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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
⊢ (weightedHomogeneousComponent w (p + q)) (f * g) =
    (weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g

Tactic:
ext d

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
⊢ (coeff d) ((weightedHomogeneousComponent w (p + q)) (f * g)) =
    (coeff d) ((weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
hd : (weight w) d = p + q
⊢ (coeff d) (f * g) = (coeff d) ((weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g)

Tactic:
apply Finset.sum_congr rfl

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
hd : (weight w) d = p + q
⊢ ∀ x ∈ Finset.antidiagonal d,
    (coeff x.1) f * (coeff x.2) g =
      (coeff x.1) ((weightedHomogeneousComponent w p) f) * (coeff x.2) ((weightedHomogeneousComponent w q) g)
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case ih
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝⁴ : Field K
inst✝³ : CommSemiring S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
v : HeightOneSpectrum R
inst✝ : Algebra S K
r : S
x : WithVal (valuation K v)
⊢ (Completion.coeRingHom.comp (algebraMap S (WithVal (valuation K v)))) r * ↑x =
    ↑x * (Completion.coeRingHom.comp (algebraMap S (WithVal (valuation K v)))) r

Tactic:
rw [mul_comm]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
r : R
x : ↥(adicCompletionIntegers K v)
⊢ { toFun := fun r ↦ ⟨{ toCompletion := ↑((WithVal.equiv (valuation K v)).symm ((algebraMap R K) r)) }, ⋯⟩,
          map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
        r *
      x =
    x *
      { toFun := fun r ↦ ⟨{ toCompletion := ↑((WithVal.equiv (valuation K v)).symm ((algebraMap R K) r)) }, ⋯⟩,
          map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
        r

Tactic:
rw [mul_comm]

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

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ MvPowerSeries.order φ ≤ φ.order

Tactic:
refine le_order φ (MvPowerSeries.order φ) fun i hi => by
      rw [← Finsupp.degree_single () i] at hi
      exact MvPowerSeries.coeff_of_lt_order hi

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

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ MvPowerSeries.order φ ≤ φ.order

Tactic:
· refine le_order φ (MvPowerSeries.order φ) fun i hi => by
      rw [← Finsupp.degree_single () i] at hi
      exact MvPowerSeries.coeff_of_lt_order hi

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
m n : ℕ
φ : R⟦X⟧
⊢ (truncAux n φ).coeff m = if m < n then (coeff m) φ else 0

Tactic:
by
  simp [truncAux, Polynomial.coeff_monomial]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
x : K
x✝¹ x✝ : R
⊢ ↑(v.intAdicAbvDef hb (x✝¹ * x✝)) = ↑(v.intAdicAbvDef hb x✝¹) * ↑(v.intAdicAbvDef hb x✝)

Tactic:
simp [intAdicAbvDef]

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ (trunc (n + 2)) X = Polynomial.X

Tactic:
ext d

Goal state after:
R : Type u_1
inst✝ : Semiring R
n d : ℕ
⊢ ((trunc (n + 2)) X).coeff d = Polynomial.X.coeff d
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
n d : ℕ
h₁ : d < n + 2
h₂ : ¬d = 1
⊢ 0 = Polynomial.X.coeff d

pp: case neg
R : Type u_1
inst✝ : Semiring R
n d : ℕ
h₁ : ¬d < n + 2
⊢ 0 = Polynomial.X.coeff d

Tactic:
· rw [coeff_X_of_ne_one h₂]

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
n d : ℕ
h₁ : ¬d < n + 2
⊢ 0 = Polynomial.X.coeff d
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: R : Type u_2
inst✝ : CommSemiring R
n m : ℕ
f : R⟦X⟧
hnm : n ≤ m
⊢ (trunc n) ↑((trunc m) f) = (trunc n) f

Tactic:
ext d

Goal state after:
R : Type u_2
inst✝ : CommSemiring R
n m : ℕ
f : R⟦X⟧
hnm : n ≤ m
d : ℕ
⊢ ((trunc n) ↑((trunc m) f)).coeff d = ((trunc n) f).coeff d
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
⊢ ∑ p ∈ antidiagonal n, (coeff p.1) φ * (coeff p.2) ψ = ∑ p ∈ antidiagonal n, 0

Tactic:
apply Finset.sum_congr rfl

Goal state after:
R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
n : ℕ
h : ↑n < ψ.order
⊢ ∀ x ∈ antidiagonal n, (coeff x.1) φ * (coeff x.2) ψ = 0
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
n : σ → ℕ := fun s ↦ sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := Finsupp.ofSupportFinite n hn
D : Set (σ →₀ ℕ) := Iic n₀
hD : D.Finite
this : ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, ∀ i ∈ D, φ (MvPolynomial.coeff i x) ∈ I
p : MvPolynomial σ R
hp : ∀ i ∈ D, φ (MvPolynomial.coeff i p) ∈ I
d : σ →₀ ℕ
a✝ : d ∈ p.support
hd : d ∉ D
⊢ φ (MvPolynomial.coeff d p) * ∏ i ∈ d.support, a i ^ d i ∈ I

Tactic:
apply Ideal.mul_mem_left

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
n : σ → ℕ := ⋯
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := ⋯
D : Set (σ →₀ ℕ) := ⋯
hD : D.Finite
this : ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, ∀ i ∈ D, φ (MvPolynomial.coeff i x) ∈ I
p : MvPolynomial σ R
hp : ∀ i ∈ D, φ (MvPolynomial.coeff i p) ∈ I
d : σ →₀ ℕ
a✝ : d ∈ p.support
hd : d ∉ D
⊢ ∏ i ∈ d.support, a i ^ d i ∈ I
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: S : Type u_2
inst✝² : CommRing S
inst✝¹ : TopologicalSpace S
inst✝ : IsLinearTopology S S
c x : S
hx : MvPowerSeries.HasEval fun x_1 ↦ x
⊢ MvPowerSeries.HasEval fun x_1 ↦ x * c

Tactic:
exact hx.mul_right _

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
this✝ : UniformSpace R := ⊥
f : MvPowerSeries σ R
this : aeval ⋯ = AlgHom.id R (MvPowerSeries σ R)
⊢ (aeval ⋯) f = id f

Tactic:
rw [DFunLike.ext_iff] at this

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
this✝ : UniformSpace R := ⊥
f : MvPowerSeries σ R
this : ∀ (x : MvPowerSeries σ R), (aeval ⋯) x = (AlgHom.id R (MvPowerSeries σ R)) x
⊢ (aeval ⋯) f = id f
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: a : ℕ → ℂ
n : ℕ
hn : n ≠ 0
h : 0 < a n
x : ℝ
⊢ 0 < term a (↑x) n

Tactic:
by
  simpa only [term_of_ne_zero hn] using! mul_pos h <| inv_natCast_cpow_ofReal_pos hn x

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

Goal state before:
pp: f g : ℕ → ℂ
s : ℂ
h : f =ᶠ[atTop] g
hf : LSeriesSummable f s
⊢ LSeriesSummable g s

Tactic:
rw [← Nat.cofinite_eq_atTop] at h

Goal state after:
f g : ℕ → ℂ
s : ℂ
h : f =ᶠ[cofinite] g
hf : LSeriesSummable f s
⊢ LSeriesSummable g s
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
⊢ (truncTotal k) (subst a f) = (truncTotal k) (subst (fun i ↦ ↑((truncTotal (x i)) (a i))) f)

Tactic:
ext d

Goal state after:
σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
⊢ MvPolynomial.coeff d ((truncTotal k) (subst a f)) =
    MvPolynomial.coeff d ((truncTotal k) (subst (fun i ↦ ↑((truncTotal (x i)) (a i))) f))
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case pos.a.a
σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
n : σ →₀ ℕ
l : σ →₀ τ →₀ ℕ
hl : l ∈ n.support.finsuppAntidiag d
i : σ
hi : i ∈ n.support
⊢ (coeff (l i)) (a i ^ n i) = (coeff (l i)) (↑((truncTotal (x i)) (a i)) ^ n i)

Tactic:
obtain ⟨hl₁, -⟩ := mem_finsuppAntidiag.mp hl

Goal state after:
case pos.a.a
σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
n : σ →₀ ℕ
l : σ →₀ τ →₀ ℕ
hl : l ∈ n.support.finsuppAntidiag d
i : σ
hi : i ∈ n.support
hl₁ : n.support.sum ⇑l = d
⊢ (coeff (l i)) (a i ^ n i) = (coeff (l i)) (↑((truncTotal (x i)) (a i)) ^ n i)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
k : ℕ
ha : HasSubst a
ha₁ : ∀ (i : σ), constantCoeff (a i) = 0
⊢ (truncTotal k) (subst a f) = (truncTotal k) (subst a (∑ i ∈ range k, (homogeneousComponent i) f))

Tactic:
ext d

Goal state after:
σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
k : ℕ
ha : HasSubst a
ha₁ : ∀ (i : σ), constantCoeff (a i) = 0
d : τ →₀ ℕ
⊢ MvPolynomial.coeff d ((truncTotal k) (subst a f)) =
    MvPolynomial.coeff d ((truncTotal k) (subst a (∑ i ∈ range k, (homogeneousComponent i) f)))
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
A : Type u_2
inst✝²¹ : CommSemiring A
R✝ : Type u_3
inst✝²⁰ : CommRing R✝
inst✝¹⁹ : Algebra A R✝
τ : Type u_4
S : Type u_5
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra A S
inst✝¹⁶ : Algebra R✝ S
inst✝¹⁵ : IsScalarTower A R✝ S
a✝¹ a✝ : σ → MvPowerSeries τ S
T✝ : Type u_6
inst✝¹⁴ : CommRing T✝
inst✝¹³ : UniformSpace T✝
inst✝¹² : T2Space T✝
inst✝¹¹ : CompleteSpace T✝
inst✝¹⁰ : IsUniformAddGroup T✝
inst✝⁹ : IsTopologicalRing T✝
inst✝⁸ : IsLinearTopology T✝ T✝
inst✝⁷ : Algebra R✝ T✝
ε : MvPowerSeries τ S →ₐ[R✝] T✝
inst✝⁶ : Algebra S T✝
inst✝⁵ : IsScalarTower R✝ S T✝
υ : Type u_7
T : Type u_8
inst✝⁴ : CommRing T
inst✝³ : Algebra R✝ T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R✝ S T
b : τ → MvPowerSeries υ T
R : Type u_9
inst✝ : CommSemiring R
a : σ → R
f g : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∑ i ∈ Finset.antidiagonal n, (n.prod fun s m ↦ a s ^ m) * ((coeff i.1) f * (coeff i.2) g) =
    ∑ p ∈ Finset.antidiagonal n,
      ((coeff p.1) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) *
        (coeff p.2) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g

Tactic:
apply Finset.sum_congr rfl

Goal state after:
σ : Type u_1
A : Type u_2
inst✝²¹ : CommSemiring A
R✝ : Type u_3
inst✝²⁰ : CommRing R✝
inst✝¹⁹ : Algebra A R✝
τ : Type u_4
S : Type u_5
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra A S
inst✝¹⁶ : Algebra R✝ S
inst✝¹⁵ : IsScalarTower A R✝ S
a✝¹ a✝ : σ → MvPowerSeries τ S
T✝ : Type u_6
inst✝¹⁴ : CommRing T✝
inst✝¹³ : UniformSpace T✝
inst✝¹² : T2Space T✝
inst✝¹¹ : CompleteSpace T✝
inst✝¹⁰ : IsUniformAddGroup T✝
inst✝⁹ : IsTopologicalRing T✝
inst✝⁸ : IsLinearTopology T✝ T✝
inst✝⁷ : Algebra R✝ T✝
ε : MvPowerSeries τ S →ₐ[R✝] T✝
inst✝⁶ : Algebra S T✝
inst✝⁵ : IsScalarTower R✝ S T✝
υ : Type u_7
T : Type u_8
inst✝⁴ : CommRing T
inst✝³ : Algebra R✝ T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R✝ S T
b : τ → MvPowerSeries υ T
R : Type u_9
inst✝ : CommSemiring R
a : σ → R
f g : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ x ∈ Finset.antidiagonal n,
    (n.prod fun s m ↦ a s ^ m) * ((coeff x.1) f * (coeff x.2) g) =
      ((coeff x.1) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) f) *
        (coeff x.2) fun n ↦ (n.prod fun s m ↦ a s ^ m) * (coeff n) g
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, (f i) n = 1 n
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
⊢ Cauchy (map (DFunLike.coe ∘ fun s ↦ ∏ b ∈ s, f b) atTop)

Tactic:
cauchy_map_iff',

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, (f i) n = 1 n
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
⊢ Tendsto (fun p ↦ ((DFunLike.coe ∘ fun s ↦ ∏ b ∈ s, f b) p.1, (DFunLike.coe ∘ fun s ↦ ∏ b ∈ s, f b) p.2))
    (atTop ×ˢ atTop) (uniformity (ℕ → R))
Module: Mathlib.Analysis.Calculus.TangentCone.Defs

Goal state before:
pp: R : Type u
E : Type v
inst✝² : AddCommGroup E
inst✝¹ : SMul R E
inst✝ : TopologicalSpace E
s : Set E
x y : E
α : Type u_1
l : Filter α
c : α → R
d : α → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∃ᶠ (n : α) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ y ∈ tangentConeAt R s x

Tactic:
suffices Tendsto (fun n ↦ c n • d n) (l ⊓ 𝓟 {y | x + d y ∈ s}) (⊤ • 𝓝[(x + ·) ⁻¹' s] 0) by
    rw [frequently_iff_neBot] at hds
    rw [tangentConeAt_def]
    exact ClusterPt.mono (hcd.mono_left inf_le_left).mapClusterPt this

Goal state after:
R : Type u
E : Type v
inst✝² : AddCommGroup E
inst✝¹ : SMul R E
inst✝ : TopologicalSpace E
s : Set E
x y : E
α : Type u_1
l : Filter α
c : α → R
d : α → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∃ᶠ (n : α) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ Tendsto (fun n ↦ c n • d n) (l ⊓ 𝓟 {y | x + d y ∈ s}) (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖(fun x ↦ c • x) y‖ₑ / ↑r ≤ ‖c‖ₑ

Tactic:
rw [enorm_smul]

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖c‖ₑ * ‖y‖ₑ / ↑r ≤ ‖c‖ₑ
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
f : E → F
x : E
⊢ ¬AccPt x (𝓟 {x})

Tactic:
accPt_iff_clusterPt,

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
x : E
⊢ ¬ClusterPt x (𝓟 {x}ᶜ ⊓ 𝓟 {x})
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≥0
hlip : LipschitzWith C f
⊢ ‖f'‖ ≤ ↑C

Tactic:
simpa using HasFDerivAt.le_of_lipschitz hf.hasFDerivAt hlip

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

Goal state before:
pp: case inr
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
s t : Set E
hf : DifferentiableOn 𝕜 f s
hf' : DifferentiableOn 𝕜 f t
hs : IsOpen[inst✝³] s
ht : IsOpen[inst✝³] t
x : E
hx : x ∈ t
⊢ DifferentiableWithinAt 𝕜 f (s ∪ t) x

Tactic:
exact (hf' x hx).differentiableAt (ht.mem_nhds hx) |>.differentiableWithinAt

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

Goal state before:
pp: 𝕜 : Type u
E : Type v
F : Type w
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
k l : ℕ
h : k = l
h' : p k = 0
⊢ p l = 0

Tactic:
by
  subst h; exact h'

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✝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' : ℝ≥0
hr : ↑r + ↑r' < p.radius
⊢ Summable fun s ↦ ‖p (s.fst + s.snd.fst)‖₊ * r ^ s.snd.fst * r' ^ s.fst

Tactic:
rw [← changeOriginIndexEquiv.symm.summable_iff]

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
r r' : ℝ≥0
hr : ↑r + ↑r' < p.radius
⊢ Summable ((fun s ↦ ‖p (s.fst + s.snd.fst)‖₊ * r ^ s.snd.fst * r' ^ s.fst) ∘ ⇑changeOriginIndexEquiv.symm)
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 n).compContinuousLinearMap fun x ↦ u‖ ≤ ‖p n‖ * ‖u‖ ^ n

Tactic:
apply le_trans (ContinuousMultilinearMap.norm_compContinuousLinearMap_le _ _)

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
⊢ ‖p n‖ * ∏ i, ‖u‖ ≤ ‖p n‖ * ‖u‖ ^ 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
⊢ ‖p.compContinuousLinearMap u n‖ₑ ≤ ‖p n‖ₑ * ‖u‖ₑ ^ n

Tactic:
by
  rw [← ofReal_norm, ← ofReal_norm, ← ofReal_norm,
    ← ENNReal.ofReal_pow (by simp), ← ENNReal.ofReal_mul (by simp)]
  gcongr
  apply norm_compContinuousLinearMap_le

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

Goal state before:
pp: case h
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 + 1)‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖p n‖ * ↑r ^ n ≤ ‖p 0‖ ∨ ‖p n‖ * ↑r ^ n ≤ C * ↑r

Tactic:
rcases n with - | m

Goal state after:
case h.zero
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 + 1)‖ * ↑r ^ n ≤ C
⊢ ‖p 0‖ * ↑r ^ 0 ≤ ‖p 0‖ ∨ ‖p 0‖ * ↑r ^ 0 ≤ C * ↑r

case h.succ
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 + 1)‖ * ↑r ^ n ≤ C
m : ℕ
⊢ ‖p (m + 1)‖ * ↑r ^ (m + 1) ≤ ‖p 0‖ ∨ ‖p (m + 1)‖ * ↑r ^ (m + 1) ≤ C * ↑r
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
pf : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFiniteFPowerSeriesOnBall f pf x 1 r
y : E
hy : y ∈ Metric.eball x r
⊢ 0 < r

Tactic:
exact hf.r_pos

Goal state after:
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 < (ofScalars E c).radius
this : 0 < r
hc : ∀ᶠ (x : ℕ) in atTop, ‖‖ofScalars E c x‖ * ↑r ^ x‖ = 0
n : ℕ
hc' : ‖‖ofScalars E c n‖ * ↑r ^ n‖ = 0
⊢ 0 = ‖c n.succ‖ / ‖c n‖

Tactic:
ofScalars_norm,

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 < (ofScalars E c).radius
this : 0 < r
hc : ∀ᶠ (x : ℕ) in atTop, ‖‖ofScalars E c x‖ * ↑r ^ x‖ = 0
n : ℕ
hc' : ‖‖c n‖ * ↑r ^ n‖ = 0
⊢ 0 = ‖c n.succ‖ / ‖c n‖
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 g : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
hg : EqOn f g (Metric.eball x r)
y : E
hy : y ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (g (x + y))

Tactic:
by
      convert hf.hasSum hy
      apply hg.symm
      simpa [edist_eq_enorm_sub] using hy

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 g : E → F
s : Set E
x : E
hf : AnalyticWithinAt 𝕜 f s x
hs : g =ᶠ[𝓝[s] x] f
hx : g x = f x
⊢ AnalyticWithinAt 𝕜 g s x

Tactic:
rcases hf with ⟨p, hp⟩
  exact ⟨p, hp.congr hs hx⟩

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

Goal state before:
pp: case inr.inr
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∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 r)
hr : 0 < r.toReal
hr' : r ≠ 0 ∧ r ≠ ∞
hr'' : r.toNNReal ≠ 0
⊢ Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r.toNNReal)

Tactic:
simp_rw [ENNReal.coe_toNNReal_eq_toReal]
      refine Tendsto.congr' ?_ <| (tendsto_toReal hr'.2).comp hc'
      filter_upwards [hc'.eventually_ne hr'.1, hc'.eventually_ne hr'.2]
      simp

Goal state after: