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

Goal state before:
pp: 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
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y : R
⊢ seminormFromBounded' f (x + y) ≤ seminormFromBounded' f x + seminormFromBounded' f y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ ‖x‖ ≤ ‖if i = i then x else 0‖

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ B.norm ((algebraMap K L) x) = ‖x‖

Tactic:
by

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

Goal state before:
pp: 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
hna : IsNonarchimedean f
x y z : R
hf : f ((x + y) * z) / f z ≤ f (x * z) / f z ∨ f ((x + y) * z) / f z ≤ f (y * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
hna : IsNonarchimedean Norm.norm
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x + y)) i‖) = ‖(B.repr (x + y)) ixy‖
⊢ ‖(B.repr (x + y)) ixy‖ ≤ max ‖(B.repr x) ixy‖ ‖(B.repr y) ixy‖

Tactic:
by

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

Goal state before:
pp: 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
⊢ ∃ g, g.Monic ∧ f.natDegree = g.natDegree ∧ ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ < ε

Tactic:
by

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

Goal state before:
pp: 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
hna : IsNonarchimedean f
⊢ IsNonarchimedean (seminormFromBounded' f)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
hna : IsNonarchimedean Norm.norm
x y : L
⊢ B.norm (x + y) ≤ max (B.norm x) (B.norm y)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
f_ne_zero : f ≠ 0
⊢ BddAbove (Set.range fun y ↦ f x * (f y / f y))

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
hna' : IsNonarchimedean ⇑(NormedField.toMulRingNorm K)
k : ι
hk :
  ‖∑ x_1, (B.repr x) x_1 * ∑ x, (B.repr y) x * (B.repr (B x_1 * B x)) ixy‖ ≤
    ‖(B.repr x) k‖ * ‖∑ x, (B.repr y) x * (B.repr (B k * B x)) ixy‖
⊢ ∃ k', ‖∑ j, (B.repr y) j • (B.repr (B k * B j)) ixy‖ ≤ ‖(B.repr y) k' • (B.repr (B k * B k')) ixy‖

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
⊢ 0 ∈ lowerBounds (Set.range fun n ↦ μ (x ^ ↑n) ^ (1 / ↑↑n))

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
⊢ μ (x ^ n) = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
⊢ μ (x ^ n) ^ (1 / ↑n) = 0

Tactic:
by

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

Goal state before:
pp: case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
hna' : IsNonarchimedean ⇑(NormedField.toMulRingNorm K)
k : ι
hk :
  ‖∑ x_1, (B.repr x) x_1 * ∑ x, (B.repr y) x * (B.repr (B x_1 * B x)) ixy‖ ≤
    ‖(B.repr x) k‖ * ‖∑ x, (B.repr y) x * (B.repr (B k * B x)) ixy‖
k' : ι
hk' : ‖∑ j, (B.repr y) j • (B.repr (B k * B j)) ixy‖ ≤ ‖(B.repr y) k' • (B.repr (B k * B k')) ixy‖
⊢ ‖(B.repr x) k‖ * (‖(B.repr y) k'‖ * ‖(B.repr (B k * B k')) ixy‖) ≤ B.norm (B c.1 * B c.2) * (B.norm x * B.norm y)

Tactic:
mul_comm (B.norm (B c.fst * B c.snd)),

Goal state after:
case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
hna' : IsNonarchimedean ⇑(NormedField.toMulRingNorm K)
k : ι
hk :
  ‖∑ x_1, (B.repr x) x_1 * ∑ x, (B.repr y) x * (B.repr (B x_1 * B x)) ixy‖ ≤
    ‖(B.repr x) k‖ * ‖∑ x, (B.repr y) x * (B.repr (B k * B x)) ixy‖
k' : ι
hk' : ‖∑ j, (B.repr y) j • (B.repr (B k * B j)) ixy‖ ≤ ‖(B.repr y) k' • (B.repr (B k * B k')) ixy‖
⊢ ‖(B.repr x) k‖ * (‖(B.repr y) k'‖ * ‖(B.repr (B k * B k')) ixy‖) ≤ B.norm x * B.norm y * B.norm (B c.1 * B c.2)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
h0 : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) ^ (1 / ↑n) = 0
n : ℕ+
⊢ 0 ≤ μ (x ^ ↑n) ^ (1 / ↑↑n)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))

Tactic:
by

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
x✝ : R
hx : μ x✝ ≠ 0
L : ℝ := ⨅ n, μ (x✝ ^ ↑n) ^ (1 / ↑↑n)
x : ℕ+
⊢ 0 ≤ μ (x✝ ^ ↑x) ^ (1 / ↑↑x)

Tactic:
by

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

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

Tactic:
simp_rw [f_ne_zero, Pi.zero_apply, zero_div, zero_mul, ciSup_const]

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
⊢ 1 = (algebraMap K L) 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
h1 : 1 = (algebraMap K L) 1
⊢ 0 < B.norm (B i * B i)

Tactic:
rw [hBi, mul_one, h1, Basis.norm_extends hBi]

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
h1 : 1 = (algebraMap K L) 1
⊢ 0 < ‖1‖
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

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

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
⊢ 0 < B.norm (B i * B i)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
⊢ ∃ c, ∃ (_ : 0 < c), ∀ (x y : L), B.norm (x * y) ≤ c * B.norm x * B.norm y

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
k : K
y : L
⊢ B.norm ((algebraMap K L) k * y) = B.norm ((algebraMap K L) k) * B.norm y

Tactic:
by

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

Goal state before:
pp: case a
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

Tactic:
refine ciSup_le (fun y ↦ ?_)
    by_cases hy : f y = 0
    · rw [hy, div_zero]; exact f_nonneg _
    · rw [div_le_iff₀ (lt_of_le_of_ne' (f_nonneg _) hy)]; exact hx _

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
σ : L ≃ₐ[K] L
⊢ (Classical.choose ⋯) (σ 0) = 0

Tactic:
by

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
σ : L ≃ₐ[K] L
x y : L
⊢ (Classical.choose ⋯) (σ (x + y)) ≤ (Classical.choose ⋯) (σ x) + (Classical.choose ⋯) (σ y)

Tactic:
by

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
σ : L ≃ₐ[K] L
x : L
⊢ (Classical.choose ⋯) (σ (-x)) = (Classical.choose ⋯) (σ x)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
hem : Nonempty ι
h1L : 1 ∈ h1.extend ⋯
⊢ B ⟨1, h1L⟩ = 1

Tactic:
by

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
σ : L ≃ₐ[K] L
x y : L
⊢ (Classical.choose ⋯) (σ (x * y)) ≤ (Classical.choose ⋯) (σ x) * (Classical.choose ⋯) (σ y)

Tactic:
by

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
σ : L ≃ₐ[K] L
x : K
y : L
⊢ (Classical.choose ⋯) (σ (x • y)) = ‖x‖ * (Classical.choose ⋯) (σ y)

Tactic:
by

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
σ : L ≃ₐ[K] L
⊢ IsPowMul ⇑(algNormOfAlgEquiv σ)

Tactic:
by

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
⊢ BddAbove (Set.range fun y ↦ f (x * y) / f y)

Tactic:
by

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
σ : L ≃ₐ[K] L
⊢ IsNonarchimedean ⇑(algNormOfAlgEquiv σ)

Tactic:
by

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
σ : L ≃ₐ[K] L
x : K
⊢ (algNormOfAlgEquiv σ) ((algebraMap K L) x) = ‖x‖

Tactic:
by

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
⊢ ⨆ σ, (algNormOfAlgEquiv σ) 0 = 0

Tactic:
by

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
⊢ ⨆ σ, (algNormOfAlgEquiv σ) (-x) = ⨆ σ, (algNormOfAlgEquiv σ) x

Tactic:
by

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
⊢ ⨆ σ, (algNormOfAlgEquiv σ) x = 0 → x = 0

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
hem : Nonempty ι
h1L : 1 ∈ h1.extend ⋯
hB1 : B ⟨1, h1L⟩ = 1
g : L → ℝ := B.norm
hg0 : g 0 = 0
x : L
⊢ 0 ≤ g x

Tactic:
by

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
r : K
x : L
⊢ ⨆ σ, (algNormOfAlgEquiv σ) (r • x) = ‖r‖ * ⨆ σ, (algNormOfAlgEquiv σ) x

Tactic:
by

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
⊢ IsPowMul ⇑(invariantExtension K L)

Tactic:
by

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 : K
⊢ (invariantExtension K L) ((algebraMap K L) x) = ‖x‖

Tactic:
by

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
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2
hε2 : 0 < ε / 2
hL2 : ∀ ε_1 > 0, ∃ N, ∀ n ≥ N, dist ((L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n)) 1 < ε_1
δ : ℝ := ε / (2 * (L + ε / 2))
hδ_def : δ = ε / (2 * (L + ε / 2))
⊢ 0 < δ

Tactic:
by

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
h_bdd : BddAbove (Set.range fun y ↦ f (x * y) / f y)
h0 : ¬f x = 0
⊢ f 1 = 1

Tactic:
by

Goal state after:
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
⊢ Tendsto (fun n ↦ C ^ (1 / ↑n) * nα x) atTop (𝓝 (nα x))

Tactic:
by

Goal state after:
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))
n : ℕ
hn : 1 ≤ n
⊢ (C ^ (1 / ↑n)) ^ n = C

Tactic:
by

Goal state after:
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))
n : ℕ
hn : 1 ≤ n
h : (C ^ (1 / ↑n)) ^ n = C
⊢ 0 ≤ C ^ (1 / ↑n) * nα x

Tactic:
by

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:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ seminormFromBounded' f ≠ 0

Tactic:
by

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

Goal state before:
pp: 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
⊢ seminormFromBounded' f ⁻¹' {0} = f ⁻¹' {0}

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_zero : f 0 = 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
f_neg : ∀ (x : R), f (-x) = f x
h0 : ∀ (x : R), (seminormFromBounded f_zero f_nonneg f_mul f_add f_neg).toFun x = 0 → x = 0
x : R
h : x = 0
⊢ seminormFromBounded' f x = 0

Tactic:
by

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
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2
⊢ ∃ m, ∀ n ≥ m, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (L + ε / 2))

Tactic:
by

Goal state after:
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
hf : RingHom.IsBoundedWrt (⇑nα) nβ f
x : α
⊢ nβ (f x) ≤ nα x

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_zero : f 0 = 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
f_neg : ∀ (x : R), f (-x) = f x
⊢ (∀ (x : R), (seminormFromBounded f_zero f_nonneg f_mul f_add f_neg).toFun x = 0 → x = 0) ↔ f ⁻¹' {0} = {0}

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: F : Type u_1
α : Type u_2
inst✝² : Ring α
inst✝¹ : FunLike F α ℝ
inst✝ : RingSeminormClass F α ℝ
f g : F
hfpm : IsPowMul ⇑f
hgpm : IsPowMul ⇑g
hfg : ∃ r, ∃ (_ : 0 < r), ∀ (a : α), f a ≤ r * g a
hgf : ∃ r, ∃ (_ : 0 < r), ∀ (a : α), g a ≤ r * f a
⊢ f = g

Tactic:
by

Goal state after: