mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.Submodule

Goal state before:
pp: R : Type u_3
R' : Type u_4
M : Type u_5
M' : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Semiring R'
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M'
inst✝³ : Module R M
inst✝² : Module R' M'
σ₁₂ : R →+* R'
f : M →ₛₗ[σ₁₂] M'
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace M'
hf : Continuous ⇑f
p : Submodule R M
⊢ Continuous ((↑p).domRestrict ⇑f)

Tactic:
]

Goal state after:
R : Type u_3
R' : Type u_4
M : Type u_5
M' : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Semiring R'
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M'
inst✝³ : Module R M
inst✝² : Module R' M'
σ₁₂ : R →+* R'
f : M →ₛₗ[σ₁₂] M'
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace M'
hf : Continuous ⇑f
p : Submodule R M
⊢ Continuous ((↑p).domRestrict ⇑f)
Module: Mathlib.Topology.MetricSpace.Algebra

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : PseudoMetricSpace α
inst✝⁵ : PseudoMetricSpace β
inst✝⁴ : Zero α
inst✝³ : Zero β
inst✝² : SMul α β
inst✝¹ : IsBoundedSMul α β
X : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace X
F : ι → X → α
G : ι → X → β
f : X → α
g : X → β
l : Filter ι
hF : TendstoLocallyUniformlyOn F f l univ
hG : TendstoLocallyUniformlyOn G g l univ
hf : ∀ (x : X), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x) fun y ↦ dist (f y) 0
hg : ∀ (x : X), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x) fun y ↦ dist (g y) 0
⊢ TendstoLocallyUniformlyOn (F • G) (f • g) l univ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : PseudoMetricSpace α
inst✝⁵ : PseudoMetricSpace β
inst✝⁴ : Zero α
inst✝³ : Zero β
inst✝² : SMul α β
inst✝¹ : IsBoundedSMul α β
X : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace X
F : ι → X → α
G : ι → X → β
f : X → α
g : X → β
l : Filter ι
hF : TendstoLocallyUniformlyOn F f l univ
hG : TendstoLocallyUniformlyOn G g l univ
hf : ∀ (x : X), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x) fun y ↦ dist (f y) 0
hg : ∀ (x : X), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x) fun y ↦ dist (g y) 0
⊢ TendstoLocallyUniformlyOn (F • G) (f • g) l univ
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝¹ : SeminormedGroup E
inst✝ : SeminormedGroup F
f : E → F
hi : Isometry f
h₁ : f 1 = 1
x : E
⊢ ‖x‖ = ‖x‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedGroup E
a b : E
⊢ dist b (b * a) = dist b (b * a)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedGroup E
a b : E
⊢ ‖a‖ = ‖a‖

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
inst✝³ : Group G
inst✝² : PseudoEMetricSpace G
inst✝¹ : IsIsometricSMul G G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
⊢ edist a b = edist a b

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
inst✝³ : Group G
inst✝² : PseudoEMetricSpace G
inst✝¹ : IsIsometricSMul G G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
x y : G
⊢ edist x y⁻¹ = edist x y⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
inst✝³ : Group G
inst✝² : PseudoEMetricSpace G
inst✝¹ : IsIsometricSMul G G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b c : G
⊢ edist b c = edist b c

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
X : Type w
inst✝³ : PseudoEMetricSpace X
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsIsometricSMul G X
c : G
x : X
r : ℝ≥0∞
⊢ eball (c⁻¹ • x) r = eball (c⁻¹ • x) r

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
X : Type w
inst✝³ : PseudoEMetricSpace X
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsIsometricSMul G X
c : G
x : X
r : ℝ≥0∞
⊢ closedEBall (c⁻¹ • x) r = closedEBall (c⁻¹ • x) r

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
inst✝² : Group G
inst✝¹ : PseudoEMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ≥0∞
⊢ (fun x ↦ x * a) ⁻¹' eball b r = eball (b * a⁻¹) r

Tactic:
]

Goal state after:
G : Type v
inst✝² : Group G
inst✝¹ : PseudoEMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ≥0∞
⊢ (fun x ↦ x * a) ⁻¹' eball b r = eball (b * a⁻¹) r
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
inst✝² : Group G
inst✝¹ : PseudoEMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ≥0∞
⊢ (fun x ↦ x * a) ⁻¹' closedEBall b r = closedEBall (b * a⁻¹) r

Tactic:
]

Goal state after:
G : Type v
inst✝² : Group G
inst✝¹ : PseudoEMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ≥0∞
⊢ (fun x ↦ x * a) ⁻¹' closedEBall b r = closedEBall (b * a⁻¹) r
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: case refine_1
𝓕 : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : SeminormedGroup E
inst✝² : SeminormedGroup F
inst✝¹ : FunLike 𝓕 E F
inst✝ : MonoidHomClass 𝓕 E F
f : 𝓕
x✝ : ∃ K, AntilipschitzWith K ⇑f
K : ℝ≥0
hK : AntilipschitzWith K ⇑f
x : E
⊢ (↑K + 1)⁻¹ * ↑K * ‖f x‖ ≤ ‖f x‖

Tactic:
]

Goal state after:
case refine_1
𝓕 : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : SeminormedGroup E
inst✝² : SeminormedGroup F
inst✝¹ : FunLike 𝓕 E F
inst✝ : MonoidHomClass 𝓕 E F
f : 𝓕
x✝ : ∃ K, AntilipschitzWith K ⇑f
K : ℝ≥0
hK : AntilipschitzWith K ⇑f
x : E
⊢ (↑K + 1)⁻¹ * ↑K * ‖f x‖ ≤ ‖f x‖
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
X : Type w
inst✝³ : PseudoMetricSpace X
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsIsometricSMul G X
c : G
x : X
r : ℝ
⊢ ball (c⁻¹ • x) r = ball (c⁻¹ • x) r

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
X : Type w
inst✝³ : PseudoMetricSpace X
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsIsometricSMul G X
c : G
x : X
r : ℝ
⊢ closedBall (c⁻¹ • x) r = closedBall (c⁻¹ • x) r

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
X : Type w
inst✝³ : PseudoMetricSpace X
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsIsometricSMul G X
c : G
x : X
r : ℝ
⊢ sphere (c⁻¹ • x) r = sphere (c⁻¹ • x) r

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
inst✝² : Group G
inst✝¹ : PseudoMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ
⊢ (fun x ↦ x * a) ⁻¹' ball b r = ball (b * a⁻¹) r

Tactic:
]

Goal state after:
G : Type v
inst✝² : Group G
inst✝¹ : PseudoMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ
⊢ (fun x ↦ x * a) ⁻¹' ball b r = ball (b * a⁻¹) r
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: G : Type v
inst✝² : Group G
inst✝¹ : PseudoMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ
⊢ (fun x ↦ x * a) ⁻¹' closedBall b r = closedBall (b * a⁻¹) r

Tactic:
]

Goal state after:
G : Type v
inst✝² : Group G
inst✝¹ : PseudoMetricSpace G
inst✝ : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ
⊢ (fun x ↦ x * a) ⁻¹' closedBall b r = closedBall (b * a⁻¹) r
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: case refine_2
𝓕 : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : SeminormedGroup E
inst✝² : SeminormedGroup F
inst✝¹ : FunLike 𝓕 E F
inst✝ : MonoidHomClass 𝓕 E F
f : 𝓕
x✝ : ∃ c > 0, ∀ (x : E), c * ‖x‖ ≤ ‖f x‖
c : ℝ
hc0 : c > 0
hc : ∀ (x : E), c * ‖x‖ ≤ ‖f x‖
x : E
⊢ ‖x‖ ≤ ‖x‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
a b : E
⊢ dist a (a * b) = dist a (a * b)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
a b : E
⊢ ‖b‖ = ‖b‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
a b c : E
⊢ dist a (c * b) = dist a (c * b)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
a b c : E
⊢ dist (a * c) b = dist (a * c) b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Finset

Goal state before:
pp: case inr
ι : Type u_1
G₀ : Type u_3
inst✝ : LinearOrderedCommGroupWithZero G₀
a : G₀
ha : 0 ≤ a
s : Finset ι
f : ι → G₀
hs : s.Nonempty
⊢ (s.sup' hs fun i ↦ f i / a) = s.sup' hs fun i ↦ f i / a

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯
x y : α
⊢ dist x y ≤ dist (f x * g x) (f y * g y) / ↑(Kf⁻¹ - Kg)

Tactic:
]

Goal state after:
α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯
x y : α
⊢ dist x y ≤ dist (f x * g x) (f y * g y) / ↑(Kf⁻¹ - Kg)
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯
x y : α
⊢ dist x y * ↑(Kf⁻¹ - Kg) ≤ dist (f x * g x) (f y * g y)

Tactic:
]

Goal state after:
α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯
x y : α
⊢ dist x y * ↑(Kf⁻¹ - Kg) ≤ dist (f x * g x) (f y * g y)
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯
x y : α
⊢ ↑Kf⁻¹ * dist x y - ↑Kg * dist x y ≤ dist (f x * g x) (f y * g y)

Tactic:
]

Goal state after:
α : Type u_4
E : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
Kf Kg : ℝ≥0
f g : α → E
hf : AntilipschitzWith Kf f
hg : LipschitzWith Kg g
hK : Kg < Kf⁻¹
this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯
x y : α
⊢ ↑Kf⁻¹ * dist x y - ↑Kg * dist x y ≤ dist (f x * g x) (f y * g y)
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
p : E
⊢ mk p = 1 ↔ mk p = 1

Tactic:
]

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

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : NormedDivisionRing α
a b : α
⊢ ‖1‖ = ‖1‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
u v : ℕ → E
N : ℕ
huv : ∀ n ≥ N, u n = v n
hv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k
d : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k
⊢ CauchySeq (d * fun n ↦ ∏ k ∈ range (n + 1), v k)

Tactic:
]

Goal state after:
E : Type u_2
inst✝ : SeminormedCommGroup E
u v : ℕ → E
N : ℕ
huv : ∀ n ≥ N, u n = v n
hv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k
d : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k
⊢ CauchySeq (d * fun n ↦ ∏ k ∈ range (n + 1), v k)
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝ : NormedDivisionRing α
z w : α
hz : z ≠ 0
hw : w ≠ 0
⊢ ‖w - z‖ * (‖z‖⁻¹ * ‖w‖⁻¹) = ‖w - z‖ * (‖z‖⁻¹ * ‖w‖⁻¹)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
u v : ℕ → E
N : ℕ
huv : ∀ n ≥ N, u n = v n
hv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k
d : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k
n : ℕ
hn : n ≥ N
⊢ ∏ k ∈ range (N + 1), u k / v k = ∏ k ∈ range (N + 1), u k / v k

pp: E : Type u_2
inst✝ : SeminormedCommGroup E
u v : ℕ → E
N : ℕ
huv : ∀ n ≥ N, u n = v n
hv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k
d : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k
n : ℕ
hn : n ≥ N
⊢ ∀ n ≥ N + 1, u n / v n = 1

Tactic:
]

Goal state after:
E : Type u_2
inst✝ : SeminormedCommGroup E
u v : ℕ → E
N : ℕ
huv : ∀ n ≥ N, u n = v n
hv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k
d : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k
n : ℕ
hn : n ≥ N
⊢ ∀ n ≥ N + 1, u n / v n = 1
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: G : Type u_4
inst✝ : SeminormedGroup G
u : ℕ → G
hu : CauchySeq u
C : ℝ
hC : ∀ (m n : ℕ), ‖(u m)⁻¹ * u n‖ < C
n : ℕ
⊢ ‖u n‖ ≤ ‖u 0‖ + C

Tactic:
]

Goal state after:
G : Type u_4
inst✝ : SeminormedGroup G
u : ℕ → G
hu : CauchySeq u
C : ℝ
hC : ∀ (m n : ℕ), ‖(u m)⁻¹ * u n‖ < C
n : ℕ
⊢ ‖u n‖ ≤ ‖u 0‖ + C
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: G : Type u_4
inst✝ : SeminormedGroup G
u : ℕ → G
hu : CauchySeq u
C : ℝ
hC : ∀ (m n : ℕ), ‖(u m)⁻¹ * u n‖ < C
this : ∀ (n : ℕ), ‖u n‖ ≤ C + ‖u 0‖
⊢ ∃ x, ∀ y ∈ Set.range fun n ↦ ‖u n‖, y ≤ x

Tactic:
]

Goal state after:
G : Type u_4
inst✝ : SeminormedGroup G
u : ℕ → G
hu : CauchySeq u
C : ℝ
hC : ∀ (m n : ℕ), ‖(u m)⁻¹ * u n‖ < C
this : ∀ (n : ℕ), ‖u n‖ ≤ C + ‖u 0‖
⊢ ∃ x, ∀ y ∈ Set.range fun n ↦ ‖u n‖, y ≤ x
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: case mpr.inr.inl
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
hx : x ≠ 0
ε : ℝ
εpos : ε > 0
h' : ∀ (y : 𝕜), ‖y‖ < ε → y = 0
H :
  ∀ {𝕜 : Type u_5} [inst : NormedDivisionRing 𝕜] [DiscreteTopology 𝕜] {x : 𝕜},
    x ≠ 0 → (∀ (y : 𝕜), ‖y‖ < ε → y = 0) → ‖x‖ < 1 → ‖x‖ = 1
h✝ : 1 ≤ ‖x‖
h : 1 = ‖x‖
⊢ ‖x‖ = ‖x‖

Tactic:
]

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

Goal state before:
pp: case mpr.inr.inr
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
hx : x ≠ 0
ε : ℝ
εpos : ε > 0
h' : ∀ (y : 𝕜), ‖y‖ < ε → y = 0
H :
  ∀ {𝕜 : Type u_5} [inst : NormedDivisionRing 𝕜] [DiscreteTopology 𝕜] {x : 𝕜},
    x ≠ 0 → (∀ (y : 𝕜), ‖y‖ < ε → y = 0) → ‖x‖ < 1 → ‖x‖ = 1
h✝ : 1 ≤ ‖x‖
h : ‖x⁻¹‖ < 1
⊢ ‖x⁻¹‖ = 1

Tactic:
]

Goal state after:
case mpr.inr.inr
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
hx : x ≠ 0
ε : ℝ
εpos : ε > 0
h' : ∀ (y : 𝕜), ‖y‖ < ε → y = 0
H :
  ∀ {𝕜 : Type u_5} [inst : NormedDivisionRing 𝕜] [DiscreteTopology 𝕜] {x : 𝕜},
    x ≠ 0 → (∀ (y : 𝕜), ‖y‖ < ε → y = 0) → ‖x‖ < 1 → ‖x‖ = 1
h✝ : 1 ≤ ‖x‖
h : ‖x⁻¹‖ < 1
⊢ ‖x⁻¹‖ = 1
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: 𝕜✝ : Type u_5
inst✝² : NormedDivisionRing 𝕜✝
ε : ℝ
εpos : ε > 0
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
hx : x ≠ 0
h' : ∀ (y : 𝕜), ‖y‖ < ε → y = 0
h : ‖x‖ < 1
k : ℕ
hk : ‖x ^ k‖ < ε
⊢ ‖x‖ = 1

Tactic:
]

Goal state after:
𝕜✝ : Type u_5
inst✝² : NormedDivisionRing 𝕜✝
ε : ℝ
εpos : ε > 0
𝕜 : Type u_5
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : DiscreteTopology 𝕜
x : 𝕜
hx : x ≠ 0
h' : ∀ (y : 𝕜), ‖y‖ < ε → y = 0
h : ‖x‖ < 1
k : ℕ
hk : ‖x ^ k‖ < ε
⊢ ‖x‖ = 1
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝ : NontriviallyNormedField α
r : ℝ
w : α
hw : 1 < ‖w‖
n : ℕ
hn : r < ‖w‖ ^ n
⊢ r < ‖w‖ ^ n

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : NontriviallyNormedField α
r : ℝ
w : α
hw : 1 < ‖w‖
n : ℕ
hn : r < ‖w‖ ^ n
⊢ r < ‖w‖ ^ n
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝ : NontriviallyNormedField α
r : ℝ
hr : 0 < r
w : α
hw : r⁻¹ < ‖w‖
⊢ ‖w‖ ∈ Set.Ioi r⁻¹

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : NontriviallyNormedField α
r : ℝ
hr : 0 < r
w : α
hw : r⁻¹ < ‖w‖
⊢ ‖w‖ ∈ Set.Ioi r⁻¹
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝ : NontriviallyNormedField α
x : α
⊢ ∀ ε > 0, ∃ b ∈ {x}ᶜ, dist x b < ε

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : NontriviallyNormedField α
x : α
⊢ ∀ ε > 0, ∃ b ∈ {x}ᶜ, dist x b < ε
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝ : NontriviallyNormedField α
x : α
ε : ℝ
ε0 : ε > 0
b : α
hb0 : 0 < ‖b‖
hbε : ‖b‖ < ε
⊢ ‖b‖ < ε

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : NontriviallyNormedField α
x : α
ε : ℝ
ε0 : ε > 0
b : α
hb0 : 0 < ‖b‖
hbε : ‖b‖ < ε
⊢ ‖b‖ < ε
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: case h
G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
𝕜 : Type u_5
h' : NormedField 𝕜
x : 𝕜
hx : x ≠ 0
hx1 : ‖x‖ ≠ 1
hlt : ‖x‖ < 1
⊢ 1 < ‖x‖⁻¹

Tactic:
]

Goal state after:
case h
G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
𝕜 : Type u_5
h' : NormedField 𝕜
x : 𝕜
hx : x ≠ 0
hx1 : ‖x‖ ≠ 1
hlt : ‖x‖ < 1
⊢ 1 < ‖x‖⁻¹
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
x✝¹ x✝ : ℝ
h₀ : 0 ≤ x✝¹
hr : x✝¹ < x✝
x : ℝ
h : x✝¹ < x ∧ x < x✝
⊢ x✝¹ < x ∧ x < x✝

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
x✝¹ x✝ : ℝ
h₀ : 0 ≤ x✝¹
hr : x✝¹ < x✝
x : ℝ
h : x✝¹ < x ∧ x < x✝
⊢ x✝¹ < x ∧ x < x✝
Module: Mathlib.Analysis.Normed.Group.Bounded

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝ : SeminormedGroup E
f : α → E
l : Filter α
⊢ Tendsto (fun x ↦ ‖f x‖) l atTop ↔ Tendsto (norm ∘ f) l atTop

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
inst✝ : SeminormedGroup E
f : α → E
l : Filter α
⊢ Tendsto (fun x ↦ ‖f x‖) l atTop ↔ Tendsto (norm ∘ f) l atTop
Module: Mathlib.Analysis.Normed.Field.Basic

Goal state before:
pp: x y : ℝ
hy : 0 ≤ y
⊢ ‖y * x‖₊ = ‖y * x‖₊

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_4
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
⊢ (if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1 else ⋯.choose) ≠ 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
F : Type u_4
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
⊢ (if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1 else ⋯.choose) ≠ 0
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_4
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
⊢ edist (f x) (f y) = ↑(if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1 else ⋯.choose) * edist x y

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
F : Type u_4
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
⊢ edist (f x) (f y) = ↑(if ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞ then 1 else ⋯.choose) * edist x y
Module: Mathlib.Analysis.Normed.Group.Bounded

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
f : α → E
g : α → F
l : Filter α
hf : Tendsto f l (𝓝 1)
op : E → F → G
A : ℝ
h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖
C : ℝ
hC : ∀ᶠ (a : α) in l, (fun x1 x2 ↦ x1 ≤ x2) ((Norm.norm ∘ g) a) C
⊢ Tendsto (fun x ↦ op (f x) (g x)) l (𝓝 1)

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
f : α → E
g : α → F
l : Filter α
hf : Tendsto f l (𝓝 1)
op : E → F → G
A : ℝ
h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖
C : ℝ
hC : ∀ᶠ (a : α) in l, (fun x1 x2 ↦ x1 ≤ x2) ((Norm.norm ∘ g) a) C
⊢ Tendsto (fun x ↦ op (f x) (g x)) l (𝓝 1)
Module: Mathlib.Analysis.Normed.Group.Bounded

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
f : α → E
g : α → F
l : Filter α
hf : ∀ ε > 0, ∀ᶠ (x : α) in l, ‖f x‖ < ε
op : E → F → G
A : ℝ
h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖
C : ℝ
hC : ∀ᶠ (a : α) in l, (fun x1 x2 ↦ x1 ≤ x2) ((Norm.norm ∘ g) a) C
⊢ ∀ ε > 0, ∀ᶠ (x : α) in l, ‖op (f x) (g x)‖ < ε

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
f : α → E
g : α → F
l : Filter α
hf : ∀ ε > 0, ∀ᶠ (x : α) in l, ‖f x‖ < ε
op : E → F → G
A : ℝ
h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖
C : ℝ
hC : ∀ᶠ (a : α) in l, (fun x1 x2 ↦ x1 ≤ x2) ((Norm.norm ∘ g) a) C
⊢ ∀ ε > 0, ∀ᶠ (x : α) in l, ‖op (f x) (g x)‖ < ε
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α : Type u_5
β : Type u_6
F : Type u_7
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
r : ℝ≥0
hxy : nndist x y ≠ 0
hr : nndist (f x) (f y) = r * nndist x y
⊢ nndist x y ≠ 0

Tactic:
]

Goal state after:
α : Type u_5
β : Type u_6
F : Type u_7
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
r : ℝ≥0
hxy : nndist x y ≠ 0
hr : nndist (f x) (f y) = r * nndist x y
⊢ nndist x y ≠ 0
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α : Type u_5
β : Type u_6
F : Type u_7
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
r : ℝ≥0
hxy : nndist x y ≠ 0
hr : nndist (f x) (f y) = r * nndist x y
⊢ ↑r * ↑(nndist x y) = ↑r * ↑(nndist x y)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α : Type u_6
β : Type u_7
F : Type u_5
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
x y : α
r : ℝ≥0
hxy : dist x y ≠ 0
hr : dist (f x) (f y) = ↑r * dist x y
⊢ ↑r * dist x y = ↑r * dist x y

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
F : Type u_4
inst✝⁴ : PseudoEMetricSpace α✝
inst✝³ : PseudoEMetricSpace β✝
inst✝² : FunLike F α✝ β✝
α : Type ?u.16
β : Type ?u.18
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
r : ℝ≥0
hne : r ≠ 0
h : ∀ (x y : α), nndist (f x) (f y) = r * nndist x y
x y : α
⊢ ↑(r * nndist x y) = ↑(r * nndist x y)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
F : Type u_4
inst✝⁴ : PseudoEMetricSpace α✝
inst✝³ : PseudoEMetricSpace β✝
inst✝² : FunLike F α✝ β✝
α : Type ?u.16
β : Type ?u.18
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), dist (f x) (f y) = ↑r * dist x y
r : ℝ≥0
hr : r ≠ 0 ∧ ∀ (x y : α), dist (f x) (f y) = ↑r * dist x y
x y : α
⊢ ↑r * dist x y = ↑r * dist x y

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: case pos
α : Type u_1
inst✝ : PseudoEMetricSpace α
h : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
g : β →ᵈ γ
f : α →ᵈ β
x y : α
hα : edist x y ≠ 0 ∧ edist x y ≠ ∞
hgf : ↑(ratio (g.comp f)) = ↑(ratio g) * ↑(ratio f)
⊢ ↑(ratio (g.comp f)) = ↑(ratio g) * ↑(ratio f)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
g : β →ᵈ γ
f : α →ᵈ β
x y : α
hα : edist x y ≠ 0 ∧ edist x y ≠ ∞
hgf : ↑(ratio (g.comp f)) = ↑(ratio g) * ↑(ratio f)
⊢ ↑(ratio (g.comp f)) = ↑(ratio g) * ↑(ratio f)
Module: Mathlib.Topology.MetricSpace.Dilation

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
g : β →ᵈ γ
f₁ f₂ : α →ᵈ β
hg : Injective ⇑g
h : g.comp f₁ = g.comp f₂
x : α
⊢ g (f₂ x) = g (f₂ x)

Tactic:
]

Goal state after: