mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Matrix.Echelon.Pivot

Goal state before:
pp: case inl
m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : Zero R
A : Matrix m n R
l : m → WithTop n
inst✝¹ : PartialOrder m
inst✝ : LinearOrder n
x✝ :
  Monotone l ∧
    StrictMonoOn l {i | l i ≠ ⊤} ∧ ∀ (i : m), (∀ (j : n), ↑j < l i → A i j = 0) ∧ ∀ (c : n), l i = ↑c → A i c ≠ 0
hmono : Monotone l
hstrict : StrictMonoOn l {i | l i ≠ ⊤}
hlead : ∀ (i : m), (∀ (j : n), ↑j < l i → A i j = 0) ∧ ∀ (c : n), l i = ↑c → A i c ≠ 0
i₁ i₂ : m
hlt : i₁ < i₂
j₂ : n
hz : ∀ j₁ < j₂, A i₁ j₁ = 0
h₂ : l i₂ = ⊤
⊢ ↑j₂ < l i₂

Tactic:
rw [h₂]
    exact WithTop.coe_lt_top j₂

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.FixedDetMatrices

Goal state before:
pp: n : Type u_1
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type u_2
inst✝ : CommRing R
m : R
A B : FixedDetMatrix n R m
h : ∀ (i j : n), ↑A i j = ↑B i j
⊢ A = B

Tactic:
apply ext'
  ext i j
  apply h

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.FixedDetMatrices

Goal state before:
pp: case pos
m : ℤ
hm : m ≠ 0
A : FixedDetMatrix (Fin 2) ℤ m
h : ↑A 1 0 = 0
hd : ↑A 1 1 ≠ 0
h1 : 0 < ↑A 0 0
⊢ reduce A ∈ reps m

Tactic:
simp only [reduce_of_pos h h1]
      have h2 := Int.emod_def (A.1 0 1) (A.1 1 1)
      have h4 := Int.ediv_mul_le (A.1 0 1) hd
      set n : ℤ := A.1 0 1 / A.1 1 1
      have h3 := Int.emod_lt_abs (A.1 0 1) hd
      rw [← abs_eq_self.mpr <| Int.emod_nonneg _ hd] at h3
      simp only [smul_def, coe_T_zpow]
      suffices A.1 1 0 = 0 ∧ n * A.1 1 0 < A.1 0 0 ∧
          n * A.1 1 1 ≤ A.1 0 1 ∧ |A.1 0 1 + -(n * A.1 1 1)| < |A.1 1 1| by
        simpa only [reps, Fin.isValue, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd, empty_mul,
          Equiv.symm_apply_apply, Set.mem_ofPred_eq, of_apply, cons_val', vecMul, cons_dotProduct,
          vecHead, one_mul, vecTail, Function.comp_apply, Fin.succ_zero_eq_one, neg_mul,
          dotProduct_of_isEmpty, add_zero, zero_mul, zero_add, empty_val', cons_val_fin_one,
          cons_val_one, cons_val_zero, lt_add_neg_iff_add_lt, le_add_neg_iff_add_le]
      simp_all only [mul_comm n, zero_mul, ← sub_eq_add_neg, ← h2,
        Fin.isValue, and_true]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Swap

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝³ : CommRing R
inst✝² : DecidableEq n
inst✝¹ : Fintype n
S : Type u_3
inst✝ : CommRing S
f : R →+* S
i j : n
⊢ (map f) (swap R i j) = swap S i j

Tactic:
ext : 1
  simp [swap]

Goal state after:
Module: Mathlib.LinearAlgebra.PiTensorProduct.Finite

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
ι : Type u_2
inst✝³ : Finite ι
M : ι → Type u_3
inst✝² : (i : ι) → AddCommGroup (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : ∀ (i : ι), Module.Finite R (M i)
n : ι → ℕ
γ : (i : ι) → Fin (n i) → M i
hg : ∀ (i : ι), Submodule.span R (Set.range (γ i)) = ⊤
⊢ ⊤.FG

Tactic:
← submodule_span_eq_top hg

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
ι : Type u_2
inst✝³ : Finite ι
M : ι → Type u_3
inst✝² : (i : ι) → AddCommGroup (M i)
inst✝¹ : (i : ι) → Module R (M i)
inst✝ : ∀ (i : ι), Module.Finite R (M i)
n : ι → ℕ
γ : (i : ι) → Fin (n i) → M i
hg : ∀ (i : ι), Submodule.span R (Set.range (γ i)) = ⊤
⊢ (Submodule.span R (Set.range fun j ↦ ⨂ₜ[R] (i : ι), γ i (j i))).FG
Module: Mathlib.LinearAlgebra.Projectivization.Action

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Field K
inst✝ : Module K V
⊢ IsMultiplyPretransitive (SpecialLinearGroup K V) (ℙ K V) 2

Tactic:
have := linearEquiv_is_two_pretransitive K V

Goal state after:
K : Type u_1
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Field K
inst✝ : Module K V
this : IsMultiplyPretransitive (V ≃ₗ[K] V) (ℙ K V) 2
⊢ IsMultiplyPretransitive (SpecialLinearGroup K V) (ℙ K V) 2
Module: Mathlib.LinearAlgebra.QuadraticForm.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
g : ι → M
l : ι →₀ R
⊢ Q ((linearCombination R g) l) =
    (linearCombination R (polarSym2 ⇑Q ∘ Sym2.map g)) l.sym2Mul - (linearCombination R (⇑Q ∘ g)) (l * l)

Tactic:
simp_rw [linearCombination_apply, map_finsuppSum', Q.map_smul, mul_smul]

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
g : ι → M
l : ι →₀ R
⊢ (∑ p ∈ l.support.sym2, polarSym2 (⇑Q) (Sym2.map (fun i ↦ l i • g i) p) - l.sum fun i a ↦ a • a • Q (g i)) =
    (l.sym2Mul.sum fun i a ↦ a • (polarSym2 ⇑Q ∘ Sym2.map g) i) - (l * l).sum fun i a ↦ a • (⇑Q ∘ g) i
Module: Mathlib.LinearAlgebra.QuadraticForm.Basis

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : LinearOrder ι
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
bm : Basis ι R M
x : M
⊢ (Q.toBilin bm).toQuadraticMap ((Finsupp.linearCombination R ⇑bm) (bm.repr x)) =
    Q ((Finsupp.linearCombination R ⇑bm) (bm.repr x))

Tactic:
LinearMap.BilinMap.toQuadraticMap_apply,

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : LinearOrder ι
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
bm : Basis ι R M
x : M
⊢ ((Q.toBilin bm) ((Finsupp.linearCombination R ⇑bm) (bm.repr x))) ((Finsupp.linearCombination R ⇑bm) (bm.repr x)) =
    Q ((Finsupp.linearCombination R ⇑bm) (bm.repr x))
Module: Mathlib.LinearAlgebra.SpecialLinearGroup

Goal state before:
pp: case pos
R : Type u_1
V : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : Module.Free R V
inst✝ : Module.Finite R V
r : ↥(rootsOfUnity (max (Module.finrank R V) 1) R)
hR : Subsingleton R
⊢ 1 = r

Tactic:
simp [Subsingleton.eq_one r]

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.Signature

Goal state before:
pp: M : Type u_2
inst✝⁵ : AddCommGroup M
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : Module 𝕜 M
Q : QuadraticForm 𝕜 M
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : FiniteDimensional 𝕜 M
⊢ sigPos Q + sigNeg Q + Module.finrank 𝕜 ↥(radical Q) = Module.finrank 𝕜 M

Tactic:
have : Invertible (2 : 𝕜) := invertibleOfNonzero (NeZero.ne _)
  obtain ⟨w, e⟩ := Q.equivalent_weightedSumSquares
  rw [e.sigPos_eq, e.sigNeg_eq, e.rank_radical_eq]
  convert! QuadraticForm.sigPos_add_sigNeg_add_radical₁ (w := w)
  exact Eq.symm (Nat.card_fin (Module.finrank 𝕜 M))

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
inst✝¹ : P.IsCrystallographic
b : P.Base
inst✝ : DecidableEq ι
d : ι → R
g : Matrix ι ι R →ₗ[R] Matrix (↥b.support ⊕ ι) (↥b.support ⊕ ι) R :=
  { toFun := fromBlocks 0 0 0, map_add' := ⋯, map_smul' := ⋯ }
h₀ : Injective (⇑g ∘ ⇑(diagonalLinearMap ι R R))
h₁ : ∀ {d : ι → R}, Matrix.diagonal (0 ⊕ᵥ d) = g ((diagonalLinearMap ι R R) d)
x✝ : Matrix (↥b.support ⊕ ι) (↥b.support ⊕ ι) R
⊢ x✝ ∈ range h ↔ x✝ ∈ ⇑g '' ⇑(diagonalLinearMap ι R R) '' range fun i j ↦ ↑(P.pairingIn ℤ j ↑i)

Tactic:
simp [g, h_def]

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple

Goal state before:
pp: case succ
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
inst✝⁸ : CharZero R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsCrystallographic
inst✝² : P.IsReduced
b : P.Base
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ↥b.support
j : ι
h : j ≠ -↑i
this✝¹ : Module.IsReflexive R M
this✝ : IsAddTorsionFree M
this : InvolutiveNeg ι := P.indexNeg
aux : ∀ (n : ℕ), (e i ^ (n + 1)).col (Sum.inr j) = e i *ᵥ (e i ^ n).col (Sum.inr j)
n : ℕ
ih :
  (e i ^ n).col (Sum.inr j) = 0 ∨
    ∃ k x, P.root k = P.root j + n • P.root ↑i ∧ (e i ^ n).col (Sum.inr j) = x • Pi.single (Sum.inr k) 1
⊢ (e i ^ (n + 1)).col (Sum.inr j) = 0 ∨
    ∃ k x, P.root k = P.root j + (n + 1) • P.root ↑i ∧ (e i ^ (n + 1)).col (Sum.inr j) = x • Pi.single (Sum.inr k) 1

Tactic:
rcases ih with hn | ⟨k, x, hk₁, hk₂⟩

Goal state after:
case succ.inl
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
inst✝⁸ : CharZero R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsCrystallographic
inst✝² : P.IsReduced
b : P.Base
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ↥b.support
j : ι
h : j ≠ -↑i
this✝¹ : Module.IsReflexive R M
this✝ : IsAddTorsionFree M
this : InvolutiveNeg ι := P.indexNeg
aux : ∀ (n : ℕ), (e i ^ (n + 1)).col (Sum.inr j) = e i *ᵥ (e i ^ n).col (Sum.inr j)
n : ℕ
hn : (e i ^ n).col (Sum.inr j) = 0
⊢ (e i ^ (n + 1)).col (Sum.inr j) = 0 ∨
    ∃ k x, P.root k = P.root j + (n + 1) • P.root ↑i ∧ (e i ^ (n + 1)).col (Sum.inr j) = x • Pi.single (Sum.inr k) 1

case succ.inr
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
inst✝⁸ : CharZero R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsCrystallographic
inst✝² : P.IsReduced
b : P.Base
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ↥b.support
j : ι
h : j ≠ -↑i
this✝¹ : Module.IsReflexive R M
this✝ : IsAddTorsionFree M
this : InvolutiveNeg ι := P.indexNeg
aux : ∀ (n : ℕ), (e i ^ (n + 1)).col (Sum.inr j) = e i *ᵥ (e i ^ n).col (Sum.inr j)
n : ℕ
k : ι
x : ℕ
hk₁ : P.root k = P.root j + n • P.root ↑i
hk₂ : (e i ^ n).col (Sum.inr j) = x • Pi.single (Sum.inr k) 1
⊢ (e i ^ (n + 1)).col (Sum.inr j) = 0 ∨
    ∃ k x, P.root k = P.root j + (n + 1) • P.root ↑i ∧ (e i ^ (n + 1)).col (Sum.inr j) = x • Pi.single (Sum.inr k) 1
Module: Mathlib.LinearAlgebra.RootSystem.Finite.G2

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R N
P : RootPairing ι R M N
inst✝⁴ : P.EmbeddedG2
inst✝³ : Finite ι
inst✝² : CharZero R
inst✝¹ : IsDomain R
inst✝ : P.IsIrreducible
i : ι
hi : P.root i ∉ allRoots P
h₁ : P.pairing i (short P) = 0
h₂ : P.pairing i (long P) = 0
this✝ : Fintype ι
B : P.InvariantForm
this : Module.IsReflexive R M
⊢ False

Tactic:
← B.apply_root_root_zero_iff

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module R N
P : RootPairing ι R M N
inst✝⁴ : P.EmbeddedG2
inst✝³ : Finite ι
inst✝² : CharZero R
inst✝¹ : IsDomain R
inst✝ : P.IsIrreducible
i : ι
hi : P.root i ∉ allRoots P
this✝ : Fintype ι
B : P.InvariantForm
h₂ : (B.form (P.root i)) (P.root (long P)) = 0
h₁ : (B.form (P.root i)) (P.root (short P)) = 0
this : Module.IsReflexive R M
⊢ False
Module: Mathlib.LinearAlgebra.RootSystem.OfBilinear

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsReflexive R M
B : M →ₗ[R] M →ₗ[R] R
hNB : B.Nondegenerate
hSB : B.IsSymm
h2 : IsRegular 2
x y : ↑{x | B.IsReflective x}
hxy : IsReflective.coroot B ⋯ = IsReflective.coroot B ⋯
h1 : ∀ (z : M), (IsReflective.coroot B ⋯) z = (IsReflective.coroot B ⋯) z
h2y : ∀ (z : M), (B ↑y) ↑y * (IsReflective.coroot B ⋯) z = (B ↑y) ↑y * (IsReflective.coroot B ⋯) z
h2x : ∀ (z : M), 2 * (B ↑x) z = (B ↑x) ↑x * (IsReflective.coroot B ⋯) z
⊢ x = y

Tactic:
apply_self_mul_coroot_apply B y.2

Goal state after:
R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsReflexive R M
B : M →ₗ[R] M →ₗ[R] R
hNB : B.Nondegenerate
hSB : B.IsSymm
h2 : IsRegular 2
x y : ↑{x | B.IsReflective x}
hxy : IsReflective.coroot B ⋯ = IsReflective.coroot B ⋯
h1 : ∀ (z : M), (IsReflective.coroot B ⋯) z = (IsReflective.coroot B ⋯) z
h2x : ∀ (z : M), 2 * (B ↑x) z = (B ↑x) ↑x * (IsReflective.coroot B ⋯) z
h2y : ∀ (z : M), (B ↑y) ↑y * (IsReflective.coroot B ⋯) z = 2 * (B ↑y) z
⊢ x = y
Module: Mathlib.MeasureTheory.Constructions.ClosedCompactCylinders

Goal state before:
pp: ι : Type u_1
X : ι → Type u_2
inst✝³ : (i : ι) → TopologicalSpace (X i)
t : Set ((i : ι) → X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (X i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (X i)
ht : t ∈ closedCompactCylinders X
⊢ t ∈ measurableCylinders X

Tactic:
rw [mem_measurableCylinders]
  refine ⟨closedCompactCylinders.finset ht, closedCompactCylinders.set ht, ?_, ?_⟩
  · exact (closedCompactCylinders.isClosed ht).measurableSet
  · exact closedCompactCylinders.eq_cylinder ht

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set (α i)
h₁ : t₁ ∈ univ.pi C
s₂ : Finset ι
t₂ : (i : ι) → Set (α i)
h₂ : t₂ ∈ univ.pi C
t₁' : (i : ι) → Set (α i) := s₁.piecewise t₁ fun i ↦ univ
t₂' : (i : ι) → Set (α i) := s₂.piecewise t₂ fun i ↦ univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
i : ι
f : (i : ι) → α i
hf : f ∈ (↑s₁).pi t₁' ∩ (↑s₂).pi t₂'
⊢ (t₁' i ∩ t₂' i).Nonempty

Tactic:
mem_inter_iff,

Goal state after:
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set (α i)
h₁ : t₁ ∈ univ.pi C
s₂ : Finset ι
t₂ : (i : ι) → Set (α i)
h₂ : t₂ ∈ univ.pi C
t₁' : (i : ι) → Set (α i) := s₁.piecewise t₁ fun i ↦ univ
t₂' : (i : ι) → Set (α i) := s₂.piecewise t₂ fun i ↦ univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
i : ι
f : (i : ι) → α i
hf : f ∈ (↑s₁).pi t₁' ∧ f ∈ (↑s₂).pi t₂'
⊢ (t₁' i ∩ t₂' i).Nonempty
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: case a
ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
i : ι
⊢ MeasurableSpace.comap (fun b ↦ b i) (inst✝ i) ≤ generateFrom (squareCylinders fun i ↦ {s | MeasurableSet s})

Tactic:
refine (comap_eval_le_generateFrom_squareCylinders_singleton α i).trans ?_

Goal state after:
case a
ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
i : ι
⊢ generateFrom ((fun t ↦ {i}.pi t) '' univ.pi fun i ↦ {s | MeasurableSet s}) ≤
    generateFrom (squareCylinders fun i ↦ {s | MeasurableSet s})
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: case a
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
i : ι
⊢ MeasurableSpace.comap (fun b ↦ b i) (inst✝ i) ≤ generateFrom (measurableCylinders α)

Tactic:
refine (comap_eval_le_generateFrom_squareCylinders_singleton α i).trans ?_

Goal state after:
case a
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
i : ι
⊢ generateFrom ((fun t ↦ {i}.pi t) '' univ.pi fun i ↦ {s | MeasurableSet s}) ≤ generateFrom (measurableCylinders α)
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
j : Set α
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
hj : j ∈ J
⊢ ↑(hC.disjointOfUnion hJ j) ⊆ C

Tactic:
lift j to J using hj

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ↑(hC.disjointOfUnion hJ ↑j) ⊆ C
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
j : Set α
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
hj : j ∈ J
⊢ (↑(hC.disjointOfUnion hJ j)).PairwiseDisjoint id

Tactic:
lift j to J using hj

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ (↑(hC.disjointOfUnion hJ ↑j)).PairwiseDisjoint id
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
j : Set α
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
hj : j ∈ J
⊢ ⋃₀ ↑(hC.disjointOfUnion hJ j) ⊆ j

Tactic:
lift j to J using hj

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ⋃₀ ↑(hC.disjointOfUnion hJ ↑j) ⊆ ↑j
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
j : Set α
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
hj : j ∈ J
⊢ ∅ ∉ hC.disjointOfUnion hJ j

Tactic:
lift j to J using hj

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ∅ ∉ hC.disjointOfUnion hJ ↑j
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
⊢ ⊤ ∈ d ∧ (Ioi ⊤ = Ioi ⊤ ∨ Ioi ⊤ = Iio ⊤)

Tactic:
by simp [d]

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
⊢ False

Tactic:
rcases eq_or_ne c 0 with hc0 | hc0

Goal state after:
case inl
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c = 0
⊢ False

case inr
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c ≠ 0
⊢ False
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
hI_ss : ∀ t ∈ I, t ⊆ s
h_dis : (↑I).PairwiseDisjoint id
⊢ m s = ∑ i ∈ I, m i + ∑ i ∈ hC.disjointOfDiffUnion hs hI, m i

Tactic:
by
  conv_lhs => rw [← hC.sUnion_union_disjointOfDiffUnion_of_subset hs hI hI_ss]
  rw [addContent_sUnion]
  · rw [sum_union]
    exact hC.disjoint_disjointOfDiffUnion hs hI
  · rw [coe_union]
    exact Set.union_subset hI (hC.disjointOfDiffUnion_subset hs hI)
  · rw [coe_union]
    exact hC.pairwiseDisjoint_union_disjointOfDiffUnion hs hI h_dis
  · rwa [hC.sUnion_union_disjointOfDiffUnion_of_subset hs hI hI_ss]

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
⊢ ⊤ ∈ d ∧ (Iio ⊤ = Ioi ⊤ ∨ Iio ⊤ = Iio ⊤)

Tactic:
by simp [d]

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : ↑J ⊆ C
ht : t ∈ C
f : Fin #J → Set α := disjointed fun j ↦ ↑(J.equivFin.symm j)
htJ : t ⊆ ⋃ i, f i
h1 : ∀ (j : Fin #J), f j ∈ supClosure C
h2 : Pairwise (Disjoint on f)
⊢ m t ≤ ∑ u ∈ J, m u

Tactic:
have h3 : ⋃ i, f i ∈ supClosure C :=
    supClosed_supClosure.iSup_mem (subset_supClosure hC.empty_mem) h1

Goal state after:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : ↑J ⊆ C
ht : t ∈ C
f : Fin #J → Set α := disjointed fun j ↦ ↑(J.equivFin.symm j)
htJ : t ⊆ ⋃ i, f i
h1 : ∀ (j : Fin #J), f j ∈ supClosure C
h2 : Pairwise (Disjoint on f)
h3 : ⋃ i, f i ∈ supClosure C
⊢ m t ≤ ∑ u ∈ J, m u
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case inl
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i ↦ ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' :
  ∀ {ι : Type ?u.192} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j ↦ μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
j : ℕ
hf₀ : d ∈ s (f j)
hrp : 0 ≤ r₁ (f j)
hr0 : 0 = r₁ (f j)
⊢ ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))

Tactic:
exact ⟨d, hf₀, by simp [← hr0]⟩

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
⊢ (range fun a ↦ ↑a).OrdConnected

Tactic:
rw [WithTop.range_coe]

Goal state after:
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
⊢ (Iio ⊤).OrdConnected