mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case pos
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : Archimedean G
⊢ Nonempty (G ≃+o ℤ)

Tactic:
rw [LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered]

Goal state after:
case pos
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : Archimedean G
⊢ ¬DenselyOrdered G
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
⊢ Tendsto (⇑v.restrict) (Filter.comap Completion.coe' (𝓝 ↑x)) (𝓝 (v.restrict x))

Tactic:
rw [← Completion.isDenseInducing_coe.nhds_eq_comap]

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
⊢ Tendsto (⇑v.restrict) (𝓝 x) (𝓝 (v.restrict x))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDiscreteValuationRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
C : VariableChange K
hC : MaximalFor (fun C ↦ IsIntegral R (C • W)) (fun C ↦ valuation_Δ_aux R (C • W)) C
j : VariableChange K
hj : IsIntegral R ((j • C) • W)
h : (fun C ↦ valuation_Δ_aux R (C • W)) C ≤ (fun C ↦ valuation_Δ_aux R (C • W)) (j • C) →
  (fun C ↦ valuation_Δ_aux R (C • W)) (j • C) ≤ (fun C ↦ valuation_Δ_aux R (C • W)) C :=
  hC.right hj
⊢ (fun C_1 ↦ valuation_Δ_aux R (C_1 • C • W)) 1 ≤ (fun C_1 ↦ valuation_Δ_aux R (C_1 • C • W)) j →
    (fun C_1 ↦ valuation_Δ_aux R (C_1 • C • W)) j ≤ (fun C_1 ↦ valuation_Δ_aux R (C_1 • C • W)) 1

Tactic:
simp_all only [one_smul]

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDiscreteValuationRing R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
C : VariableChange K
hC : MaximalFor (fun C ↦ IsIntegral R (C • W)) (fun C ↦ valuation_Δ_aux R (C • W)) C
j : VariableChange K
hj : IsIntegral R ((j • C) • W)
h : (fun C ↦ valuation_Δ_aux R (C • W)) C ≤ (fun C ↦ valuation_Δ_aux R (C • W)) (j • C) →
  (fun C ↦ valuation_Δ_aux R (C • W)) (j • C) ≤ (fun C ↦ valuation_Δ_aux R (C • W)) C :=
  hC.right hj
⊢ valuation_Δ_aux R (C • W) ≤ valuation_Δ_aux R (j • C • W) → valuation_Δ_aux R (j • C • W) ≤ valuation_Δ_aux R (C • W)
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case mp
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : PartialOrder α
a✝ : MulArchimedean (WithZero α)
⊢ MulArchimedean α

Tactic:
exact OrderMonoidIso.unitsWithZero.mulArchimedean

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

Goal state before:
pp: case e'_2.mp
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ m < n, (coeff m) φ = 0
m : Unit →₀ ℕ
hm : m () < n
⊢ (MvPowerSeries.coeff m) φ = 0

pp: case e'_2.mpr
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ (m : Unit →₀ ℕ), m () < n → (MvPowerSeries.coeff m) φ = 0
m : ℕ
hm : m < n
⊢ (coeff m) φ = 0

Tactic:
· rw [Finsupp.unique_single m]
    convert! h _ hm

Goal state after:
case e'_2.mpr
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ (m : Unit →₀ ℕ), m () < n → (MvPowerSeries.coeff m) φ = 0
m : ℕ
hm : m < n
⊢ (coeff m) φ = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : ¬r = 0
⊢ v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}

Tactic:
rw [v.intValuation_if_neg hr, ← exp_zero, exp_lt_exp,
      neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : ¬r = 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors ≠ 0 ↔ v.asIdeal ∣ Ideal.span {r}
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq σ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∀ (d : σ →₀ ℕ), (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)
u v u' v' : σ →₀ ℕ
huv : u' + v' = u + v
h : u = u' → ¬v = v'
k l : ι →₀ σ →₀ ℕ
hkl : l.update a u' = k.update a u
⊢ u' = u

Tactic:
simpa only [Finsupp.coe_update, Function.update_self] using DFunLike.congr_fun hkl a

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff (k i)) f = constantCoeff f
hs'' : ∀ i ∈ s, k i = 0
i : ℕ
hi : i < n
hi' : i < n → ¬k i = 0
⊢ ¬degree (k i) = 0

Tactic:
degree_eq_zero_iff

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff (k i)) f = constantCoeff f
hs'' : ∀ i ∈ s, k i = 0
i : ℕ
hi : i < n
hi' : i < n → ¬k i = 0
⊢ ¬k i = 0
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : CommSemiring R
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra A R
e : σ →₀ ℕ
a : σ → A
x : σ
a✝ : x ∈ e.support
⊢ (a x • MvPowerSeries.X x) ^ e x = (MvPowerSeries.C ((algebraMap A R) (a x)) * MvPowerSeries.X x) ^ e x

Tactic:
algebra_compatible_smul R,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : CommSemiring R
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra A R
e : σ →₀ ℕ
a : σ → A
x : σ
a✝ : x ∈ e.support
⊢ ((algebraMap A R) (a x) • MvPowerSeries.X x) ^ e x =
    (MvPowerSeries.C ((algebraMap A R) (a x)) * MvPowerSeries.X x) ^ e x
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑((weight w) d) < weightedOrder w g
⊢ (coeff d) (f * (1 - g)) = (coeff d) f

Tactic:
simp only [mul_sub, mul_one, map_sub, sub_eq_self]

Goal state after:
σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑((weight w) d) < weightedOrder w g
⊢ (coeff d) (f * g) = 0
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : DecidableEq σ
inst✝ : CommSemiring R
f g : MvPowerSeries σ R
⊢ f = g ↔ ∀ (n : σ →₀ ℕ), (trunc' R n) f = (trunc' R n) g

Tactic:
by
  refine ⟨fun h => by simp [h], fun h => ?_⟩
  ext n
  specialize h n
  have {f' : MvPowerSeries σ R} : f'.coeff n = (trunc' R n f').coeff n := by
    rw [coeff_trunc', if_pos le_rfl]
  simp_rw [this, h]

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

Goal state before:
pp: case refine_1
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
s : Set (adicCompletion K v)
x✝ : ∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s
t : Set (HeightOneSpectrum.valuation K v).Completion
ht : t ∈ 𝓝 0
hts : toCompletion ⁻¹' t ⊆ s
δ : (ofClass Valued.v).ValueGroup₀ˣ
hδ : {x | Valued.v.restrict x < ↑δ} ⊆ t
x : adicCompletion K v
hx : x ∈ {x | (valuation K v).restrict x < ↑((Units.mapEquiv (valueGroupOrderIso K v).symm.toMulEquiv) δ)}
⊢ x.toCompletion ∈ {x | Valued.v.restrict x < ↑δ}

Tactic:
rw [Set.mem_ofPred_eq] at hx ⊢

Goal state after:
case refine_1
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
s : Set (adicCompletion K v)
x✝ : ∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s
t : Set (HeightOneSpectrum.valuation K v).Completion
ht : t ∈ 𝓝 0
hts : toCompletion ⁻¹' t ⊆ s
δ : (ofClass Valued.v).ValueGroup₀ˣ
hδ : {x | Valued.v.restrict x < ↑δ} ⊆ t
x : adicCompletion K v
hx : (valuation K v).restrict x < ↑((Units.mapEquiv (valueGroupOrderIso K v).symm.toMulEquiv) δ)
⊢ Valued.v.restrict x.toCompletion < ↑δ
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: case e_f
R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
u : Filter ι
g : R⟦X⟧
d : Unit →₀ ℕ
⊢ (fun i ↦ (MvPowerSeries.coeff d) (f i)) = fun i ↦ (MvPowerSeries.coeff (Finsupp.single () (d ()))) (f i)

Tactic:
ext _

Goal state after:
case e_f
R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
u : Filter ι
g : R⟦X⟧
d : Unit →₀ ℕ
x✝ : ι
⊢ (MvPowerSeries.coeff d) (f x✝) = (MvPowerSeries.coeff (Finsupp.single () (d ()))) (f x✝)
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: S : Type u_2
inst✝¹ : CommRing S
inst✝ : TopologicalSpace S
⊢ HasEval 0

Tactic:
hasEval_iff

Goal state after:
S : Type u_2
inst✝¹ : CommRing S
inst✝ : TopologicalSpace S
⊢ MvPowerSeries.HasEval fun x ↦ 0
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : TopologicalSpace R
⊢ HasEval X

Tactic:
hasEval_iff

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : TopologicalSpace R
⊢ MvPowerSeries.HasEval fun x ↦ X
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
f : Polynomial R
⊢ eval₂ φ a ↑f = Polynomial.eval₂ φ a f

Tactic:
rw [← (MvPolynomial.uniqueAlgEquiv R Unit).apply_symm_apply f]
  simp only [PowerSeries.eval₂, MvPolynomial.eval₂_const_uniqueAlgEquiv]
  rw [← MvPolynomial.toMvPowerSeries_pUnitAlgEquiv, MvPowerSeries.eval₂_coe]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑(Iio a ∪ {a} ∪ Ioi a) < 𝔠

Tactic:
grw [mk_union_le, mk_union_le]

Goal state after:
a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑(Iio a) + #↑{a} + #↑(Ioi a) < 𝔠
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff e) (MvPowerSeries.subst (fun x ↦ X₀ + X₁) f) =
    ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f

Tactic:
MvPowerSeries.coeff_subst
    (MvPowerSeries.hasSubst_of_constantCoeff_zero (fun _ ↦ by simp))

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ (X₀ + X₁) ^ e) =
    ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: case hd₀
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Semiring 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousAdd E
s : Set E
x : E
hx : ∃ᶠ (x : E) in 𝓝 x, x ∈ s
⊢ Tendsto (fun x_1 ↦ x_1 + -x) (𝓝 x) (𝓝 0)

Tactic:
exact Continuous.tendsto' (by fun_prop) _ _ (by simp)

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

Goal state before:
pp: case hcd
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Semiring 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousAdd E
s : Set E
x : E
hx : ∃ᶠ (x : E) in 𝓝 x, x ∈ s
⊢ Tendsto (fun n ↦ n + -x) (𝓝 x) (𝓝 0)

Tactic:
exact Continuous.tendsto' (by fun_prop) _ _ (by simp)

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

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 f₁ : E → F
x : E
s : Set E
hs : f₁ =ᶠ[𝓝[insert x s] x] f
⊢ fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x

Tactic:
apply Filter.EventuallyEq.fderivWithin_eq (nhdsWithin_mono _ (subset_insert x s) hs)

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 f₁ : E → F
x : E
s : Set E
hs : f₁ =ᶠ[𝓝[insert x s] x] f
⊢ f₁ x = f x
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
⊢ ¬DifferentiableWithinAt ℝ f (Ici x) x

Tactic:
rwa [differentiableWithinAt_Ioi_iff_Ici] at H

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

Goal state before:
pp: 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
h : 0 < p.radius
r : ℝ≥0
r0 : 0 < ↑r
rlt : ↑r < p.radius
rpos : 0 < ↑r
C : ℝ
Cpos : C > 0
hCp : ∀ (n : ℕ), ‖p n‖ ≤ C / ↑r ^ n
⊢ 0 < (↑r)⁻¹

Tactic:
by simp only [inv_pos, rpos]

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:
rw [← ofReal_norm, ← ofReal_norm, ← ofReal_norm,
    ← ENNReal.ofReal_pow (by simp), ← ENNReal.ofReal_mul (by simp)]

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‖ ^ n)
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
k : ℕ
⊢ HasSum (fun c ↦ f ⟨k, c⟩) ((p.changeOrigin x k) fun x ↦ y)

Tactic:
· dsimp +instances only [f]
      refine ContinuousMultilinearMap.hasSum_eval ?_ _
      have := (p.hasFPowerSeriesOnBall_changeOrigin k h.pos).hasSum x_mem_ball
      rw [zero_add] at this
      refine HasSum.sigma_of_hasSum this (fun l => ?_) ?_
      · simp only [changeOriginSeries, sum_apply]
        apply hasSum_fintype
      · refine .of_nnnorm_bounded
          (p.changeOriginSeries_summable_aux₂ (mem_eball_zero_iff.1 x_mem_ball) k)
            fun s => ?_
        refine (ContinuousMultilinearMap.le_opNNNorm _ _).trans_eq ?_
        simp

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
x y : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
i : ι
⊢ (update v (some j) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) j (z.2 j) i

Tactic:
rcases eq_or_ne i j with rfl | hij

Goal state after:
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j : Option ι
x y z : G × ((i : ι) → E i)
i : ι
⊢ (update v (some i) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) i (z.2 i) i

case inr
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
x y : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
i : ι
hij : i ≠ j
⊢ (update v (some j) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) j (z.2 j) i
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
c : 𝕜
x : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
i : ι
⊢ (update v (some j) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) j (z.2 j) i

Tactic:
rcases eq_or_ne i j with rfl | hij

Goal state after:
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j : Option ι
c : 𝕜
x z : G × ((i : ι) → E i)
i : ι
⊢ (update v (some i) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) i (z.2 i) i

case inr
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
c : 𝕜
x : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
i : ι
hij : i ≠ j
⊢ (update v (some j) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) j (z.2 j) i
Module: Mathlib.Analysis.Analytic.CPolynomial

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
pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
n m : ℕ
hf : HasFiniteFPowerSeriesOnBall f pf x n r
hg : HasFiniteFPowerSeriesOnBall g pg x m r
N : ℕ
hN : max n m ≤ N
⊢ 0 + pg N = 0

Tactic:
hg.finite _ ((le_max_right n m).trans 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 g : E → F
pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
n m : ℕ
hf : HasFiniteFPowerSeriesOnBall f pf x n r
hg : HasFiniteFPowerSeriesOnBall g pg x m r
N : ℕ
hN : max n m ≤ N
⊢ 0 + 0 = 0
Module: Mathlib.Analysis.Analytic.CPolynomial

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
pf pg : FormalMultilinearSeries 𝕜 E F
x : E
n m : ℕ
hf : HasFiniteFPowerSeriesAt f pf x n
hg : HasFiniteFPowerSeriesAt g pg x m
⊢ HasFiniteFPowerSeriesAt (f + g) (pf + pg) x (max n m)

Tactic:
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩

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 g : E → F
pf pg : FormalMultilinearSeries 𝕜 E F
x : E
n m : ℕ
hf : HasFiniteFPowerSeriesAt f pf x n
hg : HasFiniteFPowerSeriesAt g pg x m
r : ℝ≥0∞
hr : HasFiniteFPowerSeriesOnBall f pf x n r ∧ HasFiniteFPowerSeriesOnBall g pg x m r
⊢ HasFiniteFPowerSeriesAt (f + g) (pf + pg) x (max n m)
Module: Mathlib.Analysis.Analytic.CPolynomial

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
G : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
ι : Type u_5
Em : ι → Type u_6
inst✝² : (i : ι) → NormedAddCommGroup (Em i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)
inst✝ : Fintype ι
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 Em F
⊢ ∀ y ∈ Metric.eball 0 ∞,
    (∑ i ∈ Finset.range (Fintype.card (Option ι) + 1), (f.toFormalMultilinearSeriesOfMultilinear i) fun x ↦ y) =
      (f (0 + y).1) (0 + y).2

Tactic:
intro y _
    rw [Finset.sum_eq_single_of_mem _ (Finset.self_mem_range_succ _), zero_add]
    · rw [toFormalMultilinearSeriesOfMultilinear, dif_pos rfl]; rfl
    · intro m _ ne; rw [toFormalMultilinearSeriesOfMultilinear, dif_neg ne.symm]; rfl

Goal state after: