mathlib-initiative/mathlib-tactics huggingface dataset:
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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource 0 M N
i✝ : ℕ
him✝ : i✝ < ⟨k, blocks_fun⟩.fst
hin✝ : i✝ < (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.length
⊢ ((p ((compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.blocksFun ⟨↑⟨i✝, him✝⟩, ⋯⟩)) fun _j ↦ z) =
    p.applyComposition (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd (fun _j ↦ z) ⟨i✝, hin✝⟩

Tactic:
applyComposition,

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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource 0 M N
i✝ : ℕ
him✝ : i✝ < ⟨k, blocks_fun⟩.fst
hin✝ : i✝ < (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.length
⊢ ((p ((compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.blocksFun ⟨↑⟨i✝, him✝⟩, ⋯⟩)) fun _j ↦ z) =
    (p ((compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.blocksFun ⟨i✝, hin✝⟩))
      ((fun _j ↦ z) ∘ ⇑((compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.embedding ⟨i✝, hin✝⟩))
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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource 0 M N
i✝ : ℕ
him✝ : i✝ < ⟨k, blocks_fun⟩.fst
hin✝ : i✝ < (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.length
⊢ ((p (⟨k, blocks_fun⟩.snd ⟨i✝, him✝⟩)) fun _j ↦ z) =
    p.applyComposition (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd (fun _j ↦ z) ⟨i✝, hin✝⟩

Tactic:
rw [← compChangeOfVariables_blocksFun 0 M N H, applyComposition, Function.comp_def]

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
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n : ℕ
a : Composition n
b : Composition a.length
j : List ℕ
hj : j ∈ a.blocks.splitWrtComposition b
i : ℕ
hi : i ∈ j
⊢ i ∈ a.blocks

Tactic:
rw [← a.blocks.flatten_splitWrtComposition b]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n : ℕ
a : Composition n
b : Composition a.length
j : List ℕ
hj : j ∈ a.blocks.splitWrtComposition b
i : ℕ
hi : i ∈ j
⊢ i ∈ (a.blocks.splitWrtComposition b).flatten
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: n : ℕ
a : Composition n
b : Composition a.length
⊢ (List.map sum (a.blocks.splitWrtComposition b)).length = b.blocks.length

Tactic:
length_map,

Goal state after:
n : ℕ
a : Composition n
b : Composition a.length
⊢ (a.blocks.splitWrtComposition b).length = b.blocks.length
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
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n : ℕ
a : Composition n
b : Composition a.length
i✝ : Fin (a.gather b).length
i : ℕ
hi : i ∈ (a.blocks.splitWrtComposition b)[↑i✝]
⊢ i ∈ a.blocks

Tactic:
rw [← a.blocks.flatten_splitWrtComposition b]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n : ℕ
a : Composition n
b : Composition a.length
i✝ : Fin (a.gather b).length
i : ℕ
hi : i ∈ (a.blocks.splitWrtComposition b)[↑i✝]
⊢ i ∈ (a.blocks.splitWrtComposition b).flatten
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
r : ℝ≥0∞
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
h : ∀ (i : ι), r ≤ (p i).radius
r' : ℝ≥0
hr' : ↑r' < r
⊢ ↑r' ≤ (pi p).radius

Tactic:
have I i : ∃ C > 0, ∀ n, ‖p i n‖ * (r' : ℝ) ^ n ≤ C :=
    norm_mul_pow_le_of_lt_radius _ (hr'.trans_le (h i))

Goal state after:
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
r : ℝ≥0∞
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
h : ∀ (i : ι), r ≤ (p i).radius
r' : ℝ≥0
hr' : ↑r' < r
I : ∀ (i : ι), ∃ C > 0, ∀ (n : ℕ), ‖p i n‖ * ↑r' ^ n ≤ C
⊢ ↑r' ≤ (pi p).radius
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 := fun a ↦ a * z.inv
f2 : A → A := fun b ↦ (1 - b)⁻¹ʳ
f3 : A → A := fun c ↦ 1 - z.inv * c
feq : ∀ᶠ (y : A) in 𝓝 ↑z, (f1 ∘ f2 ∘ f3) y = y⁻¹ʳ
⊢ AnalyticAt 𝕜 (fun b ↦ (1 - b)⁻¹ʳ) 0

Tactic:
exact analyticAt_inverse_one_sub 𝕜 A

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
x : E
ι : Type u_4
u : Finset ι
A : ι → E → F
A' : ι → E →L[𝕜] F
h : ∀ i ∈ u, HasStrictFDerivAt (A i) (A' i) x
⊢ HasStrictFDerivAt (∑ i ∈ u, A i) (∑ i ∈ u, A' i) x

Tactic:
convert! HasStrictFDerivAt.fun_sum h; simp

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

Goal state before:
pp: case neg
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
hxs : UniqueDiffWithinAt 𝕜 s x
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ 0 = -0

pp: case neg
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
hxs : UniqueDiffWithinAt 𝕜 s x
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ ¬DifferentiableWithinAt 𝕜 (fun y ↦ -f y) s x

Tactic:
neg_zero

Goal state after:
case neg
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
hxs : UniqueDiffWithinAt 𝕜 s x
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ 0 = 0

case neg
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
hxs : UniqueDiffWithinAt 𝕜 s x
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ ¬DifferentiableWithinAt 𝕜 (fun y ↦ -f y) s x
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
f' : E →L[𝕜] G
⊢ HasFDerivWithinAt (f ∘ ⇑iso) f' (⇑iso ⁻¹' s) x ↔
    HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ↑iso.symm ∘SL ↑iso) (⇑iso ⁻¹' s) x

Tactic:
iso.coe_symm_comp_coe,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
f' : E →L[𝕜] G
⊢ HasFDerivWithinAt (f ∘ ⇑iso) f' (⇑iso ⁻¹' s) x ↔
    HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ContinuousLinearMap.id 𝕜 E) (⇑iso ⁻¹' s) x
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] F)

Tactic:
rw [(isUniformInducing_postcomp (SeparationQuotient.mkCLM _ _)
      SeparationQuotient.isUniformInducing_mk).completeSpace_congr]

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] SeparationQuotient F)

case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
⊢ Surjective (SeparationQuotient.mkCLM 𝕜 F).compContinuousAlternatingMap
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_1
α : Type u
inst✝ : PseudoMetricSpace α
x : Completion α
⊢ IsClosed {a | dist a a = 0}

Tactic:
refine isClosed_eq ?_ continuous_const
    exact Completion.continuous_dist continuous_id continuous_id

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Completion

Goal state before:
pp: case ih
E : Type u_1
inst✝ : SeminormedAddCommGroup E
a✝ b✝ : E
⊢ dist ↑a✝ ↑b✝ = ‖-↑a✝ + ↑b✝‖

Tactic:
· rw [← Completion.coe_neg, ← Completion.coe_add, norm_coe, Completion.dist_eq,
        dist_eq_norm_neg_add]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
H : ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).curry0 = f x
m : ℕ
hm : ↑m ≤ 0
⊢ ContinuousOn (fun x ↦ p x m) s

Tactic:
obtain rfl : m = 0 := mod_cast hm.antisymm zero_le

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
H : ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).curry0 = f x
hm : ↑0 ≤ 0
⊢ ContinuousOn (fun x ↦ p x 0) s
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case mpr.cont.succ
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
Hzero_eq : ∀ x ∈ s, (p x 0).curry0 = f x
Hfderiv_zero : ∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft s x
Htaylor : HasFTaylorSeriesUpToOn (↑n) (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) s
m : ℕ
hm : ↑(m + 1) ≤ ↑n.succ
⊢ ContinuousOn (fun x ↦ p x (m + 1)) s

Tactic:
refine (continuousMultilinearCurryRightEquiv' 𝕜 m E F).comp_continuousOn_iff.mp ?_
        refine Htaylor.cont _ ?_
        rw [Nat.cast_le] at hm ⊢
        exact Nat.lt_succ_iff.mp hm

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn n f p univ
⊢ ∀ (m : ℕ), ↑m ≤ n → Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ p x m

Tactic:
by simpa using H.cont

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpTo n f p
⊢ ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x ↦ p x m) univ

Tactic:
by simpa using H.cont

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
⊢ HasFTaylorSeriesUpTo 0 f p ↔
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
        f ∧
      ∀ (x : E), (p x 0).curry0 = f x

Tactic:
by
  simp [hasFTaylorSeriesUpToOn_univ_iff.symm, continuousOn_univ,
    hasFTaylorSeriesUpToOn_zero_iff]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : AnalyticOn 𝕜 f s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
n : ℕ
⊢ AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s

Tactic:
exact (h.iteratedFDerivWithin hs.uniqueDiffOn n).congr
    (fun x hx ↦ (iteratedFDerivWithin_of_isOpen n hs hx).symm)

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

Goal state before:
pp: case inl.refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x 0 r
y : E
hy : y ∈ Metric.eball x r
⊢ f =ᶠ[𝓝 y] fun x ↦ 0

Tactic:
exact Filter.eventuallyEq_iff_exists_mem.mpr ⟨Metric.eball x r,
          Metric.isOpen_eball.mem_nhds hy, fun z hz ↦ by rw [h.eq_zero_of_bound_zero z hz]⟩

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c

Tactic:
by
  by_cases hab : a = b
  · subst hab
    rw [sub_self, zero_div, zero_smul, zero_add]
    by_cases hac : a = c
    · simp [hac]
    · rw [div_self (sub_ne_zero.2 <| Ne.symm hac), one_smul]
  by_cases hbc : b = c
  · subst hbc
    simp [sub_ne_zero.2 (Ne.symm hab)]
  rw [add_comm]
  simp_rw [slope, div_eq_inv_mul, mul_smul, ← smul_add,
    smul_inv_smul₀ (sub_ne_zero.2 <| Ne.symm hab), smul_inv_smul₀ (sub_ne_zero.2 <| Ne.symm hbc),
    vsub_add_vsub_cancel]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
hab : a ≠ b
⊢ r * (a - b) = a - ((1 - r) * a + r * b)

Tactic:
sub_mul,

Goal state after:
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
hab : a ≠ b
⊢ r * (a - b) = a - (1 * a - r * a + r * b)
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x v : (i : ι) → E i
m : Fin n → (i : ι) → E i
e : Fin n.succ ↪ ι
k : ι
hke : k ∈ range ⇑e
⊢ (if h : k ∈ range ⇑e then Fin.cons v m (e.toEquivRange.symm ⟨k, ⋯⟩) k else x k) =
    if h : k ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst then
      m (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h⟩) k
    else
      Function.update ((Pi.compRightL 𝕜 E Subtype.val) x) ((Equiv.embeddingFinSucc n ι) e).snd
        (v ↑((Equiv.embeddingFinSucc n ι) e).snd) ⟨k, h⟩

Tactic:
simp only [hke, ↓reduceDIte]

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x v : (i : ι) → E i
m : Fin n → (i : ι) → E i
e : Fin n.succ ↪ ι
k : ι
hke : k ∈ range ⇑e
⊢ Fin.cons v m (e.toEquivRange.symm ⟨k, ⋯⟩) k =
    if h : k ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst then
      m (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h⟩) k
    else
      Function.update ((Pi.compRightL 𝕜 E Subtype.val) x) ((Equiv.embeddingFinSucc n ι) e).snd
        (v ↑((Equiv.embeddingFinSucc n ι) e).snd) ⟨k, h⟩
Module: Mathlib.Analysis.Analytic.Uniqueness

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 g : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ U
hfg : f =ᶠ[𝓝 z₀] g
hfg' : f - g =ᶠ[𝓝 z₀] 0
⊢ EqOn f g U

Tactic:
simpa [sub_eq_zero] using! fun z hz =>
    (hf.sub hg).eqOn_zero_of_preconnected_of_eventuallyEq_zero hU h₀ hfg' hz

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: r θ : ℝ
⊢ (starRingEnd ℂ) (circleMap 0 r θ) = circleMap 0 r (-θ)

Tactic:
simp [circleMap_zero, ← exp_conj]

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
U : Set 𝕜
A : Type u_3
inst✝⁶ : NormedRing A
inst✝⁵ : IsDomain A
inst✝⁴ : NormedAlgebra 𝕜 A
B : Type u_4
inst✝³ : NormedAddCommGroup B
inst✝² : NormedSpace 𝕜 B
inst✝¹ : Module A B
inst✝ : IsTorsionFree A B
f : 𝕜 → A
g : 𝕜 → B
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hfg : ∀ z ∈ U, f z • g z = 0
hU : IsPreconnected U
hU' : ¬U = ∅
z : 𝕜
hz : z ∈ U
hU'' : U.Nontrivial
this✝ : ∃ᶠ (w : 𝕜) in 𝓝[≠] z, w ∈ U
this : ∃ᶠ (w : 𝕜) in 𝓝[≠] z, f w = 0 ∨ g w = 0
x✝ : (∃ᶠ (x : 𝕜) in 𝓝[≠] z, f x = 0) ∨ ∃ᶠ (x : 𝕜) in 𝓝[≠] z, g x = 0
⊢ (∀ z ∈ U, f z = 0) ∨ ∀ z ∈ U, g z = 0

Tactic:
cases frequently_or_distrib.mp this with

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
⊢ ↑n * (1 / x) = ↑n / x

Tactic:
mul_one_div

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
⊢ ↑n / x = ↑n / x
Module: Mathlib.Analysis.Calculus.LogDeriv

Goal state before:
pp: case inr.mpr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
s : Set 𝕜
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hsc : IsPreconnected s
hgn : ∀ x ∈ s, g x ≠ 0
hfn : ∀ x ∈ s, f x ≠ 0
t : 𝕜
ht : t ∈ s
⊢ (∃ z, z ≠ 0 ∧ EqOn f (z • g) s) → EqOn (logDeriv f) (logDeriv g) s

Tactic:
rintro ⟨z, hz0, hz⟩ x hx

Goal state after:
case inr.mpr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
s : Set 𝕜
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hsc : IsPreconnected s
hgn : ∀ x ∈ s, g x ≠ 0
hfn : ∀ x ∈ s, f x ≠ 0
t : 𝕜
ht : t ∈ s
z : 𝕜'
hz0 : z ≠ 0
hz : EqOn f (z • g) s
x : 𝕜
hx : x ∈ s
⊢ logDeriv f x = logDeriv g x
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

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
s : Set E
x : E
n : ℕ∞ω
f : E → F
g : F →ₗᵢ[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
⊢ ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖

Tactic:
have :
    iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
      g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) :=
    g.toContinuousLinearMap.iteratedFDerivWithin_comp_left hf hs hx hi

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
s : Set E
x : E
n : ℕ∞ω
f : E → F
g : F →ₗᵢ[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
this :
  iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x =
    g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x)
⊢ ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