mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
⊢ ‖(∑ i, B.repr ((B.repr x) i • B i * ∑ i, (B.repr y) i • B i)) ixy‖ ≤ B.norm (B c.1 * B c.2) * B.norm x * B.norm y

Tactic:
[

Goal state after:
case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
⊢ ‖(∑ i, B.repr ((B.repr x) i • B i * ∑ i, (B.repr y) i • B i)) ixy‖ ≤ B.norm (B c.1 * B c.2) * B.norm x * B.norm y
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case hg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)
⊢ Tendsto (fun x_1 ↦ (μ x ^ (x_1 % m1)) ^ (1 * 1 / ↑x_1)) atTop (𝓝 1)

Tactic:
[

Goal state after:
case hg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)
⊢ Tendsto (fun x_1 ↦ (μ x ^ (x_1 % m1)) ^ (1 * 1 / ↑x_1)) atTop (𝓝 1)
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
f_ne_zero : f = 0
⊢ f x ≤ ⨆ y, f x * (f y / f y)

Tactic:
[

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
f_ne_zero : f = 0
⊢ f x ≤ ⨆ y, f x * (f y / f y)
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
⊢ seminormFromBounded' f x = f x

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
⊢ seminormFromBounded' f x = f x
Module: Mathlib.Analysis.Normed.Field.Krasner

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Normal K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
z : L := x - y
this✝¹ : FiniteDimensional K ↥K⟮y⟯
this✝ : IsUltrametricDist L
y' : ↥K⟮y⟯ := AdjoinSimple.gen K y
zsep : IsSeparable (↥K⟮y⟯) z
this : z ∈ K⟮y⟯ ↔ z ∈ ⊥
⊢ z ∈ K⟮y⟯

Tactic:
this

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Normal K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
z : L := x - y
this✝¹ : FiniteDimensional K ↥K⟮y⟯
this✝ : IsUltrametricDist L
y' : ↥K⟮y⟯ := AdjoinSimple.gen K y
zsep : IsSeparable (↥K⟮y⟯) z
this : z ∈ K⟮y⟯ ↔ z ∈ ⊥
⊢ z ∈ ⊥
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
n : ℕ
⊢ (⨆ n_1, if n_1 < (X ^ n).natDegree then ‖(X ^ n).coeff n_1‖ ^ (1 / (↑(X ^ n).natDegree - ↑n_1)) else 0) = 0

Tactic:
[

Goal state after:
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
n : ℕ
⊢ (⨆ n_1, if n_1 < (X ^ n).natDegree then ‖(X ^ n).coeff n_1‖ ^ (1 / (↑(X ^ n).natDegree - ↑n_1)) else 0) = 0
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case refine_1.inr.inr
R : Type u_1
inst✝ : NormedDivisionRing R
P : R[X]
hP : P.Monic
h : iSup (spectralValueTerms P) ≤ 1
n : ℕ
hPn : n < P.natDegree
⊢ ‖P.coeff n‖ ≤ 1

Tactic:
have : spectralValueTerms P n ≤ 1 := le_ciSup (spectralValueTerms_bddAbove P) n |>.trans h

Goal state after:
case refine_1.inr.inr
R : Type u_1
inst✝ : NormedDivisionRing R
P : R[X]
hP : P.Monic
h : iSup (spectralValueTerms P) ≤ 1
n : ℕ
hPn : n < P.natDegree
this : spectralValueTerms P n ≤ 1
⊢ ‖P.coeff n‖ ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
n : ℕ
hn : 1 ≤ n
hn1 : 0 < 1 / ↑n
⊢ μ 1 ^ (1 / ↑n) ≤ 1 ^ (1 / ↑n)

Tactic:
exact (rpow_le_rpow_iff (apply_nonneg μ _) zero_le_one hn1).mpr 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
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)
ha : ¬a = 0
ha_pos : 0 < a
h : (fun n ↦ 1 / ↑(s (ψ n)) * ↑(s (ψ n))) =ᶠ[atTop] 1
⊢ (fun n ↦ (μ (x ^ s (ψ n)) ^ (1 / ↑(s (ψ n)))) ^ (↑(s (ψ n)) / ↑(ψ n))) =ᶠ[atTop] fun n ↦
    μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)
ha : ¬a = 0
ha_pos : 0 < a
h : (fun n ↦ 1 / ↑(s (ψ n)) * ↑(s (ψ n))) =ᶠ[atTop] 1
⊢ (fun n ↦ (μ (x ^ s (ψ n)) ^ (1 / ↑(s (ψ n)))) ^ (↑(s (ψ n)) / ↑(ψ n))) =ᶠ[atTop] fun n ↦
    μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
E : IntermediateField K L
x : ↥E
⊢ spectralNorm K (↥(normalClosure K (↥E) (AlgebraicClosure ↥E)))
      ((algebraMap ↥E ↥(normalClosure K (↥E) (AlgebraicClosure ↥E))) x) =
    spectralNorm K L ((algebraMap (↥E) L) x)

Tactic:
[

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
E : IntermediateField K L
x : ↥E
⊢ spectralNorm K (↥(normalClosure K (↥E) (AlgebraicClosure ↥E)))
      ((algebraMap ↥E ↥(normalClosure K (↥E) (AlgebraicClosure ↥E))) x) =
    spectralNorm K L ((algebraMap (↥E) L) x)
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: V W : SemiNormedGrp
f g✝ : V ⟶ W
this : ∀ (c : Fork f g✝), (NormedAddGroupHom.compHom (Hom.hom (f - g✝))) (Hom.hom c.ι) = 0
c : Fork f g✝
g : c.pt ⟶ (fork f g✝).pt
h : g ≫ (fork f g✝).ι = c.ι
x : c.pt.carrier
⊢ (Hom.hom g) x = (Hom.hom (ofHom (NormedAddGroupHom.ker.lift (Hom.hom c.ι) (Hom.hom f - Hom.hom g✝) ⋯))) x

Tactic:
[

Goal state after:
V W : SemiNormedGrp
f g✝ : V ⟶ W
this : ∀ (c : Fork f g✝), (NormedAddGroupHom.compHom (Hom.hom (f - g✝))) (Hom.hom c.ι) = 0
c : Fork f g✝
g : c.pt ⟶ (fork f g✝).pt
h : g ≫ (fork f g✝).ι = c.ι
x : c.pt.carrier
⊢ (Hom.hom g) x = (Hom.hom (ofHom (NormedAddGroupHom.ker.lift (Hom.hom c.ι) (Hom.hom f - Hom.hom g✝) ⋯))) x
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y Z : SemiNormedGrp
f : X ⟶ Y
e₁ e₂ : explicitCokernel f ⟶ Z
h : explicitCokernelπ f ≫ e₁ = explicitCokernelπ f ≫ e₂
g : Y ⟶ Z := explicitCokernelπ f ≫ e₂
w : f ≫ g = 0
this : e₂ = explicitCokernelDesc w
⊢ e₁ = e₂

Tactic:
this

Goal state after:
X Y Z : SemiNormedGrp
f : X ⟶ Y
e₁ e₂ : explicitCokernel f ⟶ Z
h : explicitCokernelπ f ≫ e₁ = explicitCokernelπ f ≫ e₂
g : Y ⟶ Z := explicitCokernelπ f ≫ e₂
w : f ≫ g = 0
this : e₂ = explicitCokernelDesc w
⊢ e₁ = explicitCokernelDesc w
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
k : K
⊢ spectralNorm K L ((algebraMap K L) k) = ‖k‖

Tactic:
[

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
k : K
⊢ spectralNorm K L ((algebraMap K L) k) = ‖k‖
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_1
inst✝⁹ : Semiring R
E : Type u_2
F : Type u_4
G : Type u_6
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommMonoid E
inst✝⁶ : Module R E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
f : E →L[R] F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
g : F →L[R] G
hfg : (g ∘SL f).HasRightInverse
⊢ g.HasRightInverse

Tactic:
obtain ⟨fginv, hfginv⟩ := hfg
  exact ⟨f.comp fginv, fun y ↦ by simpa using hfginv y⟩

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
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝 ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ U • Subtype.val '' V ∈ 𝓝 ((fun x1 x2 ↦ x1 • x2) ↑x ↑y)

pp: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {0}ᶜ

pp: case ha
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ ↑((homeomorphSphereProd E r hr₀).symm (y, x)) ∈ {0}ᶜ

Tactic:
homeomorphSphereProd_symm_apply_coe

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this : Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈ 𝓝 (↑(y, x).2 • ↑(y, x).1)
⊢ U • Subtype.val '' V ∈ 𝓝 ((fun x1 x2 ↦ x1 • x2) ↑x ↑y)

case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {0}ᶜ

case ha
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ ↑((homeomorphSphereProd E r hr₀).symm (y, x)) ∈ {0}ᶜ
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
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this : Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈ 𝓝 (↑(y, x).2 • ↑(y, x).1)
a : ↑(sphere 0 r)
b : ↑(Ioi 0)
ha : (a, b).1 ∈ V
hb : (a, b).2 ∈ Ioi 0 ↓∩ U
⊢ ↑((homeomorphSphereProd E r hr₀).symm (a, b)) ∈ U • Subtype.val '' V

Tactic:
homeomorphSphereProd_symm_apply_coe

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this : Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈ 𝓝 (↑(y, x).2 • ↑(y, x).1)
a : ↑(sphere 0 r)
b : ↑(Ioi 0)
ha : (a, b).1 ∈ V
hb : (a, b).2 ∈ Ioi 0 ↓∩ U
⊢ ↑(a, b).2 • ↑(a, b).1 ∈ U • Subtype.val '' V
Module: Mathlib.Topology.Maps.Strict.Group

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : Group G
inst✝² : Group H
f : G →* H
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace H
⊢ IsStrictMap ⇑f ↔ IsEmbedding ⇑(kerLift f)

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
inst✝³ : Group G
inst✝² : Group H
f : G →* H
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace H
⊢ IsStrictMap ⇑f ↔ IsEmbedding ⇑(kerLift f)
Module: Mathlib.Topology.Maps.Strict.Group

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : Group G
inst✝² : Group H
f : G →* H
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace H
⊢ IsStrictMap ⇑f ↔ IsHomeomorph ⇑(quotientKerEquivRange f)

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
inst✝³ : Group G
inst✝² : Group H
f : G →* H
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace H
⊢ IsStrictMap ⇑f ↔ IsHomeomorph ⇑(quotientKerEquivRange f)
Module: Mathlib.Topology.Maps.Strict.Group

Goal state before:
pp: G : Type u_1
H : Type u_2
G' : Type u_3
H' : Type u_4
inst✝⁹ : Group G'
inst✝⁸ : Group H'
inst✝⁷ : Group G
inst✝⁶ : Group H
f : G →* H
g : G' →* H'
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace H
inst✝³ : IsTopologicalGroup G
inst✝² : TopologicalSpace G'
inst✝¹ : IsTopologicalGroup G'
inst✝ : TopologicalSpace H'
⊢ IsStrictMap ⇑(f.prodMap g) ↔ IsStrictMap ⇑f ∧ IsStrictMap ⇑g

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
G' : Type u_3
H' : Type u_4
inst✝⁹ : Group G'
inst✝⁸ : Group H'
inst✝⁷ : Group G
inst✝⁶ : Group H
f : G →* H
g : G' →* H'
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace H
inst✝³ : IsTopologicalGroup G
inst✝² : TopologicalSpace G'
inst✝¹ : IsTopologicalGroup G'
inst✝ : TopologicalSpace H'
⊢ IsStrictMap ⇑(f.prodMap g) ↔ IsStrictMap ⇑f ∧ IsStrictMap ⇑g
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)
p q r : ℝ≥0∞
hpqr✝ : p.HolderTriple q r
e : (i : ι) → E i
f : (i : ι) → F i
hp : 0 < p.toReal
hq : 0 < q.toReal
B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i
K C D : ℝ
hBK : ∀ (i : ι), ‖B i‖ ≤ K
hK : 0 ≤ K
hC : 0 ≤ C
hCe : ∀ (s : Finset ι), ∑ i ∈ s, ‖e i‖ ^ p.toReal ≤ C
hDf : ∀ (s : Finset ι), ∑ i ∈ s, ‖f i‖ ^ q.toReal ≤ D
s : Finset ι
hpqr : p.toReal.HolderTriple q.toReal r.toReal
hr : 0 < r.toReal
⊢ ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤ C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal)

Tactic:
calc
    _ ≤ (∑ i ∈ s, ‖e i‖ ^ p.toReal) ^ (r.toReal / p.toReal) *
        (∑ i ∈ s, ‖f i‖ ^ q.toReal) ^ (r.toReal / q.toReal) := by
      apply Real.Lr_rpow_le_Lp_mul_Lq_of_nonneg s hpqr <;> (intros; positivity)
    _ ≤ _ := by
      gcongr
      · exact hCe s
      · exact hDf s

Goal state after:
Module: Mathlib.Topology.Maps.Strict.Module

Goal state before:
pp: R : Type u_1
M : Type u_3
Nₗ : Type u_5
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup Nₗ
inst✝³ : Module R M
inst✝² : Module R Nₗ
fₗ : M →ₗ[R] Nₗ
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace Nₗ
⊢ IsStrictMap ⇑fₗ ↔ IsHomeomorph ⇑fₗ.quotKerEquivRange

Tactic:
[

Goal state after:
R : Type u_1
M : Type u_3
Nₗ : Type u_5
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup Nₗ
inst✝³ : Module R M
inst✝² : Module R Nₗ
fₗ : M →ₗ[R] Nₗ
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace Nₗ
⊢ IsStrictMap ⇑fₗ ↔ IsHomeomorph ⇑fₗ.quotKerEquivRange
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁵] ↑A
A_cofg : A.CoFG
h_ker : Disjoint (↑u).ker A
range_u : (↑u).range = ⊤
range_u_restr : IsClosed[inst✝²] ↑(↑(u.domRestrict A)).range
S : Submodule 𝕜 E
ker_le_S : (↑u).ker ≤ S
this✝ : FiniteDimensional 𝕜 ↥S
S_compl_A : IsTopCompl S A
uS_compl_uA : IsTopCompl (map (↑u) S) (map (↑u) A)
this : T2Space ↥(map (↑u) S)
uₛ : ↥S →L[𝕜] ↥(map (↑u) S) := u.restrict ⋯
uₐ : ↥A →L[𝕜] ↥(map (↑u) A) := u.restrict ⋯
Φ : (↥S × ↥A) ≃L[𝕜] E := S.prodEquivOfIsTopCompl A S_compl_A
Ψ : (↥(map (↑u) S) × ↥(map (↑u) A)) ≃L[𝕜] F := (map (↑u) S).prodEquivOfIsTopCompl (map (↑u) A) uS_compl_uA
u_eq : ⇑u = ⇑Ψ ∘ ⇑(uₛ.prodMap uₐ) ∘ ⇑Φ.symm
u_restr_eq : ⇑(u.domRestrict A) = ⇑(map (↑u) A).subtypeL ∘ ⇑uₐ
⊢ IsStrictMap ⇑(uₛ.prodMap uₐ) ↔ IsStrictMap ⇑uₐ

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁵] ↑A
A_cofg : A.CoFG
h_ker : Disjoint (↑u).ker A
range_u : (↑u).range = ⊤
range_u_restr : IsClosed[inst✝²] ↑(↑(u.domRestrict A)).range
S : Submodule 𝕜 E
ker_le_S : (↑u).ker ≤ S
this✝ : FiniteDimensional 𝕜 ↥S
S_compl_A : IsTopCompl S A
uS_compl_uA : IsTopCompl (map (↑u) S) (map (↑u) A)
this : T2Space ↥(map (↑u) S)
uₛ : ↥S →L[𝕜] ↥(map (↑u) S) := u.restrict ⋯
uₐ : ↥A →L[𝕜] ↥(map (↑u) A) := u.restrict ⋯
Φ : (↥S × ↥A) ≃L[𝕜] E := S.prodEquivOfIsTopCompl A S_compl_A
Ψ : (↥(map (↑u) S) × ↥(map (↑u) A)) ≃L[𝕜] F := (map (↑u) S).prodEquivOfIsTopCompl (map (↑u) A) uS_compl_uA
u_eq : ⇑u = ⇑Ψ ∘ ⇑(uₛ.prodMap uₐ) ∘ ⇑Φ.symm
u_restr_eq : ⇑(u.domRestrict A) = ⇑(map (↑u) A).subtypeL ∘ ⇑uₐ
⊢ IsStrictMap ⇑(uₛ.prodMap uₐ) ↔ IsStrictMap ⇑uₐ
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁶] ↑A
inst✝ : A.CoFG
h_ker : Disjoint (↑u).ker A
F' : Submodule 𝕜 F := (↑u).range
h_range : IsClosed[inst✝³] ↑F'
i : ↥F' →L[𝕜] F := F'.subtypeL
i_clemb : IsClosedEmbedding ⇑i
u' : E →L[𝕜] ↥F' := u.rangeRestrict
range_u' : (↑u').range = ⊤
eq1 : u = i ∘SL u'
eq2 : u.domRestrict A = i ∘SL u'.domRestrict A
⊢ IsStrictMap ⇑u ↔ IsStrictMap ⇑(u.domRestrict A) ∧ IsClosed[inst✝³] ↑(↑(u.domRestrict A)).range

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁶] ↑A
inst✝ : A.CoFG
h_ker : Disjoint (↑u).ker A
F' : Submodule 𝕜 F := (↑u).range
h_range : IsClosed[inst✝³] ↑F'
i : ↥F' →L[𝕜] F := F'.subtypeL
i_clemb : IsClosedEmbedding ⇑i
u' : E →L[𝕜] ↥F' := u.rangeRestrict
range_u' : (↑u').range = ⊤
eq1 : u = i ∘SL u'
eq2 : u.domRestrict A = i ∘SL u'.domRestrict A
⊢ IsStrictMap ⇑u ↔ IsStrictMap ⇑(u.domRestrict A) ∧ IsClosed[inst✝³] ↑(↑(u.domRestrict A)).range
Module: Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : AddCommGroup E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : Module 𝕜 E
inst✝⁹ : Module 𝕜 F
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space E
inst✝ : T2Space F
u : E →L[𝕜] F
tfae_1_to_4 : u.IsFredholm → Nonempty u.FredholmPackage
dec : u.FredholmPackage
⊢ ∃ v, (↑v).IsQuasiInverse ↑u

Tactic:
exact ⟨dec.quasiInverse, dec.isQuasiInverse⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsUpperSet s
hx : x ∈ closure s
h : ∀ (i : ι), x i < y i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), x i + ε < y i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
hyz✝ : ∀ (i : ι), z i < y i
δ : ℝ
hδ : 0 < δ
hyz : ∀ (i : ι), z i + δ < y i
w : ι → ℝ
hw : w ∈ ball y δ
i : ι
⊢ z i ≤ w i

Tactic:
[

Goal state after:
case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsUpperSet s
hx : x ∈ closure s
h : ∀ (i : ι), x i < y i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), x i + ε < y i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
hyz✝ : ∀ (i : ι), z i < y i
δ : ℝ
hδ : 0 < δ
hyz : ∀ (i : ι), z i + δ < y i
w : ι → ℝ
hw : w ∈ ball y δ
i : ι
⊢ z i ≤ w i
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
hyz✝ : ∀ (i : ι), y i < z i
δ : ℝ
hδ : 0 < δ
hyz : ∀ (i : ι), y i + δ < z i
w : ι → ℝ
hw : w ∈ ball y δ
i : ι
⊢ w i ≤ z i

Tactic:
[

Goal state after:
case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
hyz✝ : ∀ (i : ι), y i < z i
δ : ℝ
hδ : 0 < δ
hyz : ∀ (i : ι), y i + δ < z i
w : ι → ℝ
hw : w ∈ ball y δ
i : ι
⊢ w i ≤ z i