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

Goal state before:
pp: 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
⊢ Tendsto (⇑(eval₂Hom φ a)) (𝓝 0) (𝓝 0)

Tactic:
IsLinearTopology.hasBasis_ideal.tendsto_right_iff

Goal state after:
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), ↑i ∈ 𝓝 0 → ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑i
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
p : R⟦X⟧
T : Subring R
hp : ∀ (n : ℕ), (coeff n) p ∈ T
⊢ (p.toSubring T hp).order = p.order

Tactic:
by
  refine eq_of_le_of_ge ?_ ?_
  · refine le_order _ _ fun d hd => by simp [coeff_of_lt_order d hd, ← coeff_toSubring p T hp]
  · exact le_order _ _ fun d hd => by
      exact_mod_cast (coeff_toSubring p T hp) ▸ (coeff_of_lt_order d hd)

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

Goal state before:
pp: case mp.left
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
⊢ ↑J ⊆ ↑K

Tactic:
intro x hx

Goal state after:
case mp.left
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
x : R
hx : x ∈ ↑J
⊢ x ∈ ↑K
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 : ¬∀ i ∈ d.support, d i ≤ n₀ i
⊢ ∏ i ∈ d.support, a i ^ d i ∈ I

Tactic:
push Not at hd

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 := 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 : ∃ i ∈ d.support, n₀ i < d i
⊢ ∏ i ∈ d.support, a i ^ d i ∈ I
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
f : MvPowerSeries σ R
hf : IsTopologicallyNilpotent (constantCoeff f)
⊢ ∀ (d : σ →₀ ℕ), Tendsto (fun i ↦ (coeff d) (f ^ i)) atTop (𝓝 0)

Tactic:
IsLinearTopology.hasBasis_ideal.tendsto_right_iff

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
f : MvPowerSeries σ R
hf : IsTopologicallyNilpotent (constantCoeff f)
⊢ ∀ (d : σ →₀ ℕ) (i : Ideal R), ↑i ∈ 𝓝 0 → ∀ᶠ (x : ℕ) in atTop, (coeff d) (f ^ x) ∈ ↑i
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹⁷ : CommRing R
S : Type u_2
inst✝¹⁶ : CommRing S
φ : R →+* S
a : S
inst✝¹⁵ : UniformSpace R
inst✝¹⁴ : UniformSpace S
inst✝¹³ : IsUniformAddGroup R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : IsUniformAddGroup S
inst✝¹⁰ : T2Space S
inst✝⁹ : CompleteSpace S
inst✝⁸ : IsTopologicalRing S
inst✝⁷ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
T : Type u_3
inst✝⁶ : UniformSpace T
inst✝⁵ : CompleteSpace T
inst✝⁴ : T2Space T
inst✝³ : CommRing T
inst✝² : IsTopologicalRing T
inst✝¹ : IsLinearTopology T T
inst✝ : IsUniformAddGroup T
ε : S →+* T
hε : Continuous ⇑ε
⊢ ⇑ε ∘ eval₂ φ a = eval₂ (ε.comp φ) (ε a)

Tactic:
refine eval₂_unique (by simp only [RingHom.coe_comp, hε.comp hφ]) (ha.map hε)
    (hε.comp (continuous_eval₂ hφ ha)) (fun p ↦ ?_)
  simpa [Function.comp_apply, eval₂_coe] using p.hom_eval₂ φ ε a

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

Goal state before:
pp: case refine_1
f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)
hC₀ : 0 ≤ C
hsx : -s.re + x - 1 < -1
⊢ Summable fun n ↦ ↑n ^ (-s.re + x - 1)

pp: case refine_2
f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)
hC₀ : 0 ≤ C
hsx : -s.re + x - 1 < -1
⊢ ∀ᶠ (i : ℕ) in Filter.atTop, ‖‖↑i ^ (-(s + (1 - ↑x)))‖‖ ≤ ↑i ^ (-s.re + x - 1)

Tactic:
· simpa

Goal state after:
case refine_2
f : ℕ → ℂ
x : ℝ
s : ℂ
hs : x < s.re
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)
hC₀ : 0 ≤ C
hsx : -s.re + x - 1 < -1
⊢ ∀ᶠ (i : ℕ) in Filter.atTop, ‖‖↑i ^ (-(s + (1 - ↑x)))‖‖ ≤ ↑i ^ (-s.re + x - 1)
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
hk : k ≠ 0
f : R⟦X⟧
⊢ constantCoeff (subst (X ^ k) f) = (algebraMap R S) (constantCoeff f)

Tactic:
← coeff_zero_eq_constantCoeff,

Goal state after:
R : Type u_2
inst✝² : CommRing R
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
k : ℕ
hk : k ≠ 0
f : R⟦X⟧
⊢ (coeff 0) (subst (X ^ k) f) = (algebraMap R S) (constantCoeff 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
⊢ ∀ 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

Tactic:
intro x hx

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 : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : 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.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
r : R
f : R⟦X⟧
⊢ (rescale r) f = (MvPowerSeries.rescale fun x ↦ r) f

Tactic:
ext n
  rw [coeff_rescale, coeff, MvPowerSeries.coeff_rescale]
  simp [pow_zero, Finsupp.prod_single_index]

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

Goal state before:
pp: 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
x : E
hs : Balanced 𝕜 s
hc : egauge 𝕜 s x < ‖c‖ₑ
a : 𝕜
hxa : x ∈ a • s
ha : ‖a‖ₑ < ‖c‖ₑ
⊢ ‖a‖ ≤ ‖c‖

Tactic:
simpa [enorm] using! ha.le

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
h : x ∈ s
⊢ egauge 𝕜 s x ≤ 1

Tactic:
rw [← one_smul 𝕜 s] at h

Goal state after:
𝕜 : Type u_1
inst✝² : NormedDivisionRing 𝕜
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
h : x ∈ 1 • s
⊢ egauge 𝕜 s x ≤ 1
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝³ : NormedDivisionRing 𝕜
E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
inst✝ : (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hs₀ : 0 ∈ s
h : ∀ (c : 𝕜), c ≠ 0 → 0 ∈ c • s → r ≤ ‖c‖ₑ
hc : 0 ∈ 0 • s
b : ℝ≥0∞
hb : ‖0‖ₑ < b
⊢ r < b

Tactic:
rcases Filter.nonempty_of_mem <|
    inter_mem_nhdsWithin {(0 : 𝕜)}ᶜ (Metric.eball_mem_nhds 0 (by simpa using hb))
    with ⟨c, hc₀, hcb⟩

Goal state after:
case inr
𝕜 : Type u_1
inst✝³ : NormedDivisionRing 𝕜
E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
inst✝ : (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hs₀ : 0 ∈ s
h : ∀ (c : 𝕜), c ≠ 0 → 0 ∈ c • s → r ≤ ‖c‖ₑ
hc : 0 ∈ 0 • s
b : ℝ≥0∞
hb : ‖0‖ₑ < b
c : 𝕜
hc₀ : c ∈ {0}ᶜ
hcb : c ∈ eball 0 b
⊢ r < b
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_9
inst✝ : CommSemiring R
n : ℕ
r : R
f : MvPowerSeries σ R
hf : ∀ d ∈ Function.support f, Finsupp.degree d = n
⊢ (rescale (Function.const σ r)) f = r ^ n • f

Tactic:
by
  ext e
  simp only [MvPowerSeries.coeff_rescale, map_smul, Finsupp.prod, Function.const_apply,
    Finset.prod_pow_eq_pow_sum, smul_eq_mul]
  by_cases he : e ∈ f.support
  · rw [← hf e he, Finsupp.degree_apply]
  · simp only [Function.mem_support, ne_eq, not_not] at he
    simp [he, mul_zero, coeff_apply]

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

Goal state before:
pp: case ha
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
⊢ ∀ (x : ℕ), x ≠ 1 → (if x = 1 then (coeff x) P * k else 0) = 0

Tactic:
simp +contextual

Goal state after:
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
h : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ ∀ a ∈ (fun x_1 ↦ x + x_1) ⁻¹' s, ∀ (a_2 : R × E), a_2.2 = a → x + a_2.2 ∈ s

Tactic:
simp +contextual

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

Goal state before:
pp: 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
x✝ : ℕ
⊢ (x✝ ∈ Function.support fun d ↦ (coeff d) P * if d = 1 then k else 0) → x✝ ∈ {1}

Tactic:
simp +contextual

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

Goal state before:
pp: case hds
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
⊢ ∃ᶠ (n : E) in 𝓝 x, x + (n + -x) ∈ s

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 ↦ 1 n • (n + -x)) (𝓝 x) (𝓝 0)

Tactic:
· simpa

Goal state after:
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 ↦ 1 n • (n + -x)) (𝓝 x) (𝓝 0)
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
x : 𝕜
s : Set 𝕜
h : DifferentiableAt 𝕜 f x
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ _root_.derivWithin f s x = deriv f x

Tactic:
by
  unfold _root_.derivWithin deriv
  rw [h.fderivWithin hxs]

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 g : 𝕜 → F
s : Set 𝕜
hfg : EqOn f g s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ EqOn (_root_.deriv f) (_root_.deriv g) s

Tactic:
intro x hx

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : 𝕜 → F
s : Set 𝕜
hfg : EqOn f g s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
⊢ _root_.deriv f x = _root_.deriv g 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
s t : Set E
hf : DifferentiableOn 𝕜 f s
hf' : DifferentiableOn 𝕜 f t
hs : IsOpen[inst✝³] s
ht : IsOpen[inst✝³] t
⊢ DifferentiableOn 𝕜 f (s ∪ t)

Tactic:
intro x hx

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
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 ∈ s ∪ t
⊢ DifferentiableWithinAt 𝕜 f (s ∪ t) x
Module: Mathlib.Analysis.Calculus.FormalMultilinearSeries

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

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

Goal state after:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: case refine_2
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
ε : ℝ
hε : 0 < ε
δ : ℝ≥0
hδ : δ ≠ 0
⊢ ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ε) (f x) ≤ ↑δ * egauge 𝕜 (ball 0 1) (g x)

Tactic:
· filter_upwards [@h ↑(ε * δ / ‖c‖₊) (by positivity)] with x (hx : ‖f x‖₊ ≤ ε * δ / ‖c‖₊ * ‖g x‖₊)
    lift ε to ℝ≥0 using hε.le
    calc
      egauge 𝕜 (ball 0 ε) (f x) ≤ ‖c‖₊ * ‖f x‖₊ / ε :=
        egauge_ball_le_of_one_lt_norm hc (.inl <| ne_of_gt hε)
      _ ≤ ‖c‖₊ * (↑(ε * δ / ‖c‖₊) * ‖g x‖₊) / ε := by gcongr; exact_mod_cast hx
      _ = (‖c‖₊ / ‖c‖₊) * (ε / ε) * δ * ‖g x‖₊ := by
        simp only [div_eq_mul_inv, ENNReal.coe_inv hc₀.ne', ENNReal.coe_mul]; ring
      _ ≤ 1 * 1 * δ * ‖g x‖₊ := by gcongr <;> exact ENNReal.div_self_le_one
      _ = δ * ‖g x‖₊ := by simp
      _ ≤ δ * egauge 𝕜 (ball 0 1) (g x) := by gcongr; apply le_egauge_ball_one

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
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]

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
x y : E
n : ℕ
t : Finset (Fin n)
this :
  ∀ (m : ℕ) (hm : n = m),
    (p n) (t.piecewise (fun x_1 ↦ x) fun x ↦ y) =
      (p m) ((Finset.map (finCongr hm).toEmbedding t).piecewise (fun x_1 ↦ x) fun x ↦ y)
⊢ 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)
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✝¹⁰ : Fintype ι
inst✝⁹ : Fintype ι'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 Ei G
⊢ ‖{ toFun := curryRight, map_add' := ⋯, map_smul' := ⋯, invFun := uncurryRight, left_inv := ⋯, right_inv := ⋯ } f‖ ≤
    ‖f‖

Tactic:
by
      simp only [curryRight, LinearEquiv.coe_mk, LinearMap.coe_mk, AddHom.coe_mk]
      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _

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
s : Set E
hf : AnalyticOnNhd 𝕜 f s
y : E
⊢ AnalyticOnNhd 𝕜 (fun z ↦ f (z - y)) (s + {y})

Tactic:
intro x hx

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
s : Set E
hf : AnalyticOnNhd 𝕜 f s
y x : E
hx : x ∈ s + {y}
⊢ AnalyticAt 𝕜 (fun z ↦ f (z - y)) x
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
s : Set E
hf : AnalyticOnNhd 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x - y ∈ s

Tactic:
by convert hx; abel

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
s : Set E
hf : AnalyticOn 𝕜 f s
y : E
⊢ AnalyticOn 𝕜 (fun z ↦ f (z - y)) (s + {y})

Tactic:
intro x hx

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
s : Set E
hf : AnalyticOn 𝕜 f s
y x : E
hx : x ∈ s + {y}
⊢ AnalyticWithinAt 𝕜 (fun z ↦ f (z - y)) (s + {y}) x
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
s : Set E
hf : AnalyticOn 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x - y ∈ s

Tactic:
by convert hx; abel

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

Goal state before:
pp: case neg.refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ Tendsto (fun n ↦ ‖‖c (n + 1)‖ * ↑r' ^ (n + 1)‖ / ‖‖c n‖ * ↑r' ^ n‖) atTop (𝓝 0)

Tactic:
exact mul_zero (_ : ℝ) ▸ tendsto_succ_norm_div_norm _ hrz (NNReal.coe_zero ▸ hc')

Goal state after:
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
r : ℝ≥0∞
n : ℕ
x y : E
hf : HasFiniteFPowerSeriesOnBall f p x n r
h : ↑‖y‖₊ < r
z : E
hz : z ∈ Metric.eball 0 (r - ↑‖y‖₊)
this : f (x + y + z) = (p.changeOrigin y).sum z
⊢ HasSum (fun n ↦ (p.changeOrigin y n) fun x ↦ z) ((p.changeOrigin y).sum z)

Tactic:
apply (p.changeOrigin y).hasSum_of_finite fun _ => p.changeOrigin_finite_of_finite hf.finite

Goal state after: