mathlib-initiative/mathlib-tactics huggingface dataset:
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
ha : HasSubst a
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
⊢ ∀ (x : σ →₀ ℕ), (coeff x) f • constantCoeff (x.prod fun s e ↦ a s ^ e) = 0

Tactic:
intro d

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
ha : HasSubst a
ha' : ∀ (i : σ), constantCoeff (a i) = 0
f : MvPowerSeries σ R
hf : constantCoeff f = 0
d : σ →₀ ℕ
⊢ (coeff d) f • constantCoeff (d.prod fun s e ↦ a s ^ e) = 0
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ e → (coeff x) f • (coeff e) (x.prod fun s e ↦ X s ^ e) = 0

Tactic:
intro d

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : MvPowerSeries σ R
e d : σ →₀ ℕ
⊢ d ≠ e → (coeff d) f • (coeff e) (d.prod fun s e ↦ X s ^ e) = 0
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
f : MvPowerSeries σ R
n : τ →₀ ℕ
this : ∀ {r : R}, h r = h.toAddMonoidHom r
⊢ ∀ (x : σ →₀ ℕ),
    h.toAddMonoidHom ((coeff x) f • (coeff n) (x.prod fun s e ↦ a s ^ e)) =
      (coeff x) ((map h) f) • (coeff n) (x.prod fun s e ↦ (map h) (a s) ^ e)

Tactic:
intro d

Goal state after:
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
f : MvPowerSeries σ R
n : τ →₀ ℕ
this : ∀ {r : R}, h r = h.toAddMonoidHom r
d : σ →₀ ℕ
⊢ h.toAddMonoidHom ((coeff d) f • (coeff n) (d.prod fun s e ↦ a s ^ e)) =
    (coeff d) ((map h) f) • (coeff n) (d.prod fun s e ↦ (map h) (a s) ^ e)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a : σ → MvPowerSeries τ R
ha : HasSubst a
h : R →+* S
f : MvPowerSeries σ R
n : τ →₀ ℕ
this : ∀ {r : R}, h r = h.toAddMonoidHom r
d : σ →₀ ℕ
⊢ h.toAddMonoidHom ((coeff d) f • (coeff n) (d.prod fun s e ↦ a s ^ e)) =
    (coeff d) ((map h) f) • (coeff n) (d.prod fun s e ↦ (map h) (a s) ^ e)

Tactic:
simp [smul_eq_mul, RingHom.toAddMonoidHom_eq_coe, AddMonoidHom.coe_coe, map_mul,
    ← coeff_map, Finsupp.prod]

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

Goal state before:
pp: a b : ℝ
h : a < b
h1 : #↑((fun x ↦ x - a) '' Ioo a b) ≤ #↑(Ioo a b)
⊢ 𝔠 ≤ #↑((fun x ↦ x - a) '' Ioo a b)

Tactic:
image_sub_const_Ioo,

Goal state after:
a b : ℝ
h : a < b
h1 : #↑((fun x ↦ x - a) '' Ioo a b) ≤ #↑(Ioo a b)
⊢ 𝔠 ≤ #↑(Ioo (a - a) (b - a))
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
f : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ 0 → (coeff x) f • (coeff n) (x.prod fun s e ↦ 0 s ^ e) = 0

Tactic:
intro d

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
f : MvPowerSeries σ R
n : τ →₀ ℕ
d : σ →₀ ℕ
⊢ d ≠ 0 → (coeff d) f • (coeff n) (d.prod fun s e ↦ 0 s ^ e) = 0
Module: Mathlib.Analysis.Complex.Cardinality

Goal state before:
pp: ⊢ #↑Set.univ = 𝔠

Tactic:
by rw [mk_univ, mk_complex]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
⊢ ∀ (d : τ →₀ ℕ),
    ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d →
      (coeff d) (subst a f) = 0

Tactic:
intro d

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
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
⊢ ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d →
    (coeff d) (subst a f) = 0
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
s : ℂ
n : ℕ
⊢ ‖term f s n‖ = if n = 0 then 0 else ‖f n‖ / ↑n ^ s.re

Tactic:
by
  rcases eq_or_ne n 0 with rfl | hn
  · simp
  · simp [hn, norm_natCast_cpow_of_pos <| Nat.pos_of_ne_zero hn]

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

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
τ : Type u_3
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : MvPowerSeries τ S
ha : HasSubst a
p : Polynomial R
⊢ subst a ↑p = (Polynomial.aeval a) p

Tactic:
by
  rw [← coe_substAlgHom ha, substAlgHom_coe]

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_9
inst✝ : CommSemiring R
x : MvPowerSeries σ R
n : σ →₀ ℕ
h : n = 0
⊢ ((coeff n) fun n ↦ (n.prod fun s m ↦ 0 ^ m) * (coeff n) x) = constantCoeff x

Tactic:
simp [h, coeff_apply, ← @coeff_zero_eq_constantCoeff_apply, coeff_apply]

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

Goal state before:
pp: 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 : x ∈ closure[inst✝¹] s
⊢ 0 ∈ tangentConeAt 𝕜 s x

Tactic:
rw [mem_closure_iff_frequently] at hx
  apply mem_tangentConeAt_of_frequently (𝓝 x) 1 (· + (-x))
  · exact Continuous.tendsto' (by fun_prop) _ _ (by simp)
  · simpa
  · simp only [Pi.one_apply, one_smul]
    exact Continuous.tendsto' (by fun_prop) _ _ (by simp)

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Semiring 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
s : Set E
x : E
inst✝ : T2Space E
y : E
hy : y ∈ tangentConeAt 𝕜 s x
hy₀ : ¬AccPt x (𝓟 s)
⊢ y = 0

Tactic:
exact tangentConeAt_subset_zero hy₀ hy

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
⊢ derivWithin f (Ioi x) x = derivWithin f (Ici x) x

Tactic:
by_cases H : DifferentiableWithinAt ℝ f (Ioi x) x

Goal state after:
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x : ℝ
H : DifferentiableWithinAt ℝ f (Ioi x) x
⊢ derivWithin f (Ioi x) x = derivWithin f (Ici x) x

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
⊢ derivWithin f (Ioi x) x = derivWithin f (Ici x) x
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
x : E
s : Set E
inst✝² : ContinuousAdd E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 id s x = ContinuousLinearMap.id 𝕜 E

Tactic:
DifferentiableAt.fderivWithin differentiableAt_id hxs

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
x : E
s : Set E
inst✝² : ContinuousAdd E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderiv 𝕜 id x = ContinuousLinearMap.id 𝕜 E
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
⊢ (p.changeOrigin x).sum y = p.sum (x + y)

Tactic:
have x_mem_ball : x ∈ Metric.eball (0 : E) p.radius :=
    mem_eball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
x_mem_ball : x ∈ Metric.eball 0 p.radius
⊢ (p.changeOrigin x).sum y = p.sum (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
pf : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf s x r
v : Fin 0 → E
v_eq : v = fun i ↦ 0
⊢ 0 ∈ Metric.eball 0 r

Tactic:
simp [hf.r_pos]

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
pf : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf s x r
v : Fin 0 → E
⊢ (pf 0) v = f x

Tactic:
by
  have v_eq : v = fun i => 0 := Subsingleton.elim _ _
  have zero_mem : (0 : E) ∈ Metric.eball (0 : E) r := by simp [hf.r_pos]
  have : ∀ i, i ≠ 0 → (pf i fun _ => 0) = 0 := by
    intro i hi
    have : 0 < i := pos_iff_ne_zero.2 hi
    exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
  have A := (hf.hasSum (by simp) zero_mem).unique (hasSum_single _ this)
  simpa [v_eq] using A.symm

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

Goal state before:
pp: case pos
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 = 0
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 0)
h : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
⊢ (ofScalars E c).radius = ∞

Tactic:
apply ofScalars_radius_eq_top_of_tendsto E c h ?_
      refine Tendsto.congr' ?_ <| (tendsto_toReal zero_ne_top).comp hc'
      filter_upwards [h]
      simp

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 ∈ Metric.eball x r
⊢ CPolynomialAt 𝕜 f y

Tactic:
by
  have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_enorm_sub] using! h
  have := hf.changeOrigin this
  rw [add_sub_cancel] at this
  exact this.cpolynomialAt

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : 0 < n
v : Fin n → E
j : Fin (Composition.single n hn).length
⊢ p.applyComposition (Composition.single n hn) v j = (p n) v

Tactic:
refine p.congr (by simp) fun i hi1 hi2 => ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : CommRing 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousConstSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : 0 < n
v : Fin n → E
j : Fin (Composition.single n hn).length
i : ℕ
hi1 : i < (Composition.single n hn).blocksFun j
hi2 : i < n
⊢ (v ∘ ⇑((Composition.single n hn).embedding j)) ⟨i, hi1⟩ = v ⟨i, hi2⟩
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
k : ℕ
blocksFun : Fin k → ℕ
H : ⟨k, blocksFun⟩ ∈ compPartialSumSource 2 (n + 1) n
K : (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length = k
⊢ ∏ j, r * (a ^ ⟨k, blocksFun⟩.snd j * p (⟨k, blocksFun⟩.snd j)) =
    ∏ j,
      r *
        (a ^ (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.blocksFun j *
          p ((compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.blocksFun j))

Tactic:
congr 2

Goal state after:
case e_1
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
k : ℕ
blocksFun : Fin k → ℕ
H : ⟨k, blocksFun⟩ ∈ compPartialSumSource 2 (n + 1) n
K : (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length = k
⊢ ⟨k, blocksFun⟩.fst = (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length

case e_4.e_1
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
k : ℕ
blocksFun : Fin k → ℕ
H : ⟨k, blocksFun⟩ ∈ compPartialSumSource 2 (n + 1) n
K : (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length = k
⊢ Fin ⟨k, blocksFun⟩.fst = Fin (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length

case e_4.e_2
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
k : ℕ
blocksFun : Fin k → ℕ
H : ⟨k, blocksFun⟩ ∈ compPartialSumSource 2 (n + 1) n
K : (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length = k
⊢ Fin.fintype ⟨k, blocksFun⟩.fst ≍ Fin.fintype (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length

case e_5
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
k : ℕ
blocksFun : Fin k → ℕ
H : ⟨k, blocksFun⟩ ∈ compPartialSumSource 2 (n + 1) n
K : (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length = k
⊢ (fun j ↦ r * (a ^ ⟨k, blocksFun⟩.snd j * p (⟨k, blocksFun⟩.snd j))) ≍ fun j ↦
    r *
      (a ^ (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.blocksFun j *
        p ((compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.blocksFun j))
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
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
⊢ x ∈ insert x s ∩ Metric.eball x r

Tactic:
simp [hf.r_pos]

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

Goal state before:
pp: case mp.refine_1
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
⊢ EqOn f (fun y ↦ p.sum (y - x)) (insert x s ∩ eball x r)

Tactic:
intro y ⟨ys,yb⟩
      simp only [mem_eball, edist_eq_enorm_sub] at yb
      have e0 := p.hasSum (x := y - x) ?_
      · have e1 := (h.hasSum (y := y - x) ?_ ?_)
        · simp only [add_sub_cancel] at e1
          exact e1.unique e0
        · simpa only [add_sub_cancel]
        · simpa only [mem_eball, edist_zero_right]
      · simp only [mem_eball, edist_zero_right]
        exact lt_of_lt_of_le yb h.r_le

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
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))))
⊢ CauchySeq fun s ↦ ∑ i ∈ s, (q.compAlongComposition p i.snd) fun _j ↦ y

Tactic:
apply cauchySeq_finset_of_norm_bounded (NNReal.summable_coe.2 hr) _
      simp only [coe_nnnorm, NNReal.coe_mul, NNReal.coe_pow]
      rintro ⟨n, c⟩
      calc
        ‖(compAlongComposition q p c) fun _j : Fin n => y‖ ≤
            ‖compAlongComposition q p c‖ * ∏ _j : Fin n, ‖y‖ := by
          apply ContinuousMultilinearMap.le_opNorm
        _ ≤ ‖compAlongComposition q p c‖ * (r : ℝ) ^ n := by
          rw [Finset.prod_const, Finset.card_fin]
          gcongr
          rw [Metric.mem_eball, edist_zero_right] at hy
          have := le_trans (le_of_lt hy) (min_le_right _ _)
          rwa [enorm_le_coe, ← NNReal.coe_le_coe, coe_nnnorm] at this

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
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
m n : ℕ
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hg : HasFiniteFPowerSeriesAt g q (f x) m
hf : HasFiniteFPowerSeriesAt f p x n
hn : 0 < n
r : ℝ≥0∞
hr : HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x r
i : ℕ
c : Composition i
v : Fin i → E
hc : c.length < m
hi : ∀ (j : Fin c.length), c.blocksFun j < n
h'c : c.length = 0
⊢ ∑ j, c.blocksFun j = 0

Tactic:
apply Finset.sum_eq_zero (fun j hj ↦ ?_)

Goal state after:
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
m n : ℕ
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hg : HasFiniteFPowerSeriesAt g q (f x) m
hf : HasFiniteFPowerSeriesAt f p x n
hn : 0 < n
r : ℝ≥0∞
hr : HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x r
i : ℕ
c : Composition i
v : Fin i → E
hc : c.length < m
hi : ∀ (j : Fin c.length), c.blocksFun j < n
h'c : c.length = 0
j : Fin c.length
hj : j ∈ Finset.univ
⊢ c.blocksFun j = 0
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: n : ℕ
a : Composition n
b : Composition a.length
a' : Composition n
b' : Composition a'.length
h : ⟨a, b⟩.fst.blocks = ⟨a', b'⟩.fst.blocks
h' : ⟨a, b⟩.snd.blocks = ⟨a', b'⟩.snd.blocks
⊢ ⟨a, b⟩ = ⟨a', b'⟩

Tactic:
obtain rfl : a = a' := by ext1; exact h

Goal state after:
n : ℕ
a : Composition n
b b' : Composition a.length
h : ⟨a, b⟩.fst.blocks = ⟨a, b'⟩.fst.blocks
h' : ⟨a, b⟩.snd.blocks = ⟨a, b'⟩.snd.blocks
⊢ ⟨a, b⟩ = ⟨a, b'⟩
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
A : Type u_7
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ∀ (n : ℕ), formalMultilinearSeries_geometric 𝕜 A n = FormalMultilinearSeries.ofScalars A (fun x ↦ 1) n

Tactic:
FormalMultilinearSeries.ofScalars,

Goal state after:
𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
A : Type u_7
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ∀ (n : ℕ), formalMultilinearSeries_geometric 𝕜 A n = 1 • ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
z : Aˣ
hA : Nontrivial A
f1 : A → A := fun a ↦ a * z.inv
f2 : A → A := fun b ↦ (1 - b)⁻¹ʳ
f3 : A → A := fun c ↦ 1 - z.inv * c
this : Metric.ball ↑z ‖↑z⁻¹‖⁻¹ ∈ 𝓝 ↑z
y : A
hy : ‖y - ↑z‖ < ‖↑z⁻¹‖⁻¹
⊢ (f1 ∘ f2 ∘ f3) y = y⁻¹ʳ

Tactic:
have : y = Units.ofNearby z y hy := rfl

Goal state after:
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
z : Aˣ
hA : Nontrivial A
f1 : A → A := fun a ↦ a * z.inv
f2 : A → A := fun b ↦ (1 - b)⁻¹ʳ
f3 : A → A := fun c ↦ 1 - z.inv * c
this✝ : Metric.ball ↑z ‖↑z⁻¹‖⁻¹ ∈ 𝓝 ↑z
y : A
hy : ‖y - ↑z‖ < ‖↑z⁻¹‖⁻¹
this : y = ↑(z.ofNearby y hy)
⊢ (f1 ∘ f2 ∘ f3) y = y⁻¹ʳ
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
f' : E →L[ℝ] F
x : E
L : Filter E
⊢ Tendsto (fun x' ↦ ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) ↔
    Tendsto (fun x' ↦ ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0)

Tactic:
by
  symm
  rw [tendsto_iff_norm_sub_tendsto_zero]
  refine tendsto_congr fun x' => ?_
  simp [norm_smul]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a b : ι
heq : v a = v b
⊢ a ≠ b →
    (((f.fderivCompContinuousLinearMap fun x ↦ g) ∘SL
            ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (E →L[𝕜] F))
          dg)
        v =
      0

Tactic:
hne

Goal state after:
𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a b : ι
heq : v a = v b
hne : a ≠ b
⊢ (((f.fderivCompContinuousLinearMap fun x ↦ g) ∘SL ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (E →L[𝕜] F))
        dg)
      v =
    0