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
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:
]

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 w - φ x u‖ ^ n
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:
]

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
⊢ ‖x + 1‖ ≤ a

Tactic:
have hp : max 1 ‖x‖ ^ m = max 1 (‖x‖ ^ m) := by
    rw [pow_left_monotoneOn.map_max (by simp [zero_le_one]) (norm_nonneg x), one_pow]

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
a : ℝ
ha : max 1 ‖x‖ < a
ha' : 1 < a
m : ℕ
hm : (m + 1) • max 1 ‖x‖ ^ m < a ^ m
hp : max 1 ‖x‖ ^ m = max 1 (‖x‖ ^ m)
⊢ ‖x + 1‖ ≤ a
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
hp : max 1 ‖x‖ ^ m = max 1 (‖x‖ ^ m)
⊢ ‖x + 1‖ ≤ a

Tactic:
]

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
a : ℝ
ha : max 1 ‖x‖ < a
ha' : 1 < a
m : ℕ
hm : (m + 1) • max 1 (‖x‖ ^ m) < a ^ m
hp : max 1 ‖x‖ ^ m = max 1 (‖x‖ ^ m)
⊢ ‖x + 1‖ ≤ a
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 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (cobounded (ℝ × ℝ)) atTop

Tactic:
]

Goal state after:
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 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (cobounded (ℝ × ℝ)) atTop
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:
]

Goal state after:
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x✝ : R
m : ℕ
x : R
n : ℕ
⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖
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
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop

Tactic:
]

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
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
⊢ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k ≤ ∑ _x ∈ Finset.range (m + 1), max 1 (‖x‖ ^ m)

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
⊢ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k ≤ ∑ _x ∈ Finset.range (m + 1), max 1 (‖x‖ ^ m)
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
w : ℝ × ℝ
hw : w ∈ sᶜ ×ˢ Set.univ
⊢ ‖(algebraMap ℝ F) w.2‖ - M * ‖x‖ ≤ ‖(algebraMap ℝ F) w.2 - w.1 • x‖

Tactic:
]

Goal state after:
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
w : ℝ × ℝ
hw : w ∈ sᶜ ×ˢ Set.univ
⊢ ‖(algebraMap ℝ F) w.2‖ - M * ‖x‖ ≤ ‖(algebraMap ℝ F) w.2 - w.1 • x‖
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: case inl
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm : max 1 (‖x‖ ^ m) = 1
hx : ‖x‖ ^ m ≤ 1
⊢ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k ≤ ∑ _x ∈ Finset.range (m + 1), 1

Tactic:
]

Goal state after:
case inl
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm : max 1 (‖x‖ ^ m) = 1
hx : ‖x‖ ^ m ≤ 1
⊢ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k ≤ ∑ _x ∈ Finset.range (m + 1), 1
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: case inr
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm : max 1 (‖x‖ ^ m) = ‖x‖ ^ m
hx : 1 < ‖x‖ ^ m
⊢ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k ≤ ∑ _x ∈ Finset.range (m + 1), ‖x‖ ^ m

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm : max 1 (‖x‖ ^ m) = ‖x‖ ^ m
hx : 1 < ‖x‖ ^ m
⊢ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k ≤ ∑ _x ∈ Finset.range (m + 1), ‖x‖ ^ m
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: case h.inr
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm✝ : max 1 (‖x‖ ^ m) = 1
hx : ‖x‖ ≤ 1
i : ℕ
hi : i ∈ Finset.range (m + 1)
hm : m ≠ 0
⊢ ‖x‖ ^ i ≤ 1

Tactic:
]

Goal state after:
case h.inr
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm✝ : max 1 (‖x‖ ^ m) = 1
hx : ‖x‖ ≤ 1
i : ℕ
hi : i ∈ Finset.range (m + 1)
hm : m ≠ 0
⊢ ‖x‖ ^ i ≤ 1
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case refine_2
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 Prod.fst (cobounded ℝ ×ˢ ⊤) (cobounded ℝ)

Tactic:
]

Goal state after:
case refine_2
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 Prod.fst (cobounded ℝ ×ˢ ⊤) (cobounded ℝ)
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case inl
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
⊢ ∃ z, IsMinOn (fun x_1 ↦ ‖φ x x_1‖) univ z

Tactic:
]

Goal state after:
case inl
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
⊢ ∃ z, IsMinOn (fun x_1 ↦ ‖φ x x_1‖) univ z
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
⊢ φ x z = 0

Tactic:
]

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
this : φ x z = 0
⊢ φ x z = 0
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
⊢ ‖φ x z‖ ≤ ‖x ^ 2 - 2 * x * (algebraMap ℝ F) r + (algebraMap ℝ F) r ^ 2‖

Tactic:
]

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
H : φ x z ≠ 0
M : ℝ := ‖φ x z‖
r : ℝ
⊢ ‖φ x z‖ ≤ ‖x ^ 2 - 2 * x * (algebraMap ℝ F) r + (algebraMap ℝ F) r ^ 2‖
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
⊢ 1 ≠ 0

Tactic:
]

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

Goal state before:
pp: case mk.mk
R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f : AlgebraNorm R S
toRingNorm✝¹ : RingNorm S
smul'✝¹ : ∀ (a : R) (x : S), toRingNorm✝¹.toFun (a • x) = ‖a‖ * toRingNorm✝¹.toFun x
toRingNorm✝ : RingNorm S
smul'✝ : ∀ (a : R) (x : S), toRingNorm✝.toFun (a • x) = ‖a‖ * toRingNorm✝.toFun x
h : ⇑toRingNorm✝¹.toRingSeminorm = ⇑toRingNorm✝.toRingSeminorm
s : S
⊢ toRingNorm✝.toRingSeminorm s = toRingNorm✝ s

Tactic:
]

Goal state after:
case mk.mk
R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f : AlgebraNorm R S
toRingNorm✝¹ : RingNorm S
smul'✝¹ : ∀ (a : R) (x : S), toRingNorm✝¹.toFun (a • x) = ‖a‖ * toRingNorm✝¹.toFun x
toRingNorm✝ : RingNorm S
smul'✝ : ∀ (a : R) (x : S), toRingNorm✝.toFun (a • x) = ‖a‖ * toRingNorm✝.toFun x
h : ⇑toRingNorm✝¹.toRingSeminorm = ⇑toRingNorm✝.toRingSeminorm
s : S
⊢ toRingNorm✝.toRingSeminorm s = toRingNorm✝ s
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‖ * f 1

Tactic:
]

Goal state after:
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‖ * f 1
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.succ * f a ≤ f a ^ (n + 1) * f a

Tactic:
]

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:
]

Goal state after:
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‖
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:
]

Goal state after:
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
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)) ^ (1 / ↑(ψ m))

Tactic:
]

Goal state after:
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)) ^ (1 / ↑(ψ m))
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)) ^ (1 / ↑(ψ m))

Tactic:
]

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
⊢ (algebraMap A S) x = 0

Tactic:
]

Goal state after:
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
⊢ (algebraMap A S) x = 0
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) = f (a • 1)

Tactic:
]

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:
]

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

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

Tactic:
]

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 : (↑f.natDegree + 1) * ε * max ‖a‖ 1 ^ ↑f.natDegree ≤ (Multiset.map (fun b ↦ ‖a - b‖) g.roots).prod
⊢ False

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 :
  ((↑f.natDegree + 1) * ε) ^ ((↑f.natDegree)⁻¹ * ↑f.natDegree) * max ‖a‖ 1 ^ ↑f.natDegree ≤
    (Multiset.map (fun b ↦ ‖a - b‖) g.roots).prod
⊢ ↑f.natDegree ≠ 0

Tactic:
]

Goal state after:
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 : (↑f.natDegree + 1) * ε * max ‖a‖ 1 ^ ↑f.natDegree ≤ (Multiset.map (fun b ↦ ‖a - b‖) g.roots).prod
⊢ False

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 :
  ((↑f.natDegree + 1) * ε) ^ ((↑f.natDegree)⁻¹ * ↑f.natDegree) * max ‖a‖ 1 ^ ↑f.natDegree ≤
    (Multiset.map (fun b ↦ ‖a - b‖) g.roots).prod
⊢ ↑f.natDegree ≠ 0
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
⊢ limsup u f + liminf v f ≤ limsup (u + v) f

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
⊢ limsup u f + liminf v f ≤ limsup (u + v) f
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
a : α
a_u : a > limsup u f
b : α
b_v : b > limsup v f
⊢ ∀ y > a + b, ∀ᶠ (a : ι) in f, (u + v) a < y

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
a : α
a_u : a > limsup u f
b : α
b_v : b > limsup v f
⊢ ∀ y > a + b, ∀ᶠ (a : ι) in f, (u + v) a < y
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:
]

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.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
a : α
a_u : a > limsup u f
b : α
b_v : b > limsup v f
⊢ ∀ y > a + b, ∀ᶠ (a : ι) in f, (u + v) a < y

Tactic:
intro c c_ab

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
a : α
a_u : a > limsup u f
b : α
b_v : b > limsup v f
c : α
c_ab : c > a + b
⊢ ∀ᶠ (a : ι) in f, (u + v) a < c
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
⊢ 0 ≤ 0

Tactic:
]

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
f_mul : 1 ≤ c * f 1
hx : 0 < f x
⊢ 0 ≤ c

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
x : R
f_mul : 1 ≤ c * f 1
hx : 0 < f x
⊢ 0 ≤ c
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: R✝ : Type u_1
R : Type u_2
inst✝ : NormedRing R
x : R
hx : ‖x‖ = 0
⊢ ‖x‖ = 0

Tactic:
]

Goal state after:
R✝ : Type u_1
R : Type u_2
inst✝ : NormedRing R
x : R
hx : ‖x‖ = 0
⊢ ‖x‖ = 0
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: R✝ : Type u_1
R : Type u_2
inst✝ : NormedField R
x : R
hx : ‖x‖ = 0
⊢ ‖x‖ = 0

Tactic:
]

Goal state after:
R✝ : Type u_1
R : Type u_2
inst✝ : NormedField R
x : R
hx : ‖x‖ = 0
⊢ ‖x‖ = 0
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

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

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
x : R
f_mul : 1 / f 1 ≤ c
hx : 0 < f x
f_nonneg : 0 ≤ f 1
h1 : 0 < f 1
⊢ 0 ≤ c
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) 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 (u + v)
a : α
a_u : a < liminf u f
b : α
b_v : b < liminf v f
⊢ ∀ y < a + b, ∀ᶠ (a : ι) in f, y < (u + v) a

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) 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 (u + v)
a : α
a_u : a < liminf u f
b : α
b_v : b < liminf v f
⊢ ∀ y < a + b, ∀ᶠ (a : ι) in f, y < (u + v) a