mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k : ℕ
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L
this : IsPrimitiveRoot (ζ ^ 2 ^ k) 2
H : -1 - 1 = (algebraMap K L) (-2)
hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)
⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k

Tactic:
this.eq_neg_one_of_two_right,

Goal state after:
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k : ℕ
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L
this : IsPrimitiveRoot (ζ ^ 2 ^ k) 2
H : -1 - 1 = (algebraMap K L) (-2)
hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)
⊢ (Algebra.norm K) (-1 - 1) = (-2) ^ 2 ^ k
Module: Mathlib.FieldTheory.Finite.GaloisField

Goal state before:
pp: case succ.refine_6
n : ℕ
h : n ≠ 0
n✝ : ℕ
h_prime : Fact (Nat.Prime (n✝ + 1))
this : Fintype (GaloisField (n✝ + 1) n)
g_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X
hp : 1 < n✝ + 1
aux : X ^ (n✝ + 1) ^ n - X ≠ 0
key : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n
nat_degree_eq : g_poly.natDegree = (n✝ + 1) ^ n
x : GaloisField (n✝ + 1) n
hx :
  x ∈
    Subring.closure
      (Set.range ⇑(algebraMap (ZMod (n✝ + 1)) (GaloisField (n✝ + 1) n)) ∪
        (X ^ (n✝ + 1) ^ n - X).rootSet (X ^ (n✝ + 1) ^ n - X).SplittingField)
⊢ ∀ (x y : GaloisField (n✝ + 1) n),
    x ∈
        Subring.closure
          (Set.range ⇑(algebraMap (ZMod (n✝ + 1)) (GaloisField (n✝ + 1) n)) ∪
            (X ^ (n✝ + 1) ^ n - X).rootSet (X ^ (n✝ + 1) ^ n - X).SplittingField) →
      y ∈
          Subring.closure
            (Set.range ⇑(algebraMap (ZMod (n✝ + 1)) (GaloisField (n✝ + 1) n)) ∪
              (X ^ (n✝ + 1) ^ n - X).rootSet (X ^ (n✝ + 1) ^ n - X).SplittingField) →
        x ^ (n✝ + 1) ^ n = x → y ^ (n✝ + 1) ^ n = y → x ^ (n✝ + 1) ^ n * y ^ (n✝ + 1) ^ n = x * y

Tactic:
intro x y _ _ hx hy

Goal state after:
case succ.refine_6
n : ℕ
h : n ≠ 0
n✝ : ℕ
h_prime : Fact (Nat.Prime (n✝ + 1))
this : Fintype (GaloisField (n✝ + 1) n)
g_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X
hp : 1 < n✝ + 1
aux : X ^ (n✝ + 1) ^ n - X ≠ 0
key : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n
nat_degree_eq : g_poly.natDegree = (n✝ + 1) ^ n
x✝ : GaloisField (n✝ + 1) n
hx✝¹ :
  x✝ ∈
    Subring.closure
      (Set.range ⇑(algebraMap (ZMod (n✝ + 1)) (GaloisField (n✝ + 1) n)) ∪
        (X ^ (n✝ + 1) ^ n - X).rootSet (X ^ (n✝ + 1) ^ n - X).SplittingField)
x y : GaloisField (n✝ + 1) n
hx✝ :
  x ∈
    Subring.closure
      (Set.range ⇑(algebraMap (ZMod (n✝ + 1)) (GaloisField (n✝ + 1) n)) ∪
        (X ^ (n✝ + 1) ^ n - X).rootSet (X ^ (n✝ + 1) ^ n - X).SplittingField)
hy✝ :
  y ∈
    Subring.closure
      (Set.range ⇑(algebraMap (ZMod (n✝ + 1)) (GaloisField (n✝ + 1) n)) ∪
        (X ^ (n✝ + 1) ^ n - X).rootSet (X ^ (n✝ + 1) ^ n - X).SplittingField)
hx : x ^ (n✝ + 1) ^ n = x
hy : y ^ (n✝ + 1) ^ n = y
⊢ x ^ (n✝ + 1) ^ n * y ^ (n✝ + 1) ^ n = x * y
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case right
S : Set ℕ
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
i : L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})
f : AlgebraicClosure K →ₐ[K] AlgebraicClosure K
y x : AlgebraicClosure K
n : ℕ
hn : n ∈ S
h1 : n ≠ 0
h2 : x ^ n = 1
⊢ f (x ^ n) = f 1

Tactic:
h2

Goal state after:
case right
S : Set ℕ
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
i : L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})
f : AlgebraicClosure K →ₐ[K] AlgebraicClosure K
y x : AlgebraicClosure K
n : ℕ
hn : n ∈ S
h1 : n ≠ 0
h2 : x ^ n = 1
⊢ f 1 = f 1
Module: Mathlib.FieldTheory.Finite.GaloisField

Goal state before:
pp: K : Type u_1
K' : Type u_2
inst✝³ : Field K
inst✝² : Field K'
inst✝¹ : Algebra K K'
inst✝ : Finite K'
this : Finite K
⊢ Nat.card ↥(Units.map (Algebra.norm K)).ker ≤ (Nat.card K' - 1) / (Nat.card K - 1)

Tactic:
have := Fintype.ofFinite K'ˣ

Goal state after:
K : Type u_1
K' : Type u_2
inst✝³ : Field K
inst✝² : Field K'
inst✝¹ : Algebra K K'
inst✝ : Finite K'
this✝ : Finite K
this : Fintype K'ˣ
⊢ Nat.card ↥(Units.map (Algebra.norm K)).ker ≤ (Nat.card K' - 1) / (Nat.card K - 1)
Module: Mathlib.FieldTheory.Finite.GaloisField

Goal state before:
pp: p : ℕ
inst✝³ : Fact (Nat.Prime p)
k : Type u_3
inst✝² : AddCommGroup k
inst✝¹ : Finite k
inst✝ : Module (ZMod p) k
⊢ p ^ Module.finrank (ZMod p) k = Nat.card (ZMod p) ^ Module.finrank (ZMod p) k

Tactic:
Nat.card_zmod

Goal state after:
p : ℕ
inst✝³ : Fact (Nat.Prime p)
k : Type u_3
inst✝² : AddCommGroup k
inst✝¹ : Finite k
inst✝ : Module (ZMod p) k
⊢ p ^ Module.finrank (ZMod p) k = p ^ Module.finrank (ZMod p) k
Module: Mathlib.NumberTheory.MulChar.Basic

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

Tactic:
by
  ext1
  simp only [ringHomComp_apply, coeToFun_mul, Pi.mul_apply, map_mul]

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

Goal state before:
pp: S : Set ℕ
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
C₁ C₂ : Subalgebra A B
h₁ : IsCyclotomicExtension S A ↥C₁
h₂ : IsCyclotomicExtension S A ↥C₂
n : ℕ
hn₁ : n ∈ S
hn₂ : n ≠ 0
⊢ ∃ x, IsPrimitiveRoot x n

Tactic:
obtain ⟨ζ, hζ⟩ := h₁.1 hn₁ hn₂

Goal state after:
S : Set ℕ
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
C₁ C₂ : Subalgebra A B
h₁ : IsCyclotomicExtension S A ↥C₁
h₂ : IsCyclotomicExtension S A ↥C₂
n : ℕ
hn₁ : n ∈ S
hn₂ : n ≠ 0
ζ : ↥C₁
hζ : IsPrimitiveRoot ζ n
⊢ ∃ x, IsPrimitiveRoot x n
Module: Mathlib.Analysis.Fourier.ZMod

Goal state before:
pp: N : ℕ
inst✝² : NeZero N
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
Ψ : ZMod N → E
k : ZMod N
⊢ 𝓕.symm Ψ k = (↑N)⁻¹ • ∑ j, stdAddChar (j * k) • Ψ j

Tactic:
simp only [dft, LinearEquiv.coe_symm_mk, auxDFT, mul_neg, neg_neg]

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
hg₁ : gaussSum χ ψ₈char = 2 * (τ - τ ^ 3)
hg : gaussSum χ ψ₈char ^ 2 = χ (-1) * ↑(Fintype.card (ZMod 8))
h : (χ (-1) * ↑(Fintype.card (ZMod 8))) ^ (ringChar FF ^ ↑n / 2) = χ (↑(ringChar FF) ^ ↑n)
⊢ 2 ^ (Fintype.card F / 2) = ↑(χ₈ ↑(Fintype.card F))

Tactic:
rw [ZMod.card, ← hchar, hχ, one_mul, ← hc, ← Nat.cast_pow (ringChar F), ← hc] at h

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
hg₁ : gaussSum χ ψ₈char = 2 * (τ - τ ^ 3)
hg : gaussSum χ ψ₈char ^ 2 = χ (-1) * ↑(Fintype.card (ZMod 8))
h : ↑8 ^ (Fintype.card F / 2) = χ ↑(Fintype.card F)
⊢ 2 ^ (Fintype.card F / 2) = ↑(χ₈ ↑(Fintype.card F))
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_4
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
⊢ A.IsHermitian →
    ((∀ (x : EuclideanSpace 𝕜 n), 0 ≤ re (star x.ofLp ⬝ᵥ A.mulVec x.ofLp)) ↔ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ A.mulVec x)

Tactic:
RCLike.nonneg_iff (K := 𝕜)

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_4
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
⊢ A.IsHermitian →
    ((∀ (x : EuclideanSpace 𝕜 n), 0 ≤ re (star x.ofLp ⬝ᵥ A.mulVec x.ofLp)) ↔
      ∀ (x : n → 𝕜), 0 ≤ re (star x ⬝ᵥ A.mulVec x) ∧ im (star x ⬝ᵥ A.mulVec x) = 0)
Module: Mathlib.Analysis.InnerProductSpace.Positive

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
U : Submodule 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : E), True) ∧ ∀ (x : E), 0 ≤ ⟪U.starProjection x, T (U.starProjection x)⟫

Tactic:
hT.inner_nonneg_right,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : E), True) ∧ ∀ (y : E), True
Module: Mathlib.Analysis.InnerProductSpace.Positive

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
T : E →L[𝕜] E
hT : T.IsPositive
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : ↥U), True) ∧ ∀ (x : ↥U), 0 ≤ ⟪↑x, T ↑x⟫

Tactic:
hT.inner_nonneg_right,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : ↥U), True) ∧ ∀ (y : ↥U), True
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: case h
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : E
⊢ x ∈ f.domain → x ∈ f.closure.domain

Tactic:
intro hx

Goal state after:
case h
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
x : E
hx : x ∈ f.domain
⊢ x ∈ f.closure.domain
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
f : E →ₗ.[R] F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
hf : f.toFun.ker = ⊥
hf' : f.IsClosable
hcf : f.closure.toFun.ker = ⊥
h1 : (fun a ↦ a.swap) '' ↑f.graph = ⇑(LinearEquiv.prodComm R E F).toEquiv.symm ⁻¹' ↑f.graph
h2 :
  (fun a ↦ a.swap) '' closure[instTopologicalSpaceProd] ↑f.graph =
    ⇑(LinearEquiv.prodComm R E F).toEquiv.symm ⁻¹' closure[instTopologicalSpaceProd] ↑f.graph
⊢ closure[instTopologicalSpaceProd] (⇑(LinearEquiv.prodComm R E F).toEquiv.symm ⁻¹' ↑f.graph) ⊆
    Prod.swap '' closure[instTopologicalSpaceProd] ↑f.graph

Tactic:
h2

Goal state after:
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
f : E →ₗ.[R] F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
hf : f.toFun.ker = ⊥
hf' : f.IsClosable
hcf : f.closure.toFun.ker = ⊥
h1 : (fun a ↦ a.swap) '' ↑f.graph = ⇑(LinearEquiv.prodComm R E F).toEquiv.symm ⁻¹' ↑f.graph
h2 :
  (fun a ↦ a.swap) '' closure[instTopologicalSpaceProd] ↑f.graph =
    ⇑(LinearEquiv.prodComm R E F).toEquiv.symm ⁻¹' closure[instTopologicalSpaceProd] ↑f.graph
⊢ closure[instTopologicalSpaceProd] (⇑(LinearEquiv.prodComm R E F).toEquiv.symm ⁻¹' ↑f.graph) ⊆
    ⇑(LinearEquiv.prodComm R E F).toEquiv.symm ⁻¹' closure[instTopologicalSpaceProd] ↑f.graph
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 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
x : E
y : F
⊢ ‖x ⊗ₜ[𝕜] y‖ = ‖x‖ * ‖y‖

Tactic:
by
  simpa using congr(√(RCLike.re $(inner_tmul 𝕜 x x y y)))

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

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
F' : Type := EuclideanSpace ℝ (Fin (finrank ℝ F))
e : F ≃L[ℝ] F' := toEuclidean
C₁ : ℝ≥0 := ‖↑e.symm‖₊
C : ℝ≥0 := eLpNormLESNormFDerivOfEqInnerConst μ ↑p
C₂ : ℝ≥0 := ‖↑e‖₊
v : E → F' := ⇑e ∘ u
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this : eLpNorm v (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ v) (↑p) μ
x : E
⊢ ‖fderiv ℝ (⇑e) (u x)‖ * ‖fderiv ℝ u x‖ = ↑C₂ * ‖fderiv ℝ u x‖

Tactic:
by simp_rw [e.fderiv, C₂, coe_nnnorm]

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

Goal state before:
pp: case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
r n i j : ℕ
hj : i < j
A : (↑(2 * j + r))⁻¹ < (↑(2 * i + 1 + r))⁻¹
x : X
hx : x ∈ S (2 * i + 1 + r)
y : X
hy : y ∈ s \ S (2 * j + r)
this : infEDist y t < (↑(2 * j + r))⁻¹
⊢ (↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹ ≤ edist x y

Tactic:
rcases infEDist_lt_iff.mp this with ⟨z, hzt, hyz⟩

Goal state after:
case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
r n i j : ℕ
hj : i < j
A : (↑(2 * j + r))⁻¹ < (↑(2 * i + 1 + r))⁻¹
x : X
hx : x ∈ S (2 * i + 1 + r)
y : X
hy : y ∈ s \ S (2 * j + r)
this : infEDist y t < (↑(2 * j + r))⁻¹
z : X
hzt : z ∈ t
hyz : edist y z < (↑(2 * j + r))⁻¹
⊢ (↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹ ≤ edist x y
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
⊢ (mkMetric fun x ↦ ∞) = ⊤

Tactic:
by
  simp_rw [mkMetric, mkMetric', mkMetric'.pre, extend_top, boundedBy_top, eq_top_iff]
  rw [le_iSup_iff]
  intro b hb
  simpa using hb ⊤

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

Goal state before:
pp: case pos
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✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
h : (↑f).ker = ⊥
b : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(↑f).range
⊢ ↑((↑f).normDet ^ 2) = (ContinuousLinearMap.adjoint f ∘SL f).det

Tactic:
have hf : f = f.range.subtypeₗᵢ.toContinuousLinearMap ∘L f.rangeRestrict := rfl

Goal state after:
case pos
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✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
h : (↑f).ker = ⊥
b : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(↑f).range
hf : f = (↑f).range.subtypeₗᵢ.toContinuousLinearMap ∘SL f.rangeRestrict
⊢ ↑((↑f).normDet ^ 2) = (ContinuousLinearMap.adjoint f ∘SL f).det
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
x : X
⊢ ‖kerFun H x‖ = √(‖kerFun H x‖ ^ 2)

Tactic:
Real.sqrt_sq (norm_nonneg _)

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
x : X
⊢ ‖kerFun H x‖ = ‖kerFun H x‖
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
hε : ε > 0
hnorm : Tendsto (fun x ↦ ‖F x - f‖ * C) l (𝓝 0)
⊢ ∀ᶠ (n : ι) in l, ∀ x ∈ s, dist (f x) ((F n) x) < ε

Tactic:
filter_upwards [hnorm.eventually (gt_mem_nhds hε)] with n hn x hx

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
hnorm : Tendsto (fun x ↦ ‖F x - f‖ * C) l (𝓝 0)
n : ι
hn : ‖F n - f‖ * C < ε
x : X
hx : x ∈ s
⊢ dist (f x) ((F n) x) < ε
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
x : X
r : ℕ → ℝ≥0∞ := fun x ↦ 0
⊢ μH[0] {x} ≤ 1

Tactic:
(Tactic.tacticLet__
 "let"
 (Term.letConfig [])
 (Term.letDecl
  (Term.letIdDecl
   (Term.letId `t)
   []
   [(Term.typeSpec ":" (Term.arrow (termℕ "ℕ") "→" (Term.arrow `Unit "→" (Term.app `Set [`X]))))]
   ":="
   (Term.fun
    "fun"
    (Term.basicFun
     [(Term.hole "_") (Term.hole "_")]
     []
     "=>"
     (choice
      («term{_}» "{" [`x] "}")
      (Term.structInst
       "{"
       []
       (Term.structInstFields [(Term.structInstField (Term.structInstLVal `x []) [])])
       (Term.optEllipsis [])
       []
       "}")))))))

Goal state after:
case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
x : X
r : ℕ → ℝ≥0∞ := fun x ↦ 0
t : ℕ → Unit → Set X := fun x_1 x_2 ↦ {x}
⊢ μH[0] {x} ≤ 1
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
v : V
⊢ ‖⟪(K x x) v, v⟫_𝕜‖ ≤ (√‖K x x‖ * ‖v‖) ^ 2

Tactic:
calc
    _ ≤ ‖K x x v‖ * ‖v‖ := by simp [norm_inner_le_norm]
    _ ≤ ‖K x x‖ * ‖v‖ * ‖v‖ := by simp [mul_le_mul_of_nonneg_right, le_opNorm]
    _ ≤ _ := by simp [mul_pow, mul_assoc, ← sq]

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

Goal state before:
pp: ⊢ μH[2] = volume

Tactic:
← (volume_preserving_piFinTwo fun _ => ℝ).map_eq,

Goal state after:
⊢ μH[2] = Measure.map (⇑(MeasurableEquiv.piFinTwo fun x ↦ ℝ)) volume
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.rightAngleRotationAux₁ (o.rightAngleRotationAux₁ x) = -x

Tactic:
apply ext_inner_left ℝ

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
⊢ ∀ (v : E), ⟪v, o.rightAngleRotationAux₁ (o.rightAngleRotationAux₁ x)⟫ = ⟪v, -x⟫
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
hF : Fact (finrank ℝ F = 2)
φ : E ≃ₗᵢ[ℝ] F
x y : F
⊢ (((map (Fin 2) φ.toLinearEquiv) o).areaForm x) y = ⟪o.rightAngleRotation (φ.symm x), φ.symm y⟫

pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
hF : Fact (finrank ℝ F = 2)
φ : E ≃ₗᵢ[ℝ] F
x y : F
⊢ ⟪o.rightAngleRotation (φ.symm x), φ.symm y⟫ = ⟪φ (o.rightAngleRotation (φ.symm x)), y⟫

Tactic:
· simp [o.areaForm_map]

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
hF : Fact (finrank ℝ F = 2)
φ : E ≃ₗᵢ[ℝ] F
x y : F
⊢ ⟪o.rightAngleRotation (φ.symm x), φ.symm y⟫ = ⟪φ (o.rightAngleRotation (φ.symm x)), y⟫
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_1
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
this : f x - f 1 = f x - f 1 - x * 0
⊢ Tendsto (fun n ↦ f x - f 1 - x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f x - f 1 - x * 0))

Tactic:
exact Tendsto.sub tendsto_const_nhds (tendsto_log_nat_add_one_sub_log.const_mul _)

Goal state after: