mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
this✝ : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : ℝ) in 𝓝[≥] x, ‖f x_1 - f x - (x_1 - x) • derivWithin f (Ici x) x‖ ≤ c * ‖x_1 - x‖
m : ℝ
xm : m ∈ Ioi x
hm : Ico x m ⊆ {x_1 | (fun x_2 ↦ ‖f x_2 - f x - (x_2 - x) • derivWithin f (Ici x) x‖ ≤ ε / 2 * ‖x_2 - x‖) x_1}
r : ℝ
hr : r ∈ Ioo 0 (m - x)
this : r ∈ Ioc (r / 2) r
y : ℝ
hy : y ∈ Icc x (x + r)
z : ℝ
hz : z ∈ Icc x (x + r)
⊢ ε / 2 * r + ε / 2 * r = ε * r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
ε : ℝ
hε : 0 < ε
x : ℝ
hx : DifferentiableWithinAt ℝ f (Ici x) x
⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
H : ‖(r / 2) • (L₁ - L₂)‖ ≤ r / 2 * (4 * ε)
⊢ ‖L₁ - L₂‖ ≤ 4 * ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ‖(r / 2) • (L₁ - L₂)‖ = ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂ - (f (x + r / 2) - f x - (x + r / 2 - x) • L₁)‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂‖ + ‖f (x + r / 2) - f x - (x + r / 2 - x) • L₁‖ ≤ ε * r + ε * r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ε * r + ε * r = r / 2 * (4 * ε)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
r x : ℝ
hr : 0 < r
ε : ℝ
L₁ L₂ : F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ ‖L₁ - L₂‖ ≤ 4 * ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
⊢ 0 < 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
⊢ 1 / 2 < 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ 0 < 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝⁴ : Zero X
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : T1Space R
⊢ IsClosed (range _root_.toContinuousMap)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ 0 ≤ 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
f : C(X, Y)₀
⊢ Continuous fun g ↦ g.comp f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
g : C(Y, R)₀
⊢ Continuous g.comp

Tactic:
ContinuousMapZero.isEmbedding_toContinuousMap.continuous_iff

Goal state after:
X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
g : C(Y, R)₀
⊢ Continuous (_root_.toContinuousMap ∘ g.comp)
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ 1 / 2 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
g : C(Y, R)₀
⊢ Continuous g.comp

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
hf : Continuous[inst✝², inst✝¹] f
hf₀ : f 0 = 0
⊢ mkD f g = { toFun := f, continuous_toFun := hf, map_zero' := hf₀ }

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ n ≤ p

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
hf : ¬Continuous[inst✝², inst✝¹] f
⊢ mkD f g = g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
hf : f 0 ≠ 0
⊢ mkD f g = g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁸ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁷ : RCLike 𝕜
inst✝⁶ : RCLike 𝕜'
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜' F
inst✝¹ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
⊢ Continuous fun φ ↦ ∫ (x : X), L (↑↑φ x) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
x : X
hf : Continuous[inst✝², inst✝¹] f
hf₀ : f 0 = 0
⊢ (mkD f g) x = f x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n : ℕ
hn : (1 / 2) ^ n < R
p : ℕ
hp : p ≥ n
q : ℕ
hq : q ≥ n
⊢ n ≤ q

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
s : Set X
inst✝ : Zero ↑s
f : X → R
g : C(↑s, R)₀
hf : ¬ContinuousOn f s
⊢ mkD (s.domRestrict f) g = g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
⊢ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K} ⊆ D f K

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
n : ℕ
⊢ 0 < 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
s : Set X
inst✝ : Zero ↑s
f : X → R
g : C(↑s, R)₀
x : ↑s
hf : ContinuousOn f s
hf₀ : f ↑0 = 0
⊢ (mkD (s.domRestrict f) g) x = f ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace R
inst✝ : Zero X
f : X → R
g : C(X, R)₀
f_zero : f 0 = 0
⊢ ↑(mkD f g) = ContinuousMap.mkD f ↑g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : AddZeroClass R
inst✝ : ContinuousAdd R
f g : C(X, R)₀
⊢ (↑f + ↑g) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : NegZeroClass R
inst✝ : ContinuousNeg R
f : C(X, R)₀
⊢ (-↑f) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : SubNegZeroMonoid R
inst✝ : ContinuousSub R
f g : C(X, R)₀
⊢ (↑f - ↑g) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : MulZeroClass R
inst✝ : ContinuousMul R
f g : C(X, R)₀
⊢ (↑f * ↑g) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁵ : Zero X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace R
M : Type u_3
inst✝² : Zero R
inst✝¹ : SMulZeroClass M R
inst✝ : ContinuousConstSMul M R
m : M
f : C(X, R)₀
⊢ (m • ↑f) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁶ : Zero X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace R
inst✝³ : CommSemiring R
inst✝² : IsTopologicalSemiring R
inst✝¹ : StarRing R
inst✝ : ContinuousStar R
f : C(X, R)₀
⊢ (star ↑f) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace R
inst✝¹ : CommSemiring R
inst✝ : IsTopologicalSemiring R
f g : C(X, R)₀
⊢ ⇑(f + g) = ⇑f + ⇑g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X✝ : Type u_1
R✝ : Type u_2
inst✝⁷ : Zero X✝
inst✝⁶ : TopologicalSpace X✝
inst✝⁵ : TopologicalSpace R✝
X : Type u_3
R : Type u_4
inst✝⁴ : Zero X
inst✝³ : TopologicalSpace X
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
⊢ Continuous fun a ↦ -a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
⊢ { toFun := ⇑φ, continuous_toFun := hφ } 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
⊢ ∀ (e : ℕ),
    ∃ n, ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
r : M
f : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (r • f)) x =
      ((MonoidHom.id M) r • { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp f) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
⊢ ∀ (x : X), ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp 0) x = 0 x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
x✝¹ x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (x✝¹ + x✝)) x =
      ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝¹ +
          { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝)
        x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
x✝¹ x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (x✝¹ * x✝)) x =
      ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝¹ *
          { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝)
        x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
M : Type u_3
R : Type u_4
S : Type u_5
inst✝¹⁸ : Zero X
inst✝¹⁷ : Zero Y
inst✝¹⁶ : CommSemiring M
inst✝¹⁵ : TopologicalSpace X
inst✝¹⁴ : TopologicalSpace Y
inst✝¹³ : TopologicalSpace R
inst✝¹² : TopologicalSpace S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : StarRing R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : CommSemiring S
inst✝⁶ : StarRing S
inst✝⁵ : IsTopologicalSemiring S
inst✝⁴ : ContinuousStar S
inst✝³ : Module M R
inst✝² : Module M S
inst✝¹ : ContinuousConstSMul M R
inst✝ : ContinuousConstSMul M S
φ : R →⋆ₙₐ[M] S
hφ : Continuous[inst✝¹³, inst✝¹²] ⇑φ
x✝ : C(X, R)₀
⊢ ∀ (x : X),
    ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp (star x✝)) x =
      (star ({ toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp x✝)) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Order

Goal state before:
pp: α : Type u_1
ι : Type u_2
β : Type u_3
inst✝⁶ : UniformSpace β
inst✝⁵ : AddGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : PartialOrder β
inst✝² : OrderTopology β
inst✝¹ : AddLeftMono β
inst✝ : AddRightMono β
f : ι → α → β
g : α → β
K : Set α
p : Filter ι
u v : β
huv : u < v
hg : ∀ x ∈ K, g x ≤ u
hf :
  ∀ (a : β × β → Prop),
    (∃ v ∈ 𝓝 0, ∀ y ∈ v, ∀ (a_2 b : β), -a_2 + b = y → a (a_2, b)) → ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 K, a (g x.2, f x.1 x.2)
⊢ 0 ∈ {b | b < -u + v}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
⊢ 0 ≤ 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Order

Goal state before:
pp: α : Type u_1
ι : Type u_2
β : Type u_3
inst✝⁶ : UniformSpace β
inst✝⁵ : AddGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : PartialOrder β
inst✝² : OrderTopology β
inst✝¹ : AddLeftMono β
inst✝ : AddRightMono β
f : ι → α → β
g : α → β
K : Set α
p : Filter ι
u v : β
huv : u < v
hf : TendstoUniformlyOn f g p K
hg : ∀ x ∈ K, g x ≤ u
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ K, f i x < v

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsUniformGroup.Order

Goal state before:
pp: α : Type u_1
ι : Type u_2
β : Type u_3
inst✝⁶ : UniformSpace β
inst✝⁵ : AddGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : PartialOrder β
inst✝² : OrderTopology β
inst✝¹ : AddLeftMono β
inst✝ : AddRightMono β
f : ι → α → β
g : α → β
K : Set α
p : Filter ι
u v : β
huv : u < v
hf : TendstoUniformlyOn f g p K
hg : ∀ x ∈ K, g x ≤ u
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ K, f i x ≤ v

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
⊢ 1 / 2 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
⊢ ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e

Tactic:
have I1 : x ∈ A f (L e p r) ((1 / 2) ^ r) ((1 / 2) ^ e) := (hn e p r hp (le_max_left _ _)).2.2

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
I1 : x ∈ A f (L e p r) ((1 / 2) ^ r) ((1 / 2) ^ e)
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := max (n e) (n e')
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * (1 / 2) ^ e
⊢ ‖L e p r - L e' p' r‖ ≤ 4 * (1 / 2) ^ e

Tactic:
by

Goal state after: