mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x✝ : E
⊢ (o.rightAngleRotation.trans o.rightAngleRotation) x✝ = (LinearIsometryEquiv.neg ℝ) x✝

Tactic:
;

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x✝ : E
⊢ (o.rightAngleRotation.trans o.rightAngleRotation) x✝ = (LinearIsometryEquiv.neg ℝ) x✝
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁹ : NormedField 𝕜₁
inst✝⁸ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : Module 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜₂ F
x y : E →SWOT[σ] F
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ (match x✝ with
    | (x_1, y_1) => y_1 ((x + y) x_1)) =
    ((fun x_1 ↦
          match x_1 with
          | (x_2, y) => y (x x_2)) +
        fun x ↦
        match x with
        | (x, y_1) => y_1 (y x))
      x✝

Tactic:
;

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁹ : NormedField 𝕜₁
inst✝⁸ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : Module 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜₂ F
x y : E →SWOT[σ] F
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ (match x✝ with
    | (x_1, y_1) => y_1 ((x + y) x_1)) =
    ((fun x_1 ↦
          match x_1 with
          | (x_2, y) => y (x x_2)) +
        fun x ↦
        match x with
        | (x, y_1) => y_1 (y x))
      x✝
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁹ : NormedField 𝕜₁
inst✝⁸ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : Module 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜₂ F
x : 𝕜₂
y : E →SWOT[σ] F
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ (match x✝ with
    | (x_1, y_1) => y_1 ((x • y) x_1)) =
    ((RingHom.id 𝕜₂) x • fun x ↦
        match x with
        | (x, y_1) => y_1 (y x))
      x✝

Tactic:
;

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁹ : NormedField 𝕜₁
inst✝⁸ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : Module 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜₂ F
x : 𝕜₂
y : E →SWOT[σ] F
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ (match x✝ with
    | (x_1, y_1) => y_1 ((x • y) x_1)) =
    ((RingHom.id 𝕜₂) x • fun x ↦
        match x with
        | (x, y_1) => y_1 (y x))
      x✝
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i j : n
hij : ¬i = j
⊢ diag hS i j = (lowerInv hS * S * (lowerInv hS)ᴴ) i j

Tactic:
· simp only [LDL.diag, hij, diagonal_apply_ne, Ne, not_false_iff, mul_mul_apply]
    rw [conjTranspose, transpose_map, transpose_transpose, dotProduct_mulVec,
      (LDL.lowerInv_orthogonal hS fun h : j = i => hij h.symm).symm, ← inner_conj_symm,
      mulVec_transpose, EuclideanSpace.inner_toLp_toLp, ← RCLike.star_def, ←
      star_dotProduct_star, star_star]
    rfl

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakSpace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E →ₗ[𝕜] F
he : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
f : F →L[𝕜] 𝕜
⊢ Continuous ⇑(e.dualMap ↑f)

Tactic:
;

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E →ₗ[𝕜] F
he : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
f : F →L[𝕜] 𝕜
⊢ Continuous ⇑(e.dualMap ↑f)
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
a : E →SWOT[σ₁₂] F
| z ((g.comp a) x)

Tactic:
;

Goal state after:
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
a : E →SWOT[σ₁₂] F
| z ((g.comp a) x)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s a : ℂ
⊢ MellinConvergent (fun t ↦ ↑t ^ a • f t) s ↔ MellinConvergent f (s + a)

Tactic:
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s a : ℂ
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • (fun t ↦ ↑t ^ a • f t) t = ↑t ^ (s + a - 1) • f t
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
x : ℝ
s : ℂ
f : E
| cexp (-↑x)

Tactic:
;

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
x : ℝ
s : ℂ
f : E
| cexp (-↑x)
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
| cexp (-s * ↑u) • f (rexp (-u))

Tactic:
;

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
| cexp (-s * ↑u) • f (rexp (-u))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ IntegrableOn (fun t ↦ ↑t ^ (s - 1) • f (t ^ a)) (Ioi 0) volume ↔
    IntegrableOn (fun x ↦ x ^ (a - 1) • ↑(x ^ a) ^ (s / ↑a - 1) • f (x ^ a)) (Ioi 0) volume

Tactic:
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = t ^ (a - 1) • ↑(t ^ a) ^ (s / ↑a - 1) • f (t ^ a)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑(t ^ (a - 1)) • ↑(t ^ a) ^ (s / ↑a - 1) • f (t ^ a)

Tactic:
← mul_smul,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = (↑(t ^ (a - 1)) * ↑(t ^ a) ^ (s / ↑a - 1)) • f (t ^ a)
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x✝.comp x

Tactic:
;

Goal state after:
𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x✝.comp x
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = |a|⁻¹ • (|a| * t ^ (a - 1)) • ↑(t ^ a) ^ (s / ↑a - 1) • f (t ^ a)

Tactic:
← mul_smul,

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = (|a|⁻¹ * (|a| * t ^ (a - 1))) • ↑(t ^ a) ^ (s / ↑a - 1) • f (t ^ a)
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x.comp x✝

Tactic:
;

Goal state after:
𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x.comp x✝
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (a * t) = ↑a ^ (1 - s) • (↑a ^ (s - 1) * ↑t ^ (s - 1)) • f (a * t)

Tactic:
← mul_smul,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (a * t) = (↑a ^ (1 - s) * (↑a ^ (s - 1) * ↑t ^ (s - 1))) • f (a * t)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this : EqOn (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) (Ioi 0)
⊢ ↑a ^ (1 - s) • ↑a⁻¹ • ∫ (x : ℝ) in Ioi 0, ↑x ^ (s - 1) • f x = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t

Tactic:
← mul_smul,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this : EqOn (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) (Ioi 0)
⊢ (↑a ^ (1 - s) * ↑a⁻¹) • ∫ (x : ℝ) in Ioi 0, ↑x ^ (s - 1) • f x = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
⊢ s.GammaIntegral = mellin (fun x ↦ ↑(rexp (-x))) s

Tactic:
simp only [mellin, GammaIntegral, smul_eq_mul, mul_comm]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ ∀ (m : ℕ), ¬(False ∧ m = 0)

Tactic:
;

Goal state after:
⊢ ∀ (m : ℕ), ¬(False ∧ m = 0)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case e'_3
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
⊢ z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹ = z • ∫ (t : ℝ) in 0..1, f' (0 + t • z)

Tactic:
· simp only [f', real_smul, zero_add,
        smul_eq_mul]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
hz : z ≠ 0
x : ℝ
a : HasDerivAt (fun y ↦ y * (2 * z)) (2 * z) ↑x
b : HasDerivAt (Complex.sin ∘ fun y ↦ y * (2 * z)) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x
c : HasDerivAt (fun x ↦ (Complex.sin ∘ fun y ↦ y * 2 * z) ↑x / (2 * z)) (Complex.cos (2 * z * ↑x)) x
⊢ HasDerivAt (fun y ↦ Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x

Tactic:
;

Goal state after:
z : ℂ
hz : z ≠ 0
x : ℝ
a : HasDerivAt (fun y ↦ y * (2 * z)) (2 * z) ↑x
b : HasDerivAt (Complex.sin ∘ fun y ↦ y * (2 * z)) (Complex.cos (↑x * (2 * z)) * (2 * z)) ↑x
c : HasDerivAt (fun x ↦ (Complex.sin ∘ fun y ↦ y * 2 * z) ↑x / (2 * z)) (Complex.cos (2 * z * ↑x)) x
⊢ HasDerivAt (fun y ↦ Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
hz : z ≠ 0
x : ℝ
a : HasDerivAt (fun y ↦ y * (2 * z)) (2 * z) ↑x
b : HasDerivAt (Complex.cos ∘ fun y ↦ y * (2 * z)) (-Complex.sin (↑x * (2 * z)) * (2 * z)) ↑x
c :
  HasDerivAt (fun i ↦ -(Complex.cos (↑i * (2 * z)) / (2 * z))) (-(-(Complex.sin (↑x * (2 * z)) * (2 * z)) / (2 * z))) x
⊢ HasDerivAt (fun y ↦ -Complex.cos (2 * z * ↑y) / (2 * z)) (Complex.sin (2 * z * ↑x)) x

Tactic:
;

Goal state after:
z : ℂ
hz : z ≠ 0
x : ℝ
a : HasDerivAt (fun y ↦ y * (2 * z)) (2 * z) ↑x
b : HasDerivAt (Complex.cos ∘ fun y ↦ y * (2 * z)) (-Complex.sin (↑x * (2 * z)) * (2 * z)) ↑x
c :
  HasDerivAt (fun i ↦ -(Complex.cos (↑i * (2 * z)) / (2 * z))) (-(-(Complex.sin (↑x * (2 * z)) * (2 * z)) / (2 * z))) x
⊢ HasDerivAt (fun y ↦ -Complex.cos (2 * z * ↑y) / (2 * z)) (Complex.sin (2 * z * ↑x)) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
hz : z ≠ 0
x : ℝ
a : HasDerivAt (fun y ↦ y * (2 * z)) (2 * z) ↑x
b : HasDerivAt (Complex.cos ∘ fun y ↦ y * (2 * z)) (-Complex.sin (↑x * (2 * z)) * (2 * z)) ↑x
c : HasDerivAt (fun i ↦ -(Complex.cos (↑i * 2 * z) / (2 * z))) (Complex.sin (2 * z * ↑x)) x
⊢ HasDerivAt (fun y ↦ -(Complex.cos (2 * z * ↑y) / (2 * z))) (Complex.sin (2 * z * ↑x)) x

Tactic:
;

Goal state after:
z : ℂ
hz : z ≠ 0
x : ℝ
a : HasDerivAt (fun y ↦ y * (2 * z)) (2 * z) ↑x
b : HasDerivAt (Complex.cos ∘ fun y ↦ y * (2 * z)) (-Complex.sin (↑x * (2 * z)) * (2 * z)) ↑x
c : HasDerivAt (fun i ↦ -(Complex.cos (↑i * 2 * z) / (2 * z))) (Complex.sin (2 * z * ↑x)) x
⊢ HasDerivAt (fun y ↦ -(Complex.cos (2 * z * ↑y) / (2 * z))) (Complex.sin (2 * z * ↑x)) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_2.e'_4
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
x : ℝ
⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)

Tactic:
;

Goal state after:
case e'_2.e'_4
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
x : ℝ
⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz1 : ‖z‖ < 1
hz12 : ‖z‖ < 1 / 2
⊢ 2⁻¹ ≤ 1 - ‖z‖

Tactic:
;

Goal state after:
n : ℕ
z : ℂ
hz1 : ‖z‖ < 1
hz12 : ‖z‖ < 1 / 2
⊢ 2⁻¹ ≤ 1 - ‖z‖
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
n : ℕ
hn : n > 0
| ‖z‖ ^ n

Tactic:
;

Goal state after:
z : ℂ
hz : ‖z‖ < 1
n : ℕ
hn : n > 0
| ‖z‖ ^ n
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
x : ℕ
| ∑ i ∈ Finset.range x, (-1) ^ (i + 1) * z ^ i / ↑i - log (1 + z)

Tactic:
;

Goal state after:
z : ℂ
hz : ‖z‖ < 1
x : ℕ
| ∑ i ∈ Finset.range x, (-1) ^ (i + 1) * z ^ i / ↑i - log (1 + z)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case refine_2
z : ℂ
hz : ‖z‖ < 1
⊢ Tendsto (fun x ↦ ∑ i ∈ Finset.range x, (-1) ^ (i + 1) * z ^ i / ↑i - log (1 + z)) atTop (nhds 0)

Tactic:
conv => enter [1, x]; rw [← div_one (_ - _), ← logTaylor]

Goal state after:
case refine_2
z : ℂ
hz : ‖z‖ < 1
⊢ Tendsto (fun x ↦ (logTaylor x z - log (1 + z)) / 1) atTop (nhds 0)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
n : ℕ
| z ^ n / ↑n

Tactic:
;

Goal state after:
z : ℂ
hz : ‖z‖ < 1
n : ℕ
| z ^ n / ↑n
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
A : rexp (-x) = rexp (-a * x) * rexp (-b * x)
⊢ x ^ (a * s + (1 - a) * t - 1) = x ^ (a * (s - 1) + (1 - a) * (t - 1))

Tactic:
;

Goal state after:
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
A : rexp (-x) = rexp (-a * x) * rexp (-b * x)
⊢ x ^ (a * s + (1 - a) * t - 1) = x ^ (a * (s - 1) + (1 - a) * (t - 1))