mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
H : Set (ZMod n)ˣ
χ : DirichletCharacter R n
h : ∀ x ∈ Submonoid.closure H, χ ↑x = 1
u : (↥(Submonoid.map (Units.coeHom (ZMod n)) (Submonoid.closure H)))ˣ
⊢ χ ↑↑u = 1

Tactic:
obtain ⟨y, hy, hyu⟩ := Submonoid.mem_map.mp u.val.prop
    exact hyu ▸ h _ hy

Goal state after:
Module: Mathlib.NumberTheory.DirichletCharacter.GaussSum

Goal state before:
pp: N : ℕ
inst✝¹ : NeZero N
R : Type u_1
inst✝ : CommRing R
e : AddChar (ZMod N) R
χ : DirichletCharacter R N
d : ℕ
hd : d ∣ N
he : e.mulShift ↑d = 1
u : (ZMod N)ˣ
hu : (ZMod.unitsMap hd) u = 1
⊢ ↑d ∣ ↑(↑u).val - 1

Tactic:
rw [← Units.val_inj, Units.val_one, ZMod.unitsMap_def, Units.coe_map] at hu

Goal state after:
N : ℕ
inst✝¹ : NeZero N
R : Type u_1
inst✝ : CommRing R
e : AddChar (ZMod N) R
χ : DirichletCharacter R N
d : ℕ
hd : d ∣ N
he : e.mulShift ↑d = 1
u : (ZMod N)ˣ
hu : ↑(ZMod.castHom hd (ZMod d)) ↑u = 1
⊢ ↑d ∣ ↑(↑u).val - 1
Module: Mathlib.Order.Grade

Goal state before:
pp: α : Type u_3
inst✝ : PartialOrder α
s : Flag α
a b : ↥s
⊢ ↑a ⩿ ↑b ↔ a ⩿ b

Tactic:
refine and_congr_right' ⟨fun h c hac ↦ h hac, fun h c hac hcb ↦
    @h ⟨c, mem_iff_forall_le_or_ge.2 fun d hd ↦ ?_⟩ hac hcb⟩

Goal state after:
α : Type u_3
inst✝ : PartialOrder α
s : Flag α
a b : ↥s
h : ∀ ⦃c : ↥s⦄, a < c → ¬c < b
c : α
hac : ↑a < c
hcb : c < ↑b
d : α
hd : d ∈ s
⊢ c ≤ d ∨ d ≤ c
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: F : Type u_1
inst✝¹ : Fintype F
inst✝ : Field F
hF : ringChar F ≠ 2
hp2 : ∀ (n : ℕ), 2 ^ n ≠ 0
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
FF : Type u_1 := CyclotomicField 8 F
hchar : ringChar F = ringChar FF
FFp : Nat.Prime (ringChar FF)
this : Fact (Nat.Prime (ringChar FF))
hFF : ringChar FF ≠ 2
hu : IsUnit ↑(ringChar FF)
ψ₈ : PrimitiveAddChar (ZMod ↑8) F := primitiveZModChar 8 F ⋯
ψ₈char : AddChar (ZMod 8) FF := ψ₈.char
τ : FF := ψ₈char 1
τ_spec : τ ^ 4 = -1
χ : MulChar (ZMod 8) FF := χ₈.ringHomComp (Int.castRingHom FF)
hχ : χ (-1) = 1
hq : χ.IsQuadratic
h₁ : (fun a ↦ ↑(χ₈ a) * τ ^ ↑a) = fun a ↦ χ a * ψ₈char a
⊢ ↑(χ₈ 0) * τ ^ ↑0 + ↑(χ₈ 1) * τ ^ ↑1 + ↑(χ₈ 2) * τ ^ ↑2 + ↑(χ₈ 3) * τ ^ ↑3 + ↑(χ₈ 4) * τ ^ ↑4 + ↑(χ₈ 5) * τ ^ ↑5 +
        ↑(χ₈ 6) * τ ^ ↑6 +
      ↑(χ₈ 7) * τ ^ ↑7 =
    2 * (τ - τ ^ 3)

Tactic:
show χ₈ 0 = 0 from rfl,

Goal state after:
F : Type u_1
inst✝¹ : Fintype F
inst✝ : Field F
hF : ringChar F ≠ 2
hp2 : ∀ (n : ℕ), 2 ^ n ≠ 0
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
FF : Type u_1 := CyclotomicField 8 F
hchar : ringChar F = ringChar FF
FFp : Nat.Prime (ringChar FF)
this : Fact (Nat.Prime (ringChar FF))
hFF : ringChar FF ≠ 2
hu : IsUnit ↑(ringChar FF)
ψ₈ : PrimitiveAddChar (ZMod ↑8) F := primitiveZModChar 8 F ⋯
ψ₈char : AddChar (ZMod 8) FF := ψ₈.char
τ : FF := ψ₈char 1
τ_spec : τ ^ 4 = -1
χ : MulChar (ZMod 8) FF := χ₈.ringHomComp (Int.castRingHom FF)
hχ : χ (-1) = 1
hq : χ.IsQuadratic
h₁ : (fun a ↦ ↑(χ₈ a) * τ ^ ↑a) = fun a ↦ χ a * ψ₈char a
⊢ ↑0 * τ ^ ↑0 + ↑(χ₈ 1) * τ ^ ↑1 + ↑(χ₈ 2) * τ ^ ↑2 + ↑(χ₈ 3) * τ ^ ↑3 + ↑(χ₈ 4) * τ ^ ↑4 + ↑(χ₈ 5) * τ ^ ↑5 +
        ↑(χ₈ 6) * τ ^ ↑6 +
      ↑(χ₈ 7) * τ ^ ↑7 =
    2 * (τ - τ ^ 3)
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
⊢ T.IsSymmetric → ((∀ (x : E), 0 ≤ ↑(re ⟪T x, x⟫)) ↔ ∀ (x : E), 0 ≤ ⟪T x, x⟫)

Tactic:
intro hT

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
⊢ (∀ (x : E), 0 ≤ ↑(re ⟪T x, x⟫)) ↔ ∀ (x : E), 0 ≤ ⟪T x, x⟫
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
p q : E →ₗ[𝕜] E
hp : p.IsSymmetricProjection
hq : q.IsSymmetricProjection
⊢ p ≤ q ↔ p.range ≤ q.range

Tactic:
by
  refine ⟨fun ⟨h1, h2⟩ a ha ↦ ?_, fun hpq ↦ (hp.sub_of_range_le_range hq hpq).isPositive⟩
  specialize h2 a
  have hh {T : E →ₗ[𝕜] E} (hT : T.IsSymmetricProjection) : RCLike.re ⟪T a, a⟫_𝕜 = ‖T a‖ ^ 2 := by
    conv_lhs => rw [← hT.isIdempotentElem]
    rw [Module.End.mul_apply, hT.isSymmetric]
    exact inner_self_eq_norm_sq _
  simp_rw [sub_apply, inner_sub_left, map_sub, hh hq, hh hp,
    hp.isIdempotentElem.mem_range_iff.mp ha, sub_nonneg, sq_le_sq, abs_norm] at h2
  obtain ⟨U, _, rfl⟩ := isSymmetricProjection_iff_eq_coe_starProjection.mp hq
  simpa [Submodule.toLinearMap_starProjection_eq_isComplProjection] using
    U.mem_iff_norm_starProjection _ |>.mpr <| le_antisymm (U.norm_starProjection_apply_le a) h2

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LaxMilgram

Goal state before:
pp: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
mem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
⊢ w = 0

Tactic:
by
    rw [← norm_eq_zero, ← mul_self_eq_zero, ← mul_right_inj' C_pos.ne', mul_zero, ←
      mul_assoc]
    apply le_antisymm
    · calc
        C * ‖w‖ * ‖w‖ ≤ B w w := coercivity w
        _ = ⟪B♯ w, w⟫_ℝ := (continuousLinearMapOfBilin_apply B w w).symm
        _ = 0 := mem_w_orthogonal _ ⟨w, rfl⟩
    · positivity

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝
hp : (↑#ι).HolderConjugate p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this :
  ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * ↑(c ^ p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p
⊢ ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * ↑c ^ p * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p

Tactic:
← ENNReal.coe_rpow_of_ne_zero hc.ne'

Goal state after:
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝
hp : (↑#ι).HolderConjugate p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this :
  ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * ↑(c ^ p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p
⊢ ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * ↑(c ^ p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p
Module: Mathlib.Dynamics.BirkhoffSum.Basic

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝ : AddCommMonoid M
f : α → α
x : α
h : IsFixedPt f x
g : α → M
n : ℕ
⊢ birkhoffSum f g n x = n • g x

Tactic:
by
  simp [birkhoffSum, (h.iterate _).eq]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : 0 < r
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
ε : ℝ≥0∞
left✝ : 0 < ε
εr : ε < r
⊢ (mkMetric'.pre m ε) (s ∪ t) = (mkMetric'.pre m ε) s + (mkMetric'.pre m ε) t

Tactic:
refine boundedBy_union_of_top_of_nonempty_inter ?_

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : 0 < r
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
ε : ℝ≥0∞
left✝ : 0 < ε
εr : ε < r
⊢ ∀ (u : Set X), (s ∩ u).Nonempty → (t ∩ u).Nonempty → extend (fun s x ↦ m s) u = ∞
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : ¬f.ker = ⊥
⊢ ↑(f.normDet ^ 2) = (Matrix.gram 𝕜 fun x ↦ f.rangeRestrict (b x)).det

Tactic:
trans 0
    · simp [show f.normDet = 0 from (f.normDet_eq_zero_tfae.out 1 0).mp h]
    have hrank := (f.normDet_eq_zero_tfae.out 1 3).mp h
    symm
    contrapose! hrank with h0
    rw [finrank_eq_card_basis b.toBasis]
    exact (Matrix.linearIndependent_of_det_gram_ne_zero h0).fintype_card_le_finrank

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case refine_2
X : Type u_2
inst✝³ : EMetricSpace X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
β : Type u_4
ι : β → Type u_5
inst✝ : ∀ (n : β), Countable (ι n)
s : Set X
l : Filter β
r : β → ℝ≥0∞
hr : Tendsto r l (𝓝 0)
t : (n : β) → ι n → Set X
ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n
hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ i, t n i
m : ℝ≥0∞ → ℝ≥0∞
this : (n : β) → Encodable (ι n)
ε : ℝ≥0∞
hε : 0 < ε
c : ℝ≥0∞
hc : liminf (fun n ↦ ∑' (i : ι n), m (ediam (t n i))) l < c
n : β
hn : ∑' (i : ι n), m (ediam (t n i)) < c
hrn : r n < ε
htn : ∀ (i : ι n), ediam (t n i) ≤ r n
hstn : s ⊆ ⋃ i, t n i
u : ℕ → Set X := fun j ↦ ⋃ b ∈ decode₂ (ι n) j, t n b
⊢ ∑' (n : ℕ), ⨆ (_ : (u n).Nonempty), m (ediam (u n)) ≤ c

Tactic:
· calc
      (∑' j : ℕ, ⨆ _ : (u j).Nonempty, m (ediam (u j))) = _ :=
        tsum_iUnion_decode₂ (fun t : Set X => ⨆ _ : t.Nonempty, m (ediam t)) (by simp) _
      _ ≤ ∑' i : ι n, m (ediam (t n i)) := ENNReal.tsum_le_tsum fun b => iSup_le fun _ => le_rfl
      _ ≤ c := hn.le

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
ε : ℝ
⊢ ε > 0 → ∀ᶠ (n : ι) in l, ∀ x ∈ s, dist (f x) ((F n) x) < ε

Tactic:
hε

Goal state after:
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (n : ι) in l, ∀ x ∈ s, dist (f x) ((F n) x) < ε
Module: Mathlib.Analysis.InnerProductSpace.StarOrder

Goal state before:
pp: 𝕜 : Type u_1
H : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : CompleteSpace H
inst✝¹ : Algebra ℝ (H →L[𝕜] H)
inst✝ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
f : H →L[𝕜] H
hf : f.IsPositive
⊢ SpectrumRestricts f ⇑ContinuousMap.realToNNReal

Tactic:
rw [SpectrumRestricts.nnreal_iff]
  intro c hc
  contrapose! hc
  rw [spectrum.notMem_iff, IsUnit.sub_iff, sub_eq_add_neg, ← map_neg]
  rw [← neg_pos] at hc
  set c := -c
  exact isUnit_of_forall_le_norm_inner_map _ (c := ⟨c, hc.le⟩) hc fun x ↦ calc
    ‖x‖ ^ 2 * c = re ⟪algebraMap ℝ (H →L[𝕜] H) c x, x⟫_𝕜 := by
      rw [Algebra.algebraMap_eq_smul_one, ← algebraMap_smul 𝕜 c (1 : (H →L[𝕜] H)), smul_apply,
        one_apply_eq_self, inner_smul_left, RCLike.algebraMap_eq_ofReal, conj_ofReal, re_ofReal_mul,
        inner_self_eq_norm_sq, mul_comm]
    _ ≤ re ⟪(f + (algebraMap ℝ (H →L[𝕜] H)) c) x, x⟫_𝕜 := by
      simpa only [add_apply, inner_add_left, map_add, le_add_iff_nonneg_left]
        using hf.re_inner_nonneg_left x
    _ ≤ ‖⟪(f + (algebraMap ℝ (H →L[𝕜] H)) c) x, x⟫_𝕜‖ := RCLike.re_le_norm _

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
x : X
x✝¹ x✝ : V
⊢ ↑(Finsupp.single (x, x✝¹ + x✝) 1) = ↑(Finsupp.single (x, x✝¹) 1) + ↑(Finsupp.single (x, x✝) 1)

Tactic:
refine UniformSpace.Completion.denseRange_coe.eq_of_inner_left 𝕜 fun f ↦ ?_

Goal state after:
𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
x : X
x✝¹ x✝ : V
f : H₀ K
⊢ ⟪↑(Finsupp.single (x, x✝¹ + x✝) 1), ↑f⟫_𝕜 = ⟪↑(Finsupp.single (x, x✝¹) 1) + ↑(Finsupp.single (x, x✝) 1), ↑f⟫_𝕜
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
x : X
x✝¹ : 𝕜
x✝ : V
⊢ ↑(Finsupp.single (x, x✝¹ • x✝) 1) = (RingHom.id 𝕜) x✝¹ • ↑(Finsupp.single (x, x✝) 1)

Tactic:
refine UniformSpace.Completion.denseRange_coe.eq_of_inner_left 𝕜 fun f ↦ ?_

Goal state after:
𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
x : X
x✝¹ : 𝕜
x✝ : V
f : H₀ K
⊢ ⟪↑(Finsupp.single (x, x✝¹ • x✝) 1), ↑f⟫_𝕜 = ⟪(RingHom.id 𝕜) x✝¹ • ↑(Finsupp.single (x, x✝) 1), ↑f⟫_𝕜
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: ι : Type u_4
inst✝ : Fintype ι
s : ι → Set ℝ
a b : ι → ℚ
H : ∀ (i : ι), ∃ (_ : a i < b i), s i = Ioo ↑(a i) ↑(b i)
⊢ μH[↑(Fintype.card ι)] (univ.pi s) = ∏ x, volume (s x)

Tactic:
obtain rfl : s = fun i => Ioo (α := ℝ) (a i) (b i) := funext fun i => (H i).2

Goal state after:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), ∃ (_ : a i < b i), (fun i ↦ Ioo ↑(a i) ↑(b i)) i = Ioo ↑(a i) ↑(b i)
⊢ μH[↑(Fintype.card ι)] (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) = ∏ x, volume ((fun i ↦ Ioo ↑(a i) ↑(b i)) x)
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: case refine_1
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
⊢ ∀ (i : ℕ), True → ∃ i', True ∧ c ^ i' • b i' ⊆ b i

Tactic:
intro n _

Goal state after:
case refine_1
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
a✝ : True
⊢ ∃ i', True ∧ c ^ i' • b i' ⊆ b n
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: case refine_2
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
⊢ ∀ (i' : ℕ), True → c ^ i' • b i' ∈ 𝓝 0

Tactic:
intro n _

Goal state after:
case refine_2
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
a✝ : True
⊢ c ^ n • b n ∈ 𝓝 0
Module: Mathlib.Analysis.LocallyConvex.Montel

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹³ : NormedField 𝕜₁
inst✝¹² : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : IsTopologicalAddGroup E
inst✝⁷ : ContinuousSMul 𝕜₁ E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₂ F
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜₂ F
inst✝¹ : T1Space E
inst✝ : MontelSpace 𝕜₁ E
⊢ Continuous[topologicalSpace, UniformConvergenceCLM.instTopologicalSpace σ F {S | IsCompact S}]
    (↑(LinearEquiv.toCompactConvergenceCLM σ E F)).toFun

Tactic:
simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, LinearEquiv.coe_coe, continuous_def]
    intro s hs
    apply hs.mono
    apply UniformConvergenceCLM.topologicalSpace_mono
    intro x hx
    exact hx.isVonNBounded 𝕜₁

Goal state after:
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 y : E
hx : ¬x = 0
a : ℝ := ((o.basisRightAngleRotation x hx).repr y) 0
b : ℝ := ((o.basisRightAngleRotation x hx).repr y) 1
this : 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • o.rightAngleRotation x)
⊢ 0 ≤ ⟪x, y⟫ ∧ (o.areaForm x) y = 0 → SameRay ℝ x y

Tactic:
rw [← (o.basisRightAngleRotation x hx).sum_repr y]

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 y : E
hx : ¬x = 0
a : ℝ := ((o.basisRightAngleRotation x hx).repr y) 0
b : ℝ := ((o.basisRightAngleRotation x hx).repr y) 1
this : 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • o.rightAngleRotation x)
⊢ 0 ≤ ⟪x, ∑ i, ((o.basisRightAngleRotation x hx).repr y) i • (o.basisRightAngleRotation x hx) i⟫ ∧
      (o.areaForm x) (∑ i, ((o.basisRightAngleRotation x hx).repr y) i • (o.basisRightAngleRotation x hx) i) = 0 →
    SameRay ℝ x (∑ i, ((o.basisRightAngleRotation x hx).repr y) i • (o.basisRightAngleRotation x hx) i)
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.re * ↑u + -(↑s.im * I) * ↑u) • f (rexp (-u)) = cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))

Tactic:
Complex.exp_add,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
⊢ (cexp (-↑s.re * ↑u) * cexp (-(↑s.im * I) * ↑u)) • f (rexp (-u)) =
    cexp (-↑s.im * ↑u * I) • rexp (-s.re * 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
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (s - ↑a + (↑a - 1)) • f (t ^ a)

Tactic:
←
    add_sub_assoc,

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 ^ (s - ↑a + ↑a - 1) • f (t ^ a)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: ⊢ (fun z ↦ log (1 + z) - z) =O[𝓝 0] fun z ↦ z ^ 2

Tactic:
by
  convert! log_sub_logTaylor_isBigO 1
  simp [logTaylor_succ, logTaylor_zero]

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

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

Tactic:
n

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

Goal state before:
pp: g : ℝ → ℝ
t : ℝ
hg0 : Tendsto g atTop (𝓝 0)
hg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)
⊢ Tendsto (fun x ↦ ↑((1 + g x) ^ x)) atTop (𝓝 (Complex.exp ↑t))

Tactic:
apply (Complex.tendsto_one_add_cpow_exp_of_tendsto hg).congr'

Goal state after:
g : ℝ → ℝ
t : ℝ
hg0 : Tendsto g atTop (𝓝 0)
hg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)
⊢ (fun x ↦ (1 + ↑(g x)) ^ ↑x) =ᶠ[atTop] fun x ↦ ↑((1 + g x) ^ x)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u : ℂ
hu : 0 < u.re
⊢ ∫ (x : ℝ) in 0..1, ↑x ^ (u - 1) * (1 - ↑x) ^ 0 = 1 / u

Tactic:
cpow_zero,

Goal state after:
u : ℂ
hu : 0 < u.re
⊢ ∫ (x : ℝ) in 0..1, ↑x ^ (u - 1) * 1 = 1 / u
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
n : ℕ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
m : ℕ
hm : 1 ≤ m
IH : f ↑m = f 1 + log ↑(m - 1)!
A : 0 < ↑m
⊢ f 1 + log ↑((m - 1)! * m) = f 1 + log ↑m !

Tactic:
conv_rhs => rw [← Nat.succ_pred_eq_of_pos hm, Nat.factorial_succ, mul_comm]

Goal state after:
f : ℝ → ℝ
n : ℕ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
m : ℕ
hm : 1 ≤ m
IH : f ↑m = f 1 + log ↑(m - 1)!
A : 0 < ↑m
⊢ f 1 + log ↑((m - 1)! * m) = f 1 + log ↑(m.pred ! * (m.pred + 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_1
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) (log ∘ f)
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = y * f y
hf_pos : ∀ {y : ℝ}, 0 < y → 0 < f y
hf_one : f 1 = 1
x : ℝ
hx : x ∈ Ioi 0
⊢ ∀ {y : ℝ}, 0 < y → (log ∘ f) (y + 1) = (log ∘ f) y + log y

Tactic:
· intro y hy
    rw [Function.comp_apply, Function.comp_apply, hf_feq hy, log_mul hy.ne' (hf_pos hy).ne']
    ring

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

Goal state before:
pp: case e'_6
⊢ Ici 2 ⊆ Ioi 0

Tactic:
exact fun x hx => two_pos.trans_le <| mem_Ici.mp hx

Goal state after: