mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
s : Set ℝ
x r : ℝ
hf : HasDerivWithinAt f f' s x
hr : ‖f'‖ < r
hr₀ : 0 < r
A : ∀ᶠ (z : ℝ) in 𝓝[s \ {x}] x, ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r
⊢ x ∈ {x_1 | (fun z ↦ ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r) x_1}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a : 𝕜
⊢ dslope f a a = deriv f a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : 𝕜
f : 𝕜 → E
h : b ≠ a
⊢ dslope f a b = slope f a b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E →L[𝕜] F
g : 𝕜 → E
a b : 𝕜
H : a = b → DifferentiableAt 𝕜 g a
⊢ dslope (⇑f ∘ g) a b = f (dslope g a b)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
⊢ (b - a) • dslope f a b = f b - f a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
s : Set ℝ
x r : ℝ
hf : HasDerivWithinAt f f' s x
hr : ‖f'‖ < r
⊢ ∀ᶠ (z : ℝ) in 𝓝[s] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : 𝕜
f : 𝕜 → E
h : b ≠ a
⊢ dslope (fun x ↦ (x - a) • f x) a b = f b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a : 𝕜
⊢ ContinuousAt (dslope f a) a ↔ DifferentiableAt 𝕜 f a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
s : Set 𝕜
h : ContinuousWithinAt (dslope f a) s b
⊢ ContinuousWithinAt f s b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
s : Set ℝ
x r : ℝ
hf : HasDerivWithinAt f f' s x
hr : ‖f'‖ < r
z : ℝ
hz : ‖z - x‖⁻¹ * ‖f z - f x‖ < r
⊢ 0 ≤ ‖z - x‖⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
s : Set ℝ
x r : ℝ
hf : HasDerivWithinAt f f' s x
hr : ‖f'‖ < r
⊢ ∀ᶠ (z : ℝ) in 𝓝[s] x, ‖z - x‖⁻¹ * (‖f z‖ - ‖f x‖) < r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
s : Set 𝕜
h : b ≠ a
⊢ ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
h : b ≠ a
⊢ ContinuousAt (dslope f a) b ↔ ContinuousAt f b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a : 𝕜
s : Set 𝕜
h : s ∈ 𝓝 a
⊢ ContinuousOn (dslope f a) s ↔ ContinuousOn f s ∧ DifferentiableAt 𝕜 f a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
s : Set 𝕜
h : DifferentiableWithinAt 𝕜 (dslope f a) s b
⊢ DifferentiableWithinAt 𝕜 f s b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
s : Set 𝕜
h : b ≠ a
⊢ DifferentiableWithinAt 𝕜 (dslope f a) s b ↔ DifferentiableWithinAt 𝕜 f s b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
h : b ≠ a
⊢ DifferentiableAt 𝕜 (dslope f a) b ↔ DifferentiableAt 𝕜 f b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Slope

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
f' : E
x r : ℝ
hf : HasDerivWithinAt f f' (Ici x) x
hr : ‖f'‖ < r
⊢ ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (‖f z‖ - ‖f x‖) < r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a : 𝕜
hf : f a = 0
b : 𝕜
⊢ (b - a) • dslope f a b = f b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_3
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalSpace Z
proj : Z → B
inst✝ : Nonempty F
h : IsHomeomorphicTrivialFiberBundle F proj
⊢ Function.Surjective proj

Tactic:
by

Goal state after:
Module: Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalSpace Z
proj : Z → B
h : IsHomeomorphicTrivialFiberBundle F proj
⊢ Continuous[inst✝, inst✝²] proj

Tactic:
by

Goal state after:
Module: Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : TopologicalSpace Z
proj : Z → B
h : IsHomeomorphicTrivialFiberBundle F proj
⊢ IsOpenMap proj

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : 𝕜
n : ℕ
ih : ∀ {f : 𝕜 → E}, (∀ k < n, (swap dslope a)^[k] f a = 0) → (b - a) ^ n • (swap dslope a)^[n] f b = f b
f : 𝕜 → E
hf : ∀ k < n + 1, (swap dslope a)^[k] f a = 0
⊢ ∀ k < n, (swap dslope a)^[k] f a = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a : 𝕜
n : ℕ
hf : ∀ k < n, (swap dslope a)^[k] f a = 0
b : 𝕜
⊢ (b - a) ^ n • (swap dslope a)^[n] f b = f b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ interior {z | z.re ≤ a} = {z | z.re < a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ interior {z | z.im ≤ a} = {z | z.im < a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ interior {z | a ≤ z.re} = {z | a < z.re}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ interior {z | a ≤ z.im} = {z | a < z.im}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ closure {z | z.re < a} = {z | z.re ≤ a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ closure {z | z.im < a} = {z | z.im ≤ a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ closure {z | a < z.re} = {z | a ≤ z.re}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ closure {z | a < z.im} = {z | a ≤ z.im}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | z.re ≤ a} = {z | z.re = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | z.im ≤ a} = {z | z.im = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | a ≤ z.re} = {z | z.re = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | a ≤ z.im} = {z | z.im = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | z.re < a} = {z | z.re = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | z.im < a} = {z | z.im = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | a < z.re} = {z | z.re = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a : ℝ
⊢ frontier {z | a < z.im} = {z | z.im = a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: s t : Set ℝ
⊢ closure (s ×ℂ t) = closure s ×ℂ closure t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: s t : Set ℝ
⊢ interior (s ×ℂ t) = interior s ×ℂ interior t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: s t : Set ℝ
⊢ frontier (s ×ℂ t) = closure s ×ℂ frontier t ∪ frontier s ×ℂ closure t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a b : ℝ
⊢ frontier {z | a ≤ z.re ∧ b ≤ z.im} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ b ≤ z.im}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: a b : ℝ
⊢ frontier {z | a ≤ z.re ∧ z.im ≤ b} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ z.im ≤ b}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
K : Set α
hf : TendstoUniformlyOn f g p K
⊢ TendstoUniformlyOn (fun n x ↦ (f n x).re) (fun y ↦ (g y).re) p K

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
hf : TendstoUniformly f g p
⊢ TendstoUniformly (fun n x ↦ (f n x).re) (fun y ↦ (g y).re) p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
K : Set α
hf : TendstoUniformlyOn f g p K
⊢ TendstoUniformlyOn (fun n x ↦ (f n x).im) (fun y ↦ (g y).im) p K

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
hf : TendstoUniformly f g p
⊢ TendstoUniformly (fun n x ↦ (f n x).im) (fun y ↦ (g y).im) p

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
𝕜' : Type u_5
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : E → 𝕜'
c' : E →L[𝕜] 𝕜'
hc : HasFDerivWithinAt c c' s x
hf : HasFDerivWithinAt f f' s x
⊢ HasFDerivWithinAt (c • f) (c x • f' + c'.smulRight (f x)) s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
𝕜' : Type u_5
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : E → 𝕜'
c' : E →L[𝕜] 𝕜'
hc : HasFDerivAt c c' x
hf : HasFDerivAt f f' x
⊢ HasFDerivAt (c • f) (c x • f' + c'.smulRight (f x)) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : E
𝕜' : Type u_5
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : E → 𝕜'
c' : E →L[𝕜] 𝕜'
hc : HasStrictFDerivAt c c' x
f : F
⊢ HasStrictFDerivAt (fun y ↦ c y • f) (c'.smulRight f) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : E
s : Set E
𝕜' : Type u_5
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : E → 𝕜'
c' : E →L[𝕜] 𝕜'
hc : HasFDerivWithinAt c c' s x
f : F
⊢ HasFDerivWithinAt (fun y ↦ c y • f) (c'.smulRight f) s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
x : E
𝕜' : Type u_5
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : E → 𝕜'
c' : E →L[𝕜] 𝕜'
hc : HasFDerivAt c c' x
f : F
⊢ HasFDerivAt (fun y ↦ c y • f) (c'.smulRight f) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasStrictFDerivAt c c' x
hd : HasStrictFDerivAt d d' x
⊢ HasStrictFDerivAt (c * d) (c x •> d' + d x •> c') x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
s : Set E
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
a b : E → 𝔸
a' b' : E →L[𝕜] 𝔸
ha : HasFDerivWithinAt a a' s x
hb : HasFDerivWithinAt b b' s x
⊢ HasFDerivWithinAt (a * b) (a x •> b' + a' <• b x) s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
s : Set E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasFDerivWithinAt c c' s x
hd : HasFDerivWithinAt d d' s x
⊢ HasFDerivWithinAt (c * d) (c x •> d' + d x •> c') s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
a b : E → 𝔸
a' b' : E →L[𝕜] 𝔸
ha : HasFDerivAt a a' x
hb : HasFDerivAt b b' x
⊢ HasFDerivAt (a * b) (a x •> b' + a' <• b x) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasFDerivAt c c' x
hd : HasFDerivAt d d' x
⊢ HasFDerivAt (c * d) (c x •> d' + d x •> c') x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c : E → 𝔸'
c' : E →L[𝕜] 𝔸'
hc : HasStrictFDerivAt c c' x
d : 𝔸'
⊢ HasStrictFDerivAt (fun y ↦ c y * d) (d •> c') x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
s : Set E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c : E → 𝔸'
c' : E →L[𝕜] 𝔸'
hc : HasFDerivWithinAt c c' s x
d : 𝔸'
⊢ HasFDerivWithinAt (fun y ↦ c y * d) (d •> c') s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c : E → 𝔸'
c' : E →L[𝕜] 𝔸'
hc : HasFDerivAt c c' x
d : 𝔸'
⊢ HasFDerivAt (fun y ↦ c y * d) (d •> c') x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : Finite ι
x : ι → 𝔸
this : Fintype ι
a : ι
l : List ι
IH :
  HasStrictFDerivAt (fun x ↦ (List.map x l).prod)
    (∑ i, (List.map x (List.take (↑i) l)).prod •> (proj l[i] <• (List.map x (List.drop (↑i).succ l)).prod)) x
⊢ (proj a) x •> ∑ i, (List.map x (List.take (↑i) l)).prod •> (proj l[i] <• (List.map x (List.drop (↑i).succ l)).prod) +
      proj a <• (List.map x l).prod =
    ∑ i,
      (List.map x (List.take (↑i) (a :: l))).prod •>
        (proj (a :: l)[i] <• (List.map x (List.drop (↑i).succ (a :: l))).prod)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : Finite ι
l : List ι
x : ι → 𝔸
⊢ HasStrictFDerivAt (fun x ↦ (List.map x l).prod)
    (∑ i, (List.map x (List.take (↑i) l)).prod •> (proj l[i] <• (List.map x (List.drop (↑i).succ l)).prod)) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
n : ℕ
x : Fin n → 𝔸
⊢ ∀ (i : Fin (List.finRange n).length), i ∈ Finset.univ ↔ (finCongr ⋯) i ∈ Finset.univ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
n : ℕ
x : Fin n → 𝔸
⊢ ∀ i ∈ Finset.univ,
    (List.map x (List.take (↑i) (List.finRange n))).prod •>
        (proj (List.finRange n)[i] <• (List.map x (List.drop (↑i).succ (List.finRange n))).prod) =
      (List.map x (List.take (↑((finCongr ⋯) i)) (List.finRange n))).prod •>
        (proj ((finCongr ⋯) i) <• (List.map x (List.drop (↑((finCongr ⋯) i)).succ (List.finRange n))).prod)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ ∀ (i : Fin l.length), i ∈ Finset.univ ↔ (finCongr ⋯) i ∈ Finset.univ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ ∀ i ∈ Finset.univ,
    (List.map x (List.take (↑i) l.attach)).prod •>
        (proj l.attach[Fin.cast ⋯ i] <• (List.map x (List.drop (↑i).succ l.attach)).prod) =
      (List.map x (List.take (↑((finCongr ⋯) i)) l.attach)).prod •>
        (proj l.attach[(finCongr ⋯) i] <• (List.map x (List.drop (↑((finCongr ⋯) i)).succ l.attach)).prod)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ HasStrictFDerivAt (fun x ↦ (List.map x l.attach).prod)
    (∑ i,
      (List.map x (List.take (↑i) l.attach)).prod •>
        (proj l.attach[Fin.cast ⋯ i] <• (List.map x (List.drop (↑i).succ l.attach)).prod))
    x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ HasFDerivAt (fun x ↦ (List.map x l.attach).prod)
    (∑ i,
      (List.map x (List.take (↑i) l.attach)).prod •>
        (proj l.attach[Fin.cast ⋯ i] <• (List.map x (List.drop (↑i).succ l.attach)).prod))
    x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: c : ℂ
R θ : ℝ
⊢ circleMap c R θ - c = circleMap 0 R θ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: R θ : ℝ
⊢ ‖circleMap 0 R θ‖ = |R|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: c : ℂ
R θ : ℝ
⊢ circleMap c R θ ∉ ball c R

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: c : ℂ
R θ : ℝ
⊢ circleMap c R θ ∈ sphere c |R|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: c : ℂ
R : ℝ
hR : 0 ≤ R
θ : ℝ
⊢ circleMap c R θ ∈ sphere c R

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: c : ℂ
R θ : ℝ
⊢ circleMap c R θ = c ↔ R = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: R₁ R₂ θ₁ θ₂ : ℝ
⊢ circleMap 0 R₁ θ₁ * circleMap 0 R₂ θ₂ = circleMap 0 (R₁ * R₂) (θ₁ + θ₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: R₁ R₂ θ₁ θ₂ : ℝ
⊢ circleMap 0 R₁ θ₁ / circleMap 0 R₂ θ₂ = circleMap 0 (R₁ / R₂) (θ₁ - θ₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: R θ : ℝ
⊢ (circleMap 0 R θ)⁻¹ = circleMap 0 R⁻¹ (-θ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: n : ℕ
R θ : ℝ
⊢ circleMap 0 R θ ^ n = circleMap 0 (R ^ n) (↑n * θ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: n : ℤ
R θ : ℝ
⊢ circleMap 0 R θ ^ n = circleMap 0 (R ^ n) (↑n * θ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleMap

Goal state before:
pp: r θ : ℝ
⊢ (starRingEnd ℂ) (circleMap 0 r θ) = circleMap 0 r (-θ)

Tactic:
by

Goal state after: