mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: 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
l : S
⊢ UniformContinuous fun x ↦ WithVal.toVal (valuation K v) (l • x.ofVal)

Tactic:
Algebra.smul_def

Goal state after:
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
l : S
⊢ UniformContinuous fun x ↦ WithVal.toVal (valuation K v) ((algebraMap S K) l * x.ofVal)
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)
⊢ r • ↑x ∈ adicCompletionIntegers K v

Tactic:
Algebra.smul_def

Goal state after:
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)
⊢ (algebraMap R (adicCompletion K v)) r * ↑x ∈ adicCompletionIntegers K v
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: case pos
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f g : R⟦X⟧
m : ℕ
h : m < n
x✝ : ℕ × ℕ
hab : x✝ ∈ antidiagonal m
⊢ (coeff x✝.1) ↑((trunc n) f) * (coeff x✝.2) g = (coeff x✝.1) f * (coeff x✝.2) g

Tactic:
have ha := lt_of_le_of_lt (antidiagonal.fst_le hab) h

Goal state after:
case pos
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f g : R⟦X⟧
m : ℕ
h : m < n
x✝ : ℕ × ℕ
hab : x✝ ∈ antidiagonal m
ha : x✝.1 < n
⊢ (coeff x✝.1) ↑((trunc n) f) * (coeff x✝.2) g = (coeff x✝.1) f * (coeff x✝.2) g
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: R : Type u_2
inst✝ : CommSemiring R
n a b : ℕ
f g : R⟦X⟧
ha : n < a
hb : n < b
⊢ (coeff n) ↑((trunc n.succ) (↑((trunc n.succ) f) * ↑((trunc n.succ) g))) = (coeff n) (f * g)

Tactic:
trunc_trunc_mul_trunc,

Goal state after:
R : Type u_2
inst✝ : CommSemiring R
n a b : ℕ
f g : R⟦X⟧
ha : n < a
hb : n < b
⊢ (coeff n) ↑((trunc n.succ) (f * g)) = (coeff n) (f * g)
Module: Mathlib.Topology.Algebra.LinearTopology

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
⊢ IsLinearTopology Rᵐᵒᵖ R

Tactic:
IsCentralScalar.isLinearTopology_iff

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
⊢ IsLinearTopology R R
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
⊢ cantorFunction c f < cantorFunction c g

Tactic:
have hf_max : ∀ n, f n → f_max n := by
      intro n hn
      cases n
      · rw [fn] at hn
        contradiction
      simp [f_max]

Goal state after:
case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
⊢ cantorFunction c f < cantorFunction c g
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0

Tactic:
by_cases ha : ∃ i, q ^ i = a

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ∃ i, q ^ i = a
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0

case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ¬∃ i, q ^ i = a
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0

Tactic:
by_cases ha : ∃ i, q ^ i = a

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ i, q ^ i = a
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0

case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ¬∃ i, q ^ i = a
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (a, b).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (a, b).2 =
    0
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: τ : Type u_3
S : Type u_4
inst✝ : CommRing S
t : τ
⊢ HasSubst (MvPowerSeries.X t)

Tactic:
by
  simp [HasSubst]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NormedDivisionRing 𝕜
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
U : Set E
V : Set F
hU : Balanced 𝕜 U
hV : Balanced 𝕜 V
a : E
b : F
r : ℝ≥0∞
x : 𝕜
hx : a ∈ x • U
hxr : ‖x‖ₑ < r
y : 𝕜
hy : b ∈ y • V
hyr : ‖y‖ₑ < r
x✝ : ‖x‖ ≤ ‖y‖ ∨ ‖y‖ ≤ ‖x‖
⊢ ∃ c, (a, b) ∈ (c • U) ×ˢ (c • V) ∧ ‖c‖ₑ < r

Tactic:
cases le_total ‖x‖ ‖y‖ with

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

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

Tactic:
suffices ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k by
    rw [mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun k hk ↦ ?_
    rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]

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

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

Tactic:
this (u \ t) hu k hk,

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

Goal state before:
pp: case pos
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), (f i).IsMultiplicative
hf' : Multipliable f
⊢ (eulerProduct f).IsMultiplicative

