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

Goal state before:
pp: case neg
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n ↦ f n x₀
this : (fun x ↦ ∑' (n : α), f n x) = 0
⊢ Differentiable 𝕜 0

Tactic:
exact differentiable_const 0

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : p ≠ 0
f : ↥(lp E p)
i : α
⊢ ‖↑f i‖ ≤ ‖f‖

Tactic:
rcases eq_or_ne p ∞ with (rfl | hp')
  · have : Nonempty α := ⟨i⟩
    exact (isLUB_norm f).1 ⟨i, rfl⟩
  have hp'' : 0 < p.toReal := ENNReal.toReal_pos hp hp'
  have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i ↦ by positivity
  rw [← Real.rpow_le_rpow_iff (norm_nonneg _) (norm_nonneg' _) hp'']
  convert! le_hasSum (hasSum_norm hp'' f) i fun i _ => this i

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inl.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
c : 𝕜
hp : ∞ ≠ 0
f : ↥(lp E ∞)
h✝ : Nonempty α
hfc : IsLUB (Set.range ((fun b ↦ ‖c‖ * b) ∘ fun i ↦ ‖↑f i‖)) (‖c‖ * ‖f‖)
⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖

Tactic:
Function.comp_def

Goal state after:
case inr.inl.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
c : 𝕜
hp : ∞ ≠ 0
f : ↥(lp E ∞)
h✝ : Nonempty α
hfc : IsLUB (Set.range fun x ↦ ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)
⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: I : Type u_5
B : I → Type u_6
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ∞
hg : Memℓp g ∞
⊢ BddAbove (Set.range fun i ↦ ‖(f * g) i‖)

Tactic:
obtain ⟨⟨Cf, hCf⟩, ⟨Cg, hCg⟩⟩ := hf.bddAbove, hg.bddAbove

Goal state after:
I : Type u_5
B : I → Type u_6
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ∞
hg : Memℓp g ∞
Cf : ℝ
hCf : Cf ∈ upperBounds (Set.range fun i ↦ ‖f i‖)
Cg : ℝ
hCg : Cg ∈ upperBounds (Set.range fun i ↦ ‖g i‖)
⊢ BddAbove (Set.range fun i ↦ ‖(f * g) i‖)
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: case pos
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
h✝ : 0 < r
⊢ (univUnitBall.trans' (unitBallBall c r h✝) ⋯).source = univ

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: case neg
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
h✝ : ¬0 < r
⊢ (IsometryEquiv.vaddConst c).toHomeomorph.toOpenPartialHomeomorph.source = univ

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
hr : 0 < r
⊢ (univUnitBall.trans' (unitBallBall c r hr) ⋯).target = ball c r

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
t : Set H
y : H
⊢ (∀ (i : ι), ContDiffWithinAt 𝕜 n (fun x ↦ (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) x i) t y) ↔
    ∀ (i : ι), ContDiffWithinAt 𝕜 n (fun x ↦ (f x).ofLp i) t y

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
y : H
⊢ (∀ (i : ι), ContDiffAt 𝕜 n (fun x ↦ (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) x i) y) ↔
    ∀ (i : ι), ContDiffAt 𝕜 n (fun x ↦ (f x).ofLp i) y

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
t : Set H
⊢ (∀ (i : ι), ContDiffOn 𝕜 n (fun x ↦ (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) x i) t) ↔
    ∀ (i : ι), ContDiffOn 𝕜 n (fun x ↦ (f x).ofLp i) t

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
⊢ (∀ (i : ι), ContDiff 𝕜 n fun x ↦ (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) x i) ↔
    ∀ (i : ι), ContDiff 𝕜 n fun x ↦ (f x).ofLp i

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
t : Set H
y : H
this : Fintype ι
⊢ (∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x ↦ (⇑(continuousLinearEquiv p 𝕜 E) ∘ f) x i) t y) ↔
    ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x ↦ (f x).ofLp i) t y

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
y : H
this : Fintype ι
⊢ (∀ (i : ι), DifferentiableAt 𝕜 (fun x ↦ (⇑(continuousLinearEquiv p 𝕜 E) ∘ f) x i) y) ↔
    ∀ (i : ι), DifferentiableAt 𝕜 (fun x ↦ (f x).ofLp i) y

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
t : Set H
this : Fintype ι
⊢ (∀ (i : ι), DifferentiableOn 𝕜 (fun x ↦ (⇑(continuousLinearEquiv p 𝕜 E) ∘ f) x i) t) ↔
    ∀ (i : ι), DifferentiableOn 𝕜 (fun x ↦ (f x).ofLp i) t

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
this : Fintype ι
⊢ (∀ (i : ι), Differentiable 𝕜 fun x ↦ (⇑(continuousLinearEquiv p 𝕜 E) ∘ f) x i) ↔
    ∀ (i : ι), Differentiable 𝕜 fun x ↦ (f x).ofLp i

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
f' : H →L[𝕜] PiLp p E
y : H
this : Fintype ι
⊢ (∀ (i : ι),
      HasStrictFDerivAt (fun x ↦ (⇑(continuousLinearEquiv p 𝕜 E) ∘ f) x i)
        (ContinuousLinearMap.proj i ∘SL ↑(continuousLinearEquiv p 𝕜 E) ∘SL f') y) ↔
    ∀ (i : ι), HasStrictFDerivAt (fun x ↦ (f x).ofLp i) (PiLp.proj p E i ∘SL f') y

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
f' : H →L[𝕜] PiLp p E
t : Set H
y : H
this : Fintype ι
⊢ (∀ (i : ι),
      HasFDerivWithinAt (fun x ↦ (⇑(continuousLinearEquiv p 𝕜 E) ∘ f) x i)
        (ContinuousLinearMap.proj i ∘SL ↑(continuousLinearEquiv p 𝕜 E) ∘SL f') t y) ↔
    ∀ (i : ι), HasFDerivWithinAt (fun x ↦ (f x).ofLp i) (PiLp.proj p E i ∘SL f') t y

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p✝ q✝ : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q r : ℝ≥0∞
h : p ≤ q
x✝² x✝¹ : ↥(lp E p)
x✝ : α
⊢ ↑⟨↑(x✝² + x✝¹), ⋯⟩ x✝ = ↑(⟨↑x✝², ⋯⟩ + ⟨↑x✝¹, ⋯⟩) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p✝ q✝ : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q r : ℝ≥0∞
h : p ≤ q
x✝² : 𝕜
x✝¹ : ↥(lp E p)
x✝ : α
⊢ ↑⟨↑(x✝² • x✝¹), ⋯⟩ x✝ = ↑((RingHom.id 𝕜) x✝² • ⟨↑x✝¹, ⋯⟩) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q : ℝ≥0∞
h : p ≤ q
f : ↥(lp E p)
x✝ : α
⊢ ↑((linearMapOfLE 𝕜 E h) f) x✝ = ↑f x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q : ℝ≥0∞
h : p ≤ q
x✝¹ : ↥(lp E p)
x✝ : α
⊢ ↑((linearMapOfLE 𝕜 E h).toAddMonoidHom x✝¹) x✝ = ↑((AddSubgroup.inclusion ⋯) x✝¹) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q r : ℝ≥0∞
hpq : p ≤ q
hqr : q ≤ r
x✝¹ : ↥(lp E p)
x✝ : α
⊢ ↑((linearMapOfLE 𝕜 E hqr ∘ₗ linearMapOfLE 𝕜 E hpq) x✝¹) x✝ = ↑((linearMapOfLE 𝕜 E ⋯) x✝¹) x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: p : ℝ[X]
this : Tendsto (fun x ↦ Polynomial.eval x⁻¹ p / rexp x⁻¹) (𝓝[>] 0) (𝓝 0)
⊢ Tendsto (fun x ↦ Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | 0 < x}) (𝓝 0)

Tactic:
refine this.congr' <| mem_of_superset self_mem_nhdsWithin fun x hx ↦ ?_

Goal state after:
p : ℝ[X]
this : Tendsto (fun x ↦ Polynomial.eval x⁻¹ p / rexp x⁻¹) (𝓝[>] 0) (𝓝 0)
x : ℝ
hx : x ∈ {x | 0 < x}
⊢ x ∈ {x | (fun x ↦ Polynomial.eval x⁻¹ p / rexp x⁻¹ = (fun x ↦ Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) x) x}
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: case succ
n : ℕ∞
m : ℕ
ihm : ∀ (p : ℝ[X]), ContDiff ℝ ↑m fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x
p : ℝ[X]
⊢ ContDiff ℝ ↑(m + 1) fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x

Tactic:
show ((m + 1 : ℕ) : WithTop ℕ∞) = m + 1 from rfl

Goal state after:
case succ
n : ℕ∞
m : ℕ
ihm : ∀ (p : ℝ[X]), ContDiff ℝ ↑m fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x
p : ℝ[X]
⊢ ContDiff ℝ (↑m + 1) fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x
Module: Mathlib.Analysis.InnerProductSpace.Calculus

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
x : G
hf : DifferentiableAt ℝ f x
hg : DifferentiableAt ℝ g x
y : G
⊢ (fderivInnerCLM 𝕜 (f x, g x) ∘SL (fderiv ℝ f x).prod (fderiv ℝ g x)) y =
    ⟪f x, (fderiv ℝ g x) y⟫ + ⟪(fderiv ℝ f x) y, g x⟫

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case e'_7.inr.inl
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : AddGroup G
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
t : G
h : f t = 0
⊢ (L (f t)) (g (x - t)) = 0

pp: case e'_7.inr.inr
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : AddGroup G
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
t : G
h : ¬x - t + t = x
⊢ (L (f t)) (g (x - t)) = 0

Tactic:
· rw [h, L.map_zero₂]

Goal state after:
case e'_7.inr.inr
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : AddGroup G
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
t : G
h : ¬x - t + t = x
⊢ (L (f t)) (g (x - t)) = 0
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case e'_7.inr.inr
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : AddGroup G
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
t : G
h : ¬x - t + t = x
⊢ (L (f t)) (g (x - t)) = 0

Tactic:
· exact (h <| sub_add_cancel x t).elim

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

Goal state before:
pp: case pos.h_bound.h₁.hb
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖x - x₀‖

Tactic:
rfl

Goal state after:
case pos.h_bound.h₁.hb
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖x - x₀‖
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case pos.h_bound.h₁.h
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ b a * ‖x - x₀‖ ≤ b a * ‖x - x₀‖

Tactic:
rfl

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

Goal state before:
pp: case pos.h_bound.h₂.hb
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖x - x₀‖

Tactic:
rfl

Goal state after:
case pos.h_bound.h₂.hb
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ 0 ≤ ‖x - x₀‖