‖c‖ₑ) (⨅ c, ⨅ (_ : x ∈ c • t), ‖c‖ₑ)

Tactic:
simp [smul_set_union, iInf_or, iInf_inf_eq]

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

Goal state before:
pp: case e_a.succ.succ
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
hB' : constantCoeff B = 0
k : R
hk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k
i : ℕ
⊢ (coeff (n + 2)) ((B + -(C k * X ^ (n + 2))) ^ (i + 2)) = (coeff (n + 2)) (B ^ (i + 2)) - if i + 2 = 1 then k else 0

Tactic:
add_comm B,

Goal state after:
case e_a.succ.succ
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
hB' : constantCoeff B = 0
k : R
hk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k
i : ℕ
⊢ (coeff (n + 2)) ((-(C k * X ^ (n + 2)) + B) ^ (i + 2)) = (coeff (n + 2)) (B ^ (i + 2)) - if i + 2 = 1 then k else 0
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP' : IsUnit ((coeff 1) P)
this : Invertible ((coeff 1) P) := hP'.invertible
⊢ (coeff 1) (P.substInvOfIsUnit hP') = ↑hP'.unit⁻¹

Tactic:
substInvOfIsUnit_eq_substInv,

Goal state after:
R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP' : IsUnit ((coeff 1) P)
this : Invertible ((coeff 1) P) := hP'.invertible
⊢ (coeff 1) P.substInv = ↑hP'.unit⁻¹
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f f₁ : 𝕜 → F
x : 𝕜
s t : Set 𝕜
h : f₁ =ᶠ[𝓝[s] x] f
ht : t ⊆ s
⊢ derivWithin f₁ t =ᶠ[𝓝[s] x] derivWithin f t

Tactic:
unfold derivWithin
  exact h.fderivWithin' ht |>.fun_comp (fun a ↦ a 1)

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x₀ : 𝕜
hf : HasDerivAt f f' x₀
C : ℝ
hC₀ : 0 ≤ C
hlip : ∀ᶠ (x : 𝕜) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖
⊢ ‖f'‖ ≤ C

Tactic:
by
  simpa using HasFDerivAt.le_of_lip' hf.hasFDerivAt hC₀ hlip

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
⊢ HasFDerivWithinAt f f' (insert x s) x ↔ HasFDerivWithinAt f f' s x

Tactic:
hasFDerivWithinAt_iff_isLittleOTVS

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
⊢ ((fun x' ↦ f x' - f x - f' (x' - x)) =o[𝕜; 𝓝[insert x s] x] fun x' ↦ x' - x) ↔
    (fun x' ↦ f x' - f x - f' (x' - x)) =o[𝕜; 𝓝[s] x] fun x' ↦ x' - x
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
ι : Type u_4
s : ι → Set E
hf : ∀ (i : ι), DifferentiableOn 𝕜 f (s i)
hs : ∀ (i : ι), IsOpen[inst✝³] (s i)
⊢ DifferentiableOn 𝕜 f (⋃ i, s i)

Tactic:
rintro x ⟨si, ⟨i, rfl⟩, hxsi⟩
  exact (hf i).differentiableAt ((hs i).mem_nhds hxsi) |>.differentiableWithinAt

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
inst✝³ : ContinuousAdd E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousSMul 𝕜 F
L : Filter E
hL : L ≤ 𝓝 x
h : HasFDerivAtFilter f f' (L ×ˢ pure x)
⊢ Tendsto f L (𝓝 (f x))

Tactic:
have : (f · - f x) =o[𝕜; L] (1 : E → 𝕜) := by
    refine h.isBigOTVS_sub |>.comp_tendsto prod_pure.ge |>.trans_isLittleOTVS ?_
    rw [isLittleOTVS_one]
    simpa [sub_eq_add_neg] using! (tendsto_id'.mpr hL).add_const (-x)

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
inst✝³ : ContinuousAdd E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousSMul 𝕜 F
L : Filter E
hL : L ≤ 𝓝 x
h : HasFDerivAtFilter f f' (L ×ˢ pure x)
this : (fun x_1 ↦ f x_1 - f x) =o[𝕜; L] 1
⊢ Tendsto f L (𝓝 (f x))
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: case neg
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
x₀ : E
C : ℝ
hC₀ : 0 ≤ C
hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖
hf : ¬DifferentiableAt 𝕜 f x₀
⊢ ‖0‖ ≤ C

Tactic:
simp [hC₀]

Goal state after:
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
k n : ℕ
⊢ (fslope^[k] p).coeff n = p.coeff (n + k)

Tactic:
induction k generalizing p with

Goal state after:
Module: Mathlib.Analysis.Calculus.FormalMultilinearSeries

Goal state before:
pp: 𝕜 : Type u
E : Type v
F : Type w
G : Type x
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
⊢ p.compContinuousLinearMap 0 = constFormalMultilinearSeries 𝕜 E ((p 0) 0)

Tactic:
by
  ext n v
  cases n with
  | zero =>
    simp only [FormalMultilinearSeries.compContinuousLinearMap_apply, Matrix.zero_empty,
      constFormalMultilinearSeries_apply_zero, ContinuousMultilinearMap.uncurry0_apply]
    congr
    apply Subsingleton.allEq
  | succ =>
    simp [FunLike.coe_zero]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
x_mem_ball : x ∈ Metric.eball 0 p.radius
y_mem_ball : y ∈ Metric.eball 0 (p.changeOrigin x).radius
x_add_y_mem_ball : x + y ∈ Metric.eball 0 p.radius
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
hsf : Summable f
hf : HasSum f ((p.changeOrigin x).sum y)
⊢ (p.changeOrigin x).sum y = p.sum (x + y)

Tactic:
refine hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 ?_)

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
x_mem_ball : x ∈ Metric.eball 0 p.radius
y_mem_ball : y ∈ Metric.eball 0 (p.changeOrigin x).radius
x_add_y_mem_ball : x + y ∈ Metric.eball 0 p.radius
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
hsf : Summable f
hf : HasSum f ((p.changeOrigin x).sum y)
⊢ HasSum (f ∘ ⇑changeOriginIndexEquiv.symm) (p.sum (x + y))
Module: Mathlib.Analysis.Analytic.CPolynomialDef

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
x : E
r : ℝ≥0∞
n : ℕ
hf : HasFiniteFPowerSeriesOnBall f p x n r
y : E
hy : y ∈ Metric.eball 0 r
m : ℕ
hm : n ≤ m
N : ℕ
hN : m ≤ N
⊢ ((p N) fun x ↦ y) = 0

Tactic:
hf.finite _ (le_trans hm hN),

Goal state after:
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
x : E
r : ℝ≥0∞
n : ℕ
hf : HasFiniteFPowerSeriesOnBall f p x n r
y : E
hy : y ∈ Metric.eball 0 r
m : ℕ
hm : n ≤ m
N : ℕ
hN : m ≤ N
⊢ (0 fun x ↦ y) = 0
Module: Mathlib.Analysis.Analytic.CPolynomialDef

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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x y : E
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
finsupp : (Function.support f).Finite
hfkl : ∀ (k l : ℕ), HasSum (fun x ↦ f ⟨k, ⟨l, x⟩⟩) (((p.changeOriginSeries k l) fun x_1 ↦ x) fun x ↦ y)
hfk : ∀ (k : ℕ), HasSum (fun x ↦ f ⟨k, x⟩) ((p.changeOrigin x k) fun x ↦ y)
hf : HasSum f ((p.changeOrigin x).sum y)
⊢ (p.changeOrigin x).sum y = p.sum (x + y)

Tactic:
refine hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 ?_)

Goal state after:
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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x y : E
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
finsupp : (Function.support f).Finite
hfkl : ∀ (k l : ℕ), HasSum (fun x ↦ f ⟨k, ⟨l, x⟩⟩) (((p.changeOriginSeries k l) fun x_1 ↦ x) fun x ↦ y)
hfk : ∀ (k : ℕ), HasSum (fun x ↦ f ⟨k, x⟩) ((p.changeOrigin x k) fun x ↦ y)
hf : HasSum f ((p.changeOrigin x).sum y)
⊢ HasSum (f ∘ ⇑changeOriginIndexEquiv.symm) (p.sum (x + y))
Module: Mathlib.Analysis.Analytic.Inverse

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
j : ℕ
hj1 : j < (Composition.ones (n + 2)).length
hj2 : j < n + 2
k : ℕ
x✝¹ x✝ : k < 1
⊢ v (Fin.castLE ⋯ ⟨j, hj1⟩) = v ⟨j, hj2⟩

Tactic:
by congr

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

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ (p.leftInv i x).comp (id 𝕜 F ((p 0) 0)) = (p.leftInv i x).comp (p.comp (p.rightInv i x))

Tactic:
comp_rightInv p i _ h

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ (p.leftInv i x).comp (id 𝕜 F ((p 0) 0)) = (p.leftInv i x).comp (id 𝕜 F ((p 0) 0))
Module: Mathlib.Analysis.Analytic.Inverse

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ 0 < (p.leftInv i x).radius

Tactic:
rw [leftInv_eq_rightInv _ _ _ h]

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ 0 < (p.rightInv i x).radius