mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ 2 ≤ 2 * n

Tactic:
by

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

Goal state before:
pp: q : ℍ
hq : q.re = 0
c s : ℝ
hc : HasSum (fun n ↦ (-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!) c
hs : HasSum (fun n ↦ (-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)!) s
⊢ HasSum (fun n ↦ (expSeries ℝ ℍ n) fun x ↦ q) (↑c + (s / ‖q‖) • q)

Tactic:
by

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

Goal state before:
pp: q : ℍ
hq : q.re = 0
⊢ exp q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: q : ℍ
⊢ (exp q).re = exp q.re * Real.cos ‖q - ↑q.re‖

Tactic:
by

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

Goal state before:
pp: q : ℍ
⊢ (exp q).im = (exp q.re * (Real.sin ‖q.im‖ / ‖q.im‖)) • q.im

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : NormedField K
inst✝² : SeminormedRing L
inst✝¹ : NormOneClass L
inst✝ : NormedAlgebra K L
h : IsUltrametricDist L
x y z : K
⊢ dist x z ≤ max (dist x y) (dist y z)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedDivisionRing L
inst✝ : NormedAlgebra K L
h : IsUltrametricDist K
⊢ IsUltrametricDist L

Tactic:
by

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

Goal state before:
pp: q : ℍ
⊢ normSq (exp q.re • (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im)) =
    exp q.re ^ 2 * normSq (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im)

Tactic:
by

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

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

Tactic:
by

Goal state after:
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
H : 1 < ‖x‖
⊢ 0 < ‖x‖

Tactic:
by

Goal state after:
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
H : 1 < ‖x‖
⊢ x ≠ 0

Tactic:
by

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

Goal state before:
pp: case inr
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1
x : R
H : 1 < ‖x‖
⊢ ‖x / x + 1 / x‖ ≤ 1

Tactic:
div_self (by simpa using H.trans' zero_lt_one),

Goal state after:
case inr
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1
x : R
H : 1 < ‖x‖
⊢ ‖1 + 1 / x‖ ≤ 1
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
⊢ IsUltrametricDist R

Tactic:
by

Goal state after:
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:
by

Goal state after:
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
⊢ IsUltrametricDist R

Tactic:
by

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
⊢ 0 < max 1 ‖x‖

Tactic:
by

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
⊢ ∃ m, ↑(m + 1) < (a / max 1 ‖x‖) ^ m

Tactic:
by

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

Goal state before:
pp: q : ℍ
⊢ exp q.re ^ 2 * normSq (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im) =
    exp q.re ^ 2 * (Real.cos ‖q.im‖ ^ 2 + Real.sin ‖q.im‖ ^ 2)

Tactic:
by

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
m : ℕ
hm : ↑(m + 1) < a ^ m / max 1 ‖x‖ ^ m
⊢ 0 < max 1 ‖x‖ ^ m

Tactic:
by

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

Goal state before:
pp: q : ℍ
⊢ exp q.re ^ 2 * (Real.cos ‖q.im‖ ^ 2 + Real.sin ‖q.im‖ ^ 2) = exp q.re ^ 2

Tactic:
by

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

Goal state before:
pp: q : ℍ
⊢ ‖exp q‖ = ‖exp q.re‖

Tactic:
by

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
m : ℕ
hm : (m + 1) • max 1 ‖x‖ ^ m < a ^ m
⊢ 1 ∈ {x | 0 ≤ x}

Tactic:
by

Goal state after:
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)‖
⊢ ‖(aeval x) (q w ^ n)‖ + ‖(aeval x) ((q w - q u) ^ n)‖ ≤ M ^ n + ‖φ x w - φ x u‖ ^ n

Tactic:
by

Goal state after:
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
H : ‖φ x z‖ ≠ 0
w : ℝ × ℝ
⊢ ‖φ x w‖ = ‖φ x z‖

Tactic:
by

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
m : ℕ
hm : (m + 1) • max 1 ‖x‖ ^ m < a ^ m
⊢ max 1 ‖x‖ ^ m = max 1 (‖x‖ ^ m)

Tactic:
by

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)
⊢ IsUltrametricDist R

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: 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
this : Tendsto (fun x_1 ↦ ‖(algebraMap ℝ F) x_1.2‖ - M * ‖x‖) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop

Tactic:
by

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

Goal state before:
pp: 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‖
this : Tendsto (fun y ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop
y : ℝ × ℝ
hy : y ∈ {0}ᶜ ×ˢ Set.univ
⊢ ‖y.1‖ * c ≤ ‖y.1‖ * ‖x - (algebraMap ℝ F) (y.1⁻¹ * y.2)‖

Tactic:
by

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

Goal state before:
pp: 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‖
this : Tendsto (fun y ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop
y : ℝ × ℝ
hy : y ∈ {0}ᶜ ×ˢ Set.univ
⊢ ‖y.1‖ * ‖x - (algebraMap ℝ F) (y.1⁻¹ * y.2)‖ = ‖y.1 • x - (algebraMap ℝ F) y.2‖

Tactic:
by

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

Goal state before:
pp: 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‖
this : Tendsto (fun y ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (cobounded ℝ ×ˢ ⊤) atTop

Tactic:
by

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

Goal state before:
pp: 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‖
⊢ Tendsto (fun x_1 ↦ φ x x_1) (cobounded (ℝ × ℝ)) (cobounded F)

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
u : ℝ
hu : IsMinOn (fun x_1 ↦ ‖x - (algebraMap ℝ F) x_1‖) univ u
hc₀ : x = (algebraMap ℝ F) u
x✝ : ℝ × ℝ
⊢ x✝ ∈ univ → x✝ ∈ {x_1 | (fun x_2 ↦ (fun x_3 ↦ ‖φ x x_3‖) (u, 0) ≤ (fun x_3 ↦ ‖φ x x_3‖) x_2) x_1}

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
⊢ IsUltrametricDist R

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
⊢ ∃ z, IsMinOn (fun x_1 ↦ ‖φ x x_1‖) univ z

Tactic:
by

Goal state after:
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
this : φ x z = 0
⊢ (aeval x) (X ^ 2 - C z.1 * X + C z.2) = 0

Tactic:
by

Goal state after:
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
H : φ x z ≠ 0
M : ℝ := ‖φ x z‖
r : ℝ
⊢ √M ≤ ‖x - (algebraMap ℝ F) r‖

Tactic:
by

Goal state after:
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
H : φ x z ≠ 0
M : ℝ := ‖φ x z‖
h' : ∀ (r : ℝ), √M ≤ ‖x - (algebraMap ℝ F) r‖
⊢ 0 < √M

Tactic:
by

Goal state after:
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
⊢ ∃ p, p.IsMonicOfDegree 2 ∧ (aeval x) p = 0

Tactic:
by

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

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedField F
inst✝ : NormedAlgebra ℝ F
⊢ Algebra.IsAlgebraic ℝ F

Tactic:
by

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

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedField F
inst✝ : NormedAlgebra ℝ F
⊢ Nonempty (F ≃ₐ[ℝ] ℝ) ∨ Nonempty (F ≃ₐ[ℝ] ℂ)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalRing R
f g : RingSeminorm R
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalRing R
p : RingSeminorm R
⊢ p ≠ 0 ↔ ∃ x, p x ≠ 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹ : NonUnitalRing R
inst✝ : DecidableEq R
x y : R
⊢ AddGroupSeminorm.toFun 1 (x * y) ≤ AddGroupSeminorm.toFun 1 x * AddGroupSeminorm.toFun 1 y

Tactic:
by

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:
by

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
⊢ p 1 = 1 ↔ p ≠ 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : RingSeminorm R
hna : IsNonarchimedean ⇑p
x y : R
n m : ℕ
hm_lt : m < n + 1
hm : p ((x + y) ^ n) ≤ p (x ^ m) * p (y ^ (n - m))
⊢ p ((x + y) ^ n) ^ (1 / ↑n) ≤ (p (x ^ m) * p (y ^ (n - m))) ^ (1 / ↑n)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : RingSeminorm R
hna : IsNonarchimedean ⇑p
x y : R
n : ℕ
⊢ ∃ m < n + 1, p ((x + y) ^ n) ^ (1 / ↑n) ≤ (p (x ^ m) * p (y ^ (n - m))) ^ (1 / ↑n)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f✝ f f' : AlgebraNorm R S
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) f'
⊢ f = f'

Tactic:
by

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

Goal state before:
pp: F : Type u_2
α : Type u_3
inst✝² : Ring α
inst✝¹ : FunLike F α ℝ
inst✝ : RingSeminormClass F α ℝ
f : F
a : α
x✝ : 1 ≠ 0
⊢ f (a ^ 1) ≤ f a ^ 1

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : NormedField K
f g : K[X]
ε : ℝ
hε : 0 < ε
a : K
ha : eval a f = 0
hfm : f.Monic
hgm : g.Monic
hdeg : g.natDegree = f.natDegree
hcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε
hg : g.Splits
this : (Multiset.map (fun x ↦ ‖a - x‖) g.roots).prod < (↑f.natDegree + 1) * ε * max ‖a‖ 1 ^ ↑f.natDegree
h : ∀ b ∈ g.roots, ((↑f.natDegree + 1) * ε) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1 ≤ ‖a - b‖
⊢ ∀ i ∈ g.roots, 0 ≤ ((↑f.natDegree + 1) * ε) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f : AlgebraNorm R S
hf1 : f 1 = 1
a : R
⊢ f (a • 1) = ‖a‖

Tactic:
by

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

Goal state before:
pp: F : Type u_2
α : Type u_3
inst✝² : Ring α
inst✝¹ : FunLike F α ℝ
inst✝ : RingSeminormClass F α ℝ
f : F
a : α
n : ℕ
x✝ : n + 2 ≠ 0
⊢ f (a ^ (n + 2)) ≤ f a ^ (n + 2)

Tactic:
by

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

Goal state before:
pp: F : Type u_2
α : Type u_3
inst✝² : Ring α
inst✝¹ : FunLike F α ℝ
inst✝ : RingSeminormClass F α ℝ
f : F
hf1 : f 1 ≤ 1
a : α
⊢ f (a ^ 0) ≤ f a ^ 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f : AlgebraNorm R S
hf1 : f 1 = 1
a : R
⊢ f (a • 1) = ‖a‖

Tactic:
exact extends_norm' hf1 _

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

Goal state before:
pp: R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f : AlgebraNorm R S
hf1 : f 1 = 1
a : R
⊢ f ((algebraMap R S) a) = ‖a‖

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f✝ : AlgebraNorm R S
A : Subalgebra R S
f : AlgebraNorm R S
x : ↥A
hx : f ↑x = 0
⊢ x = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : SeminormedCommRing R
S : Type u_2
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
f✝ : AlgebraNorm R S
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra A S
inst✝ : IsScalarTower R A S
hinj : Function.Injective ⇑(algebraMap A S)
f : AlgebraNorm R S
⊢ f ((algebraMap A S) 0) = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : SeminormedCommRing R
S : Type u_2
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
f✝ : AlgebraNorm R S
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra A S
inst✝ : IsScalarTower R A S
hinj : Function.Injective ⇑(algebraMap A S)
f : AlgebraNorm R S
x y : A
⊢ f ((algebraMap A S) (x + y)) ≤ f ((algebraMap A S) x) + f ((algebraMap A S) y)

Tactic:
by

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

Goal state before:
pp: K : Type u_1
inst✝ : NormedField K
f g : K[X]
ε : ℝ
hε : 0 < ε
a : K
ha : eval a f = 0
hfm : f.Monic
hgm : g.Monic
hdeg : g.natDegree = f.natDegree
hcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε
hg : g.Splits
this✝ : (Multiset.map (fun x ↦ ‖a - x‖) g.roots).prod < (↑f.natDegree + 1) * ε * max ‖a‖ 1 ^ ↑f.natDegree
h : ∀ b ∈ g.roots, ((↑f.natDegree + 1) * ε) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1 ≤ ‖a - b‖
this :
  (Multiset.map (fun b ↦ ((↑f.natDegree + 1) * ε) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1) g.roots).prod ≤
    (Multiset.map (fun b ↦ ‖a - b‖) g.roots).prod
⊢ (↑f.natDegree + 1) * ε > 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : SeminormedCommRing R
S : Type u_2
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
f✝ : AlgebraNorm R S
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra A S
inst✝ : IsScalarTower R A S
hinj : Function.Injective ⇑(algebraMap A S)
f : AlgebraNorm R S
x : A
⊢ f ((algebraMap A S) (-x)) = f ((algebraMap A S) x)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : SeminormedCommRing R
S : Type u_2
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
f✝ : AlgebraNorm R S
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra A S
inst✝ : IsScalarTower R A S
hinj : Function.Injective ⇑(algebraMap A S)
f : AlgebraNorm R S
x y : A
⊢ f ((algebraMap A S) (x * y)) ≤ f ((algebraMap A S) x) * f ((algebraMap A S) y)

Tactic:
by

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
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
⊢ ∀ (m : ℕ), p (x ^ s (ψ m)) ^ (1 / ↑(ψ m)) ≤ p x ^ (↑(s (ψ m)) / ↑(ψ m))

Tactic:
by

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
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
h_le : ∀ (m : ℕ), p (x ^ s (ψ m)) ^ (1 / ↑(ψ m)) ≤ p x ^ (↑(s (ψ m)) / ↑(ψ m))
hfx : p x ≤ 1
m : ℕ
⊢ 0 ≤ p x

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : SeminormedCommRing R
S : Type u_2
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
f✝ : AlgebraNorm R S
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra A S
inst✝ : IsScalarTower R A S
hinj : Function.Injective ⇑(algebraMap A S)
f : AlgebraNorm R S
x : A
hx : f ((algebraMap A S) x) = 0
⊢ x = 0

Tactic:
by

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
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
h_le : ∀ (m : ℕ), p (x ^ s (ψ m)) ^ (1 / ↑(ψ m)) ≤ p x ^ (↑(s (ψ m)) / ↑(ψ m))
hfx : p x ≤ 1
m : ℕ
⊢ 0 ≤ ↑(s (ψ m)) / ↑(ψ m)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : SeminormedCommRing R
S : Type u_2
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
f✝ : AlgebraNorm R S
A : Type u_3
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra A S
inst✝ : IsScalarTower R A S
hinj : Function.Injective ⇑(algebraMap A S)
f : AlgebraNorm R S
r : R
x : A
⊢ f ((algebraMap A S) (r • x)) = ‖r‖ * f ((algebraMap A S) x)

Tactic:
by

Goal state after:
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✝ : AlgebraNorm R S
f f' : MulAlgebraNorm R S
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) f'
⊢ f = f'

Tactic:
by

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
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
⊢ IsBoundedUnder LE.le atTop fun n ↦ p (x ^ s (ψ n)) ^ (1 / ↑(ψ n))

Tactic:
by

Goal state after:
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 (a • 1) = ‖a‖

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalRing R
f g : RingNorm R
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedField L
inst✝ : NormedAlgebra K L
r : K
x : L
⊢ (NormedField.toMulRingNorm L).toFun (r • x) = ‖r‖ * (NormedField.toMulRingNorm L).toFun 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
⊢ f 1 ≠ 0

Tactic:
by

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

Goal state before:
pp: A : Type u_2
inst✝ : Ring A
f : MulRingNorm A
x : A
n : ℕ
hn : 1 ≤ n
⊢ f (x ^ n) = f x ^ n

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : NonAssocRing R
f g : MulRingSeminorm R
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

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
x : R
hx : IsUnit x
hfx : f x ≠ 0
n : ℕ
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
h1 : f 1 ≠ 0
hxn : f (x ^ n) = 0
⊢ f 1 ≤ 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : DecidableEq R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
x y : R
⊢ AddGroupSeminorm.toFun 1 (x * y) = AddGroupSeminorm.toFun 1 x * AddGroupSeminorm.toFun 1 y

Tactic:
by

Goal state after: