mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.RootSystem.Finite.G2

Goal state before:
pp: case refine_1
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : P.IsCrystallographic
inst✝⁴ : P.IsReduced
inst✝³ : P.IsIrreducible
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : IsDomain R
h : ∀ (x x_1 : ι), ¬P.pairingIn ℤ x x_1 = -3
i j : ι
⊢ P.pairingIn ℤ i j = -2 ∨
    P.pairingIn ℤ i j = -1 ∨ P.pairingIn ℤ i j = 0 ∨ P.pairingIn ℤ i j = 1 ∨ P.pairingIn ℤ i j = 2

pp: case refine_2
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : P.IsCrystallographic
inst✝⁴ : P.IsReduced
inst✝³ : P.IsIrreducible
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : IsDomain R
h :
  ∀ (i j : ι),
    P.pairingIn ℤ i j = -2 ∨
      P.pairingIn ℤ i j = -1 ∨ P.pairingIn ℤ i j = 0 ∨ P.pairingIn ℤ i j = 1 ∨ P.pairingIn ℤ i j = 2
i j : ι
⊢ ¬P.pairingIn ℤ i j = -3

Tactic:
· have hij := h (P.reflectionPerm i i) j
    have := P.pairingIn_pairingIn_mem_set_of_isCrystal_of_isRed i j
    aesop

Goal state after:
case refine_2
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
inst✝⁵ : P.IsCrystallographic
inst✝⁴ : P.IsReduced
inst✝³ : P.IsIrreducible
inst✝² : Finite ι
inst✝¹ : CharZero R
inst✝ : IsDomain R
h :
  ∀ (i j : ι),
    P.pairingIn ℤ i j = -2 ∨
      P.pairingIn ℤ i j = -1 ∨ P.pairingIn ℤ i j = 0 ∨ P.pairingIn ℤ i j = 1 ∨ P.pairingIn ℤ i j = 2
i j : ι
⊢ ¬P.pairingIn ℤ i j = -3
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✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : P.EmbeddedG2
⊢ P.pairing (long P) (short P) = -3

Tactic:
by
  rw [← P.algebraMap_pairingIn ℤ, pairingIn_long_short]
  simp

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Finite ι
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CharZero R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R N
P : RootPairing ι R M N
inst✝² : P.IsCrystallographic
b : P.Base
inst✝¹ : Fintype ι
i : ↥b.support
inst✝ : P.IsReduced
k : ι
hki : k ≠ ↑i
hki' : k ≠ (P.reflectionPerm ↑i) ↑i
h_lin_ind : LinearIndependent R ![P.root ↑i, P.root k]
this :
  ((∑ x, if P.root k = P.root ↑i + P.root x then (↑(chainBotCoeff (↑i) x) + 1) * (↑(chainTopCoeff (↑i) k) + 1) else 0) -
      ∑ x,
        if P.root k = P.root x - P.root ↑i then (↑(chainTopCoeff (↑i) x) + 1) * (↑(chainBotCoeff (↑i) k) + 1) else 0) =
    ↑(chainBotCoeff (↑i) k) - ↑(chainTopCoeff (↑i) k)
aux : ∀ (x : ι), P.root x = P.root k - P.root ↑i ↔ P.root k = P.root ↑i + P.root x
aux' : ∀ (x : ι), P.root x = P.root ↑i + P.root k ↔ P.root k = P.root x - P.root ↑i
⊢ ⁅e i, f i⁆ (Sum.inr k) (Sum.inr k) = h i (Sum.inr k) (Sum.inr k)

Tactic:
simpa [e, f, h, hki, hki', aux, aux', ← ite_and, ← P.chainBotCoeff_sub_chainTopCoeff h_lin_ind]

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CharZero R
inst✝⁸ : IsDomain 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 : Finite ι
inst✝² : P.IsCrystallographic
b : P.Base
i j l m : ι
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
hi : i ∈ b.support
hj : j ∈ b.support
hij : i ≠ j
this✝² : Module.IsReflexive R M := Module.IsReflexive.of_isPerfPair P.toLinearMap
him_mem : P.root i + P.root m ∈ range ⇑P.root
hjl_mem : P.root j + P.root (-l) ∈ range ⇑P.root
this✝¹ : P.IsNotG2
this✝ : InvolutiveNeg ι := P.indexNeg
this : IsAddTorsionFree M
n : ι
h₁ : P.root n + P.root i = P.root l
h₂ : P.root n - P.root j = P.root m
hki : P.pairingIn ℤ n i = 0
hjk_mem : P.root j + P.root (-n) ∈ range ⇑P.root
hn : P.root n = P.root n + P.root i - P.root j
⊢ False

Tactic:
simp [sub_eq_zero, hij, add_sub_assoc] at hn

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

Goal state before:
pp: ι : Type u_1
K : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : CharZero K
inst✝⁸ : DecidableEq ι
inst✝⁷ : Fintype ι
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module K M
inst✝⁴ : AddCommGroup N
inst✝³ : Module K N
P : RootPairing ι K M N
inst✝² : P.IsCrystallographic
b : P.Base
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
U : LieSubmodule K (↥(lieAlgebra b)) (↥b.support ⊕ ι → K)
_i : InvolutiveNeg ι := P.indexNeg
hωu : ∀ (i : ↥b.support), ω b *ᵥ u i = u i
hωv : ∀ (i : ι), ω b *ᵥ v b i = v b (-i)
aux : ∀ (k : ↥b.support), u k ∈ U
this : ∀ (j : ι), v b j ∈ U
⊢ span K (range ⇑(Pi.basisFun K (↥b.support ⊕ ι))) ≤ ↑U

Tactic:
Submodule.span_le,

Goal state after:
ι : Type u_1
K : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : CharZero K
inst✝⁸ : DecidableEq ι
inst✝⁷ : Fintype ι
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module K M
inst✝⁴ : AddCommGroup N
inst✝³ : Module K N
P : RootPairing ι K M N
inst✝² : P.IsCrystallographic
b : P.Base
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
U : LieSubmodule K (↥(lieAlgebra b)) (↥b.support ⊕ ι → K)
_i : InvolutiveNeg ι := P.indexNeg
hωu : ∀ (i : ↥b.support), ω b *ᵥ u i = u i
hωv : ∀ (i : ι), ω b *ᵥ v b i = v b (-i)
aux : ∀ (k : ↥b.support), u k ∈ U
this : ∀ (j : ι), v b j ∈ U
⊢ range ⇑(Pi.basisFun K (↥b.support ⊕ ι)) ⊆ ↑↑U
Module: Mathlib.LinearAlgebra.SModEq.Pow

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
p : ℕ
hpI : ↑p ∈ I
x y : R
h : x ≡ y [SMOD I]
m : ℕ
⊢ x ^ p ^ m ≡ y ^ p ^ m [SMOD I ^ (m + 1)]

Tactic:
induction m with
  | zero => simpa
  | succ m ih =>
    simp_rw [pow_succ _ m, pow_mul]
    exact ih.pow_add_one hpI m.succ_ne_zero

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

Goal state before:
pp: ι : Type u_1
K : Type u_2
M : Type u_3
N : Type u_4
inst✝¹² : Field K
inst✝¹¹ : CharZero K
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : Fintype ι
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module K M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module K N
P : RootPairing ι K M N
inst✝⁴ : P.IsRootSystem
inst✝³ : P.IsCrystallographic
b : P.Base
inst✝² : P.IsReduced
inst✝¹ : P.IsIrreducible
inst✝ : Nonempty ι
U : LieSubmodule K (↥(lieAlgebra b)) (↥b.support ⊕ ι → K)
U' : LieSubmodule K (↥H) (↥b.support ⊕ ι → K) := { toSubmodule := ↑U, lie_mem := ⋯ }
hU : ↑U ≤ span K (range u)
x : ↥b.support ⊕ ι → K
hx : x ∈ U
c : ↥b.support → K
hc : ∑ i, c i • u i = x
hCM : c ≠ 0
j : ↥b.support
this : ∑ k, c k * ↑|b.cartanMatrix j k| = 0
⊢ ((b.cartanMatrix.map abs).map Int.cast *ᵥ c) j = 0 j

Tactic:
simpa [mulVec_eq_sum, -Base.cartanMatrix_map_abs]

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

Goal state before:
pp: case inr.inr
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CharZero R
inst✝⁸ : IsDomain 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 : Finite ι
inst✝² : P.IsCrystallographic
b : P.Base
i j k l m : ι
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
hi : i ∈ b.support
hj : j ∈ b.support
hij : i ≠ j
h₁ : P.root k + P.root i = P.root l
h₂ : P.root k - P.root j = P.root m
h₃ : P.root k + P.root i - P.root j ∈ range ⇑P.root
this✝ : P.IsNotG2
this : InvolutiveNeg ι := P.indexNeg
him_mem : P.root i + P.root m ∈ range ⇑P.root
hik_mem : P.root i + P.root k ∈ range ⇑P.root
hjk_mem : P.root (-k) + P.root j ∈ range ⇑P.root
hjl_mem : P.root j + P.root (-l) ∈ range ⇑P.root
h₁' : P.root (-k) - P.root i = P.root (-l)
h₂' : P.root (-k) + P.root j = P.root (-m)
h₃' : P.root (-k) + P.root j - P.root i ∈ range ⇑P.root
hki : P.pairingIn ℤ k i < 0
hik : chainBotCoeff i k = 0
hkj : P.pairingIn ℤ (-k) j < 0
hjk : chainBotCoeff j (-k) = 0
aux₁ : ¬(chainBotCoeff i m = 1 ∧ chainBotCoeff j (-l) = 0)
⊢ chainBotCoeff i m = chainBotCoeff j (-l)

Tactic:
have aux₂ : ¬(P.chainBotCoeff j (-l) = 1 ∧ P.chainBotCoeff i m = 0) := by
    simpa using aux_2 hj hi hij.symm h₂' h₁' h₃' hkj (by simpa)
      hjl_mem (by simpa only [neg_neg]) (by simpa only [neg_neg])

Goal state after:
case inr.inr
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CharZero R
inst✝⁸ : IsDomain 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 : Finite ι
inst✝² : P.IsCrystallographic
b : P.Base
i j k l m : ι
inst✝¹ : P.IsReduced
inst✝ : P.IsIrreducible
hi : i ∈ b.support
hj : j ∈ b.support
hij : i ≠ j
h₁ : P.root k + P.root i = P.root l
h₂ : P.root k - P.root j = P.root m
h₃ : P.root k + P.root i - P.root j ∈ range ⇑P.root
this✝ : P.IsNotG2
this : InvolutiveNeg ι := P.indexNeg
him_mem : P.root i + P.root m ∈ range ⇑P.root
hik_mem : P.root i + P.root k ∈ range ⇑P.root
hjk_mem : P.root (-k) + P.root j ∈ range ⇑P.root
hjl_mem : P.root j + P.root (-l) ∈ range ⇑P.root
h₁' : P.root (-k) - P.root i = P.root (-l)
h₂' : P.root (-k) + P.root j = P.root (-m)
h₃' : P.root (-k) + P.root j - P.root i ∈ range ⇑P.root
hki : P.pairingIn ℤ k i < 0
hik : chainBotCoeff i k = 0
hkj : P.pairingIn ℤ (-k) j < 0
hjk : chainBotCoeff j (-k) = 0
aux₁ : ¬(chainBotCoeff i m = 1 ∧ chainBotCoeff j (-l) = 0)
aux₂ : ¬(chainBotCoeff j (-l) = 1 ∧ chainBotCoeff i m = 0)
⊢ chainBotCoeff i m = chainBotCoeff j (-l)
Module: Mathlib.LinearAlgebra.SymmetricAlgebra.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : M →ₗ[R] A
e : SymmetricAlgebra R M ≃ₐ[R] A
he : ↑↑e ∘ₗ SymmetricAlgebra.ι R M = f
⊢ IsSymmetricAlgebra f

Tactic:
suffices h : e = SymmetricAlgebra.lift f by
    change Function.Bijective _
    exact h ▸ e.bijective
  ext x
  simpa using congr($he x)

Goal state after:
Module: Mathlib.Logic.Hydra

Goal state before:
pp: α : Type u_1
r : α → α → Prop
inst✝ : Std.Irrefl r
a : α
hacc : Acc r a
⊢ Acc (CutExpand r) {a}

Tactic:
induction hacc with | _ a h ih
  refine Acc.intro _ fun s ↦ ?_
  classical
  simp only [cutExpand_iff, mem_singleton]
  rintro ⟨t, a, hr, rfl, rfl⟩
  refine acc_of_singleton fun a' ↦ ?_
  rw [erase_singleton, zero_add]
  exact ih a' ∘ hr a'

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝ : (n : ℕ) → MeasurableSpace (X n)
s : Set ((i : ℕ) → X i)
⊢ s ∈ measurableCylinders X ↔ s ∈ ⋃ a, ⋃ S, ⋃ (_ : MeasurableSet S), {cylinder (Finset.Iic a) S}

Tactic:
simp only [mem_measurableCylinders, exists_prop, mem_iUnion]

Goal state after:
X : ℕ → Type u_1
inst✝ : (n : ℕ) → MeasurableSpace (X n)
s : Set ((i : ℕ) → X i)
⊢ (∃ s_1 S, MeasurableSet S ∧ s = cylinder s_1 S) ↔ ∃ i i_1, MeasurableSet i_1 ∧ s ∈ {cylinder (Finset.Iic i) i_1}
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a
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
⊢ ∀ (a : Set (WithTop ι)) (x : WithTop ι),
    a = Ioi x ∨ a = Iio x → IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] a

Tactic:
rintro u a (rfl | rfl)

Goal state after:
case a.inl
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
a : WithTop ι
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi a)

case a.inr
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
a : WithTop ι
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio a)
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
f : (i : Set (Set α)) → i ∈ A → Set α
hf : ∀ (i : Set (Set α)) (i_1 : i ∈ A), f i i_1 ∈ i ∧ x ∉ f i i_1
x✝ : Set α
a : Set (Set α)
ha : a ∈ A
h :
  (↑((fun x_1 ↦
            match x_1 with
            | ⟨a, ha⟩ => ⟨f a ha, ⋯⟩)
          ⟨a, ⋯⟩))ᶜ =
    x✝
⊢ x ∈ x✝

Tactic:
by rw [← h]; exact (hf a ha).2

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl.refine_2.mp
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
hx : ∀ i ∈ A, ∃ i_2 ∈ i, x ∉ i_2
⊢ ∃ i, ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A), (↑(i ⟨i_2, ⋯⟩))ᶜ = i_1 → x ∈ i_1

pp: case compl.refine_2.mpr
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
hx : ∃ i, ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A), (↑(i ⟨i_2, ⋯⟩))ᶜ = i_1 → x ∈ i_1
⊢ ∀ i ∈ A, ∃ i_2 ∈ i, x ∉ i_2

Tactic:
· choose f hf using hx
        exact ⟨fun ⟨a, ha⟩ ↦ ⟨f a ha, (hf a ha).1⟩, fun _ a ha h ↦ by rw [← h]; exact (hf a ha).2⟩

Goal state after:
case compl.refine_2.mpr
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
x : α
hx : ∃ i, ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A), (↑(i ⟨i_2, ⋯⟩))ᶜ = i_1 → x ∈ i_1
⊢ ∀ i ∈ A, ∃ i_2 ∈ i, x ∉ i_2
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case convert_2.«1».«0»
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ (Disjoint on ![s, t]) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
exact h_dis.symm

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
hs : s ∈ C
⊢ ↑{s} ⊆ C

Tactic:
simp [hs]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: case neg.refine_1
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ ∀ (n : ℕ), Integrable (fun ω ↦ (B ((fs n) ω)) (g ω)) μ

Tactic:
exact fun n ↦ B.integrable_of_bilin_of_bdd_left c
      ((fs n).stronglyMeasurable.mono hm).aestronglyMeasurable (ae_of_all _ <| hfs_bound n) hg

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: case neg.refine_3
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n ↦ (B ((fs n) x)) (g x)) atTop (𝓝 ((B (f x)) (g x)))

Tactic:
filter_upwards [hfs_tendsto] with ω hω
    exact ((by fun_prop : Continuous (fun x ↦ B x (g ω))).tendsto (f ω)).comp hω

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ < ↑‖f x‖₊

Tactic:
← ENNReal.toReal_lt_toReal hbdd.ne ENNReal.coe_lt_top.ne,

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ (eLpNormEssSup f μ).toReal < (↑‖f x‖₊).toReal
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: 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
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[>] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
i : ℕ
hi : 0 < r i
⊢ cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)

Tactic:
simp only [r', hi, if_true]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
X : Type u_5
mX : MeasurableSpace X
μ ν : VectorMeasure X M
C : Set (Set X)
hμν : ∀ s ∈ C, μ s = ν s
hA : mX = MeasurableSpace.generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set X
hs : MeasurableSet s
⊢ μ s = ν s

Tactic:
induction s, hs using MeasurableSpace.induction_on_inter hA hC with

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : AEStronglyMeasurable f μ
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
a : Ω
ha : f a = AEStronglyMeasurable.mk f hf a
⊢ (B (f a)) (g a) = (B (AEStronglyMeasurable.mk f hf a)) (g a)

Tactic:
rw [ha]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
⊢ ennrealToMeasure 0 = 0

Tactic:
by
  simp [ennrealToMeasure]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝ : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg : Integrable (fun ω ↦ (B (f ω)) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ∞
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
this : ∀ (n : ℕ), ∀ᵐ (ω : Ω) ∂μ, ω ∈ sets n → μ[fun ω ↦ (B (f ω)) (g ω) | m] ω = (B (f ω)) (μ[g | m] ω)
⊢ μ[fun ω ↦ (B (f ω)) (g ω) | m] =ᵐ[μ] fun ω ↦ (B (f ω)) (μ[g | m] ω)

Tactic:
← ae_all_iff

Goal state after:
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝ : CompleteSpace E
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hfg : Integrable (fun ω ↦ (B (f ω)) (g ω)) μ
hg : Integrable g μ
hm : m ≤ mΩ
hμm : SigmaFinite (μ.trim hm)
sets : ℕ → Set Ω
h_univ : ⋃ i, sets i = Set.univ
h_meas : ∀ (x : ℕ), MeasurableSet (sets x)
h_finite : ∀ (x : ℕ), μ (sets x) < ∞
h_norm : ∀ (x : ℕ), ∀ x_1 ∈ sets x, ‖f x_1‖ ≤ ↑x
this : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), a ∈ sets i → μ[fun ω ↦ (B (f ω)) (g ω) | m] a = (B (f a)) (μ[g | m] a)
⊢ μ[fun ω ↦ (B (f ω)) (g ω) | m] =ᵐ[μ] fun ω ↦ (B (f ω)) (μ[g | m] ω)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case inl
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h✝ : n ≠ m
h : n < m
⊢ s.restrictNonposSeq i n ∩ s.restrictNonposSeq i m = ∅

pp: case inr
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h✝ : n ≠ m
h : m < n
⊢ s.restrictNonposSeq i n ∩ s.restrictNonposSeq i m = ∅

Tactic:
· rw [restrictNonposSeq_disjoint' h]

Goal state after:
case inr
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
n m : ℕ
h✝ : n ≠ m
h : m < n
⊢ s.restrictNonposSeq i n ∩ s.restrictNonposSeq i m = ∅