Tactic:
have h (s : Finset ι) : (∏ b ∈ s, f b).IsMultiplicative :=
      isMultiplicative_finsetProd f s fun i a ↦ hf i
    have key := tendsto_iff.mp hf'.hasProd
    refine (forall_and.mp h).imp (fun h ↦ ?_) fun h m n hmn ↦ ?_
    · specialize key 1
      simp_rw [h] at key
      rwa [eventually_const, eq_comm] at key
    · replace h s : (∏ b ∈ s, f b) (m * n) = (∏ b ∈ s, f b) m * (∏ b ∈ s, f b) n := h s hmn
      have h2 := key (m * n)
      simp_rw [h] at h2
      exact eventually_const.mp (EventuallyEq.trans (.symm h2) (.mul (key m) (key n)))

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)
this : Invertible ((coeff 1) P.substInv)
Q : R⟦X⟧ := P.substInv.substInv
hQ : HasSubst Q
eq_aux : subst Q P.substInv = X
⊢ Q = subst Q (subst P.substInv P)

Tactic:
rw [subst_substInv_right _ hP, subst_X hQ]

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

Goal state before:
pp: R : Type u_2
inst✝⁶ : CommRing R
τ : Type u_3
S : Type u_4
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : ℕ → R⟦X⟧
a : MvPowerSeries τ S
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ subst a (∑' (i : ℕ), x i) = ∑' (i : ℕ), subst a (x i)

Tactic:
rw [← coe_substAlgHom ha, substAlgHom_eq_aeval ha, hx.map_tsum _]

Goal state after:
R : Type u_2
inst✝⁶ : CommRing R
τ : Type u_3
S : Type u_4
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : ℕ → R⟦X⟧
a : MvPowerSeries τ S
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ Continuous ⇑(aeval ⋯)
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
x : E
s : Set E
c : F
⊢ fderivWithin 𝕜 (fun x ↦ c) s x = 0

Tactic:
by
  rw [fderivWithin, if_pos]
  apply hasFDerivWithinAt_const

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : Module 𝕜 F
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
f₁ f₂ f₃ : α → E
g : α → F
l : Filter α
h₁ : (f₁ - f₂) =o[𝕜; l] g
h₂ : (f₂ - f₃) =o[𝕜; l] g
⊢ (f₁ - f₃) =o[𝕜; l] g

Tactic:
by
  simpa using h₁.add h₂

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : Module 𝕜 F
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul 𝕜 E
f₁ f₂ f₃ : α → E
g : α → F
l : Filter α
h₁ : (f₁ - f₂) =O[𝕜; l] g
h₂ : (f₂ - f₃) =O[𝕜; l] g
⊢ (f₁ - f₃) =O[𝕜; l] g

Tactic:
by
  simpa using h₁.add h₂

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 : 𝕜 → F
s : Set F
x : 𝕜
⊢ deriv f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s

Tactic:
by
  by_cases hx : DifferentiableAt 𝕜 f x <;> simp [deriv_zero_of_not_differentiableAt, *]

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
σ σ' : 𝕜 →+* 𝕜
inst✝³ : RingHomIsometric σ
inst✝² : RingHomInvPair σ σ'
F' : Type u_1
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜 F'
L : F →SL[σ] F'
hf : HasDerivAt f f' x
this : RingHomIsometric σ'
R : 𝕜 →SL[σ'] 𝕜 := { toLinearMap := σ'.toSemilinearMap, cont := ⋯ }
⊢ HasDerivAt (⇑L ∘ f ∘ ⇑σ') (L f') (σ x)

Tactic:
have hR (k : 𝕜) : R k = σ' k := rfl

Goal state after:
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
σ σ' : 𝕜 →+* 𝕜
inst✝³ : RingHomIsometric σ
inst✝² : RingHomInvPair σ σ'
F' : Type u_1
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜 F'
L : F →SL[σ] F'
hf : HasDerivAt f f' x
this : RingHomIsometric σ'
R : 𝕜 →SL[σ'] 𝕜 := { toLinearMap := σ'.toSemilinearMap, cont := ⋯ }
hR : ∀ (k : 𝕜), R k = σ' k
⊢ HasDerivAt (⇑L ∘ f ∘ ⇑σ') (L f') (σ 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
x : E
h : ¬DifferentiableAt 𝕜 f x
⊢ fderiv 𝕜 f x = 0

Tactic:
rw [fderiv, fderivWithin_zero_of_not_differentiableWithinAt]
  rwa [differentiableWithinAt_univ]

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✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Nontrivial E
p : FormalMultilinearSeries 𝕜 F G
u : E ≃ₗᵢ[𝕜] F
⊢ (p.compContinuousLinearMap u.toLinearIsometry.toContinuousLinearMap).radius = p.radius

Tactic:
refine le_antisymm ?_ <| le_radius_compContinuousLinearMap _ _
  have _ : Nontrivial F := u.symm.toEquiv.nontrivial
  convert! radius_compContinuousLinearMap_le p u.toContinuousLinearEquiv
  have : u.toContinuousLinearEquiv.symm.toContinuousLinearMap =
    u.symm.toLinearIsometry.toContinuousLinearMap := rfl
  simp [this]

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

Goal state before:
pp: case base
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ⋯
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := ⋯
⊢ S 1 ≤ (I + 1) * a

Tactic:
simp only [S]

Goal state after:
case base
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
⊢ ∑ k ∈ Ico 1 1, a ^ k * ‖p.rightInv i x k‖ ≤ (I + 1) * a
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
x : E
h : {x} ∈ 𝓝[s] x
t : Set E
ot : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
xt : x ∈ t
st : t ∩ s ⊆ {x}
r : ℝ
r0 : r > 0
rt : ball x r ⊆ t
⊢ ∀ {y : E},
    x + y ∈ insert x s →
      y ∈ eball 0 (ENNReal.ofReal r) → HasSum (fun n ↦ (constFormalMultilinearSeries 𝕜 E (f x) n) fun x ↦ y) (f (x + y))

Tactic:
intro y ys yr
      simp only [subset_singleton_iff, mem_inter_iff, and_imp] at st
      simp only [mem_insert_iff, add_eq_left] at ys
      have : x + y = x := by
        rcases ys with rfl | ys
        · simp
        · exact st (x + y) (rt (by simpa using yr)) ys
      simp only [this]
      apply (hasFPowerSeriesOnBall_const (e := 0)).hasSum
      simp only [eball_top, mem_univ]

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

Goal state before:
pp: n : ℕ
a : Composition n
b : Composition a.length
i : Fin b.length
j : Fin (b.blocksFun i)
⊢ { blocks := (a.blocks.splitWrtComposition b)[↑⟨↑i, ⋯⟩], blocks_pos := ⋯, blocks_sum := ⋯ }.blocks[↑⟨↑j, ⋯⟩] =
    a.blocksFun ((b.embedding i) j)

Tactic:
getElem_of_eq (getElem_splitWrtComposition _ _ _ _),

Goal state after:
n : ℕ
a : Composition n
b : Composition a.length
i : Fin b.length
j : Fin (b.blocksFun i)
⊢ (drop (b.sizeUpTo ↑⟨↑i, ⋯⟩) (take (b.sizeUpTo (↑⟨↑i, ⋯⟩ + 1)) a.blocks))[↑⟨↑j, ⋯⟩] = a.blocksFun ((b.embedding i) j)
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  j < b.blocksFun ⟨i, hi⟩ →
    a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < (a.sigmaCompositionAux b ⟨i, ⋯⟩).length
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ (a.gather b).sizeUpTo i +
      ({ blocks := (a.blocks.splitWrtComposition b)[i], blocks_pos := ⋯, blocks_sum := ⋯ }.sizeUpTo j +
        a.blocks[b.sizeUpTo i + j]) =
    (a.gather b).sizeUpTo i +
      ({ blocks := (a.blocks.splitWrtComposition b)[i], blocks_pos := ⋯, blocks_sum := ⋯ }.sizeUpTo j +
        (a.blocks.splitWrtComposition b)[i][j])

Tactic:
getElem_of_eq (getElem_splitWrtComposition _ _ _ _),

Goal state after:
case succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
  j < b.blocksFun ⟨i, hi⟩ →
    a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < (a.sigmaCompositionAux b ⟨i, ⋯⟩).length
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ (a.gather b).sizeUpTo i +
      ({ blocks := (a.blocks.splitWrtComposition b)[i], blocks_pos := ⋯, blocks_sum := ⋯ }.sizeUpTo j +
        a.blocks[b.sizeUpTo i + j]) =
    (a.gather b).sizeUpTo i +
      ({ blocks := (a.blocks.splitWrtComposition b)[i], blocks_pos := ⋯, blocks_sum := ⋯ }.sizeUpTo j +
        (drop (b.sizeUpTo i) (take (b.sizeUpTo (i + 1)) a.blocks))[j])
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: case hg
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 := ⋯
f2 : A → A := ⋯
f3 : A → A := ⋯
feq : ∀ᶠ (y : A) in 𝓝 ↑z, (f1 ∘ f2 ∘ f3) y = y⁻¹ʳ
⊢ AnalyticAt 𝕜 f2 (f3 ↑z)

Tactic:
simp only [Units.inv_eq_val_inv, Units.inv_mul, sub_self, f2, f3]
      exact analyticAt_inverse_one_sub 𝕜 A

Goal state after:
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
⊢ AnalyticAt 𝕜 Ring.inverse ↑z

Tactic:
rcases subsingleton_or_nontrivial A with hA | hA
  · convert! analyticAt_const (v := (0 : A))
  · let f1 : A → A := fun a ↦ a * z.inv
    let f2 : A → A := fun b ↦ (1 - b)⁻¹ʳ
    let f3 : A → A := fun c ↦ 1 - z.inv * c
    have feq : ∀ᶠ y in 𝓝 (z : A), (f1 ∘ f2 ∘ f3) y = y⁻¹ʳ := by
      have : Metric.ball (z : A) (‖(↑z⁻¹ : A)‖⁻¹) ∈ 𝓝 (z : A) := by
        apply Metric.ball_mem_nhds
        simp
      filter_upwards [this] with y hy
      simp only [Metric.mem_ball, dist_eq_norm] at hy
      have : y = Units.ofNearby z y hy := rfl
      rw [this, Eq.comm]
      simp only [Ring.inverse_unit, Function.comp_apply]
      simp only [Units.ofNearby, Units.add, mul_sub, Units.inv_mul, neg_sub, add_sub_cancel,
        mul_inv_rev, Units.val_mul, Units.val_inv_copy, Units.inv_eq_val_inv, Units.val_copy,
        _root_.sub_sub_cancel, Units.mul_left_inj, f1, f2, f3]
      rw [← Ring.inverse_unit]
      congr
      simp
    apply AnalyticAt.congr _ feq
    apply (analyticAt_id.mul analyticAt_const).comp
    apply AnalyticAt.comp
    · simp only [Units.inv_eq_val_inv, Units.inv_mul, sub_self, f2, f3]
      exact analyticAt_inverse_one_sub 𝕜 A
    · exact analyticAt_const.sub (analyticAt_const.mul analyticAt_id)

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕝 : Type u_8
inst✝⁴ : NormedDivisionRing 𝕝
inst✝³ : NormedAlgebra 𝕜 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : IsBoundedSMul 𝕝 F
inst✝ : IsScalarTower 𝕜 𝕝 F
f : E → 𝕝
g : E → F
z : E
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
⊢ AnalyticAt 𝕜 g z ↔ AnalyticAt 𝕜 (fun z ↦ f z • g z) z

Tactic:
by
  constructor
  · exact fun a ↦ h₁f.smul a
  · intro hprod
    rw [analyticAt_congr (g := (f⁻¹ • f) • g), smul_assoc]
    · exact (h₁f.inv h₂f).fun_smul hprod
    · filter_upwards [h₁f.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.2 h₂f)]
      intro y hy
      rw [Set.preimage_compl, Set.mem_compl_iff, Set.mem_preimage, Set.mem_singleton_iff] at hy
      simp [hy]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
L : Filter (E × E)
c : F
⊢ HasFDerivAtFilter (fun x ↦ c + f x) f' L ↔ HasFDerivAtFilter f f' L

Tactic:
simpa only [add_comm] using hasFDerivAtFilter_add_const_iff c

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
h : DifferentiableWithinAt 𝕜 (-f) s x
⊢ DifferentiableWithinAt 𝕜 f s x

Tactic:
simpa only [neg_neg] using h.neg

Goal state after: