mathlib-initiative/mathlib-tactics huggingface dataset:
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
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n
nu : ℕ → ℕ := fun n ↦ n - mu n
hnu : nu = fun n ↦ n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
m : ℕ
hm : 1 ≤ m
h0 : ↑(ψ m) ≠ 0
⊢ (↑(ψ m) - ↑(mu (ψ m))) / ↑(ψ m) = (↑(ψ m) - ↑(mu (ψ m))) / ↑(ψ m)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.ControlledClosure

Goal state before:
pp: G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : f.SurjectiveOnWith K C
h : H
h_in : h ∈ K.topologicalClosure
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i ↦ (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n ↦ ∑ k ∈ range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖-v 0 + h‖ < b 0
hv : ∀ n > 0, ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n ↦ ∑ k ∈ range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k ∈ range (n + 1), C * b k ≤ ε * ‖h‖
⊢ C * ‖h‖ + ∑ k ∈ range (n + 1), C * b k ≤ C * ‖h‖ + ε * ‖h‖

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : f.SurjectiveOnWith K C
h : H
h_in : h ∈ K.topologicalClosure
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i ↦ (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n ↦ ∑ k ∈ range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖-v 0 + h‖ < b 0
hv : ∀ n > 0, ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n ↦ ∑ k ∈ range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k ∈ range (n + 1), C * b k ≤ ε * ‖h‖
⊢ C * ‖h‖ + ∑ k ∈ range (n + 1), C * b k ≤ C * ‖h‖ + ε * ‖h‖
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝ : SeminormedAddCommGroup G
x : Completion G
⊢ _root_.id x = (id (Completion G)) x

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : SeminormedAddCommGroup G
x : Completion G
⊢ _root_.id x = (id (Completion G)) x
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝² : SeminormedAddCommGroup G
H : Type u_2
inst✝¹ : SeminormedAddCommGroup H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
f : NormedAddGroupHom G H
g : NormedAddGroupHom H K
x : Completion G
⊢ Completion.map (⇑g ∘ ⇑f) x = Completion.map (⇑(g.comp f)) x

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : SeminormedAddCommGroup G
H : Type u_2
inst✝¹ : SeminormedAddCommGroup H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
f : NormedAddGroupHom G H
g : NormedAddGroupHom H K
x : Completion G
⊢ Completion.map (⇑g ∘ ⇑f) x = Completion.map (⇑(g.comp f)) x
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
⊢ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ ↑(toCompl.comp (incl f.ker)).range, ‖hatg - b‖ < ε

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
⊢ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ ↑(toCompl.comp (incl f.ker)).range, ‖hatg - b‖ < ε
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 ≤ f y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.ControlledClosure

Goal state before:
pp: G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖
k : K
h_in : j k ∈ j.range
⊢ ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖
k : K
h_in : j k ∈ j.range
⊢ ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
g' : G
hgg' : f g' = f g
hfg : ‖g'‖ ≤ C' * ‖f g‖
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: V W : SemiNormedGrp
i : V ≅ W
h1 : (Hom.hom i.hom).NormNoninc
h2 : (Hom.hom i.inv).NormNoninc
v : V.carrier
⊢ ‖v‖ = ‖v‖

Tactic:
]

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

Goal state before:
pp: M N : SemiNormedGrp₁
e : M ≅ N
r : M.carrier
⊢ ↑(Hom.hom (e.hom ≫ e.inv)) r = r

Tactic:
]

Goal state after:
M N : SemiNormedGrp₁
e : M ≅ N
r : M.carrier
⊢ ↑(Hom.hom (e.hom ≫ e.inv)) r = r
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp

Goal state before:
pp: M N : SemiNormedGrp₁
e : M ≅ N
s : N.carrier
⊢ ↑(Hom.hom (e.inv ≫ e.hom)) s = s

Tactic:
]

Goal state after:
M N : SemiNormedGrp₁
e : M ≅ N
s : N.carrier
⊢ ↑(Hom.hom (e.inv ≫ e.hom)) s = s
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
g' : G
hgg' : f g' = f g
hfg : ‖g'‖ ≤ C' * ‖f g‖
mem_ker : g - g' ∈ f.ker
this : ‖f g‖ ≤ ‖f‖ * δ
⊢ ‖hatg - (↑g - ↑g')‖ = ‖hatg - (↑g - ↑g')‖

Tactic:
]

Goal state after:
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
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
g' : G
hgg' : f g' = f g
hfg : ‖g'‖ ≤ C' * ‖f g‖
mem_ker : g - g' ∈ f.ker
this : ‖f g‖ ≤ ‖f‖ * δ
⊢ ‖hatg - ↑g‖ + ‖g'‖ = ‖hatg - ↑g‖ + ‖g'‖

Tactic:
]

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

Goal state before:
pp: V W : SemiNormedGrp₁
i : V ≅ W
v : V.carrier
⊢ ‖v‖ = ‖v‖

Tactic:
]

Goal state after:
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
y : L
hy : y ≠ 0
hy_alg : IsAlgebraic K y
⊢ ¬minpoly K y = X ^ (minpoly K y).natDegree

Tactic:
]

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
y : L
hy : y ≠ 0
hy_alg : IsAlgebraic K y
⊢ ¬minpoly K y = X ^ (minpoly K y).natDegree
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
y : L
hy : y ≠ 0
hy_alg : IsAlgebraic K y
h : minpoly K y = X ^ (minpoly K y).natDegree
⊢ 0 = 0

Tactic:
]

Goal state after:
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
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
x : L
hx_alg : IsAlgebraic K x
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: X Y : SemiNormedGrp₁
f : X ⟶ Y
x : X.carrier
⊢ ∃ w, ↑(Hom.hom f) w = ↑(Hom.hom f) x

Tactic:
]

Goal state after:
X Y : SemiNormedGrp₁
f : X ⟶ Y
x : X.carrier
⊢ ∃ w, ↑(Hom.hom f) w = ↑(Hom.hom f) x
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
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
⊢ ‖1‖ = 1

Tactic:
]

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
⊢ ‖1‖ = 1
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝³ : SeminormedAddCommGroup G
H : Type u_2
inst✝² : SeminormedAddCommGroup H
inst✝¹ : T0Space H
inst✝ : CompleteSpace H
f : NormedAddGroupHom G H
g : NormedAddGroupHom (Completion G) H
hg : ∀ (v : G), f v = g ↑v
v : Completion G
⊢ g v = g v

Tactic:
]

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

Goal state before:
pp: V W : SemiNormedGrp
f g : V ⟶ W
c : Fork f g
⊢ (NormedAddGroupHom.compHom (Hom.hom f)) (Hom.hom c.ι) = (NormedAddGroupHom.compHom (Hom.hom g)) (Hom.hom c.ι)

Tactic:
]

Goal state after:
V W : SemiNormedGrp
f g : V ⟶ W
c : Fork f g
⊢ (NormedAddGroupHom.compHom (Hom.hom f)) (Hom.hom c.ι) = (NormedAddGroupHom.compHom (Hom.hom g)) (Hom.hom c.ι)
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: V W : SemiNormedGrp
f g : V ⟶ W
c : Fork f g
⊢ (NormedAddGroupHom.compHom (Hom.hom (f - g))) (Hom.hom c.ι) = 0

Tactic:
rw [hom_sub, map_sub, AddMonoidHom.sub_apply, sub_eq_zero]

Goal state after:
V W : SemiNormedGrp
f g : V ⟶ W
c : Fork f g
⊢ (NormedAddGroupHom.compHom (Hom.hom f)) (Hom.hom c.ι) = (NormedAddGroupHom.compHom (Hom.hom g)) (Hom.hom c.ι)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n
nu : ℕ → ℕ := fun n ↦ n - mu n
hnu : nu = fun n ↦ n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n ↦ ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n ↦ μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
ε : ℝ
hε : ε > 0
⊢ smoothingFun μ (x + y) ≤ max (smoothingFun μ x) (smoothingFun μ y) + ε

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n
nu : ℕ → ℕ := fun n ↦ n - mu n
hnu : nu = fun n ↦ n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n ↦ ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n ↦ μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
ε : ℝ
hε : ε > 0
⊢ smoothingFun μ (x + y) ≤ max (smoothingFun μ x) (smoothingFun μ y) + ε
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
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs :
  Polynomial.map (algebraMap K L) (minpoly K x) =
    C (Polynomial.map (algebraMap K L) (minpoly K x)).leadingCoeff * (Multiset.map (fun x ↦ X - C x) s).prod
⊢ 1 = 1

Tactic:
]

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
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n
nu : ℕ → ℕ := fun n ↦ n - mu n
hnu : nu = fun n ↦ n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n ↦ ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n ↦ μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
ε : ℝ
hε : ε > 0
⊢ smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε ≤
    (if smoothingFun μ x ≤ smoothingFun μ y then smoothingFun μ y else smoothingFun μ x) + ε

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n
nu : ℕ → ℕ := fun n ↦ n - mu n
hnu : nu = fun n ↦ n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n ↦ ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n ↦ μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
ε : ℝ
hε : ε > 0
⊢ smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε ≤
    (if smoothingFun μ x ≤ smoothingFun μ y then smoothingFun μ y else smoothingFun μ x) + ε
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y : SemiNormedGrp
f : X ⟶ Y
x : X.carrier
⊢ (ConcreteCategory.hom 0) x = 0

Tactic:
]

Goal state after:
X Y : SemiNormedGrp
f : X ⟶ Y
x : X.carrier
⊢ (ConcreteCategory.hom 0) x = 0
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
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
⊢ spectralNorm K L x ≤ ⨆ σ, f (σ 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
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
⊢ spectralNorm K L x ≤ ⨆ σ, f (σ x)
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
x : Y.carrier
⊢ (ConcreteCategory.hom g) x = (ConcreteCategory.hom g) x

Tactic:
]

Goal state after:
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
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
⊢ (⨆ x, if x ∈ s then f x else 0) ≤ ⨆ σ, f (σ 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
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
⊢ (⨆ x, if x ∈ s then f x else 0) ≤ ⨆ σ, f (σ x)
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y Z W : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ W
cond : f ≫ g = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: X Y Z W : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ W
cond : f ≫ g = 0
⊢ g ≫ h = g ≫ h

Tactic:
]

Goal state after:
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 = explicitCokernelDesc w

Tactic:
]

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.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y : SemiNormedGrp
f : X ⟶ Y
Z : SemiNormedGrp
g h : explicitCokernel f ⟶ Z
H : explicitCokernelπ f ≫ g = explicitCokernelπ f ≫ h
x : Y.carrier
⊢ (Hom.hom (explicitCokernelπ f ≫ h)) x = (Hom.hom (explicitCokernelπ f ≫ h)) x

Tactic:
]

Goal state after:
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:
]

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.SpectralNorm

Goal state before:
pp: case pos
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
y : L
h : y ∈ s
σ : Gal(L/K)
hσ : σ x = y
⊢ f (σ x) ≤ ⨆ σ, f (σ x)

Tactic:
]

Goal state after:
case pos
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
y : L
h : y ∈ s
σ : Gal(L/K)
hσ : σ x = y
⊢ f (σ x) ≤ ⨆ σ, f (σ x)
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y Z W : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ W
cond : f ≫ g = 0
cond2 : g ≫ h = 0
⊢ g ≫ h = explicitCokernelπ f ≫ 0

Tactic:
]

Goal state after:
X Y Z W : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
h : Z ⟶ W
cond : f ≫ g = 0
cond2 : g ≫ h = 0
⊢ g ≫ h = explicitCokernelπ f ≫ 0