mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Lp.LpEquiv

Goal state before:
pp: case inl
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Fintype α
f : ↥(lp E 0)
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖

pp: case inr.inl
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Fintype α
f : ↥(lp E ∞)
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖

pp: case inr.inr
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
inst✝ : Fintype α
f : ↥(lp E p)
h : 0 < p.toReal
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖

Tactic:
·

Goal state after:
case inl
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Fintype α
f : ↥(lp E 0)
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖
Module: Mathlib.Analysis.Normed.Lp.LpEquiv

Goal state before:
pp: case inr.inl
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Fintype α
f : ↥(lp E ∞)
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖

pp: case inr.inr
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
inst✝ : Fintype α
f : ↥(lp E p)
h : 0 < p.toReal
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖

Tactic:
·

Goal state after:
case inr.inl
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Fintype α
f : ↥(lp E ∞)
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖
Module: Mathlib.Analysis.Normed.Lp.LpEquiv

Goal state before:
pp: case inr.inr
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
inst✝ : Fintype α
f : ↥(lp E p)
h : 0 < p.toReal
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖

Tactic:
·

Goal state after:
case inr.inr
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
inst✝ : Fintype α
f : ↥(lp E p)
h : 0 < p.toReal
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑ x ∈ T, (f n x - g x)‖ < ε / 3

pp: case refine_2
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (x : ↑(↑T)ᶜ), (f n ↑x - g ↑x)‖ < ε / 3 + ε / 3

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑ x ∈ T, (f n x - g x)‖ < ε / 3
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (x : ↑(↑T)ᶜ), (f n ↑x - g ↑x)‖ < ε / 3 + ε / 3

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (x : ↑(↑T)ᶜ), (f n ↑x - g ↑x)‖ < ε / 3 + ε / 3
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
hx : x = 0
⊢ (spectralAlgNorm K L) (x * y) = (spectralAlgNorm K L) x * (spectralAlgNorm K L) y

pp: case neg
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
hx : ¬x = 0
⊢ (spectralAlgNorm K L) (x * y) = (spectralAlgNorm K L) x * (spectralAlgNorm K L) y

Tactic:
·

Goal state after:
case pos
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
hx : x = 0
⊢ (spectralAlgNorm K L) (x * y) = (spectralAlgNorm K L) x * (spectralAlgNorm K L) y
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case neg
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
hx : ¬x = 0
⊢ (spectralAlgNorm K L) (x * y) = (spectralAlgNorm K L) x * (spectralAlgNorm K L) y

Tactic:
·

Goal state after:
case neg
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
hx : ¬x = 0
⊢ (spectralAlgNorm K L) (x * y) = (spectralAlgNorm K L) x * (spectralAlgNorm K L) y
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x✝ y✝ : L
hxy : spectralNorm K L (x✝ - y✝) = 0
⊢ x✝ = y✝

Tactic:
rw [← sub_eq_zero]
      exact (map_eq_zero_iff_eq_zero (spectralMulAlgNorm K L)).mp hxy

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

Goal state before:
pp: case refine_2.refine_1
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (b : { x // x ∈ (↑T)ᶜ }), f n ↑b‖ < ε / 3

pp: case refine_2.refine_2
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (b : { x // x ∈ (↑T)ᶜ }), g ↑b‖ < ε / 3

Tactic:
·

Goal state after:
case refine_2.refine_1
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (b : { x // x ∈ (↑T)ᶜ }), f n ↑b‖ < ε / 3
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: case refine_2.refine_2
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (b : { x // x ∈ (↑T)ᶜ }), g ↑b‖ < ε / 3

Tactic:
·

Goal state after:
case refine_2.refine_2
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma✝ : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
h2 : ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑ k ∈ T, f x k) (T.sum g) < ε
n : α
hn : dist (∑ k ∈ T, f n k) (T.sum g) < ε / 3
h_suma : Summable fun x ↦ ‖f n x‖
h_bound : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ‖∑' (b : { x // x ∈ (↑T)ᶜ }), g ↑b‖ < ε / 3
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).natDegree

pp: K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ ((mapAlg K E) (minpoly K x)).natDegree = ((mapAlg K E) (minpoly K x)).roots.card

Tactic:
·

Goal state after:
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).natDegree
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ ((mapAlg K E) (minpoly K x)).natDegree = ((mapAlg K E) (minpoly K x)).roots.card

Tactic:
·

Goal state after:
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ ((mapAlg K E) (minpoly K x)).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
h_deg : (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
r : ℝ
hr : (spectralMulAlgNorm K E) ((algebraMap L E) x) = r
⊢ (minpoly K x).natDegree =
    Multiset.count r (Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots)

pp: case neg
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
h_deg : (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
r : ℝ
hr : ¬(spectralMulAlgNorm K E) ((algebraMap L E) x) = r
⊢ 0 = Multiset.count r (Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots)

Tactic:
·

Goal state after:
case pos
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
h_deg : (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
r : ℝ
hr : (spectralMulAlgNorm K E) ((algebraMap L E) x) = r
⊢ (minpoly K x).natDegree =
    Multiset.count r (Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case neg
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
h_deg : (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
r : ℝ
hr : ¬(spectralMulAlgNorm K E) ((algebraMap L E) x) = r
⊢ 0 = Multiset.count r (Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots)

Tactic:
·

Goal state after:
case neg
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
h_deg : (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
r : ℝ
hr : ¬(spectralMulAlgNorm K E) ((algebraMap L E) x) = r
⊢ 0 = Multiset.count r (Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ (spectralMulAlgNorm K E) ((algebraMap L E) x) ^ (minpoly K x).natDegree =
    (spectralMulAlgNorm K E) ((mapAlg K E) (minpoly K x)).roots.prod

Tactic:
have h_deg : (minpoly K x).natDegree = Multiset.card ((mapAlg K E) (minpoly K x)).roots := by
    trans (mapAlg K E (minpoly K x)).natDegree
    · rw [mapAlg_eq_map, natDegree_map]
    · rw [eq_comm, ← splits_iff_card_roots]
      exact IsSplittingField.IsScalarTower.splits (K := L) E (minpoly K x)
  rw [map_multiset_prod, ← Multiset.prod_replicate]
  apply congr_arg
  ext r
  rw [Multiset.count_replicate]
  split_ifs with hr
  · have h : ∀ s ∈ Multiset.map (spectralMulAlgNorm K E) ((mapAlg K E) (minpoly K x)).roots,
        r = s := by
      intro s hs
      obtain ⟨a, ha, has⟩ := Multiset.mem_map.mp hs
      rw [← hr, ← has, spectralMulAlgNorm_eq_of_mem_roots K L x ha]
    rwa [Multiset.count_eq_card.mpr h, Multiset.card_map]
  · rw [Multiset.count_eq_zero_of_notMem]
    intro hr_mem
    obtain ⟨e, he, her⟩ := Multiset.mem_map.mp hr_mem
    rw [spectralMulAlgNorm_eq_of_mem_roots K L x he] at her
    exact hr her

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

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x : L
E : Type v := ((mapAlg K L) (minpoly K x)).SplittingField
hspl : ((mapAlg K E) (minpoly K x)).Splits
this✝ : Algebra.IsAlgebraic L E
this : Algebra.IsAlgebraic K E
⊢ ((mapAlg K E) (minpoly K x)).Monic

pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x : L
E : Type v := ((mapAlg K L) (minpoly K x)).SplittingField
hspl : ((mapAlg K E) (minpoly K x)).Splits
this✝ : Algebra.IsAlgebraic L E
this : Algebra.IsAlgebraic K E
⊢ ↑(minpoly K x).natDegree ≠ 0

Tactic:
·

Goal state after:
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x : L
E : Type v := ((mapAlg K L) (minpoly K x)).SplittingField
hspl : ((mapAlg K E) (minpoly K x)).Splits
this✝ : Algebra.IsAlgebraic L E
this : Algebra.IsAlgebraic K E
⊢ ((mapAlg K E) (minpoly K x)).Monic
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x : L
E : Type v := ((mapAlg K L) (minpoly K x)).SplittingField
hspl : ((mapAlg K E) (minpoly K x)).Splits
this✝ : Algebra.IsAlgebraic L E
this : Algebra.IsAlgebraic K E
⊢ ↑(minpoly K x).natDegree ≠ 0

Tactic:
·

Goal state after:
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x : L
E : Type v := ((mapAlg K L) (minpoly K x)).SplittingField
hspl : ((mapAlg K E) (minpoly K x)).Splits
this✝ : Algebra.IsAlgebraic L E
this : Algebra.IsAlgebraic K E
⊢ ↑(minpoly K x).natDegree ≠ 0
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x : L
⊢ spectralNorm K L x = ‖(minpoly K x).coeff 0‖ ^ (1 / ↑(minpoly K x).natDegree)

Tactic:
set E := (mapAlg K L (minpoly K x)).SplittingField
  have hspl : Splits (mapAlg K E (minpoly K x)) :=
    IsSplittingField.IsScalarTower.splits (K := L) E (minpoly K x)
  have : Algebra.IsAlgebraic L E :=
    IsSplittingField.IsScalarTower.isAlgebraic E (mapAlg K L (minpoly K x))
  have : Algebra.IsAlgebraic K E := Algebra.IsAlgebraic.trans K L E
  rw [one_div, Real.eq_rpow_inv (spectralNorm_nonneg x) (norm_nonneg ((minpoly K x).coeff 0)),
    Real.rpow_natCast, @spectralNorm.eq_of_tower K _ E,
    ← @spectralNorm_extends K _ L _ _ ((minpoly K x).coeff 0),
    @spectralNorm.eq_of_tower K _ E _ _ L, ← spectralMulAlgNorm_def,
    ← spectralMulAlgNorm_def, Polynomial.coeff_zero_of_isScalarTower,
    hspl.coeff_zero_eq_prod_roots_of_monic _, map_mul, map_pow,
    map_neg_eq_map, map_one, one_pow, one_mul, spectralNorm_pow_natDegree_eq_prod_roots _ _ x]
  · simp [monic_mapAlg_iff, minpoly.monic (Algebra.IsAlgebraic.isAlgebraic x).isIntegral]
  · exact_mod_cast (minpoly.natDegree_pos (Algebra.IsIntegral.isIntegral x)).ne'

Goal state after:
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
hg : g.HasRightInverse
hf : f.HasRightInverse
⊢ (g ∘SL f).HasRightInverse

Tactic:
obtain ⟨finv, hfinv⟩ := hf
  obtain ⟨ginv, hginv⟩ := hg
  refine ⟨finv.comp ginv, fun x ↦ ?_⟩
  simp only [comp_apply]
  rw [hfinv, hginv]

Goal state after: