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

Goal state before:
pp: case mk
p n : ℕ
inst✝¹² : NeZero n
A : Type w
B : Type z
K : Type u
L : Type v
C✝ : Type w
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsCyclotomicExtension {n} A B
inst✝⁷ : Field K
inst✝⁶ : CommRing L
inst✝⁵ : IsDomain L
inst✝⁴ : Algebra K L
inst✝³ : IsCyclotomicExtension {n} K L
ζ : L
hζ : IsPrimitiveRoot ζ n
C : Type u_1
inst✝² : CommRing C
inst✝¹ : IsDomain C
inst✝ : Algebra K C
hirr : Irreducible (cyclotomic n K)
this : NeZero ↑n
hn : NeZero ↑n
val✝ : C
property✝ : (aeval val✝) (minpoly K (IsPrimitiveRoot.powerBasis K hζ).gen) = 0
⊢ (cyclotomic n C).IsRoot ↑⟨val✝, property✝⟩

Tactic:
IsRoot.def,

Goal state after:
case mk
p n : ℕ
inst✝¹² : NeZero n
A : Type w
B : Type z
K : Type u
L : Type v
C✝ : Type w
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsCyclotomicExtension {n} A B
inst✝⁷ : Field K
inst✝⁶ : CommRing L
inst✝⁵ : IsDomain L
inst✝⁴ : Algebra K L
inst✝³ : IsCyclotomicExtension {n} K L
ζ : L
hζ : IsPrimitiveRoot ζ n
C : Type u_1
inst✝² : CommRing C
inst✝¹ : IsDomain C
inst✝ : Algebra K C
hirr : Irreducible (cyclotomic n K)
this : NeZero ↑n
hn : NeZero ↑n
val✝ : C
property✝ : (aeval val✝) (minpoly K (IsPrimitiveRoot.powerBasis K hζ).gen) = 0
⊢ eval (↑⟨val✝, property✝⟩) (cyclotomic n C) = 0
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: p n : ℕ
inst✝¹² : NeZero n
A : Type w
B : Type z
K : Type u
L : Type v
C✝ : Type w
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsCyclotomicExtension {n} A B
inst✝⁷ : Field K
inst✝⁶ : CommRing L
inst✝⁵ : IsDomain L
inst✝⁴ : Algebra K L
inst✝³ : IsCyclotomicExtension {n} K L
ζ : L
hζ : IsPrimitiveRoot ζ n
C : Type u_1
inst✝² : CommRing C
inst✝¹ : IsDomain C
inst✝ : Algebra K C
hirr : Irreducible (cyclotomic n K)
x : ↥(primitiveRoots n C)
⊢ { y // (aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K hζ).gen) = 0 }

Tactic:
haveI := IsCyclotomicExtension.neZero' n K L

Goal state after:
p n : ℕ
inst✝¹² : NeZero n
A : Type w
B : Type z
K : Type u
L : Type v
C✝ : Type w
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra A B
inst✝⁸ : IsCyclotomicExtension {n} A B
inst✝⁷ : Field K
inst✝⁶ : CommRing L
inst✝⁵ : IsDomain L
inst✝⁴ : Algebra K L
inst✝³ : IsCyclotomicExtension {n} K L
ζ : L
hζ : IsPrimitiveRoot ζ n
C : Type u_1
inst✝² : CommRing C
inst✝¹ : IsDomain C
inst✝ : Algebra K C
hirr : Irreducible (cyclotomic n K)
x : ↥(primitiveRoots n C)
this : NeZero ↑n
⊢ { y // (aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K hζ).gen) = 0 }
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_1
m k p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
hp : Fact (Nat.Prime p)
hchar : CharP R p
μ : R
inst✝ : NeZero ↑m
hk : k > 0
h : (cyclotomic (p ^ k * m) R).IsRoot μ
⊢ IsPrimitiveRoot μ m

Tactic:
IsRoot.def,

Goal state after:
case inr.refine_1
m k p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
hp : Fact (Nat.Prime p)
hchar : CharP R p
μ : R
inst✝ : NeZero ↑m
hk : k > 0
h : eval μ (cyclotomic (p ^ k * m) R) = 0
⊢ IsPrimitiveRoot μ m
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_1.inl
S T : Set ℕ
A : Type u
B : Type v
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
C : Type w
inst✝³ : CommRing C
inst✝² : Algebra A C
inst✝¹ : Algebra B C
inst✝ : IsScalarTower A B C
hS : IsCyclotomicExtension S A B
hT : IsCyclotomicExtension T B C
h : Function.Injective ⇑(algebraMap B C)
n✝ : ℕ
hn' : n✝ ≠ 0
hn : n✝ ∈ S
⊢ ∃ r, IsPrimitiveRoot r n✝

Tactic:
obtain ⟨b, hb⟩ := ((isCyclotomicExtension_iff _ _ _).1 hS).1 hn hn'
      refine ⟨algebraMap B C b, ?_⟩
      exact hb.map_of_injective h

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: case inr.refine_2
m k p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
hp : Fact (Nat.Prime p)
hchar : CharP R p
μ : R
inst✝ : NeZero ↑m
hk : k > 0
h : eval μ (cyclotomic m R) = 0
⊢ (cyclotomic m R ^ (p ^ k - p ^ (k - 1))).IsRoot μ

Tactic:
IsRoot.def,

Goal state after:
case inr.refine_2
m k p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
hp : Fact (Nat.Prime p)
hchar : CharP R p
μ : R
inst✝ : NeZero ↑m
hk : k > 0
h : eval μ (cyclotomic m R) = 0
⊢ eval μ (cyclotomic m R ^ (p ^ k - p ^ (k - 1))) = 0
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case neg
S✝ : Set ℕ
n : ℕ
S : Set ℕ
a✝ : n ∉ S
hs✝ : S.Finite
H :
  ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [IsDomain B]
    [h₂ : IsCyclotomicExtension S A B], Module.Finite A B
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
h₂ : IsCyclotomicExtension (insert n S) A B
hn : ¬n = 0
this✝¹ : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
this✝ : Module.Finite A ↥(adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})
this : Module.Finite (↥(adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1})) B
⊢ Module.Finite A B

Tactic:
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ, n ∈ S ∧ n ≠ 0 ∧ b ^ n = 1}) _

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

Goal state before:
pp: case refine_1
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
ζ : B
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
x : B
hx : x ∈ (cyclotomic n A).rootSet B
⊢ x ∈ {b | n ≠ 0 ∧ b ^ n = 1}

Tactic:
rw [mem_rootSet'] at hx

Goal state after:
case refine_1
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
ζ : B
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
x : B
hx : map (algebraMap A B) (cyclotomic n A) ≠ 0 ∧ (aeval x) (cyclotomic n A) = 0
⊢ x ∈ {b | n ≠ 0 ∧ b ^ n = 1}
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_1
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
ζ : B
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
x : B
hx : map (algebraMap A B) (cyclotomic n A) ≠ 0 ∧ (aeval x) (cyclotomic n A) = 0
⊢ (cyclotomic n B).IsRoot x

Tactic:
IsRoot.def,

Goal state after:
case refine_1
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
ζ : B
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
x : B
hx : map (algebraMap A B) (cyclotomic n A) ≠ 0 ∧ (aeval x) (cyclotomic n A) = 0
⊢ eval x (cyclotomic n B) = 0
Module: Mathlib.Data.ZMod.Units

Goal state before:
pp: case inr
N : ℕ
a : ZMod N
hN : N ≠ 0
this : NeZero N
d : ℕ := a.val.gcd N
hd : d ≠ 0
a₀ : ℕ
ha₀ : a.val = d * a₀
N₀ : ℕ
hN₀ : N = d * N₀
hu₀ : IsUnit ↑a₀
⊢ ∃ u, IsUnit u ∧ a = u * ↑d

Tactic:
obtain ⟨u, hu⟩ := (unitsMap_surjective (⟨d, mul_comm d N₀ ▸ hN₀⟩ : N₀ ∣ N)) hu₀.unit

Goal state after:
case inr
N : ℕ
a : ZMod N
hN : N ≠ 0
this : NeZero N
d : ℕ := a.val.gcd N
hd : d ≠ 0
a₀ : ℕ
ha₀ : a.val = d * a₀
N₀ : ℕ
hN₀ : N = d * N₀
hu₀ : IsUnit ↑a₀
u : (ZMod N)ˣ
hu : (unitsMap ⋯) u = hu₀.unit
⊢ ∃ u, IsUnit u ∧ a = u * ↑d
Module: Mathlib.FieldTheory.Finite.GaloisField

Goal state before:
pp: case succ.refine_5
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 ↑((X ^ (n✝ + 1) ^ n - X).rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n
nat_degree_eq : (X ^ (n✝ + 1) ^ n - X).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 : 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)
hx : x ^ (n✝ + 1) ^ n = x
⊢ (-x) ^ (n✝ + 1) ^ n + x = 0

Tactic:
rw [neg_pow, hx, neg_one_pow_char_pow]

Goal state after:
case succ.refine_5
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 ↑((X ^ (n✝ + 1) ^ n - X).rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n
nat_degree_eq : (X ^ (n✝ + 1) ^ n - X).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 : 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)
hx : x ^ (n✝ + 1) ^ n = x
⊢ -1 * x + x = 0
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: p : ℕ
K : Type u
L : Type v
inst✝² : Field L
ζ : L
inst✝¹ : Field K
inst✝ : Algebra K L
hpri : Fact (Nat.Prime p)
hcyc : IsCyclotomicExtension {p} K L
hζ : IsPrimitiveRoot ζ p
hirr : Irreducible (cyclotomic p K)
h : p ≠ 2
⊢ (Algebra.norm K) (ζ - 1) = ↑p

Tactic:
by
  replace hirr : Irreducible (cyclotomic (p ^ (0 + 1)) K) := by simp [hirr]
  replace hζ : IsPrimitiveRoot ζ (p ^ (0 + 1)) := by simp [hζ]
  have : IsCyclotomicExtension {p ^ (0 + 1)} K L := by simp [hcyc]
  simpa using norm_sub_one_of_prime_ne_two hζ hirr h

Goal state after:
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 = 1

Tactic:
rw [← map_pow, ← map_one f, h2]

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

Goal state before:
pp: case right.add
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 y : AlgebraicClosure K
hx : x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
hy : y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihx : f x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihy : f y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
⊢ f (x + y) ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}

Tactic:
rw [map_add]

Goal state after:
case right.add
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 y : AlgebraicClosure K
hx : x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
hy : y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihx : f x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihy : f y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
⊢ f x + f y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
hm : n ∣ m
⊢ toUnitHom ((changeLevel hm) χ) = (toUnitHom χ).comp (ZMod.unitsMap hm)

Tactic:
by
  simp [changeLevel]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
f : Rˣ →* R'ˣ
⊢ (if hx : IsUnit 1 then ↑(f hx.unit) else 0) = 1

Tactic:
by
    have h1 : (isUnit_one.unit : Rˣ) = 1 := Units.ext rfl
    simp only [h1, dif_pos, Units.val_eq_one, map_one, isUnit_one]

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

Goal state before:
pp: A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
n₁ n₂ : ℕ
C₁ C₂ : Subalgebra A B
h₁ : IsCyclotomicExtension {n₁} A ↥C₁
h₂ : IsCyclotomicExtension {n₂} A ↥C₂
inst✝ : NeZero n₂
this : NeZero n₁
ζ₂ : ↥C₂
hζ₂ : IsPrimitiveRoot ((algebraMap (↥C₂) B) ζ₂) n₂
d : ℕ
hd : n₂ = n₁ * d
⊢ n₂ = ?m.112 * ?m.113

Tactic:
by rwa [mul_comm]

Goal state after:
Module: Mathlib.Analysis.Hofer

Goal state before:
pp: X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n ↦ Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
lim_top : Tendsto (ϕ ∘ u) atTop atTop
lim : Tendsto (ϕ ∘ u) atTop (𝓝 (ϕ y))
⊢ False

Tactic:
exact not_tendsto_atTop_of_tendsto_nhds lim lim_top

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : H →L[𝕜] H
c : ℝ≥0
hc : 0 < c
h : ∀ (x : H), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖
⊢ AntilipschitzWith c⁻¹ ⇑f

Tactic:
by
  refine f.antilipschitz_of_bound (K := c⁻¹) fun x ↦ ?_
  rw [NNReal.coe_inv, inv_mul_eq_div, le_div_iff₀ (by exact_mod_cast hc)]
  simp_rw [sq, mul_assoc] at h
  by_cases hx0 : x = 0
  · simp [hx0]
  · apply (map_le_map_iff <| OrderIso.mulLeft₀ ‖x‖ (norm_pos_iff.mpr hx0)).mp
    exact (h x).trans <| (norm_inner_le_norm _ _).trans <| (mul_comm _ _).le

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
inst✝¹ : U.HasOrthogonalProjection
inst✝ : V.HasOrthogonalProjection
⊢ U.starProjection ≤ V.starProjection ↔ U ≤ V

Tactic:
simp_rw [← coe_le_coe_iff, isSymmetricProjection_starProjection _
      |>.le_iff_range_le_range <| isSymmetricProjection_starProjection _,
    toLinearMap_starProjection_eq_isComplProjection, range_projection]

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
⊢ 1 / ↑n' - 1 / q = 1 / ↑p'

Tactic:
simp_rw -zeta [one_div, hp']

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
⊢ (↑n')⁻¹ - q⁻¹ = ↑p⁻¹ - (↑n)⁻¹
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
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)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
have h5u : (∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ) ^ (1 / q) ≠ 0 :=
    ENNReal.rpow_pos (pos_iff_ne_zero.mpr h3u) h4u |>.ne'

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'
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)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
g : E →L[𝕜] F
⊢ ‖(commIsometry 𝕜 F G).toLinearIsometry.toContinuousLinearMap ∘SL
        rTensor G g ∘SL (commIsometry 𝕜 G E).toLinearIsometry.toContinuousLinearMap‖ ≤
    ‖g‖

Tactic:
grw [opNorm_comp_le, opNorm_comp_le, LinearIsometry.norm_toContinuousLinearMap_le,
    LinearIsometry.norm_toContinuousLinearMap_le, mul_one, one_mul, norm_rTensor_le]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: case e_f
V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
x : ↥s.direction
⊢ volume (Prod.mk x ⁻¹' ⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑((IsometryEquiv.vaddConst p) x)) s.directionᗮ))

Tactic:
let u : Set (mk' (x +ᵥ p).val s.directionᗮ) := Subtype.val ⁻¹' (t ∩ mk' (x +ᵥ p).val s.directionᗮ)

Goal state after:
case e_f
V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
x : ↥s.direction
u : Set ↥(mk' (↑(x +ᵥ p)) s.directionᗮ) := Subtype.val ⁻¹' (t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ))
⊢ volume (Prod.mk x ⁻¹' ⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑((IsometryEquiv.vaddConst p) x)) s.directionᗮ))
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
⊢ Monotone m → Monotone fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s

Tactic:
intro h_mono

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
h_mono : Monotone m
⊢ Monotone fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hcvx : Convex ℝ s
hne : s.Nonempty
this : Nonempty ↑s
φ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩
hs_eq : s = Subtype.val '' Subtype.val ⁻¹' s
⊢ dimH s = ↑(finrank ℝ ↥(vectorSpan ℝ s))

Tactic:
rw [hs_eq, isometry_subtype_coe.dimH_image, ← φ.isometry.dimH_image,
      Real.dimH_of_nonempty_interior, direction_affineSpan ℝ s, ← hs_eq]

Goal state after:
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hcvx : Convex ℝ s
hne : s.Nonempty
this : Nonempty ↑s
φ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩
hs_eq : s = Subtype.val '' Subtype.val ⁻¹' s
⊢ (interior (⇑φ '' Subtype.val ⁻¹' s)).Nonempty
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet

Tactic:
by_cases hf : f.ker = ⊥

Goal state after:
case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet

case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : ¬f.ker = ⊥
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet
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 y : X
v w : V
⊢ ⟪(kernel H x y) v, w⟫_𝕜 = ⟪(kerFun H y) v, (kerFun H x) w⟫_𝕜

Tactic:
by
  simp [← adjoint_inner_left, kernel]

Goal state after: