mathlib-initiative/mathlib-tactics huggingface dataset:
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 + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j + a.blocks[b.sizeUpTo i + j] =
    (a.gather b).sizeUpTo i + ((a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j + (a.sigmaCompositionAux b ⟨i, ⋯⟩).blocks[j])

Tactic:
simp only [sigmaCompositionAux, add_assoc]

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 +
        (a.blocks.splitWrtComposition b)[i][j])
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
C : ι → ℝ
C_pos : ∀ (i : ι), C i > 0
hC : ∀ (i : ι) (n : ℕ), ‖p i n‖ * ↑r' ^ n ≤ C i
D : ℝ
D_nonneg : D ≥ 0
hD : ∀ (i : ι), C i ≤ D
⊢ ↑r' ≤ (pi p).radius

Tactic:
apply le_radius_of_bound _ D (fun n ↦ ?_)

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
C : ι → ℝ
C_pos : ∀ (i : ι), C i > 0
hC : ∀ (i : ι) (n : ℕ), ‖p i n‖ * ↑r' ^ n ≤ C i
D : ℝ
D_nonneg : D ≥ 0
hD : ∀ (i : ι), C i ≤ D
n : ℕ
⊢ ‖pi p n‖ * ↑r' ^ n ≤ D
Module: Mathlib.Analysis.Analytic.Inverse

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 : OpenPartialHomeomorph E F
a : E
i : E ≃L[𝕜] F
h0 : a ∈ f.source
p : FormalMultilinearSeries 𝕜 E F
h : HasFPowerSeriesAt (↑f) p a
hp : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
A : HasFPowerSeriesAt (↑f.symm ∘ ↑f) ((p.leftInv i a).comp p) a
B : ∀ᶠ (y : E) in 𝓝 (a - a), HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f (a + y) - ↑f a) (↑f.symm (↑f (a + y)))
⊢ ContinuousAt (fun x ↦ x - a) a

Tactic:
fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
f' : E →L[ℝ] F
x : E
hf : HasFDerivAt f f' x
v : E
⊢ Tendsto (fun c ↦ c • (f (x + c⁻¹ • v) - f x)) atTop (𝓝 (f' v))

Tactic:
apply hf.lim v

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
f' : E →L[ℝ] F
x : E
hf : HasFDerivAt f f' x
v : E
⊢ Tendsto (fun n ↦ ‖n‖) atTop atTop
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Finite ι
val✝ : Fintype ι
⊢ Continuous
    (⇑(ContinuousLinearMap.precomp (ContinuousMultilinearMap 𝕜 (fun x ↦ E) G) (toContinuousMultilinearMapCLM 𝕜)) ∘
      ⇑(ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear 𝕜 (fun x ↦ E) (fun x ↦ F) G) ∘ fun f x ↦
        f)

Tactic:
fun_prop

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

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_6
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
i : ι
f : (i : ι) → F' i
s' : Set ((i : ι) → F' i)
⊢ HasFDerivWithinAt (fun f ↦ f i) ?f' s' f

pp: case f'
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_6
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
i : ι
f : (i : ι) → F' i
s' : Set ((i : ι) → F' i)
⊢ ((i : ι) → F' i) →L[𝕜] F' i

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: case mpr
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}
⊢ s ∈ 𝓤 (Completion α)

Tactic:
refine mem_of_superset ht1 ?_

Goal state after:
case mpr
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}
⊢ t1 ⊆ s
Module: Mathlib.Analysis.Normed.Module.Completion

Goal state before:
pp: case hp.hf
𝕜 : Type u_1
E : Type u_2
A : Type u_3
inst✝ : SeminormedRing A
⊢ Continuous fun b ↦ ‖b.1 * b.2‖

Tactic:
fun_prop

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

Goal state before:
pp: case hp.hg
𝕜 : Type u_1
E : Type u_2
A : Type u_3
inst✝ : SeminormedRing A
⊢ Continuous fun b ↦ ‖b.1‖ * ‖b.2‖

Tactic:
fun_prop

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : NormedField A
inst✝ : CompletableTopField A
x y : Completion A
⊢ Continuous fun x ↦ ‖x.1 * x.2‖

Tactic:
fun_prop

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
A : Type u_3
inst✝¹ : NormedField A
inst✝ : CompletableTopField A
x y : Completion A
⊢ Continuous fun x ↦ ‖x.1‖ * ‖x.2‖

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝⁵ : PseudoMetricSpace α
M : Type u_1
inst✝⁴ : Zero M
inst✝³ : Zero α
inst✝² : SMul M α
inst✝¹ : PseudoMetricSpace M
inst✝ : IsBoundedSMul M α
c : M
⊢ Continuous fun x ↦ dist (c • x.1) (c • x.2)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝⁵ : PseudoMetricSpace α
M : Type u_1
inst✝⁴ : Zero M
inst✝³ : Zero α
inst✝² : SMul M α
inst✝¹ : PseudoMetricSpace M
inst✝ : IsBoundedSMul M α
c : M
⊢ Continuous fun x ↦ dist c 0 * dist x.1 x.2

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝⁵ : PseudoMetricSpace α
M : Type u_1
inst✝⁴ : Zero M
inst✝³ : Zero α
inst✝² : SMul M α
inst✝¹ : PseudoMetricSpace M
inst✝ : IsBoundedSMul M α
c₁ c₂ : M
⊢ Continuous fun a ↦ dist (c₁ • a) (c₂ • a)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝⁵ : PseudoMetricSpace α
M : Type u_1
inst✝⁴ : Zero M
inst✝³ : Zero α
inst✝² : SMul M α
inst✝¹ : PseudoMetricSpace M
inst✝ : IsBoundedSMul M α
c₁ c₂ : M
⊢ Continuous fun a ↦ dist c₁ c₂ * dist a 0

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : MetricSpace β
inst✝ : CompleteSpace β
f : α → β
K : ℝ≥0
h : LipschitzWith K f
x y : Completion α
⊢ Continuous fun x ↦ dist (Completion.extension f x.1) (Completion.extension f x.2)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : MetricSpace β
inst✝ : CompleteSpace β
f : α → β
K : ℝ≥0
h : LipschitzWith K f
x y : Completion α
⊢ Continuous fun x ↦ ↑K * dist x.1 x.2

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α → β
h : Isometry f
x y : Completion α
⊢ Continuous fun x ↦ dist (Completion.extension f x.1) (Completion.extension f x.2)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α → β
h : Isometry f
x y : Completion α
⊢ Continuous fun x ↦ dist x.1 x.2

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝⁷ : PseudoMetricSpace α
inst✝⁶ : Ring α
inst✝⁵ : IsTopologicalRing α
inst✝⁴ : IsUniformAddGroup α
inst✝³ : Ring β
inst✝² : PseudoMetricSpace β
inst✝¹ : IsUniformAddGroup β
inst✝ : IsTopologicalRing β
f : α →+* β
h : Isometry ⇑f
⊢ Continuous fun x ↦ dist x.1 x.2

Tactic:
fun_prop

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 t : Set E
f : E → F
x y : E
h : s =ᶠ[𝓝[{y}ᶜ] x] t
n : ℕ
⊢ iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t

Tactic:
induction n generalizing x with
  | zero => rfl
  | succ n ihn =>
    refine (eventually_nhds_nhdsWithin.2 h).mono fun y hy => ?_
    simp only [iteratedFDerivWithin_succ_eq_comp_left, (· ∘ ·)]
    rw [(ihn hy).fderivWithin_eq_of_nhds, fderivWithin_congr_set' _ hy]

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
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ (fun y ↦ ‖y - (x, x)‖ * ‖y.1 - y.2‖) =o[𝓝[insert x s ×ˢ insert x s] (x, x)] fun x ↦ ‖x.1 - x.2‖

Tactic:
refine isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, ?_, EventuallyEq.rfl⟩

Goal state after:
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
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ Tendsto (fun y ↦ ‖y - (x, x)‖) (𝓝[insert x s ×ˢ insert x s] (x, x)) (𝓝 0)
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: 𝕜 : Type u
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : DecidableEq 𝕜
f : 𝕜 → 𝕜
c a : 𝕜
hf : HasDerivAt f a c
⊢ ContinuousAt (Function.update (slope f c) c a) c

Tactic:
exact continuousAt_update_same.mpr <| hasDerivAt_iff_tendsto_slope.mp hf

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | a < z.re} = {z | z.re = a}

Tactic:
by
  simpa only [frontier_Ioi] using! frontier_preimage_re (Ioi a)

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

Goal state before:
pp: case inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
k : ℕ
hnk : n < k
⊢ (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ (↑n - ↑k) = (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ ↑(n - k)

Tactic:
have : (∏ i ∈ Finset.range k, (n - i : 𝕜)) = 0 :=
      Finset.prod_eq_zero (Finset.mem_range.2 hnk) (sub_self _)

Goal state after:
case inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
k : ℕ
hnk : n < k
this : ∏ i ∈ Finset.range k, (↑n - ↑i) = 0
⊢ (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ (↑n - ↑k) = (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ ↑(n - k)
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case neg.inr
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
h : ∃ᶠ (x : 𝕜) in 𝓝[≠] z, g x = 0
⊢ (∀ z ∈ U, f z = 0) ∨ ∀ z ∈ U, g z = 0

Tactic:
| inr h =>

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
⊢ DifferentiableAt 𝕜 (fun x ↦ x) x

Tactic:
fun_prop

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

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a)
x : ℝ
hx : x ∈ Icc a b
⊢ ContinuousWithinAt (fun r ↦ B x + r * (x - a)) (Ioi 0) 0

Tactic:
fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → G
s : Set E
x : E
hs : Convex ℝ s
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs' : UniqueDiffOn 𝕜 s
hf' : EqOn (fderivWithin 𝕜 f s) (fderivWithin 𝕜 g s) s
hx : x ∈ s
hfgx : f x = g x
y : E
hy : y ∈ s
⊢ f y = g y

Tactic:
suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this
  refine hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => ?_) hx hy
  rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' hz]

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hs' : IsPreconnected s
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hf' : EqOn (fderiv 𝕜 f) (fderiv 𝕜 g) s
⊢ ∃ a, EqOn f (fun x ↦ g x + a) s

Tactic:
Set.EqOn,

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hs' : IsPreconnected s
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hf' : EqOn (fderiv 𝕜 f) (fderiv 𝕜 g) s
⊢ ∃ a, ∀ ⦃x : E⦄, x ∈ s → f x = g x + a
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
m n : ℕ∞ω
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
H : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hmn : m + 1 ≤ n
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 m (fun f ↦ f 1) univ (fderivWithin 𝕜 f s x)

Tactic:
fun_prop

Goal state after:
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
h : ContDiff 𝕜 2 f
⊢ Differentiable 𝕜 fun x ↦ (fderiv 𝕜 f x) 1

Tactic:
fun_prop

Goal state after:
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
n : ℕ∞ω
f : 𝕜 → F
h : ContDiff 𝕜 (n + 1) f
⊢ ContDiff 𝕜 n fun x ↦ (fderiv 𝕜 f x) 1

Tactic:
fun_prop

Goal state after:
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✝ c : OrderedFinpartition n
⊢ Subsingleton (Fin (cons 1 c.partSize 0))

Tactic:
by simp; infer_instance

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
⊢ iteratedFDerivWithin 𝕜 n f s = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F) ∘ iteratedDerivWithin n f s

Tactic:
rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp_assoc, LinearIsometryEquiv.self_comp_symm,
    Function.id_comp]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
⊢ iteratedDerivWithin 1 f s = derivWithin f s

Tactic:
by
  ext x
  by_cases hsx : AccPt x (𝓟 s)
  · simp only [iteratedDerivWithin, iteratedFDerivWithin_one_apply hsx.uniqueDiffWithinAt,
      derivWithin]
  · simp [derivWithin_zero_of_not_accPt hsx, iteratedDerivWithin, iteratedFDerivWithin,
      fderivWithin_zero_of_not_accPt hsx]

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

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
⊢ Nonempty (Fin (cons 1 c.partSize 0))

Tactic:
by simp; infer_instance

Goal state after:
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
f : F [×c.length]→L[𝕜] G
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
⊢ Continuous fun v m ↦ (p m) (v ∘ c.emb m)

Tactic:
fun_prop

Goal state after:
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
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x ↦ c * x) s s
n : ℕ
ih :
  ∀ {x : 𝕜},
    x ∈ s →
      ContDiffOn 𝕜 (↑n) f s → iteratedDerivWithin n (fun x ↦ f (c * x)) s x = c ^ n • iteratedDerivWithin n f s (c * x)
x : 𝕜
hx : x ∈ s
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hcx : c * x ∈ s
h₀ : Set.EqOn (iteratedDerivWithin n (fun x ↦ f (c * x)) s) (fun x ↦ c ^ n • iteratedDerivWithin n f s (c * x)) s
h₁ : DifferentiableWithinAt 𝕜 (iteratedDerivWithin n f s) s (c * x)
⊢ DifferentiableWithinAt 𝕜 (fun x ↦ c * x) s x

Tactic:
fun_prop

Goal state after:
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
s : Set 𝕜
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
c : 𝕜
hs : Set.MapsTo (fun x ↦ c * x) s s
n : ℕ
ih :
  ∀ {x : 𝕜},
    x ∈ s →
      ContDiffOn 𝕜 (↑n) f s → iteratedDerivWithin n (fun x ↦ f (c * x)) s x = c ^ n • iteratedDerivWithin n f s (c * x)
x : 𝕜
hx : x ∈ s
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hcx : c * x ∈ s
h₀ : Set.EqOn (iteratedDerivWithin n (fun x ↦ f (c * x)) s) (fun x ↦ c ^ n • iteratedDerivWithin n f s (c * x)) s
h₁ : DifferentiableWithinAt 𝕜 (iteratedDerivWithin n f s) s (c * x)
h₂ : DifferentiableWithinAt 𝕜 (fun x ↦ iteratedDerivWithin n f s (c * x)) s x
⊢ DifferentiableWithinAt 𝕜 (HMul.hMul c) s x

Tactic:
fun_prop

Goal state after:
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
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
e : E
v : Fin m → E
⊢ ((q.compAlongOrderedFinpartition p (c.extend none)).curryLeft e) v =
    ((((compAlongOrderedFinpartitionL 𝕜 E F G c).flipMultilinear fun i ↦ p (c.partSize i)) ∘SL
          (q (c.length + 1)).curryLeft ∘SL (continuousMultilinearCurryFin1 𝕜 E F) (p 1))
        e)
      v

Tactic:
simp only [Nat.succ_eq_add_one, OrderedFinpartition.extend, extendLeft,
    ContinuousMultilinearMap.curryLeft_apply,
    FormalMultilinearSeries.compAlongOrderedFinpartition_apply, applyOrderedFinpartition_apply,
    ContinuousLinearMap.comp_apply, continuousMultilinearCurryFin1_apply,
    Matrix.zero_empty, ContinuousLinearMap.flipMultilinear_apply_apply,
    compAlongOrderedFinpartitionL_apply, compAlongOrderFinpartition_apply]

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
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
e : E
v : Fin m → E
⊢ ((q (c.length + 1)) fun m_1 ↦
      (p (cons 1 c.partSize m_1))
        (cons e v ∘
          cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (m + 1)) (fun x ↦ 0) (fun m_2 ↦ succ ∘ c.emb m_2)
            m_1)) =
    (q (c.length + 1)) (cons ((p 1) (snoc ![] e)) fun m_1 ↦ (p (c.partSize m_1)) (v ∘ c.emb m_1))