mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : FiniteDimensional ℝ V
s : Set P
⊢ Bornology.IsBounded s ↔ asymptoticCone ℝ s ⊆ {0}

Tactic:
refine ⟨asymptoticCone_subset_singleton_of_bounded, fun h => ?_⟩
  simp_rw [isBounded_def, cobounded_eq_iSup_sphere_asymptoticNhds, mem_iSup]
  intro v hv
  by_contra h'
  exact Metric.ne_of_mem_sphere hv one_ne_zero (h h')

Goal state after:
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set P
x✝ : s.Nonempty
⊢ 0 ∈ asymptoticCone k s

Tactic:
simpa [mem_asymptoticCone_iff]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
n : ℕ
⊢ IntervalIntegrable (fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) volume 0 ↑n

Tactic:
apply IntervalIntegrable.continuousOn_mul

Goal state after:
case hf
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := ⋯
n : ℕ
⊢ IntervalIntegrable (fun x ↦ ↑x ^ (s - 1)) volume 0 ↑n

case hg
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := ⋯
n : ℕ
⊢ ContinuousOn (fun x ↦ ↑((1 - x / ↑n) ^ n)) (uIcc 0 ↑n)
Module: Mathlib.Analysis.Polynomial.Factorization

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
inst✝ : IsAlgClosed F
f : F[X]
n : ℕ
hf : f.IsMonicOfDegree (n + 1)
⊢ ∃ f₁ f₂, f₁.IsMonicOfDegree 1 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂

Tactic:
obtain ⟨f₁, hf₁m, hf₁i, f₂, hf₂⟩ :=
    exists_monic_irreducible_factor f <| not_isUnit_of_natDegree_pos f <|
      by grind [IsMonicOfDegree.natDegree_eq]
  rw [hf₂, add_comm] at hf
  have hf₁ : IsMonicOfDegree f₁ 1 :=
    ⟨natDegree_eq_of_degree_eq_some <| IsAlgClosed.degree_eq_one_of_irreducible F hf₁i, hf₁m⟩
  exact ⟨f₁, f₂, hf₁, hf₁.of_mul_left hf, hf₂⟩

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

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
⊢ (∑' (n : ℕ), (expSeries ℚ (tsze R M) n) fun x_1 ↦ x) =
    inl (∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) + inr ((∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) • x.snd)

Tactic:
by_cases h : Summable (fun (n : ℕ) => (expSeries ℚ R n) fun _ ↦ fst x)

Goal state after:
case pos
R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
h : Summable fun n ↦ (expSeries ℚ R n) fun x_1 ↦ x.fst
⊢ (∑' (n : ℕ), (expSeries ℚ (tsze R M) n) fun x_1 ↦ x) =
    inl (∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) + inr ((∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) • x.snd)

case neg
R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
h : ¬Summable fun n ↦ (expSeries ℚ R n) fun x_1 ↦ x.fst
⊢ (∑' (n : ℕ), (expSeries ℚ (tsze R M) n) fun x_1 ↦ x) =
    inl (∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) + inr ((∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) • x.snd)
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: case e_a.inl
q : ℍ
hv : ‖q.im‖ = 0
⊢ normSq (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im) = Real.cos ‖q.im‖ ^ 2 + Real.sin ‖q.im‖ ^ 2

Tactic:
simp [hv]

Goal state after:
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x✝ : R
m : ℕ
x : R
n : ℕ
⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖

Tactic:
rcases (norm_nonneg x).eq_or_lt with hx | hx

Goal state after:
case inl
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x✝ : R
m : ℕ
x : R
n : ℕ
hx : 0 = ‖x‖
⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖

case inr
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x✝ : R
m : ℕ
x : R
n : ℕ
hx : 0 < ‖x‖
⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case refine_2
F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
c : ℝ
hc₀ : 0 < c
hbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖
⊢ Tendsto Prod.fst (cobounded ℝ ×ˢ ⊤) (cobounded ℝ)

Tactic:
exact tendsto_fst

Goal state after:
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
⊢ (aeval a) f = 0

Tactic:
← eval_map_algebraMap

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
⊢ eval a (map (algebraMap L f.SplittingField) f) = 0
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
h✝ : 0 < ↑p.natDegree - ↑m
hps : s.card = p.natDegree
hc : ‖p.coeff m‖ = f (((mapAlg K L) p).coeff m)
h : f ((-1) ^ (s.card - m) * s.esymm (s.card - m)) = f (s.esymm (s.card - m))
t : Multiset L
ht_card : t.card = s.card - m
hts : ∀ x ∈ t, x ∈ s
ht_ge : f (Multiset.map prod (powersetCard (s.card - m) s)).sum ≤ f t.prod
h_pr : f t.prod ≤ (Multiset.map (⇑f) t).prod
hs_ne : s ≠ 0
y : L
hyx : y ∈ s
hy_max : ∀ z ∈ s, f z ≤ f y
this : (Multiset.map (⇑f) t).prod ≤ f y ^ (p.natDegree - m)
h_bdd : BddAbove (Set.range fun x ↦ if x ∈ s then f x else 0)
⊢ f y ≤ if y ∈ s then f y else 0

Tactic:
rw [if_pos hyx]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y : SemiNormedGrp
f : X ⟶ Y
⊢ ∀ {Z : SemiNormedGrp} (g h : explicitCokernel f ⟶ Z), explicitCokernelπ f ≫ g = explicitCokernelπ f ≫ h → g = h

Tactic:
intro Z

Goal state after:
X Y : SemiNormedGrp
f : X ⟶ Y
Z : SemiNormedGrp
⊢ ∀ (g h : explicitCokernel f ⟶ Z), explicitCokernelπ f ≫ g = explicitCokernelπ f ≫ h → g = h
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
cond : f ≫ g = 0
hg : (Hom.hom g).NormNoninc
⊢ ‖Hom.hom (explicitCokernelDesc cond)‖ ≤ 1

Tactic:
← NNReal.coe_one

Goal state after:
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
cond : f ≫ g = 0
hg : (Hom.hom g).NormNoninc
⊢ ‖Hom.hom (explicitCokernelDesc cond)‖ ≤ ↑1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
m : ℕ
hm : 1 ≤ m
hlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))
h_eq : ∀ (n : ℕ), smoothingSeminormSeq μ x (m * n) ^ m = smoothingSeminormSeq μ (x ^ m) n
⊢ Tendsto (fun x_1 ↦ smoothingSeminormSeq μ x (m * x_1) ^ m) atTop (𝓝 (smoothingFun μ (x ^ m)))

Tactic:
simpa [h_eq] using tendsto_smoothingFun_of_map_one_le_one μ hμ1 _

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
y : R
⊢ smoothingFun μ (x * y) = smoothingFun μ x * smoothingFun μ y

Tactic:
have hlim : Tendsto (fun n => μ x * smoothingSeminormSeq μ y n) atTop
      (𝓝 (smoothingFun μ x * smoothingFun μ y)) := by
    rw [smoothingFun_apply_of_map_mul_eq_mul μ hμ1 hx]
    exact Tendsto.const_mul _ (tendsto_smoothingFun_of_map_one_le_one μ hμ1 y)
  apply tendsto_nhds_unique_of_eventuallyEq (tendsto_smoothingFun_of_map_one_le_one μ hμ1 (x * y))
    hlim
  simp only [EventuallyEq, eventually_atTop]
  use 1
  intro n hn1
  have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (_root_.ne_of_gt (lt_of_lt_of_le zero_lt_one hn1))
  simp only [smoothingSeminormSeq]
  rw [mul_pow, pow_mul_apply_eq_pow_mul μ hx, mul_rpow (pow_nonneg (apply_nonneg μ _) _)
    (apply_nonneg μ _), ← rpow_natCast, ← rpow_mul (apply_nonneg μ _), mul_one_div_cancel hn0,
    rpow_one]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.RadialEquiv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
U : Set ℝ
V : Set ↑(sphere 0 r)
hU : IsOpen U
hU₀ : 0 ∉ U
hV : IsOpen[instTopologicalSpaceSubtype] V
x : ℝ
hxU : x ∈ U
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
this : ∀ {U : Set ℝ}, IsOpen U → 0 ∉ U → ∀ x ∈ U, 0 < x → U • Subtype.val '' V ∈ 𝓝 ((fun x1 x2 ↦ x1 • x2) x ↑y)
hx₀ : x = 0 ∨ x < 0
⊢ 0 < -x

Tactic:
← neg_pos

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
U : Set ℝ
V : Set ↑(sphere 0 r)
hU : IsOpen U
hU₀ : 0 ∉ U
hV : IsOpen[instTopologicalSpaceSubtype] V
x : ℝ
hxU : x ∈ U
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
this : ∀ {U : Set ℝ}, IsOpen U → 0 ∉ U → ∀ x ∈ U, 0 < x → U • Subtype.val '' V ∈ 𝓝 ((fun x1 x2 ↦ x1 • x2) x ↑y)
hx₀ : x = 0 ∨ 0 < -x
⊢ 0 < -x
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
f : ι → ℝ
hf : Summable f
⊢ Summable fun x ↦ ↑(log (1 + f x))

Tactic:
apply (Complex.summable_log_one_add_of_summable (summable_ofReal.mpr hf)).congr_cofinite

Goal state after:
ι : Type u_1
f : ι → ℝ
hf : Summable f
⊢ (fun i ↦ Complex.log (1 + ↑(f i))) =ᶠ[cofinite] fun x ↦ ↑(log (1 + f x))
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
hf : Summable fun i ↦ ‖f i‖
| Multipliable fun i ↦ ‖1 + f i‖

Tactic:
enter [1, i]

Goal state after:
ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
hf : Summable fun i ↦ ‖f i‖
i : ι
| ‖1 + f i‖
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
hf : Summable fun i ↦ ‖f i‖
ε : ℝ
hε : 0 < ε
⊢ ∃ s, ∀ (t : Finset ι), Disjoint t s → Real.exp (∑ i ∈ t, ‖f i‖) - 1 < ε

Tactic:
suffices {x | Real.exp x - 1 < ε} ∈ 𝓝 0 from hf.vanishing this

Goal state after:
ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
hf : Summable fun i ↦ ‖f i‖
ε : ℝ
hε : 0 < ε
⊢ {x | Real.exp x - 1 < ε} ∈ 𝓝 0
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f✝ : ι → R
hf : Summable fun i ↦ ‖f✝ i‖
ε : ℝ
hε : 0 < ε
f : ℝ → ℝ := fun x ↦ Real.exp x - 1
⊢ Set.Iio ε ∈ 𝓝 (f 0)

Tactic:
simpa [f] using Iio_mem_nhds hε

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpHolder

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : ι → Type u_3
F : ι → Type u_4
G : ι → Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : (i : ι) → NormedAddCommGroup (F i)
inst✝² : (i : ι) → NormedSpace 𝕜 (F i)
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → NormedSpace 𝕜 (G i)
B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i
e : (i : ι) → E i
f : (i : ι) → F i
he : {i | e i ≠ 0}.Finite
⊢ {i | ((B i) (e i)) (f i) ≠ 0}.Finite

Tactic:
exact he.subset fun i hi h ↦ hi <| by simp [h]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
𝕜 : Type u_2
E : ι → Type u_3
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : NormedField 𝕜
p q : FreeAddMonoid (𝕜 × ((i : ι) → E i))
⊢ projectiveSeminormAux (p + q) ≤ projectiveSeminormAux p + projectiveSeminormAux q

Tactic:
by
  simp [projectiveSeminormAux]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 E
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
E₁ : Submodule 𝕜 E
E₁_closed : IsClosed[inst✝⁶] ↑E₁
E₁_coFG : E₁.CoFG
F₁ : Submodule 𝕜 F
F₁_closed : IsClosed[inst✝⁵] ↑F₁
F₁_coFG : F₁.CoFG
h_mapsto : MapsTo ⇑u ↑E₁ ↑F₁
e : ↥E₁ ≃L[𝕜] ↥F₁
he : ↑e = u.restrict h_mapsto
eqL : u.domRestrict E₁ = F₁.subtypeL ∘SL ↑e
⊢ u.IsFredholm

Tactic:
have eqₗ : u.toLinearMap.domRestrict E₁ = F₁.subtype ∘ₗ e := congr(($eqL).toLinearMap)

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 E
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
E₁ : Submodule 𝕜 E
E₁_closed : IsClosed[inst✝⁶] ↑E₁
E₁_coFG : E₁.CoFG
F₁ : Submodule 𝕜 F
F₁_closed : IsClosed[inst✝⁵] ↑F₁
F₁_coFG : F₁.CoFG
h_mapsto : MapsTo ⇑u ↑E₁ ↑F₁
e : ↥E₁ ≃L[𝕜] ↥F₁
he : ↑e = u.restrict h_mapsto
eqL : u.domRestrict E₁ = F₁.subtypeL ∘SL ↑e
eqₗ : (↑u).domRestrict E₁ = F₁.subtype ∘ₗ ↑↑e
⊢ u.IsFredholm
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: α : Type u_1
inst✝² : NormedCommGroup α
inst✝¹ : Preorder α
inst✝ : IsOrderedMonoid α
s : Set α
hs : IsLowerSet s
ε : ℝ
⊢ IsLowerSet (cthickening ε s)

Tactic:
by
  rw [cthickening_eq_iInter_thickening'']
  exact isLowerSet_iInter₂ fun δ _ => hs.thickening' _

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case mk.mk
E : Type u_1
inst✝ : NormedAddCommGroup E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
r L : ℝ≥0
toFun✝¹ : ↑(Icc tmin tmax) → E
lipschitzWith✝¹ : LipschitzWith L toFun✝¹
mem_closedBall₀✝¹ : toFun✝¹ t₀ ∈ closedBall x₀ ↑r
toFun✝ : ↑(Icc tmin tmax) → E
lipschitzWith✝ : LipschitzWith L toFun✝
mem_closedBall₀✝ : toFun✝ t₀ ∈ closedBall x₀ ↑r
h :
  ∀ (t : ↑(Icc tmin tmax)),
    { toFun := toFun✝¹, lipschitzWith := lipschitzWith✝¹, mem_closedBall₀ := mem_closedBall₀✝¹ }.toFun t =
      { toFun := toFun✝, lipschitzWith := lipschitzWith✝, mem_closedBall₀ := mem_closedBall₀✝ }.toFun t
⊢ toFun✝¹ = toFun✝

Tactic:
ext t

Goal state after:
case mk.mk
E : Type u_1
inst✝ : NormedAddCommGroup E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
r L : ℝ≥0
toFun✝¹ : ↑(Icc tmin tmax) → E
lipschitzWith✝¹ : LipschitzWith L toFun✝¹
mem_closedBall₀✝¹ : toFun✝¹ t₀ ∈ closedBall x₀ ↑r
toFun✝ : ↑(Icc tmin tmax) → E
lipschitzWith✝ : LipschitzWith L toFun✝
mem_closedBall₀✝ : toFun✝ t₀ ∈ closedBall x₀ ↑r
h :
  ∀ (t : ↑(Icc tmin tmax)),
    { toFun := toFun✝¹, lipschitzWith := lipschitzWith✝¹, mem_closedBall₀ := mem_closedBall₀✝¹ }.toFun t =
      { toFun := toFun✝, lipschitzWith := lipschitzWith✝, mem_closedBall₀ := mem_closedBall₀✝ }.toFun t
t : ↑(Icc tmin tmax)
⊢ toFun✝¹ t = toFun✝ t
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
dt : ℝ
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s)
⊢ γ = (γ ∘ fun x ↦ x + dt) ∘ fun x ↦ x + -dt

Tactic:
ext t

Goal state after:
case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
dt : ℝ
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s)
t : ℝ
⊢ γ t = ((γ ∘ fun x ↦ x + dt) ∘ fun x ↦ x + -dt) t
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
dt : ℝ
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s)
⊢ v = (v ∘ fun x ↦ x + dt) ∘ fun x ↦ x + -dt

Tactic:
ext t

Goal state after:
case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
dt : ℝ
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s)
t : ℝ
x✝ : E
⊢ v t x✝ = ((v ∘ fun x ↦ x + dt) ∘ fun x ↦ x + -dt) t x✝
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
a : ℝ
ha : a ≠ 0
⊢ a⁻¹ • s = {t | t * a ∈ s}

Tactic:
ext t

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
a : ℝ
ha : a ≠ 0
t : ℝ
⊢ t ∈ a⁻¹ • s ↔ t ∈ {t | t * a ∈ s}
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
a : ℝ
ha : a ≠ 0
heq : a⁻¹ • s = {t | t * a ∈ s}
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)
⊢ γ = (γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹

Tactic:
ext t

Goal state after:
case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
a : ℝ
ha : a ≠ 0
heq : a⁻¹ • s = {t | t * a ∈ s}
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)
t : ℝ
⊢ γ t = ((γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹) t
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
a : ℝ
ha : a ≠ 0
heq : a⁻¹ • s = {t | t * a ∈ s}
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)
⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹

Tactic:
ext t

Goal state after:
case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
a : ℝ
ha : a ≠ 0
heq : a⁻¹ • s = {t | t * a ∈ s}
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)
t : ℝ
x✝ : E
⊢ v t x✝ = (a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹) t x✝
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
t₀ a : ℝ
ha : a ≠ 0
ε : ℝ
hε : ε > 0
h : IsIntegralCurveOn γ v (Metric.ball t₀ ε)
⊢ Metric.ball (t₀ / a) (ε / |a|) = {t | t * a ∈ Metric.ball t₀ ε}

Tactic:
ext t

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
t₀ a : ℝ
ha : a ≠ 0
ε : ℝ
hε : ε > 0
h : IsIntegralCurveOn γ v (Metric.ball t₀ ε)
t : ℝ
⊢ t ∈ Metric.ball (t₀ / a) (ε / |a|) ↔ t ∈ {t | t * a ∈ Metric.ball t₀ ε}
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
t₀ a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurveAt (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (t₀ / a)
⊢ γ = (γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹

Tactic:
ext t

Goal state after:
case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
t₀ a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurveAt (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (t₀ / a)
t : ℝ
⊢ γ t = ((γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹) t
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
t₀ a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurveAt (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (t₀ / a)
⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹

Tactic:
ext t

Goal state after:
case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
t₀ a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurveAt (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (t₀ / a)
t : ℝ
x✝ : E
⊢ v t x✝ = (a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹) t x✝
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)
⊢ γ = (γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹

Tactic:
ext t

Goal state after:
case e'_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)
t : ℝ
⊢ γ t = ((γ ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹) t
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)
⊢ v = a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹

Tactic:
ext t

Goal state after:
case e'_5
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
a : ℝ
ha : a ≠ 0
hγ : IsIntegralCurve (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a)
t : ℝ
x✝ : E
⊢ v t x✝ = (a⁻¹ • (a • v ∘ fun x ↦ x * a) ∘ fun x ↦ x * a⁻¹) t x✝
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
h : ∀ (t : ↑(Icc tmin tmax)), α.toFun t = picard f (↑t₀) x α.compProj ↑t
⊢ IsFixedPt (next hf hx) α

Tactic:
ext t

Goal state after:
case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
h : ∀ (t : ↑(Icc tmin tmax)), α.toFun t = picard f (↑t₀) x α.compProj ↑t
t : ↑(Icc tmin tmax)
⊢ (next hf hx α).toFun t = α.toFun t