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

Goal state before:
pp: case succ
R : Type u_1
inst✝² : CommMonoid R
R' : Type u_2
inst✝¹ : CommRing R'
R'' : Type u_3
inst✝ : CommRing R''
χ : MulChar R R'
f : R' →+* R''
n : ℕ
ih : χ.ringHomComp f ^ n = (χ ^ n).ringHomComp f
⊢ χ.ringHomComp f ^ (n + 1) = (χ ^ (n + 1)).ringHomComp f

Tactic:
| succ n ih =>

Goal state after:
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: case inr.inr
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommRing R'
χ : MulChar R R'
hχ : χ.IsQuadratic
x : Rˣ
h₂ : χ ↑x = -1
this : -1 = ↑(-1)
⊢ (↑(-1))⁻¹ʳ = ↑(-1)

Tactic:
Ring.inverse_unit (-1 : R'ˣ),

Goal state after:
case inr.inr
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommRing R'
χ : MulChar R R'
hχ : χ.IsQuadratic
x : Rˣ
h₂ : χ ↑x = -1
this : -1 = ↑(-1)
⊢ ↑(-1)⁻¹ = ↑(-1)
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ χ.primitiveCharacter.IsPrimitive

Tactic:
by
  by_cases h : χ.conductor = 0
  · rw [isPrimitive_def]
    convert! conductor_eq_zero_iff_level_eq_zero.mpr h
  · exact le_antisymm (Nat.le_of_dvd (Nat.pos_of_ne_zero h) (conductor_dvd_level _)) <|
      conductor_le_conductor_mem_conductorSet <| conductor_mem_conductorSet χ

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

Goal state before:
pp: R : Type u_1
inst✝² : CommMonoidWithZero R
n : ℕ
inst✝¹ : Nontrivial R
m : ℕ
inst✝ : NeZero m
hm : n ∣ m
χ : DirichletCharacter R n
a : ℤ
⊢ ((changeLevel hm) χ).primitiveCharacter ↑a = χ.primitiveCharacter ↑a

Tactic:
by_cases ha : IsCoprime a χ.conductor
  · suffices changeLevel (dvd_of_eq <| conductor_changeLevel ..)
        (changeLevel hm χ).primitiveCharacter = χ.primitiveCharacter by
      have := DFunLike.congr_fun this (a : ZMod _)
      rwa [changeLevel_eq_cast_of_dvd' _ _ ha] at this
    apply changeLevel_injective (χ.conductor_dvd_level.trans hm)
    rw [← changeLevel_trans, changeLevel_primitiveCharacter,
      χ.primitiveCharacter.changeLevel_trans χ.conductor_dvd_level, changeLevel_primitiveCharacter]
  · rw [(apply_eq_zero_iff ..).mpr ha, (apply_eq_zero_iff ..).mpr (by rwa [conductor_changeLevel])]

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
⊢ e.mulShift (↑u - 1 + 1) = e

Tactic:
← mulShift_mul,

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.unitsMap hd) u = 1
⊢ e.mulShift (↑u - 1) * e.mulShift 1 = e
Module: Mathlib.NumberTheory.DirichletCharacter.GaussSum

Goal state before:
pp: N : ℕ
inst✝² : NeZero N
R : Type u_1
inst✝¹ : CommRing R
e : AddChar (ZMod N) R
inst✝ : IsDomain R
χ : DirichletCharacter R N
d : ℕ
hd : d ∣ N
he : e.mulShift ↑d = 1
h_ne : gaussSum χ e ≠ 0
x✝ : (ZMod N)ˣ
⊢ x✝ ∈ (ZMod.unitsMap hd).ker → x✝ ∈ (MulChar.toUnitHom χ).ker

Tactic:
hu

Goal state after:
N : ℕ
inst✝² : NeZero N
R : Type u_1
inst✝¹ : CommRing R
e : AddChar (ZMod N) R
inst✝ : IsDomain R
χ : DirichletCharacter R N
d : ℕ
hd : d ∣ N
he : e.mulShift ↑d = 1
h_ne : gaussSum χ e ≠ 0
x✝ : (ZMod N)ˣ
hu : x✝ ∈ (ZMod.unitsMap hd).ker
⊢ x✝ ∈ (MulChar.toUnitHom χ).ker
Module: Mathlib.Order.Grade

Goal state before:
pp: case inr.inl
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
had : a < ⟨d, hd⟩
hbd : b ≤ ⟨d, hd⟩
⊢ c ≤ d ∨ d ≤ c

Tactic:
exact .inl (hcb.le.trans hbd)

Goal state after:
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: case succ
R : Type u
inst✝² : CommRing R
inst✝¹ : Fintype R
R' : Type v
inst✝ : CommRing R'
p : ℕ
fp : Fact (Nat.Prime p)
hch : CharP R' p
hp : IsUnit ↑p
χ : MulChar R R'
hχ : χ.IsQuadratic
ψ : AddChar R R'
n : ℕ
ih : gaussSum χ ψ ^ p ^ n = χ (↑p ^ n) * gaussSum χ ψ
⊢ gaussSum χ ψ ^ p ^ (n + 1) = χ (↑p ^ (n + 1)) * gaussSum χ ψ

Tactic:
| succ n ih =>

Goal state after:
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
⊢ ↑0 * τ ^ ↑0 + ↑1 * τ ^ ↑1 + ↑0 * τ ^ ↑2 + ↑(-1) * τ ^ ↑3 + ↑0 * τ ^ ↑4 + ↑(-1) * τ ^ ↑5 + ↑0 * τ ^ ↑6 +
      ↑(χ₈ 7) * τ ^ ↑7 =
    2 * (τ - τ ^ 3)

Tactic:
show χ₈ 7 = 1 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 + ↑1 * τ ^ ↑1 + ↑0 * τ ^ ↑2 + ↑(-1) * τ ^ ↑3 + ↑0 * τ ^ ↑4 + ↑(-1) * τ ^ ↑5 + ↑0 * τ ^ ↑6 + ↑1 * τ ^ ↑7 =
    2 * (τ - τ ^ 3)
Module: Mathlib.Analysis.InnerProductSpace.Affine

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
⊢ (AffineMap.lineMap a b) t₁ -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)

Tactic:
rw [AffineMap.lineMap_apply, AffineMap.lineMap_apply, vadd_vsub_vadd_cancel_right]

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

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : f.IsPositive
h : ∃ s, Nonempty (Module.Basis (↥s) 𝕜 E)
this : FiniteDimensional 𝕜 E
⊢ 0 ≤ (traceAux 𝕜 (stdOrthonormalBasis 𝕜 E).toBasis) f

Tactic:
exact posSemidef_toMatrix_iff (stdOrthonormalBasis 𝕜 E) |>.mpr hf |>.trace_nonneg

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

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
hA : A.IsSymmetric
hB : B.IsSymmetric
i j : 𝕜 × 𝕜
hij : i ≠ j
v : E
x✝ : v ∈ eigenspace A i.2 ⊓ eigenspace B i.1
hv1 : v ∈ ↑(eigenspace A i.2)
hv2 : v ∈ ↑(eigenspace B i.1)
h₂ : i.2 ≠ j.2
w : E
hw1 : w ∈ ↑(eigenspace A j.2)
hw2 : w ∈ ↑(eigenspace B j.1)
⊢ inner 𝕜 w v = 0

Tactic:
· exact hA.orthogonalFamily_eigenspaces.pairwise h₂ hv1 w hw1

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
hA : A.IsSymmetric
hB : B.IsSymmetric
hAB : Commute A B
⊢ DirectSum.IsInternal fun i ↦ eigenspace A i.2 ⊓ eigenspace B i.1

Tactic:
by
  apply (orthogonalFamily_eigenspace_inf_eigenspace hA hB).isInternal_iff.mpr
  rw [Submodule.orthogonal_eq_bot_iff, iSup_prod, iSup_comm]
  exact iSup_iSup_eigenspace_inf_eigenspace_eq_top_of_commute hA hB hAB

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
coercive : IsCoercive B
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
⊢ (↑(continuousLinearMapOfBilin B)).rangeᗮᗮ = ⊤

Tactic:
rw [Submodule.eq_top_iff']

Goal state after:
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
⊢ ∀ (x : V), x ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮᗮ
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case neg
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
have hn : NNReal.HolderConjugate n n' := .conjExponent (by norm_cast)

Goal state after:
case neg
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p' : ℝ≥0
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
n' : ℝ≥0 := (↑n).conjExponent
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hp : 1 ≤ 1
hp' : (↑p')⁻¹ = ↑1⁻¹ - (↑n)⁻¹
h2p : ↑1 < ↑n
hnp : 0 < ↑n - ↑1
this : ↑p' = ↑n'
⊢ ↑p' = ↑n'

Tactic:
by simpa using this

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

Goal state before:
pp: E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
⊢ 1 < ↑γ

Tactic:
rwa [h0γ, one_lt_div hnp, mul_sub, mul_one, sub_lt_sub_iff_right, lt_mul_iff_one_lt_left]

Goal state after:
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
⊢ 0 < ↑n
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
g : Submodule 𝕜 (E × F)
x : F × E
h : ∀ (a : E) (b : F), (a, b) ∈ g → ⟪b, x.1⟫ - ⟪a, x.2⟫ = 0
u : WithLp 2 (F × E)
a : E
b : F
hab : (a, b) ∈ g
⊢ toLp 2 (b, -a) = u → ⟪u.fst, (toLp 2 x).fst⟫ + ⟪u.snd, (toLp 2 x).snd⟫ = 0

Tactic:
hu

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
g : Submodule 𝕜 (E × F)
x : F × E
h : ∀ (a : E) (b : F), (a, b) ∈ g → ⟪b, x.1⟫ - ⟪a, x.2⟫ = 0
u : WithLp 2 (F × E)
a : E
b : F
hab : (a, b) ∈ g
hu : toLp 2 (b, -a) = u
⊢ ⟪u.fst, (toLp 2 x).fst⟫ + ⟪u.snd, (toLp 2 x).snd⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
h : T.singularValues = 0
⊢ finrank 𝕜 ↥T.range = 0

Tactic:
← Nat.le_zero,

Goal state after:
case mp
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
h : T.singularValues = 0
⊢ finrank 𝕜 ↥T.range ≤ 0
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
ι₁ : Type u_6
ι₂ : Type u_7
inst✝¹ : Fintype ι₁
inst✝ : Fintype ι₂
b₁ : OrthonormalBasis ι₁ 𝕜 E
b₂ : OrthonormalBasis ι₂ 𝕜 F
i : ι₁
j : ι₂
⊢ (b₁.tensorProduct b₂) (i, j) = b₁ i ⊗ₜ[𝕜] b₂ j

Tactic:
by simp [OrthonormalBasis.tensorProduct]

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: case refine_2
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
⊢ dimH s ≤ ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets

Tactic:
refine le_of_forall_lt_imp_le_of_dense fun r hr => ?_
    rcases exists_mem_nhdsWithin_lt_dimH_of_lt_dimH hr with ⟨x, hxs, hxr⟩
    refine le_iSup₂_of_le x hxs ?_; rw [limsup_eq]; refine le_sInf fun b hb => ?_
    rcases eventually_smallSets.1 hb with ⟨t, htx, ht⟩
    exact (hxr t htx).le.trans (ht t Subset.rfl)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
hI : ¬I = 0
hI' : I * I = -1
I_smul : ∀ (v : E), ‖I • v‖ = ‖v‖
h₁ : ‖I • y - x‖ = ‖I • x + y‖
⊢ I • x - y = I • (I • y + x)

Tactic:
linear_combination (norm := module) -hI' • y

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

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ f.normDet = ∏ i ∈ Finset.range (finrank 𝕜 U), f.singularValues i

Tactic:
rw [← sq_eq_sq₀ f.normDet_nonneg (Finset.prod_nonneg fun i _ ↦ f.singularValues_nonneg i),
    ← RCLike.ofReal_inj (K := 𝕜), ← Finset.prod_pow, ← Fin.prod_univ_eq_prod_range, normDet_sq]

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ LinearMap.det (adjoint f ∘ₗ f) = ↑(∏ i, f.singularValues ↑i ^ 2)
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inl
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞
c : ℝ≥0
hc : c ≠ 0
this : 0 < ↑c ^ (d₂ - d₁)⁻¹
hr₀ : 0 ≤ ↑0
hrc : ↑0 < ↑c ^ (d₂ - d₁)⁻¹
⊢ ↑0 ^ d₂ / ↑0 ^ d₁ ≤ ↑c

pp: case inr
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞
c : ℝ≥0
hc : c ≠ 0
this : 0 < ↑c ^ (d₂ - d₁)⁻¹
r : ℝ≥0
hr₀✝ : 0 ≤ ↑r
hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹
hr₀ : r ≠ 0
⊢ ↑r ^ d₂ / ↑r ^ d₁ ≤ ↑c

Tactic:
· rcases lt_or_ge 0 d₂ with (h₂ | h₂)
    · simp only [h₂, ENNReal.zero_rpow_of_pos, zero_le, ENNReal.zero_div, ENNReal.coe_zero]
    · simp only [h.trans_le h₂, ENNReal.div_top, zero_le, ENNReal.zero_rpow_of_neg,
        ENNReal.coe_zero]

Goal state after:
case inr
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞
c : ℝ≥0
hc : c ≠ 0
this : 0 < ↑c ^ (d₂ - d₁)⁻¹
r : ℝ≥0
hr₀✝ : 0 ≤ ↑r
hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹
hr₀ : r ≠ 0
⊢ ↑r ^ d₂ / ↑r ^ d₁ ≤ ↑c
Module: Mathlib.Analysis.InnerProductSpace.StandardSubspace

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
x : H
⊢ x ∈ S ↔ x ∈ (↑S).carrier

Tactic:
by
  exact Eq.to_iff rfl

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

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
⊢ Sᗮ.mulI = S.mulIᗮ

Tactic:
mulI_orthogonal_eq_symplComp

Goal state after:
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
⊢ S.symplComp = S.mulIᗮ
Module: Mathlib.Analysis.InnerProductSpace.StandardSubspace

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
⊢ S.mulIᗮ.mulI = S.mulI.mulIᗮ

Tactic:
mulI_orthogonal_eq_symplComp

Goal state after:
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
⊢ S.mulI.symplComp = S.mulI.mulIᗮ
Module: Mathlib.Analysis.InnerProductSpace.StandardSubspace

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S T : ClosedSubmodule ℝ H
⊢ (S ⊔ T).symplComp = S.symplComp ⊓ T.symplComp

Tactic:
rw [symplComp, symplComp, symplComp, mulI_sup]

Goal state after:
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S T : ClosedSubmodule ℝ H
⊢ (S.mulI ⊔ T.mulI)ᗮ = S.mulIᗮ ⊓ T.mulIᗮ
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inr.refine_2
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
hCd : ↑C ^ d ≠ ∞
R : ℝ≥0∞
hR : 0 < R
this : Tendsto (fun d ↦ ↑C * d ^ ↑r) (𝓝 0) (𝓝 0)
δ : ℝ≥0∞
δ0 : 0 < δ
H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R
t : ℕ → Set X
hst : s ⊆ ⋃ n, t n
htδ : ∀ (n : ℕ), ediam (t n) ≤ δ
n : ℕ
hft : (t n ∩ s).Nonempty
⊢ ediam (f '' (t n ∩ s)) ^ d ≤ ↑C ^ d * (ediam (t n) ^ ↑r) ^ d

Tactic:
← ENNReal.mul_rpow_of_nonneg _ _ hd

Goal state after:
case inr.refine_2
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
hCd : ↑C ^ d ≠ ∞
R : ℝ≥0∞
hR : 0 < R
this : Tendsto (fun d ↦ ↑C * d ^ ↑r) (𝓝 0) (𝓝 0)
δ : ℝ≥0∞
δ0 : 0 < δ
H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R
t : ℕ → Set X
hst : s ⊆ ⋃ n, t n
htδ : ∀ (n : ℕ), ediam (t n) ≤ δ
n : ℕ
hft : (t n ∩ s).Nonempty
⊢ ediam (f '' (t n ∩ s)) ^ d ≤ (↑C * ediam (t n) ^ ↑r) ^ d
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inr.refine_2
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
K : ℝ≥0
d : ℝ
hf : AntilipschitzWith K f
hd : 0 ≤ d
s : Set Y
h0 : K ≠ 0
hKd0 : ↑K ^ d ≠ 0
hKd : ↑K ^ d ≠ ∞
ε : ℝ≥0∞
ε0 : 0 < ε
t : ℕ → Set Y
htε : ∀ (n : ℕ), ediam (t n) ≤ ε / ↑K
hst : f ⁻¹' s ⊆ ⋃ i, f ⁻¹' t i
n : ℕ
hft : ((fun n ↦ f ⁻¹' t n) n).Nonempty
⊢ ediam (f ⁻¹' t n) ^ d ≤ ↑K ^ d * ediam (t n) ^ d

Tactic:
← ENNReal.mul_rpow_of_nonneg _ _ hd

Goal state after:
case inr.refine_2
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
K : ℝ≥0
d : ℝ
hf : AntilipschitzWith K f
hd : 0 ≤ d
s : Set Y
h0 : K ≠ 0
hKd0 : ↑K ^ d ≠ 0
hKd : ↑K ^ d ≠ ∞
ε : ℝ≥0∞
ε0 : 0 < ε
t : ℕ → Set Y
htε : ∀ (n : ℕ), ediam (t n) ≤ ε / ↑K
hst : f ⁻¹' s ⊆ ⋃ i, f ⁻¹' t i
n : ℕ
hft : ((fun n ↦ f ⁻¹' t n) n).Nonempty
⊢ ediam (f ⁻¹' t n) ^ d ≤ (↑K * ediam (t n)) ^ d
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: E : Type u_5
P : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace P
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : BorelSpace P
x y : P
s : Set ℝ
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ ‖y -ᵥ x‖₊ • μH[1] s = nndist x y • μH[1] s

Tactic:
nndist_eq_nnnorm_vsub' E

Goal state after:
E : Type u_5
P : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace P
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : BorelSpace P
x y : P
s : Set ℝ
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ ‖y -ᵥ x‖₊ • μH[1] s = ‖y -ᵥ x‖₊ • μH[1] s
Module: Mathlib.Analysis.LocallyConvex.PointwiseConvergence

Goal state before:
pp: α : Type u_1
R : Type u_2
𝕜₁ : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₃ →+* 𝕜₂
D : Type u_6
E : Type u_7
F : Type u_8
G : Type u_9
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : Module 𝕜₁ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜₂ F
x : E
A B : E →SLₚₜ[σ] F
⊢ ‖(A + B) x‖ ≤ ‖A x‖ + ‖B x‖

Tactic:
simpa only using! norm_add_le _ _

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

Goal state before:
pp: case succ
n : ℕ
ih : logTaylor n 0 = 0
⊢ logTaylor (n + 1) 0 = 0

Tactic:
| succ n ih =>

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

Goal state before:
pp: case succ
z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
⊢ HasDerivAt (logTaylor (n + 1 + 1)) (∑ j ∈ Finset.range (n + 1), (-1) ^ j * z ^ j) z

Tactic:
| succ n ih =>

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

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ (n : ℕ) (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
⊢ -↑(⌊-s.re⌋₊ + 1) < s.re

Tactic:
grind [Nat.lt_floor_add_one (-s.re)]

Goal state after: