mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
s' : Set 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
y : 𝕜'
hh₂ : DifferentiableWithinAt 𝕜' h₂ s' y
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s s'
hy : h x = y
⊢ derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x

Tactic:
subst hy; exact derivWithin_comp x hh₂ hh hs

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
m n : ℤ
hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z
h_le : n ≤ m
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g z₀
left✝ : g z₀ ≠ 0
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
j : 𝕜 → E
hj_an : AnalyticAt 𝕜 j z₀
hj_eq : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • j z
hj_ne : m ≠ n
this : ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, j z = (z - z₀) ^ (m - n) • g z
⊢ AnalyticAt 𝕜 (fun z ↦ (z - z₀) ^ (m - n).toNat • g z) z₀

Tactic:
exact ((analyticAt_id.sub analyticAt_const).pow _).smul hg_an

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

Goal state before:
pp: case cons.hdg
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_3
f : ι → 𝕜 → 𝕜'
x : 𝕜
a : ι
s : Finset ι
ha : a ∉ s
ih :
  (∀ i ∈ s, f i x ≠ 0) →
    (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → logDeriv (fun x ↦ ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x
hf : f a x ≠ 0 ∧ ∀ x_1 ∈ s, f x_1 x ≠ 0
hd : DifferentiableAt 𝕜 (f a) x ∧ ∀ x_1 ∈ s, DifferentiableAt 𝕜 (f x_1) x
⊢ DifferentiableAt 𝕜 (fun x ↦ ∏ i ∈ s, f i x) x

Tactic:
· exact .fun_finsetProd hd.2

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
f : 𝕜 → 𝕜
x : 𝕜
hf : AnalyticAt 𝕜 f x
hfx : f x = 0
hf' : deriv f x ≠ 0
h_slope : Tendsto (slope f x) (𝓝[≠] x) (𝓝 (deriv f x))
x✝ : 𝕜
⊢ (x✝ - x) * (deriv f x✝ / f x✝) = deriv f x✝ / ((x✝ - x)⁻¹ * f x✝)

Tactic:
field

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

Goal state before:
pp: case mpr
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
f : E → F
x : E
n : ℕ∞ω
e : G ≃L[𝕜] E
H : ContDiffWithinAt 𝕜 n f s x
⊢ ContDiffWithinAt 𝕜 n (f ∘ ⇑e) (⇑e ⁻¹' s) (e.symm x)

Tactic:
← e.apply_symm_apply x,

Goal state after:
case mpr
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
f : E → F
x : E
n : ℕ∞ω
e : G ≃L[𝕜] E
H : ContDiffWithinAt 𝕜 n f s (e (e.symm x))
⊢ ContDiffWithinAt 𝕜 n (f ∘ ⇑e) (⇑e ⁻¹' s) (e.symm x)
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

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
x : E
n✝ : ℕ∞ω
t : Set E
hst : s ∈ 𝓝[t] x
n : ℕ∞
h : ContDiffWithinAt 𝕜 (↑n) f s x
m : ℕ
hm : ↑m ≤ n
⊢ ∃ u ∈ 𝓝[insert x t] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u

Tactic:
rcases h m hm with ⟨u, hu, p, H⟩

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
x : E
n✝ : ℕ∞ω
t : Set E
hst : s ∈ 𝓝[t] x
n : ℕ∞
h : ContDiffWithinAt 𝕜 (↑n) f s x
m : ℕ
hm : ↑m ≤ n
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn (↑m) f p u
⊢ ∃ u ∈ 𝓝[insert x t] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u
Module: Mathlib.Analysis.Calculus.ContDiff.Deriv

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : UniqueDiffOn 𝕜 s
⊢ ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (derivWithin f s) s

Tactic:
show ∞ = ∞ + 1 by rfl,

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : UniqueDiffOn 𝕜 s
⊢ ContDiffOn 𝕜 (∞ + 1) f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (∞ + 1) (derivWithin f s) s
Module: Mathlib.Analysis.Calculus.ContDiff.Deriv

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen s
⊢ ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (deriv f) s

Tactic:
show ∞ = ∞ + 1 by rfl,

Goal state after:
𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen s
⊢ ContDiffOn 𝕜 (∞ + 1) f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (∞ + 1) (deriv f) s
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
m : Fin c.length
⊢ c.partSize m ≤ n

Tactic:
by
  simpa only [Fintype.card_fin] using Fintype.card_le_of_injective _ (c.emb_strictMono m).injective

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

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
j : Fin n
⊢ ∃ p, c.emb p.fst p.snd = j

Tactic:
by
  rcases c.cover j with ⟨m, r, hmr⟩
  exact ⟨⟨m, r⟩, hmr⟩

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

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
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this : ↑(m + 1) ≤ n
u : Set E
hu : u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑(m + 1)) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ p x m.succ = ftaylorSeriesWithin 𝕜 f s x m.succ

Tactic:
change p x m.succ = iteratedFDerivWithin 𝕜 m.succ f s x

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
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this : ↑(m + 1) ≤ n
u : Set E
hu : u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑(m + 1)) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ p x m.succ = iteratedFDerivWithin 𝕜 m.succ f s x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case cont
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
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m ≤ n
x : E
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑m) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ ∃ t,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧
      x ∈ t ∧ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ t)

Tactic:
refine ⟨o, o_open, xo, ?_⟩

Goal state after:
case cont
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
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m ≤ n
x : E
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑m) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ o)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) = {0}
i : Fin (c.length - 1)
x✝¹ : i ∈ univ
j : Fin (c.length - 1)
x✝ : j ∈ univ
hij : i ≠ j
⊢ ∀ ⦃a : Fin n⦄,
    a ∈
        (fun m ↦
            range fun j ↦
              have this := ⋯;
              (c.emb (Fin.cast this m.succ) j).pred ⋯)
          i →
      ∀ ⦃b : Fin n⦄,
        b ∈
            (fun m ↦
                range fun j ↦
                  have this := ⋯;
                  (c.emb (Fin.cast this m.succ) j).pred ⋯)
              j →
          a ≠ b

Tactic:
simp only [mem_range, ne_eq, forall_exists_index, forall_apply_eq_imp_iff, pred_inj]

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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) = {0}
i : Fin (c.length - 1)
x✝¹ : i ∈ univ
j : Fin (c.length - 1)
x✝ : j ∈ univ
hij : i ≠ j
⊢ ∀ (a : Fin (c.partSize (Fin.cast ⋯ i.succ))) (a_1 : Fin (c.partSize (Fin.cast ⋯ j.succ))),
    ¬c.emb (Fin.cast ⋯ i.succ) a = c.emb (Fin.cast ⋯ j.succ) a_1
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

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
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ ContDiffOn 𝕜 (∞ + 1) f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (∞ + 1) (fderiv 𝕜 f) s

Tactic:
contDiffOn_succ_iff_fderiv_of_isOpen hs

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
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ DifferentiableOn 𝕜 f s ∧ (∞ = ω → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 ∞ (fderiv 𝕜 f) s ↔
    DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (∞ + 1) (fderiv 𝕜 f) s
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

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
x : E
n : ℕ∞ω
⊢ ContDiffWithinAt 𝕜 n f {x}ᶜ x ↔ ContDiffAt 𝕜 n f x

Tactic:
rw [compl_eq_univ_sdiff, contDiffWithinAt_sdiff_singleton, contDiffWithinAt_univ]

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

Goal state before:
pp: case inl
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
h : i ≠ c.index 0
x✝ : c.index 0 ∈ univ
hij : i ≠ c.index 0
⊢ ∀ ⦃a : Fin n⦄,
    (a ∈ range fun j ↦ (c.emb i (Fin.cast ⋯ j)).pred ⋯) →
      ∀ ⦃b : Fin n⦄, (b ∈ range fun j ↦ (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯) → a ≠ b

Tactic:
simp only [mem_range, ne_eq, forall_exists_index, forall_apply_eq_imp_iff, pred_inj]

Goal state after:
case inl
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
h : i ≠ c.index 0
x✝ : c.index 0 ∈ univ
hij : i ≠ c.index 0
⊢ ∀ (a : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i))
    (a_1 : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0))),
    ¬c.emb i (Fin.cast ⋯ a) = c.emb (c.index 0) (Fin.cast ⋯ a_1.succ)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case inr
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
j : Fin c.length
x✝ : j ∈ univ
hij : i ≠ j
h : i ≠ c.index 0
hj : j ≠ c.index 0
⊢ ∀ ⦃a : Fin n⦄,
    (a ∈ range fun j ↦ (c.emb i (Fin.cast ⋯ j)).pred ⋯) →
      ∀ ⦃b : Fin n⦄, (b ∈ range fun j_1 ↦ (c.emb j (Fin.cast ⋯ j_1)).pred ⋯) → a ≠ b

Tactic:
simp only [mem_range, ne_eq, forall_exists_index, forall_apply_eq_imp_iff, pred_inj]

Goal state after:
case inr
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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
i : Fin c.length
x✝¹ : i ∈ univ
j : Fin c.length
x✝ : j ∈ univ
hij : i ≠ j
h : i ≠ c.index 0
hj : j ≠ c.index 0
⊢ ∀ (a : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i))
    (a_1 : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) j)),
    ¬c.emb i (Fin.cast ⋯ a) = c.emb j (Fin.cast ⋯ a_1)
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case h
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
H : ContDiffOn 𝕜 ω f univ
i : ℕ
⊢ AnalyticOnNhd 𝕜 (fun x ↦ ftaylorSeriesWithin 𝕜 f univ x i) univ

Tactic:
rw [← analyticOn_univ]

Goal state after:
case h
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
H : ContDiffOn 𝕜 ω f univ
i : ℕ
⊢ AnalyticOn 𝕜 (fun x ↦ ftaylorSeriesWithin 𝕜 f univ x i) univ
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ AnalyticAt 𝕂 exp x

Tactic:
by
  by_cases h : (expSeries 𝕂 𝔸).radius = 0
  · rw [h] at hx; exact (ENNReal.not_lt_zero hx).elim
  · have h := pos_iff_ne_zero.mpr h
    exact (hasFPowerSeriesOnBall_exp_of_radius_pos h).analyticAt_of_mem hx

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case f
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x y : 𝔸
hxy : Commute x y
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
x✝ : ℕ
| (↑x✝!)⁻¹ • (x + y) ^ x✝

pp: case L
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x y : 𝔸
hxy : Commute x y
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
| SummationFilter.unconditional ℕ

Tactic:
rw [hxy.add_pow' _, Finset.smul_sum]

Goal state after:
case f
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x y : 𝔸
hxy : Commute x y
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
x✝ : ℕ
| ∑ x_1 ∈ Finset.antidiagonal x✝, (↑x✝!)⁻¹ • x✝.choose x_1.1 • (x ^ x_1.1 * y ^ x_1.2)

case L
𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x y : 𝔸
hxy : Commute x y
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
| SummationFilter.unconditional ℕ
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: case succ
𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
ih : exp (n • x) = exp x ^ n
⊢ exp (n • x + x) = exp x ^ n * exp x

Tactic:
exp_add_of_commute ((Commute.refl x).smul_left n),

Goal state after:
case succ
𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
n : ℕ
ih : exp (n • x) = exp x ^ n
⊢ exp (n • x) * exp x = exp x ^ n * exp x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
c : F
n : ℕ
hn : 0 < n.succ
⊢ iteratedDerivWithin n.succ (fun z ↦ c + f z) s x = iteratedDerivWithin n.succ f s x

Tactic:
rw [iteratedDerivWithin_succ', iteratedDerivWithin_succ']

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
c : F
n : ℕ
hn : 0 < n.succ
⊢ iteratedDerivWithin n (derivWithin (fun z ↦ c + f z) s) s x = iteratedDerivWithin n (derivWithin f s) s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
c : F
n : ℕ
hn : 0 < n.succ
⊢ iteratedDerivWithin n.succ (fun z ↦ c - f z) s x = iteratedDerivWithin n.succ (fun z ↦ -f z) s x

Tactic:
rw [iteratedDerivWithin_succ', iteratedDerivWithin_succ']

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
c : F
n : ℕ
hn : 0 < n.succ
⊢ iteratedDerivWithin n (derivWithin (fun z ↦ c - f z) s) s x = iteratedDerivWithin n (derivWithin (fun z ↦ -f z) s) s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case e_f
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
c : F
n : ℕ
hn : 0 < n.succ
y : 𝕜
⊢ derivWithin (fun z ↦ c - f z) s y = derivWithin (fun z ↦ -f z) s y

Tactic:
derivWithin.fun_neg

Goal state after:
case e_f
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
f : 𝕜 → F
c : F
n : ℕ
hn : 0 < n.succ
y : 𝕜
⊢ derivWithin (fun z ↦ c - f z) s y = -derivWithin f s y
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
f : 𝕜 → 𝔸
hf : ContDiffWithinAt 𝕜 (↑n) f s x
v : F
⊢ iteratedDerivWithin n (fun y ↦ f y • v) s x = iteratedDerivWithin n f s x • v

Tactic:
by
  simp [iteratedDerivWithin, iteratedFDerivWithin_smul_const_apply hf h hx]

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

Goal state before:
pp: s : Set ℝ
a : ℝ
h : 1 ∈ posTangentConeAt s a
ι : Type
l : Filter ι
hl : l.NeBot
c : ι → ℝ≥0
d : ι → ℝ
hd₀ : Tendsto d l (𝓝 0)
hd : ∀ᶠ (n : ι) in l, a + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 1)
this : Tendsto (fun x ↦ a + d x) l (𝓝 a)
n : ι
hcdn : 0 < c n • d n
hdn : a + d n ∈ s
⊢ a + d n ∈ Ioi a

Tactic:
simpa using pos_of_mul_pos_right hcdn

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : T2Space β
hab : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
⊢ extendFrom (Ioo a b) f a = la

Tactic:
apply extendFrom_eq

Goal state after:
case hx
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : T2Space β
hab : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
⊢ a ∈ closure[inst✝⁵] (Ioo a b)

case hf
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la : β
inst✝ : T2Space β
hab : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
⊢ Tendsto f (𝓝[Ioo a b] a) (𝓝 la)
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
lb : β
inst✝ : T2Space β
hab : a < b
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ extendFrom (Ioo a b) f b = lb

Tactic:
apply extendFrom_eq

Goal state after:
case hx
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
lb : β
inst✝ : T2Space β
hab : a < b
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ b ∈ closure[inst✝⁵] (Ioo a b)

case hf
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
lb : β
inst✝ : T2Space β
hab : a < b
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
⊢ Tendsto f (𝓝[Ioo a b] b) (𝓝 lb)
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case hf
𝕜 : Type u_1
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
i✝ : ℕ
a : 𝕜
i : ℕ
hi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦ a ^ i • iteratedFDeriv 𝕜 i f (a • x)
hf : ContDiff 𝕜 (↑(i + 1)) f
v : E
x✝ : Fin (i + 1) → E
⊢ DifferentiableAt 𝕜 (fun x ↦ a • x) v

Tactic:
· exact differentiableAt_id.const_smul _

Goal state after: