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

Goal state before:
pp: case h₂
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖r₁ • m₂ + MulOpposite.op r₂ • m₁‖ ≤ ‖r₁‖ * ‖m₂‖ + ‖r₂‖ * ‖m₁‖

Tactic:
refine norm_add_le_of_le ?_ ?_

Goal state after:
case h₂.refine_1
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖r₁ • m₂‖ ≤ ‖r₁‖ * ‖m₂‖

case h₂.refine_2
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖MulOpposite.op r₂ • m₁‖ ≤ ‖r₂‖ * ‖m₁‖
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case inl
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
⊢ ∃ z, ‖x - (algebraMap ℂ F) z‖ = 0

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
⊢ ∃ z, ‖x - (algebraMap ℂ F) z‖ = 0

Tactic:
· exact ⟨z, hM₀.symm⟩

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
⊢ ∃ z, ‖x - (algebraMap ℂ F) z‖ = 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

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

Tactic:
mul_assoc

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

Goal state before:
pp: q : ℍ
this : NormedAlgebra ℚ ℍ
⊢ exp q = exp q.re • (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im)

Tactic:
← exp_of_re_eq_zero q.im q.re_im,

Goal state after:
q : ℍ
this : NormedAlgebra ℚ ℍ
⊢ exp q = exp q.re • exp q.im
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
hsub : q w - q u = (C u.1 - C w.1) * X + C w.2 - C u.2
hdvd : q u ∣ q w ^ n - (q w - q u) ^ n
H' : ((q w - q u) ^ n).natDegree < 2 * n
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * n - 2)
hrel : q w ^ n = p * q u + (q w - q u) ^ n
⊢ ‖φ x u‖ * M ^ (n - 1) ≤ M ^ n + ‖φ x u - φ x w‖ ^ n

Tactic:
clear H' hdvd hsub

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 - 2)
hrel : q w ^ n = p * q u + (q w - q u) ^ n
⊢ ‖φ x u‖ * M ^ (n - 1) ≤ M ^ n + ‖φ x u - φ x w‖ ^ n
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (x : R), ‖x‖ ≤ 1 → ‖x - 1‖ ≤ 1
x : R
hx : ‖x‖ ≤ 1
⊢ ‖x + 1‖ ≤ 1

Tactic:
simpa only [← neg_add', norm_neg] using h (-x) (norm_neg x ▸ hx)

Goal state after:
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
a : ℝ
ha : max 1 ‖x‖ < a
ha' : 1 < a
⊢ 1 < a / max 1 ‖x‖

Tactic:
rwa [one_lt_div (by positivity)]

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

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p 1 = 1
⊢ p 1 ≠ 0

Tactic:
rw [h]; exact one_ne_zero

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

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p ≠ 0
⊢ p 1 = 1

Tactic:
obtain hp0 | hp0 := (apply_nonneg p (1 : R)).eq_or_lt'

Goal state after:
case inl
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p ≠ 0
hp0 : p 1 = 0
⊢ p 1 = 1

case inr
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p ≠ 0
hp0 : 0 < p 1
⊢ p 1 = 1
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: case inl
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p ≠ 0
hp0 : p 1 = 0
⊢ False

Tactic:
refine h (ext fun x => (apply_nonneg _ _).antisymm' ?_)

Goal state after:
case inl
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
h : p ≠ 0
hp0 : p 1 = 0
x : R
⊢ p x ≤ 0
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: R : Type u_1
K : Type u_2
inst✝ : Field K
f : RingSeminorm K
hnt : f ≠ 0
x : K
hx : f.toFun x = 0
c : K
hc : f c ≠ 0
hn0 : ¬x = 0
⊢ f (x * c * x⁻¹) = 0

Tactic:
mul_assoc

Goal state after:
R : Type u_1
K : Type u_2
inst✝ : Field K
f : RingSeminorm K
hnt : f ≠ 0
x : K
hx : f.toFun x = 0
c : K
hc : f c ≠ 0
hn0 : ¬x = 0
⊢ f (x * (c * x⁻¹)) = 0
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 y : R
hy : ¬seminormFromBounded' f y = 0
z : R
hz : ¬f z = 0
hxz : ¬f (x * z) = 0
⊢ f (y * x * z) ≤ f (y * (x * z))

Tactic:
mul_assoc

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 y : R
hy : ¬seminormFromBounded' f y = 0
z : R
hz : ¬f z = 0
hxz : ¬f (x * z) = 0
⊢ f (y * (x * z)) ≤ f (y * (x * z))
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: case h
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : f.natDegree = 0
⊢ Monic 1 ∧ natDegree 1 = natDegree 1 ∧ ∀ (n : ℕ), ‖(map (algebraMap K L) 1).coeff n - coeff 1 n‖ < ε

Tactic:
simp only [monic_one, natDegree_one, Polynomial.map_one, sub_self, norm_zero, hε,
      implies_true, and_self]

Goal state after:
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:
simp_rw [seminormFromBounded']

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
⊢ ⨆ y, f (x * y) / f y = f x
Module: Mathlib.Analysis.Normed.Unbundled.IsPowMulFaithful

Goal state before:
pp: F : Type u_1
α : Type u_2
inst✝³ : Ring α
inst✝² : FunLike F α ℝ
inst✝¹ : RingSeminormClass F α ℝ
β : Type u_3
inst✝ : Ring β
nα : F
nβ : β → ℝ
hβ : IsPowMul nβ
f : α →+* β
x : α
C : ℝ
hC0 : 0 < C
hC : ∀ (x : α), nβ (f x) ≤ C * nα x
hlim : Tendsto (fun n ↦ C ^ (1 / ↑n) * nα x) atTop (𝓝 (nα x))
⊢ ∀ᶠ (c : ℕ) in atTop, nβ (f x) ≤ C ^ (1 / ↑c) * nα x

Tactic:
simp only [eventually_atTop]

Goal state after:
F : Type u_1
α : Type u_2
inst✝³ : Ring α
inst✝² : FunLike F α ℝ
inst✝¹ : RingSeminormClass F α ℝ
β : Type u_3
inst✝ : Ring β
nα : F
nβ : β → ℝ
hβ : IsPowMul nβ
f : α →+* β
x : α
C : ℝ
hC0 : 0 < C
hC : ∀ (x : α), nβ (f x) ≤ C * nα x
hlim : Tendsto (fun n ↦ C ^ (1 / ↑n) * nα x) atTop (𝓝 (nα x))
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → nβ (f x) ≤ C ^ (1 / ↑b) * nα x
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
x : R
⊢ ∀ᶠ (c_1 : ℕ) in atTop, seminormFromConst_seq c f x c_1 ≤ f x

Tactic:
simp only [eventually_atTop]

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
x : R
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → seminormFromConst_seq c f x b ≤ f x
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case neg
R : Type u_1
inst✝ : SeminormedRing R
p : R[X]
n : ℕ
h : ¬n < p.natDegree
⊢ 0 ≤ 0

Tactic:
exact le_refl _

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

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

Tactic:
exact le_refl _

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

Goal state before:
pp: R : Type u_1
inst✝ : NormedDivisionRing R
P : R[X]
hP : P.Monic
h : ∀ (n : ℕ), ‖P.coeff n‖ ≤ 1
n : ℕ
⊢ (if n < P.natDegree then ‖P.coeff n‖ ^ (1 / (↑P.natDegree - ↑n)) else 0) ≤ 1

Tactic:
split_ifs with hn

Goal state after:
case pos
R : Type u_1
inst✝ : NormedDivisionRing R
P : R[X]
hP : P.Monic
h : ∀ (n : ℕ), ‖P.coeff n‖ ≤ 1
n : ℕ
hn : n < P.natDegree
⊢ ‖P.coeff n‖ ^ (1 / (↑P.natDegree - ↑n)) ≤ 1

case neg
R : Type u_1
inst✝ : NormedDivisionRing R
P : R[X]
hP : P.Monic
h : ∀ (n : ℕ), ‖P.coeff n‖ ≤ 1
n : ℕ
hn : ¬n < P.natDegree
⊢ 0 ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
⊢ ∀ᶠ (c : ℕ) in atTop, smoothingSeminormSeq μ 1 c ≤ 1

Tactic:
simp only [eventually_atTop]

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → smoothingSeminormSeq μ 1 b ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
n : ℕ
hn0 : n = 0
⊢ ↑(mu μ hn n) / ↑n ≤ 1

Tactic:
rw [hn0, cast_zero, div_zero]

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
n : ℕ
hn0 : n = 0
⊢ 0 ≤ 1
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 (explicitCokernelπ f ≫ explicitCokernelDesc cond)) x = (ConcreteCategory.hom g) x

Tactic:
rw [explicitCokernelπ_desc]

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

Goal state before:
pp: X Y : SemiNormedGrp
f : X ⟶ Y
⊢ explicitCokernelπ f ≫ (explicitCokernelIso f).hom = cokernel.π f

Tactic:
by
  simp [explicitCokernelπ, explicitCokernelIso, IsColimit.coconePointUniqueUpToIso]

Goal state after:
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
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
hy : -y = (algebraMap (↥K⟮y⟯) L) (-g)
⊢ spectralNorm K L ((algebraMap (↥K⟮y⟯) L) (-g)) =
    spectralNorm K (↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯)))
      ((algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g)

Tactic:
← spectralNorm.eq_of_normalClosure (-g) hy,

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
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
hy : -y = (algebraMap (↥K⟮y⟯) L) (-g)
⊢ spectralNorm K (↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯)))
      ((algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) (-g)) =
    spectralNorm K (↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯)))
      ((algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
n✝ : R
n : ℕ+
hpos : (-1) ^ ↑n = 1
⊢ μ ((-1) ^ ↑n * n✝ ^ ↑n) ^ (1 / ↑↑n) = μ (n✝ ^ ↑n) ^ (1 / ↑↑n)

Tactic:
hpos,

Goal state after:
case inl
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
n✝ : R
n : ℕ+
hpos : (-1) ^ ↑n = 1
⊢ μ (1 * n✝ ^ ↑n) ^ (1 / ↑↑n) = μ (n✝ ^ ↑n) ^ (1 / ↑↑n)
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
N2 : NormedSpace K ↥K⟮x⟯ :=
  { toSMul := K⟮x⟯.algebra'.toSMul, mul_smul := ⋯, one_smul := ⋯, smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯,
    zero_smul := ⋯, norm_smul_le := ⋯ }
⊢ ∃ C₁ C₂,
    ∃ (_ : 0 < C₁) (_ : 0 < C₂),
      ∀ (x_3 : ↥K[x]), f ↑x_3 ≤ C₁ * (spectralAlgNorm K L) ↑x_3 ∧ (spectralAlgNorm K L) ↑x_3 ≤ C₂ * f ↑x_3

Tactic:
have hKx_fin : FiniteDimensional K ↥K⟮x⟯ :=
    IntermediateField.adjoin.finiteDimensional (Algebra.IsAlgebraic.isAlgebraic x).isIntegral

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
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
N2 : NormedSpace K ↥K⟮x⟯ :=
  { toSMul := K⟮x⟯.algebra'.toSMul, mul_smul := ⋯, one_smul := ⋯, smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯,
    zero_smul := ⋯, norm_smul_le := ⋯ }
hKx_fin : FiniteDimensional K ↥K⟮x⟯
⊢ ∃ C₁ C₂,
    ∃ (_ : 0 < C₁) (_ : 0 < C₂),
      ∀ (x_3 : ↥K[x]), f ↑x_3 ≤ C₁ * (spectralAlgNorm K L) ↑x_3 ∧ (spectralAlgNorm K L) ↑x_3 ≤ C₂ * f ↑x_3
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: case empty
ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
⊢ ‖∏ i ∈ ∅, (1 + f i) - 1‖ ≤ Real.exp (∑ i ∈ ∅, ‖f i‖) - 1

Tactic:
| empty =>

Goal state after:
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + (↑Q - ↑P * ↑Q) = ↑P * ↑R + (↑Q * ↑R - ↑P * ↑Q * ↑R)

Tactic:
mul_assoc

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + (↑Q - ↑P * ↑Q) = ↑P * ↑R + (↑Q * ↑R - ↑P * (↑Q * ↑R))
Module: Mathlib.Analysis.Normed.Lp.lpHolder

Goal state before:
pp: case h₁
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
this : ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤ C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal)
i : ι
hi : i ∈ s
⊢ ‖((B i) (e i)) (f i)‖ ≤ K * ‖e i‖ * ‖f i‖

Tactic:
exact (B i (e i)).le_of_opNorm_le ((B i).le_of_opNorm_le (hBK i) _) _

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : ι → Type u_3
F : ι → Type u_4
G : ι → Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : (i : ι) → NormedAddCommGroup (F i)
inst✝² : (i : ι) → NormedSpace 𝕜 (F i)
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → NormedSpace 𝕜 (G i)
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
this : ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤ C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal)
⊢ K ^ r.toReal * ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤
    K ^ r.toReal * C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal)

Tactic:
mul_assoc

Goal state after:
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
this : ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤ C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal)
⊢ K ^ r.toReal * ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤
    K ^ r.toReal * (C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal))
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
⊢ w ∈ s

Tactic:
refine hs (fun i => ?_) hz

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
⊢ w ∈ s

Tactic:
refine hs (fun i => ?_) hz

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
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: case neg
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this : IsUnit (adjoint ↑y ∘SL ↑y)
hα_re : α = ↑(RCLike.re α)
hα_ne_zero : α ≠ 0
hα_nonneg : 0 ≤ α
hα_pos : 0 < RCLike.re α
h_comp_adjoint : ↑y ∘SL adjoint ↑y = α • ContinuousLinearMap.id 𝕜 W
β : 𝕜 := ↑(RCLike.re α ^ (-(1 / 2)))
hβ : β * β = α⁻¹
U : V ≃ₗᵢ[𝕜] W := auxIsometry y hα_ne_zero hβ ⋯ h_comp_adjoint ⋯
⊢ ∃ U, f = U.conjStarAlgEquiv

Tactic:
use U

Goal state after:
case h
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this : IsUnit (adjoint ↑y ∘SL ↑y)
hα_re : α = ↑(RCLike.re α)
hα_ne_zero : α ≠ 0
hα_nonneg : 0 ≤ α
hα_pos : 0 < RCLike.re α
h_comp_adjoint : ↑y ∘SL adjoint ↑y = α • ContinuousLinearMap.id 𝕜 W
β : 𝕜 := ⋯
hβ : β * β = α⁻¹
U : V ≃ₗᵢ[𝕜] W := ⋯
⊢ f = U.conjStarAlgEquiv