mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case inr
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormOneClass F
inst✝¹ : NormMulClass F
inst✝ : NormedAlgebra ℂ F
x : F
z : ℂ
hz : IsMinOn (fun x_1 ↦ ‖x - (algebraMap ℂ F) x_1‖) Set.univ z
M : ℝ := ‖x - (algebraMap ℂ F) z‖
hM : M = ‖x - (algebraMap ℂ F) z‖
hM₀ : 0 < M
H : ∀ (z : ℂ), ‖x - (algebraMap ℂ F) z‖ ≠ 0
key : ‖x - (algebraMap ℂ F) (↑‖x‖ + ↑M + 1)‖ = ‖x - (algebraMap ℂ F) z‖
⊢ False

Tactic:
← hM,

Goal state after:
case inr
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormOneClass F
inst✝¹ : NormMulClass F
inst✝ : NormedAlgebra ℂ F
x : F
z : ℂ
hz : IsMinOn (fun x_1 ↦ ‖x - (algebraMap ℂ F) x_1‖) Set.univ z
M : ℝ := ‖x - (algebraMap ℂ F) z‖
hM : M = ‖x - (algebraMap ℂ F) z‖
hM₀ : 0 < M
H : ∀ (z : ℂ), ‖x - (algebraMap ℂ F) z‖ ≠ 0
key : ‖x - (algebraMap ℂ F) (↑‖x‖ + ↑M + 1)‖ = M
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z * I) ^ (2 * k + 1) / ↑(2 * k + 1)! / I =
    ((-z * I) ^ (2 * k) / ↑(2 * k)! - (z * I) ^ (2 * k) / ↑(2 * k)!) * I / 2 +
      ((-z * I) ^ (2 * k + 1) / ↑(2 * k + 1)! - (z * I) ^ (2 * k + 1) / ↑(2 * k + 1)!) * I / 2

Tactic:
pow_succ,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) * (z * I) / ↑(2 * k + 1)! / I =
    ((-z * I) ^ (2 * k) / ↑(2 * k)! - (z * I) ^ (2 * k) / ↑(2 * k)!) * I / 2 +
      ((-z * I) ^ (2 * k) * (-z * I) / ↑(2 * k + 1)! - (z * I) ^ (2 * k) * (z * I) / ↑(2 * k + 1)!) * I / 2
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case succ
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
M : ℝ
hM : 0 ≤ M
h : ∀ (z : ℝ × ℝ), M ≤ ‖φ x z‖
n : ℕ
ih : ∀ {p : ℝ[X]}, p.IsMonicOfDegree (2 * n) → M ^ n ≤ ‖(aeval x) p‖
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * n + 2)
f₁ f₂ : ℝ[X]
hf₁ : f₁.IsMonicOfDegree 2
hf₂ : f₂.IsMonicOfDegree (2 * n)
H : p = f₁ * f₂
a b : ℝ
hab : f₁ = X ^ 2 - C a * X + C b
⊢ M ^ (n + 1) ≤ ‖(aeval x) f₂‖ * ‖(aeval x) f₁‖

Tactic:
pow_succ,

Goal state after:
case succ
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
M : ℝ
hM : 0 ≤ M
h : ∀ (z : ℝ × ℝ), M ≤ ‖φ x z‖
n : ℕ
ih : ∀ {p : ℝ[X]}, p.IsMonicOfDegree (2 * n) → M ^ n ≤ ‖(aeval x) p‖
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * n + 2)
f₁ f₂ : ℝ[X]
hf₁ : f₁.IsMonicOfDegree 2
hf₂ : f₂.IsMonicOfDegree (2 * n)
H : p = f₁ * f₂
a b : ℝ
hab : f₁ = X ^ 2 - C a * X + C b
⊢ M ^ n * M ≤ ‖(aeval x) f₂‖ * ‖(aeval x) f₁‖
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ • q ^ (2 * n + 1) = k⁻¹ • ((-↑(normSq q)) ^ n * q)

Tactic:
pow_succ,

Goal state after:
q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ • (q ^ (2 * n) * q) = k⁻¹ • ((-↑(normSq q)) ^ n * q)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: z : ℂ
⊢ (fun n ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) = fun n ↦ z ^ (2 * n + 1) * I ^ (2 * n + 1) / ↑(2 * n + 1)! / I

Tactic:
pow_succ,

Goal state after:
z : ℂ
⊢ (fun n ↦ (-1) ^ n * (z ^ (2 * n) * z) / ↑(2 * n + 1)!) = fun n ↦
    z ^ (2 * n) * z * (I ^ (2 * n) * I) / ↑(2 * n + 1)! / I
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: case inr.calc_2.e_a
q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖

Tactic:
pow_succ,

Goal state after:
case inr.calc_2.e_a
q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * (‖q‖ ^ (2 * n) * ‖q‖) / k / ‖q‖
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)
x : R
⊢ ‖x + 1‖ ≤ max ‖x‖ 1

Tactic:
rw [max_comm]

Goal state after:
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)
x : R
⊢ ‖x + 1‖ ≤ max 1 ‖x‖
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * (n - 1))
hrel : ‖φ x u‖ * ‖(aeval x) p‖ = ‖(aeval x) (q w ^ n) - (aeval x) ((q w - q u) ^ n)‖
⊢ ‖φ x u‖ * ‖(aeval x) p‖ ≤ M ^ n + ‖φ x u - φ x w‖ ^ n

Tactic:
hrel,

Goal state after:
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * (n - 1))
hrel : ‖φ x u‖ * ‖(aeval x) p‖ = ‖(aeval x) (q w ^ n) - (aeval x) ((q w - q u) ^ n)‖
⊢ ‖(aeval x) (q w ^ n) - (aeval x) ((q w - q u) ^ n)‖ ≤ M ^ n + ‖φ x u - φ x w‖ ^ n
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case refine_1
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‖
s : Set ℝ
hs : Bornology.IsBounded sᶜ
M : ℝ
hM_pos : M > 0
hM : ∀ y ∈ sᶜ, ‖y‖ ≤ M
⊢ Tendsto (fun x_1 ↦ ‖(algebraMap ℝ F) x_1.2‖ - M * ‖x‖) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop

Tactic:
simp only [norm_algebraMap', sub_eq_add_neg]

Goal state after:
case refine_1
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‖
s : Set ℝ
hs : Bornology.IsBounded sᶜ
M : ℝ
hM_pos : M > 0
hM : ∀ y ∈ sᶜ, ‖y‖ ≤ M
⊢ Tendsto (fun x_1 ↦ ‖x_1.2‖ + -(M * ‖x‖)) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop
Module: Mathlib.Analysis.Normed.Unbundled.AlgebraNorm

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : SeminormedCommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
f : MulAlgebraNorm R S
a : R
⊢ f ((algebraMap R S) a) = ‖a‖

Tactic:
rw [Algebra.algebraMap_eq_smul_one]; exact extends_norm' _ _

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

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
x : R
f_mul : 1 ≤ c * f 1
hx : 0 < f x
f_nonneg : 0 ≤ f 1
h1 : f 1 = 0
⊢ 0 ≤ c

Tactic:
linarith [show (1 : ℝ) ≤ 0 by simpa [h1] using f_mul]

Goal state after:
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
f : Filter ι
u v : ι → ℝ
inst✝ : f.NeBot
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u * v)
h' : IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) f fun x ↦ u x * v x
⊢ liminf u f * liminf v f ≤ liminf (u * v) f

Tactic:
apply mul_le_of_forall_lt_of_nonneg (le_liminf_of_le h₂.isCoboundedUnder_ge h₁)
    (le_liminf_of_le h ((h₁.and h₃).mono fun x ⟨u0, v0⟩ ↦ mul_nonneg u0 v0))

Goal state after:
ι : Type u_1
f : Filter ι
u v : ι → ℝ
inst✝ : f.NeBot
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u * v)
h' : IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) f fun x ↦ u x * v x
⊢ ∀ a' ≥ 0, a' < liminf u f → ∀ b' ≥ 0, b' < liminf v f → a' * b' ≤ liminf (u * v) f
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
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 y : R
hy : ¬seminormFromBounded' f y = 0
z : R
hz : ¬f z = 0
hxz : f (x * z) = 0
⊢ f (x * y * z) ≤ seminormFromBounded' f y * f (x * z)

Tactic:
rw [mul_comm x y, mul_assoc, mul_comm y, map_mul_zero_of_map_zero f_nonneg f_mul hxz y]
        exact mul_nonneg (seminormFromBounded_nonneg f_nonneg f_mul y) (f_nonneg _)

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

Goal state before:
pp: K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
x : L
n : ℕ
hn : 1 ≤ n
⊢ (invariantExtension K L) (x ^ n) = (invariantExtension K L) x ^ n

Tactic:
invariantExtension_apply,

Goal state after:
K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
x : L
n : ℕ
hn : 1 ≤ n
⊢ ⨆ σ, (algNormOfAlgEquiv σ) (x ^ n) = (invariantExtension K L) x ^ n
Module: Mathlib.Analysis.Normed.Unbundled.InvariantExtension

Goal state before:
pp: K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
x : L
n : ℕ
hn : 1 ≤ n
⊢ ⨆ σ, (algNormOfAlgEquiv σ) (x ^ n) = (invariantExtension K L) x ^ n

Tactic:
invariantExtension_apply,

Goal state after:
K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
x : L
n : ℕ
hn : 1 ≤ n
⊢ ⨆ σ, (algNormOfAlgEquiv σ) (x ^ n) = (⨆ σ, (algNormOfAlgEquiv σ) x) ^ n
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
n : ℕ
⊢ f c ^ (n + 1) / f c ^ n = f c

Tactic:
pow_succ,

Goal state after:
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
n : ℕ
⊢ f c ^ n * f c / f c ^ n = f c
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
r : R
n : ℕ
h✝ : ¬n = 0
⊢ 0 ≤ ‖r‖

Tactic:
· exact norm_nonneg _

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

Goal state before:
pp: case pos
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 ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : S.Nonempty
⊢ (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ ≤ ‖a - a'‖

Tactic:
apply Finset.min'_le (S.image fun x => ‖a - x‖) (‖a - a'‖)

Goal state after:
case pos
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 := ⋯
this✝ : NormedField F := ⋯
this : NormedAlgebra L F := ⋯
a : f.SplittingField := ⋯
fa0 : (aeval a) f = 0
S : Finset F := ⋯
δ : ℝ := ⋯
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : S.Nonempty
⊢ ‖a - a'‖ ∈ Finset.image (fun x ↦ ‖a - x‖) S
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
g : L → NNReal := fun x ↦ ⟨f x, ⋯⟩
h_card : p.natDegree - m = (Multiset.map g t).card
r : NNReal
hr : r ∈ Multiset.map g t
⊢ r ≤ g y

Tactic:
obtain ⟨_, hzt, hzr⟩ := mem_map.mp hr

Goal state after:
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
g : L → NNReal := fun x ↦ ⟨f x, ⋯⟩
h_card : p.natDegree - m = (Multiset.map g t).card
r : NNReal
hr : r ∈ Multiset.map g t
w✝ : L
hzt : w✝ ∈ t
hzr : g w✝ = r
⊢ r ≤ g y
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
x : L
hx : spectralNorm K L x = 0
hx_alg : IsAlgebraic K x
⊢ x = 0

Tactic:
by_contra h0

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
hx : spectralNorm K L x = 0
hx_alg : IsAlgebraic K x
h0 : ¬x = 0
⊢ False
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✝ : IsUltrametricDist K
k : K
y : L
hy : IsAlgebraic K y
E : IntermediateField K L := K⟮y⟯
h_finiteDimensional_E : FiniteDimensional K ↥E
g : ↥K⟮y⟯ := AdjoinSimple.gen K y
⊢ spectralNorm K L (k • y) = ↑‖k‖₊ * spectralNorm K L y

Tactic:
have hgy : k • y = (algebraMap (↥K⟮y⟯) L) (k • g) := rfl

Goal state after:
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
k : K
y : L
hy : IsAlgebraic K y
E : IntermediateField K L := K⟮y⟯
h_finiteDimensional_E : FiniteDimensional K ↥E
g : ↥K⟮y⟯ := AdjoinSimple.gen K y
hgy : k • y = (algebraMap (↥K⟮y⟯) L) (k • g)
⊢ spectralNorm K L (k • y) = ↑‖k‖₊ * spectralNorm K L y
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
n : R
⊢ smoothingFun μ (-n) = smoothingFun μ n

Tactic:
simp only [smoothingFun]

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
n : R
⊢ ⨅ n_1, μ ((-n) ^ ↑n_1) ^ (1 / ↑↑n_1) = ⨅ n_1, μ (n ^ ↑n_1) ^ (1 / ↑↑n_1)
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
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
x : L
E : Type v := id ↥K⟮x⟯
this✝ : Field E :=
  id
    (have this := inferInstance;
    this)
this : Module K E :=
  id
    (have this := inferInstance;
    this)
id1 : ↥K⟮x⟯ →ₗ[K] E := LinearMap.id
id2 : E →ₗ[K] ↥K⟮x⟯ := LinearMap.id
hs_norm : RingNorm E :=
  { toFun := fun y ↦ spectralNorm K L ↑(id2 y), map_zero' := ⋯, add_le' := ⋯, neg' := ⋯, mul_le' := ⋯,
    eq_zero_of_map_eq_zero' := ⋯ }
n1 : NormedRing E := hs_norm.toNormedRing
e : E
⊢ 1 • e = e

Tactic:
by simp [one_smul]

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
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
x : L
E : Type v := id ↥K⟮x⟯
this✝ : Field E :=
  id
    (have this := inferInstance;
    this)
this : Module K E :=
  id
    (have this := inferInstance;
    this)
id1 : ↥K⟮x⟯ →ₗ[K] E := LinearMap.id
id2 : E →ₗ[K] ↥K⟮x⟯ := LinearMap.id
hs_norm : RingNorm E :=
  { toFun := fun y ↦ spectralNorm K L ↑(id2 y), map_zero' := ⋯, add_le' := ⋯, neg' := ⋯, mul_le' := ⋯,
    eq_zero_of_map_eq_zero' := ⋯ }
n1 : NormedRing E := hs_norm.toNormedRing
N1 : NormedSpace K E :=
  { toSMul := DistribMulAction.toDistribSMul.toSMul, mul_smul := ⋯, one_smul := ⋯, smul_zero := ⋯, smul_add := ⋯,
    add_smul := ⋯, zero_smul := ⋯, norm_smul_le := ⋯ }
hf_norm : RingNorm ↥K⟮x⟯ :=
  { toFun := fun y ↦ f ((algebraMap (↥K⟮x⟯) L) y), map_zero' := ⋯, add_le' := ⋯, neg' := ⋯, mul_le' := ⋯,
    eq_zero_of_map_eq_zero' := ⋯ }
n2 : NormedRing ↥K⟮x⟯ := hf_norm.toNormedRing
e : ↥K⟮x⟯
⊢ 1 • e = e

Tactic:
by simp [one_smul]

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

Goal state before:
pp: 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)
⊢ ‖Equiv.lpPiLp f‖ = ‖f‖

Tactic:
rcases p.trichotomy with (rfl | rfl | h)

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‖

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‖

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.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)).roots.card = ((mapAlg K E) (minpoly K x)).natDegree

Tactic:
← splits_iff_card_roots

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)).Splits
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
hr_mem : r ∈ Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots
e : E
he : e ∈ ((mapAlg K E) (minpoly K x)).roots
her : (spectralMulAlgNorm K E) e = r
⊢ False

Tactic:
spectralMulAlgNorm_eq_of_mem_roots K L x he

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
hr_mem : r ∈ Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots
e : E
he : e ∈ ((mapAlg K E) (minpoly K x)).roots
her : (spectralMulAlgNorm K E) ((algebraMap L E) x) = r
⊢ False
Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm

Goal state before:
pp: ι : Type u_1
inst✝⁷ : Fintype ι
𝕜 : Type u_2
E : ι → Type u_3
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
E' : ι → Type u_5
E'' : ι → Type u_6
inst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E' i)
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E'' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E'' i)
g : (i : ι) → E' i →L[𝕜] E'' i
f : (i : ι) → E i →L[𝕜] E' i
⊢ (mapL fun i ↦ g i ∘SL f i) = mapL g ∘SL mapL f

Tactic:
by
  apply ContinuousLinearMap.coe_injective
  ext; simp

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

Goal state before:
pp: ι : Type u_1
inst✝⁷ : Fintype ι
𝕜 : Type u_2
E : ι → Type u_3
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
F : Type u_4
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E' : ι → Type u_5
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E' i)
f : (i : ι) → E i →L[𝕜] E' i
h : ContinuousMultilinearMap 𝕜 E' F
⊢ (liftIsometry 𝕜 E' F) h ∘SL mapL f = (liftIsometry 𝕜 E F) (h.compContinuousLinearMap f)

Tactic:
by
  apply ContinuousLinearMap.coe_injective
  ext; simp

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

Goal state before:
pp: ι : Type u_1
inst✝³ : Fintype ι
𝕜 : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → SeminormedAddCommGroup (E i)
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ (mapL fun i ↦ ContinuousLinearMap.id 𝕜 (E i)) = ContinuousLinearMap.id 𝕜 (⨂[𝕜] (i : ι), E i)

Tactic:
by
  apply ContinuousLinearMap.coe_injective
  ext; simp

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

Goal state before:
pp: case inr.inl
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)
q r : ℝ≥0∞
e : (i : ι) → E i
f : (i : ι) → F i
hf : Memℓp f q
B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i
K : ℝ
hBK : ∀ (i : ι), ‖B i‖ ≤ K
h : Nonempty ι
hK : 0 ≤ K
hpqr : HolderTriple 0 q r
he : Memℓp e 0
hpqr' : r⁻¹ = 0⁻¹ + q⁻¹
⊢ Memℓp (fun i ↦ ((B i) (e i)) (f i)) r

Tactic:
simp_all only [ENNReal.inv_zero, top_add, inv_eq_top]

Goal state after:
case inr.inl
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)
q r : ℝ≥0∞
e : (i : ι) → E i
f : (i : ι) → F i
hf : Memℓp f q
B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i
K : ℝ
hBK : ∀ (i : ι), ‖B i‖ ≤ K
h : Nonempty ι
hK : 0 ≤ K
hpqr : HolderTriple 0 q 0
he : Memℓp e 0
hpqr' : r = 0
⊢ Memℓp (fun i ↦ ((B i) (e i)) (f i)) 0