mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource m M N
k' : ℕ
blocks_fun' : Fin k' → ℕ
H' : ⟨k', blocks_fun'⟩ ∈ compPartialSumSource m M N
heq : compChangeOfVariables m M N ⟨k, blocks_fun⟩ H = compChangeOfVariables m M N ⟨k', blocks_fun'⟩ H'
⊢ k = k'

Tactic:
have := (compChangeOfVariables_length m M N H).symm

Goal state after:
α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource m M N
k' : ℕ
blocks_fun' : Fin k' → ℕ
H' : ⟨k', blocks_fun'⟩ ∈ compPartialSumSource m M N
heq : compChangeOfVariables m M N ⟨k, blocks_fun⟩ H = compChangeOfVariables m M N ⟨k', blocks_fun'⟩ H'
this : ⟨k, blocks_fun⟩.fst = (compChangeOfVariables m M N ⟨k, blocks_fun⟩ H).snd.length
⊢ k = k'
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: 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
j : Fin k
⊢ r * (a ^ ⟨k, blocksFun⟩.snd j * p (⟨k, blocksFun⟩.snd 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:
rw [compChangeOfVariables_blocksFun]

Goal state after:
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
h : ∀ x ∈ s, ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧ x ∈ u ∧ AnalyticOn 𝕜 f (s ∩ u)
x : E
m : x ∈ s
u : Set E
ou : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
fu : AnalyticOn 𝕜 f (s ∩ u)
r : ℝ
r0 : r > 0
ru : ball x r ⊆ u
p : FormalMultilinearSeries 𝕜 E F
t : ℝ≥0∞
fp : HasFPowerSeriesWithinOnBall f p (s ∩ u) x t
y : E
ys : x + y ∈ insert x s
yr : y ∈ eball 0 (min (ENNReal.ofReal r) t)
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (f (x + y))

Tactic:
simp only [mem_eball, lt_min_iff, edist_lt_ofReal, dist_zero_right] at yr

Goal state after:
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
h : ∀ x ∈ s, ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧ x ∈ u ∧ AnalyticOn 𝕜 f (s ∩ u)
x : E
m : x ∈ s
u : Set E
ou : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
fu : AnalyticOn 𝕜 f (s ∩ u)
r : ℝ
r0 : r > 0
ru : ball x r ⊆ u
p : FormalMultilinearSeries 𝕜 E F
t : ℝ≥0∞
fp : HasFPowerSeriesWithinOnBall f p (s ∩ u) x t
y : E
ys : x + y ∈ insert x s
yr : ‖y‖ < r ∧ edist y 0 < t
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (f (x + y))
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case zero.zero
n : ℕ
a : Composition n
b : Composition a.length
hi : 0 < b.length
hj : 0 < b.blocksFun ⟨0, hi⟩
⊢ (take (take 0 b.blocks).sum a.blocks).sum = (take 0 (List.map sum (a.blocks.splitWrtComposition b))).sum

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
L : Filter (E × E)
f : E → E
f' : E →L[𝕜] E
hf : HasFDerivAtFilter f f' L
hL : Tendsto (Prod.map f f) L L
⊢ HasFDerivAtFilter f^[0] (f' ^ 0) L

Tactic:
| zero =>

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

Goal state before:
pp: case zero
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
hj : 0 < b.blocksFun ⟨i, hi⟩
⊢ a.sizeUpTo (b.sizeUpTo i + 0) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo 0

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
A : Type u_7
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f : E → A
z : E
s : Set E
hf : AnalyticWithinAt 𝕜 f s z
⊢ AnalyticWithinAt 𝕜 (f ^ 0) s z

Tactic:
| zero =>

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

Goal state before:
pp: case hasSum
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
y : A
hy : y ∈ Metric.eball 0 1
⊢ HasSum (fun n ↦ (formalMultilinearSeries_geometric 𝕜 A n) fun x ↦ y) (1 - (0 + y))⁻¹ʳ

Tactic:
simp only [Metric.mem_eball, edist_dist, dist_zero_right, ofReal_lt_one] at hy

Goal state after:
case hasSum
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
y : A
hy : ‖y‖ < 1
⊢ HasSum (fun n ↦ (formalMultilinearSeries_geometric 𝕜 A n) fun x ↦ y) (1 - (0 + y))⁻¹ʳ
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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
i : ℕ
⊢ ∀ (him : i < b.length) (hin : i < (a.gather b).length),
    q.applyComposition ⟨a, b⟩.snd (p.applyComposition ⟨a, b⟩.fst v) ⟨i, him⟩ =
      (q (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩).length)
        (p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩)
          (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hin⟩)))

Tactic:
hi1

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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
i : ℕ
hi1 : i < b.length
⊢ ∀ (hin : i < (a.gather b).length),
    q.applyComposition ⟨a, b⟩.snd (p.applyComposition ⟨a, b⟩.fst v) ⟨i, hi1⟩ =
      (q (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩).length)
        (p.applyComposition (⟨a.gather b, a.sigmaCompositionAux b⟩.snd ⟨i, hin⟩)
          (v ∘ ⇑(⟨a.gather b, a.sigmaCompositionAux b⟩.fst.embedding ⟨i, hin⟩)))
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: case hbc
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
g : E →L[𝕜] F
f : F [⋀^ι]→L[𝕜] G
dg : E →L[𝕜] F
v : ι → E
i : ι
⊢ ∏ i_1 ∈ {i}ᶜ, ‖(Function.update (fun x ↦ g) i dg i_1) (v i_1)‖ ≤ (∏ _x ∈ {i}ᶜ, ‖g‖) * ∏ i ∈ {i}ᶜ, ‖v i‖

Tactic:
← Finset.prod_mul_distrib

Goal state after:
case hbc
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
g : E →L[𝕜] F
f : F [⋀^ι]→L[𝕜] G
dg : E →L[𝕜] F
v : ι → E
i : ι
⊢ ∏ i_1 ∈ {i}ᶜ, ‖(Function.update (fun x ↦ g) i dg i_1) (v i_1)‖ ≤ ∏ x ∈ {i}ᶜ, ‖g‖ * ‖v x‖
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x : Completion α
⊢ ∀ (a : α), dist ↑a ↑a = 0

Tactic:
intro a

Goal state after:
case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x : Completion α
a : α
⊢ dist ↑a ↑a = 0
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y : Completion α
⊢ ∀ (a b : α), dist ↑a ↑b = dist ↑b ↑a

Tactic:
intro a

Goal state after:
case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y : Completion α
a : α
⊢ ∀ (b : α), dist ↑a ↑b = dist ↑b ↑a
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
⊢ ∀ (a b c : α), dist ↑a ↑c ≤ dist ↑a ↑b + dist ↑b ↑c

Tactic:
intro a

Goal state after:
case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
a : α
⊢ ∀ (b c : α), dist ↑a ↑c ≤ dist ↑a ↑b + dist ↑b ↑c
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case refine_1
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
this : {x | ε ≤ dist x.1 x.2 ∨ (x.1, x.2) ∈ t} = {p | ε ≤ dist p.1 p.2} ∪ t
⊢ IsClosed ({p | ε ≤ dist p.1 p.2} ∪ t)

Tactic:
apply IsClosed.union _ tclosed

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
this : {x | ε ≤ dist x.1 x.2 ∨ (x.1, x.2) ∈ t} = {p | ε ≤ dist p.1 p.2} ∪ t
⊢ IsClosed {p | ε ≤ dist p.1 p.2}
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
ε : ℝ
εpos : ε > 0
hε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s
r : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}
this : r ∈ 𝓤 ℝ
t1 : Set (Completion α × Completion α)
ht1 : t1 ∈ 𝓤 (Completion α)
t2 : Set (Completion α × Completion α)
ht2 : t2 ∈ 𝓤 (Completion α)
ht : t1 ×ˢ t2 ⊆ (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' {x | (dist x.1.1 x.1.2, dist x.2.1 x.2.2) ∈ r}
⊢ ∀ (a b : Completion α), (a, b) ∈ t1 → dist a b < ε

Tactic:
intro a

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
ε : ℝ
εpos : ε > 0
hε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s
r : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}
this : r ∈ 𝓤 ℝ
t1 : Set (Completion α × Completion α)
ht1 : t1 ∈ 𝓤 (Completion α)
t2 : Set (Completion α × Completion α)
ht2 : t2 ∈ 𝓤 (Completion α)
ht : t1 ×ˢ t2 ⊆ (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' {x | (dist x.1.1 x.1.2, dist x.2.1 x.2.2) ∈ r}
a : Completion α
⊢ ∀ (b : Completion α), (a, b) ∈ t1 → dist a b < ε
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case zero
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 : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
m : Fin (0 + 1) → E
⊢ (iteratedFDerivWithin 𝕜 (0 + 1) f s x) m =
    ((iteratedFDerivWithin 𝕜 0 (fun y ↦ fderivWithin 𝕜 f s y) s x) (init m)) (m (last 0))

Tactic:
| zero =>

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

Goal state before:
pp: case zero
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 t : Set E
f f₁ : E → F
x : E
h : f₁ =ᶠ[𝓝[s] x] f
ht : t ⊆ s
⊢ iteratedFDerivWithin 𝕜 0 f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 0 f t

Tactic:
| zero =>

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

Goal state before:
pp: case zero
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 t : Set E
f : E → F
y x : E
h : s =ᶠ[𝓝[{y}ᶜ] x] t
⊢ iteratedFDerivWithin 𝕜 0 f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 0 f t

Tactic:
| zero =>

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

Goal state before:
pp: case zero
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
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpToOn n f p s
hs : UniqueDiffOn 𝕜 s
x : E
hmn : ↑0 ≤ n
hx : x ∈ s
⊢ p x 0 = iteratedFDerivWithin 𝕜 0 f s x

Tactic:
| zero =>

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

Goal state before:
pp: case zero
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
a : E
⊢ iteratedFDerivWithin 𝕜 0 (fun z ↦ f (a + z)) s = fun x ↦ iteratedFDerivWithin 𝕜 0 f (a +ᵥ s) (a + x)

Tactic:
| zero =>

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

Goal state before:
pp: case zero
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
⊢ tsupport (iteratedFDeriv 𝕜 0 f) ⊆ tsupport f

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
a : 𝕜
⊢ iteratedFDerivWithin 𝕜 0 (fun x ↦ f (-x)) s a = (-1) ^ 0 • iteratedFDerivWithin 𝕜 0 f (-s) (-a)

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOnNhd 𝕜 f s
⊢ AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 0 f) s

Tactic:
| zero =>

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

Goal state before:
pp: case pos
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hf : DifferentiableWithinAt 𝕜 f s x
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ c y • f y) s x = c x • derivWithin f s x + derivWithin c s x • f x

pp: case neg
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hf : DifferentiableWithinAt 𝕜 f s x
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ c y • f y) s x = c x • derivWithin f s x + derivWithin c s x • f x

Tactic:
· exact (hc.hasDerivWithinAt.smul hf.hasDerivWithinAt).derivWithin hsx

Goal state after:
case neg
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hf : DifferentiableWithinAt 𝕜 f s x
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun y ↦ c y • f y) s x = c x • derivWithin f s x + derivWithin c s x • f x
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕝 : Type u_3
inst✝³ : DivisionSemiring 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : SMulCommClass 𝕜 𝕝 F
inst✝ : ContinuousConstSMul 𝕝 F
c : 𝕝
f : 𝕜 → F
⊢ derivWithin (fun y ↦ c • f y) s x = c • derivWithin f s x

Tactic:
by
  by_cases hsx : UniqueDiffWithinAt 𝕜 s x
  · simp [← fderivWithin_derivWithin, ← Pi.smul_def, fderivWithin_const_smul_field c hsx]
  · simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx]

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

Goal state before:
pp: 𝕜 : Type u_1
𝔸 : Type u_2
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedSpace 𝕜 E
f : E → 𝔸
f' : E →L[𝕜] 𝔸
x : E
n : ℕ
⊢ ∀ a ∈ Finset.range n, f x ^ (a + (n.pred - a)) = f x ^ (n - 1)

Tactic:
intro a

Goal state after:
𝕜 : Type u_1
𝔸 : Type u_2
E : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : NormedSpace 𝕜 E
f : E → 𝔸
f' : E →L[𝕜] 𝔸
x : E
n a : ℕ
⊢ a ∈ Finset.range n → f x ^ (a + (n.pred - a)) = f x ^ (n - 1)
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case zero
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
hu : UniqueDiffOn 𝕜 s
⊢ AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 0 f s) s

Tactic:
| zero =>

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
d : 𝕜 → 𝔸
d' c : 𝔸
hd : HasStrictDerivAt d d' x
⊢ HasStrictDerivAt (fun y ↦ c * d y) (c * d') x

Tactic:
convert! (hasStrictDerivAt_const _ _).mul hd using 1
  rw [zero_mul, zero_add]

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
inst✝ : CompleteSpace F
h : AnalyticOnNhd 𝕜 f s
⊢ AnalyticOnNhd 𝕜 (deriv^[0] f) s

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
inst✝ : CompleteSpace F
h : AnalyticAt 𝕜 f x
⊢ AnalyticAt 𝕜 (deriv^[0] f) x

Tactic:
| zero =>

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
c : F
⊢ derivWithin (fun y ↦ f y - c) s x = derivWithin f s x

Tactic:
by
  simp only [derivWithin, fderivWithin_sub_const]

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

Goal state before:
pp: case inl.refine_2
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
n : ℕ
⊢ ∀ (x : Fin n → E), (p.derivSeries n) x = 0 x

Tactic:
intro a

Goal state after:
case inl.refine_2
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
n : ℕ
a : Fin n → E
⊢ (p.derivSeries n) a = 0 a
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case zero
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 : CPolynomialOn 𝕜 f s
⊢ CPolynomialOn 𝕜 (_root_.iteratedFDeriv 𝕜 0 f) s

Tactic:
| zero =>

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
h : CPolynomialOn 𝕜 f s
⊢ CPolynomialOn 𝕜 (deriv^[0] f) s

Tactic:
| zero =>

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
⊢ (f.toFormalMultilinearSeries.changeOrigin x 1) (Fin.snoc 0 y) = ∑ i, f (Function.update x i (y i))

Tactic:
changeOrigin,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
⊢ ((f.toFormalMultilinearSeries.changeOriginSeries 1).sum x) (Fin.snoc 0 y) = ∑ i, f (Function.update x i (y i))
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
r : ℝ≥0∞
y : E
⊢ ∀ {F : Type (max u v)} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace 𝕜 F] [CompleteSpace F]
    {p : FormalMultilinearSeries 𝕜 E F} {f : E → F},
    HasFPowerSeriesOnBall f p x r → (0! • (p 0) fun x ↦ y) = (iteratedFDeriv 𝕜 0 f x) fun x ↦ y

Tactic:
| zero =>

Goal state after:
case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
r : ℝ≥0∞
y : E
⊢ ∀ {F : Type (max u v)} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace 𝕜 F] [CompleteSpace F]
    {p : FormalMultilinearSeries 𝕜 E F} {f : E → F},
    HasFPowerSeriesOnBall f p x r → (0! • (p 0) fun x ↦ y) = (iteratedFDeriv 𝕜 0 f x) fun x ↦ y
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: case zero
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : 𝕜
f : 𝕜 → E
hf : ∀ k < 0, (swap dslope a)^[k] f a = 0
⊢ (b - a) ^ 0 • (swap dslope a)^[0] f b = f b

Tactic:
| zero =>

Goal state after: