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

Goal state before:
pp: case inr
σ : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
hempty : Nonempty ι
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)
d : σ →₀ ℕ
i : ι
hi : ∀ (b : ι), i ≤ b → ↑((Finsupp.weight w) d) < weightedOrder w (f b)
⊢ ∀ (x : ι), (coeff d) (f x) ≠ 0 → x ≤ i

Tactic:
intro k

Goal state after:
case inr
σ : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
w : σ → ℕ
hempty : Nonempty ι
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < weightedOrder w (f b)
d : σ →₀ ℕ
i : ι
hi : ∀ (b : ι), i ≤ b → ↑((Finsupp.weight w) d) < weightedOrder w (f b)
k : ι
⊢ (coeff d) (f k) ≠ 0 → k ≤ i
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : Semiring R
n : ℕ
a : R
inst✝ : Decidable (a = 0)
h : a = 0
⊢ ((monomial n) a).order = ⊤

Tactic:
h,

Goal state after:
case pos
R : Type u_1
inst✝¹ : Semiring R
n : ℕ
a : R
inst✝ : Decidable (a = 0)
h : a = 0
⊢ ((monomial n) 0).order = ⊤
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_4
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
inst✝¹ : Algebra R ℚ
inst✝ : IsFractionRing R ℚ
𝔭 : HeightOneSpectrum R
x : ℚ
⊢ (valuation ℚ 𝔭) x ≤ 1 ↔ ↑x.den ∉ 𝔭.asIdeal

Tactic:
have : CharZero R := ⟨.of_comp (f := algebraMap R ℚ) (by simpa using Nat.cast_injective)⟩
  have : (x.den : R) ≠ 0 := by simp
  simp [x.num_div_den, ← 𝔭.valuation_div_le_one_iff ℚ x.num this
    (Ideal.IsPrime.notMem_of_isCoprime_of_mem (mod_cast x.isCoprime_num_den.symm.intCast))]

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

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
hempty : Nonempty ι
n : ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order
i : ι
hi : ∀ (b : ι), i ≤ b → ↑n < (f b).order
⊢ ∀ (x : ι), (coeff n) (f x) ≠ 0 → x ≤ i

Tactic:
intro k

Goal state after:
case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
hempty : Nonempty ι
n : ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order
i : ι
hi : ∀ (b : ι), i ≤ b → ↑n < (f b).order
k : ι
⊢ (coeff n) (f k) ≠ 0 → k ≤ i
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: R : Type u_2
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : T2Space R
inst✝ : Nontrivial R
⊢ ∏' (i : ℕ), (1 - X ^ (i + 1)) ≠ 0

Tactic:
by
  by_contra! h
  obtain h := PowerSeries.ext_iff.mp h 0
  simp [coeff_zero_eq_constantCoeff, (multipliable_one_sub_X_pow R).map_tprod _
    (continuous_constantCoeff R)] at h

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
r : R
⊢ eval₂ φ a ↑(MvPolynomial.C r) = φ r

Tactic:
eval₂_coe,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
r : R
⊢ MvPolynomial.eval₂ φ a (MvPolynomial.C r) = φ r
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
s : σ
⊢ eval₂ φ a ↑(MvPolynomial.X s) = a s

Tactic:
eval₂_coe,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
s : σ
⊢ MvPolynomial.eval₂ φ a (MvPolynomial.X s) = a s
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
r : R
⊢ eval₂ φ a ↑(Polynomial.C r) = φ r

Tactic:
eval₂_coe,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
r : R
⊢ Polynomial.eval₂ φ a (Polynomial.C r) = φ r
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
⊢ eval₂ φ a ↑Polynomial.X = a

Tactic:
eval₂_coe,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
⊢ Polynomial.eval₂ φ a Polynomial.X = a
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
f : R⟦X⟧
⊢ HasSum (fun d ↦ φ ((coeff d) f) * a ^ d) (eval₂ φ a f)

Tactic:
have := MvPowerSeries.hasSum_eval₂ hφ (hasEval ha) f
  simp only [PowerSeries.eval₂]
  rw [← (Finsupp.single_injective ()).hasSum_iff] at this
  · convert this; simp
  · intro d hd
    exact False.elim (hd ⟨d (), by ext; simp⟩)

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

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
ε : R⟦X⟧ → S
hε : Continuous ε
h : ∀ (p : Polynomial R), ε ↑p = Polynomial.eval₂ φ a p
p : MvPolynomial Unit R
⊢ ε ↑((MvPolynomial.uniqueAlgEquiv R PUnit.{1}) p) = MvPolynomial.eval₂ φ (fun x ↦ a) p

Tactic:
h,

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
ε : R⟦X⟧ → S
hε : Continuous ε
h : ∀ (p : Polynomial R), ε ↑p = Polynomial.eval₂ φ a p
p : MvPolynomial Unit R
⊢ Polynomial.eval₂ φ a ((MvPolynomial.uniqueAlgEquiv R PUnit.{1}) p) = MvPolynomial.eval₂ φ (fun x ↦ a) p
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
p : MvPolynomial σ R
this✝ : UniformSpace R := ⊥
this : UniformSpace S := ⊥
⊢ eval₂ (algebraMap R (MvPowerSeries τ S)) a ↑p = (MvPolynomial.aeval a) p

Tactic:
eval₂_coe,

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
p : MvPolynomial σ R
this✝ : UniformSpace R := ⊥
this : UniformSpace S := ⊥
⊢ MvPolynomial.eval₂ (algebraMap R (MvPowerSeries τ S)) a p = (MvPolynomial.aeval a) p
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 =
    ∑ x ∈ n.divisorsAntidiagonal,
      { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0, map_zero' := ⋯ } x.1 *
        { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0, map_zero' := ⋯ } x.2

Tactic:
coe_mk

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
⊢ 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: 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
⊢ ∀ k < n, f k = g k

Tactic:
intro k

Goal state after:
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
k : ℕ
⊢ k < n → f k = g k
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
h : f 1 = 1
⊢ f 1 • δ = δ

Tactic:
h,

Goal state after:
f : ℕ → ℂ
h : f 1 = 1
⊢ 1 • δ = δ
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

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

Tactic:
simp [ofPowerSeries, dif_neg hq]

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

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s : Set E
h : c = 0 → s.Nonempty
x : E
hc : c ≠ 0
⊢ ‖c⁻¹‖ₑ * egauge 𝕜 s (c • x) ≤ egauge 𝕜 s x

Tactic:
refine (le_egauge_smul_right _ _ _).trans_eq ?_

Goal state after:
case inr
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
c : 𝕜
s : Set E
h : c = 0 → s.Nonempty
x : E
hc : c ≠ 0
⊢ egauge 𝕜 s (c⁻¹ • c • x) = egauge 𝕜 s x
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝¹ : UniformSpace R := ⊥
this✝ : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
this : ∀ (w : Finset ι), (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = 1 k
⊢ ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k

Tactic:
intro k

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝¹ : UniformSpace R := ⊥
this✝ : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
this : ∀ (w : Finset ι), (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = 1 k
k : ℕ
⊢ k ≤ n → (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' : (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = 1 k
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
⊢ ∀ k ≤ n, (∏ x ∈ insert i w, f x) k = 1 k

Tactic:
intro k

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' : (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = 1 k
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
⊢ k ≤ n → (∏ x ∈ insert i w, f x) k = 1 k
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 ∈ antidiagonal e,
    b ≠ (single 0 (e 0), single 1 (e 1)) →
      (MvPowerSeries.coeff b.1) (subst X₀ f) * (MvPowerSeries.coeff b.2) (subst X₁ f) = 0

Tactic:
intro b hb hb'
    by_contra hmul_ne_zero
    rcases ne_zero_and_ne_zero_of_mul hmul_ne_zero with ⟨h0, h1⟩
    simp only [Fin.isValue, coeff_subst_single, ne_eq, ite_eq_right_iff,
      not_forall, exists_prop] at h0 h1
    apply hb'
    rw [Prod.ext_iff, ← mem_antidiagonal.mp hb, h0.1, h1.1]
    simp

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

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
s : Set E
c : F
⊢ fderivWithin 𝕜 (fun x ↦ c) s = 0

Tactic:
by
  ext
  rw [fderivWithin_const_apply, Pi.zero_apply]

Goal state after:
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
L : Filter (E × E)
⊢ ((fun x ↦ ‖f x.1 - f x.2 - f' (x.1 - x.2)‖) =o[L] fun p ↦ p.1 - p.2) ↔
    Tendsto (fun p ↦ ‖p.1 - p.2‖⁻¹ * ‖f p.1 - f p.2 - f' (p.1 - p.2)‖) L (𝓝 0)

Tactic:
← isLittleO_norm_right,

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
L : Filter (E × E)
⊢ ((fun x ↦ ‖f x.1 - f x.2 - f' (x.1 - x.2)‖) =o[L] fun x ↦ ‖x.1 - x.2‖) ↔
    Tendsto (fun p ↦ ‖p.1 - p.2‖⁻¹ * ‖f p.1 - f p.2 - f' (p.1 - p.2)‖) L (𝓝 0)
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
x₀ : E
s : Set E
n : ℕ
h : f =O[𝓝[s] x₀] fun x ↦ ‖x - x₀‖ ^ n
hx₀ : x₀ ∈ s
hn : 1 < n
⊢ (fun x' ↦ f x' - f x₀ - 0 (x' - x₀)) =o[𝓝[s] x₀] fun x' ↦ x' - x₀

Tactic:
h.eq_zero_of_norm_pow_within hx₀ hn.ne_bot,

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
x₀ : E
s : Set E
n : ℕ
h : f =O[𝓝[s] x₀] fun x ↦ ‖x - x₀‖ ^ n
hx₀ : x₀ ∈ s
hn : 1 < n
⊢ (fun x' ↦ f x' - 0 - 0 (x' - x₀)) =o[𝓝[s] x₀] fun x' ↦ x' - x₀
Module: Mathlib.Analysis.Analytic.ChangeOrigin

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
k l : ℕ
s : Finset (Fin (k + l))
hs : s.card = l
x y : E
⊢ ‖((p.changeOriginSeriesTerm k l s hs) fun x_1 ↦ x) fun x ↦ y‖₊ ≤
    ‖p.changeOriginSeriesTerm k l s hs‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k

Tactic:
← Fin.prod_const,

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
k l : ℕ
s : Finset (Fin (k + l))
hs : s.card = l
x y : E
⊢ ‖((p.changeOriginSeriesTerm k l s hs) fun x_1 ↦ x) fun x ↦ y‖₊ ≤
    (‖p.changeOriginSeriesTerm k l s hs‖₊ * ∏ _i, ‖x‖₊) * ‖y‖₊ ^ k
Module: Mathlib.Analysis.Analytic.ChangeOrigin

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
x y : E
n : ℕ
t : Finset (Fin n)
⊢ let s := changeOriginIndexEquiv.symm ⟨n, t⟩;
  (((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y) =
    (p n) (t.piecewise (fun x_1 ↦ x) fun x ↦ y)

Tactic:
have : ∀ (m) (hm : n = m), p n (t.piecewise (fun _ ↦ x) fun _ ↦ y) =
      p m ((t.map (finCongr hm).toEmbedding).piecewise (fun _ ↦ x) fun _ ↦ y) := by
    rintro m rfl
    simp +unfoldPartialApp [Finset.piecewise]
  simp_rw [changeOriginSeriesTerm_apply, eq_comm]; apply this

Goal state after:
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
x : E
hf : HasFPowerSeriesAt f p x
y : E
⊢ HasFPowerSeriesAt (fun z ↦ f (z - y)) p (x + y)

Tactic:
obtain ⟨r, hf⟩ := hf

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 y : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
⊢ HasFPowerSeriesAt (fun z ↦ f (z - y)) p (x + y)
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
hf : HasFPowerSeriesWithinAt f p s x
y : E
⊢ HasFPowerSeriesWithinAt (fun z ↦ f (z - y)) p (s + {y}) (x + y)

Tactic:
obtain ⟨r, hf⟩ := hf

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
s : Set E
x y : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
⊢ HasFPowerSeriesWithinAt (fun z ↦ f (z - y)) p (s + {y}) (x + y)
Module: Mathlib.Analysis.Analytic.Inverse

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
n : ℕ
hn : 0 < n
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 F E
v : Fin n → F
c : Composition n
x✝ : c ∈ univ
h : ¬1 < c.length
this : c.length = 1
⊢ c ∈ {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}

Tactic:
rw [← Composition.eq_single_iff_length hn] at this

Goal state after:
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
n : ℕ
hn : 0 < n
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 F E
v : Fin n → F
c : Composition n
x✝ : c ∈ univ
h : ¬1 < c.length
this : c = Composition.single n hn
⊢ c ∈ {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}
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
hf : HasFPowerSeriesWithinAt f p s x
⊢ (fun y ↦ f y.1 - f y.2 - (p 1) fun x ↦ y.1 - y.2) =O[𝓝[insert x s ×ˢ insert x s] (x, x)] fun y ↦
    ‖y - (x, x)‖ * ‖y.1 - y.2‖

Tactic:
by
  rcases hf with ⟨r, hf⟩
  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
  refine (hf.isBigO_image_sub_image_sub_deriv_principal h).mono ?_
  rw [inter_comm]
  exact le_principal_iff.2 (inter_mem_nhdsWithin _ (Metric.eball_mem_nhds _ r'0))

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

Goal state before:
pp: case refine_1
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
u : Set (F × F)
hu : u ∈ uniformity F
y : E
hy : y ∈ (fun x_1 ↦ x + x_1) ⁻¹' insert x s ∩ Metric.eball 0 r
r' : ℝ≥0
yr' : edist y 0 < ↑r'
hr' : ↑r' < r
this : Metric.eball 0 ↑r' ∈ 𝓝 y
⊢ Metric.eball 0 r ∈ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' insert x s] y

Tactic:
apply mem_nhdsWithin_of_mem_nhds

Goal state after:
case refine_1
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
u : Set (F × F)
hu : u ∈ uniformity F
y : E
hy : y ∈ (fun x_1 ↦ x + x_1) ⁻¹' insert x s ∩ Metric.eball 0 r
r' : ℝ≥0
yr' : edist y 0 < ↑r'
hr' : ↑r' < r
this : Metric.eball 0 ↑r' ∈ 𝓝 y
⊢ Metric.eball 0 r ∈ 𝓝 y
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
A : Tendsto (fun n ↦ (n, ∑ a ∈ Finset.Ico 1 n, (p a) fun x ↦ y)) atTop (atTop ×ˢ 𝓝 (f (x + y) - f x))
B : Tendsto (fun n ↦ q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b ↦ y)) atTop (𝓝 (g (f (x + y))))
C :
  Tendsto (fun n ↦ ∑ i ∈ compPartialSumTarget 0 n n, (q.compAlongComposition p i.snd) fun _j ↦ y) atTop
    (𝓝 (g (f (x + y))))
D : HasSum (fun i ↦ (q.compAlongComposition p i.snd) fun _j ↦ y) (g (f (x + y)))
n : ℕ
⊢ HasSum (fun c ↦ (q c.length) (p.applyComposition c fun _j ↦ y)) (∑ c, (q c.length) (p.applyComposition c fun _j ↦ y))

Tactic:
exact hasSum_fintype _

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
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n✝ n : ℕ
i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
⊢ ∀ {i_1 : ℕ}, (i_1 ∈ ofFn fun j ↦ (i.snd j).length) → 0 < i_1

Tactic:
intro k

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n✝ n : ℕ
i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
k : ℕ
⊢ (k ∈ ofFn fun j ↦ (i.snd j).length) → 0 < k