mathlib-initiative/mathlib-tactics huggingface dataset:
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 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
hki' : P.pairingIn ℤ k i < 0
hkj' : 0 < P.pairingIn ℤ k 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
hjk_mem : P.root j + P.root (-k) ∈ range ⇑P.root
this✝¹ : InvolutiveNeg ι := P.indexNeg
this✝ : P.IsNotG2
this : IsAddTorsionFree M
hkj_ne : k ≠ j ∧ P.root k ≠ -P.root j
hlj_mem : P.root l - P.root j ∈ range ⇑P.root
hmi : P.pairingIn ℤ m i = 0
hlj : P.pairingIn ℤ l j ≠ 0
aux₀ : P.pairingIn ℤ j i = P.pairingIn ℤ k i
⊢ False

Tactic:
obtain ⟨aux₁, aux₂⟩ : P.pairingIn ℤ i j = -1 ∧ P.pairingIn ℤ k j = 2 := by
    suffices 0 < - P.pairingIn ℤ i j ∧ - P.pairingIn ℤ i j < P.pairingIn ℤ k j ∧
      P.pairingIn ℤ k j ≤ 2 by lia
    refine ⟨?_, ?_, ?_⟩
    · rwa [neg_pos, P.pairingIn_lt_zero_iff, aux₀]
    · suffices P.pairingIn ℤ l j = P.pairingIn ℤ i j + P.pairingIn ℤ k j by
        have := zero_le_pairingIn_of_root_sub_mem hlj_mem; lia
      suffices P.pairing l j = P.pairing i j + P.pairing k j from
        algebraMap_injective ℤ R <| by simpa only [algebraMap_pairingIn, map_add]
      simp [← P.root_coroot_eq_pairing l, ← h₁, add_comm]
    · have := IsNotG2.pairingIn_mem_zero_one_two (P := P) k j
      grind

Goal state after:
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
hki' : P.pairingIn ℤ k i < 0
hkj' : 0 < P.pairingIn ℤ k 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
hjk_mem : P.root j + P.root (-k) ∈ range ⇑P.root
this✝¹ : InvolutiveNeg ι := P.indexNeg
this✝ : P.IsNotG2
this : IsAddTorsionFree M
hkj_ne : k ≠ j ∧ P.root k ≠ -P.root j
hlj_mem : P.root l - P.root j ∈ range ⇑P.root
hmi : P.pairingIn ℤ m i = 0
hlj : P.pairingIn ℤ l j ≠ 0
aux₀ : P.pairingIn ℤ j i = P.pairingIn ℤ k i
aux₁ : P.pairingIn ℤ i j = -1
aux₂ : P.pairingIn ℤ k j = 2
⊢ False
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
_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)
j k l m : ι
hm : P.root m = P.root l + P.root k
hk : ∀ {U : LieSubmodule K (↥(lieAlgebra b)) (↥b.support ⊕ ι → K)}, (∀ (k : ↥b.support), u k ∈ U) → v b k ∈ U
hl : l ∈ b.support
U : LieSubmodule K (↥(lieAlgebra b)) (↥b.support ⊕ ι → K)
aux : ∀ (k : ↥b.support), u k ∈ U
e' : ↥(lieAlgebra b) := ⟨e ⟨l, hl⟩, ⋯⟩
this : (↑(chainBotCoeff l k) + 1) • v b m ∈ U
⊢ ↑(chainBotCoeff l k) + 1 ≠ 0

Tactic:
by norm_cast

Goal state after:
Module: Mathlib.LinearAlgebra.TensorAlgebra.Basis

Goal state before:
pp: κ : Type uκ
R : Type uR
M : Type uM
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis κ R M
⊢ ((lift R) (Finsupp.linearCombination R (FreeAlgebra.ι R) ∘ₗ ↑b.repr)).comp ((FreeAlgebra.lift R) (⇑(ι R) ∘ ⇑b)) =
    AlgHom.id R (FreeAlgebra R κ)

Tactic:
ext

Goal state after:
κ : Type uκ
R : Type uR
M : Type uM
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis κ R M
x✝ : κ
⊢ (⇑(((lift R) (Finsupp.linearCombination R (FreeAlgebra.ι R) ∘ₗ ↑b.repr)).comp ((FreeAlgebra.lift R) (⇑(ι R) ∘ ⇑b))) ∘
        FreeAlgebra.ι R)
      x✝ =
    (⇑(AlgHom.id R (FreeAlgebra R κ)) ∘ FreeAlgebra.ι R) x✝
Module: Mathlib.LinearAlgebra.SymmetricAlgebra.Basic

Goal state before:
pp: case algebraMap
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
motive : SymmetricAlgebra R M → Prop
algebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)
ι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)
mul : ∀ (a b : SymmetricAlgebra R M), motive a → motive b → motive (a * b)
add : ∀ (a b : SymmetricAlgebra R M), motive a → motive b → motive (a + b)
r : R
⊢ motive ((algHom R M) ((Algebra.algebraMap R (TensorAlgebra R M)) r))

Tactic:
rw [AlgHom.commutes]; exact algebraMap r

Goal state after:
Module: Mathlib.LinearAlgebra.SymmetricAlgebra.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ι R M = ↑(AlgHom.id R (SymmetricAlgebra R M)) ∘ₗ ι R M

Tactic:
ext

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x✝ : M
⊢ (ι R M) x✝ = (↑(AlgHom.id R (SymmetricAlgebra R M)) ∘ₗ ι R M) x✝
Module: Mathlib.LinearAlgebra.TensorPower.Symmetric

Goal state before:
pp: case of.perm.right
R ι : Type u
inst✝² : CommSemiring R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
x y : ⨂[R] (x : ι), M
e : Perm ι
f : ι → M
h : Nonempty ι
⊢ (addConGen (Rel R ι M)) (r • (PiTensorProduct.tprod R) fun i ↦ f i) (r • (PiTensorProduct.tprod R) fun i ↦ f (e i))

Tactic:
let i := Nonempty.some h

Goal state after:
case of.perm.right
R ι : Type u
inst✝² : CommSemiring R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r : R
x y : ⨂[R] (x : ι), M
e : Perm ι
f : ι → M
h : Nonempty ι
i : ι := h.some
⊢ (addConGen (Rel R ι M)) (r • (PiTensorProduct.tprod R) fun i ↦ f i) (r • (PiTensorProduct.tprod R) fun i ↦ f (e i))
Module: Mathlib.LinearAlgebra.TensorProduct.Matrix

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
M' : Type u_5
N' : Type u_6
ι : Type u_7
κ : Type u_8
ι' : Type u_10
κ' : Type u_11
inst✝¹⁴ : DecidableEq ι
inst✝¹³ : DecidableEq κ
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype κ
inst✝¹⁰ : Finite ι'
inst✝⁹ : Finite κ'
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : AddCommGroup M'
inst✝⁴ : AddCommGroup N'
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R M'
inst✝ : Module R N'
bM : Basis ι R M
bN : Basis κ R N
bM' : Basis ι' R M'
bN' : Basis κ' R N'
f : M →ₗ[R] M'
g : N →ₗ[R] N'
⊢ (toMatrix (bM.tensorProduct bN) (bM'.tensorProduct bN')) (map f g) =
    kroneckerMap (fun x1 x2 ↦ x1 * x2) ((toMatrix bM bM') f) ((toMatrix bN bN') g)

Tactic:
by
  ext ⟨i, j⟩ ⟨i', j'⟩
  simp_rw [Matrix.kroneckerMap_apply, toMatrix_apply, Basis.tensorProduct_apply,
    TensorProduct.map_tmul, Basis.tensorProduct_repr_tmul_apply]
  exact mul_comm _ _

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 : Dual K V
v : V
hv : v ∉ (↑e).fixedSubmodule
hf : map f (↑e).fixedSubmodule = ⊥
hfv : f (v - e v) = 0
hfv' : f (e v) = 1
ht : (↑e).fixedSubmodule ≤ (↑(transvection hfv)).fixedSubmodule
⊢ (↑e).fixedSubmodule ≤ (↑(transvection hfv * e)).fixedSubmodule ∧ v ∈ (↑(transvection hfv * e)).fixedSubmodule

Tactic:
constructor

Goal state after:
case left
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 : Dual K V
v : V
hv : v ∉ (↑e).fixedSubmodule
hf : map f (↑e).fixedSubmodule = ⊥
hfv : f (v - e v) = 0
hfv' : f (e v) = 1
ht : (↑e).fixedSubmodule ≤ (↑(transvection hfv)).fixedSubmodule
⊢ (↑e).fixedSubmodule ≤ (↑(transvection hfv * e)).fixedSubmodule

case right
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 : Dual K V
v : V
hv : v ∉ (↑e).fixedSubmodule
hf : map f (↑e).fixedSubmodule = ⊥
hfv : f (v - e v) = 0
hfv' : f (e v) = 1
ht : (↑e).fixedSubmodule ≤ (↑(transvection hfv)).fixedSubmodule
⊢ v ∈ (↑(transvection hfv * e)).fixedSubmodule
Module: Mathlib.MeasureTheory.Constructions.Projective

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
h : IsEmpty ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
I : Finset ι
i : ι
hi : IsEmpty (α i)
⊢ P I = 0

Tactic:
hP (insert i I) I (I.subset_insert i)

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
h : IsEmpty ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
I : Finset ι
i : ι
hi : IsEmpty (α i)
⊢ Measure.map (Finset.restrict₂ ⋯) (P (insert i I)) = 0
Module: Mathlib.MeasureTheory.Constructions.Projective

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
I J : Finset ι
hP : IsProjectiveMeasureFamily P
S : Set ((i : ↥I) → α ↑i)
T : Set ((i : ↥J) → α ↑i)
hS : MeasurableSet S
hT : MeasurableSet T
h_eq : cylinder I S = cylinder J T
U : Set ((i : ↥(I ∪ J)) → α ↑i) := Finset.restrict₂ ⋯ ⁻¹' S ∩ Finset.restrict₂ ⋯ ⁻¹' T
⊢ (P (I ∪ J)) U = (P I) S ∧ (P (I ∪ J)) U = (P J) T

Tactic:
constructor

Goal state after:
case left
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
I J : Finset ι
hP : IsProjectiveMeasureFamily P
S : Set ((i : ↥I) → α ↑i)
T : Set ((i : ↥J) → α ↑i)
hS : MeasurableSet S
hT : MeasurableSet T
h_eq : cylinder I S = cylinder J T
U : Set ((i : ↥(I ∪ J)) → α ↑i) := ⋯
⊢ (P (I ∪ J)) U = (P I) S

case right
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
I J : Finset ι
hP : IsProjectiveMeasureFamily P
S : Set ((i : ↥I) → α ↑i)
T : Set ((i : ↥J) → α ↑i)
hS : MeasurableSet S
hT : MeasurableSet T
h_eq : cylinder I S = cylinder J T
U : Set ((i : ↥(I ∪ J)) → α ↑i) := ⋯
⊢ (P (I ∪ J)) U = (P J) T
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
n✝ : ℕ
e : V ≃ₗ[K] V
he : e.fixedReduce = 1
n : ℕ
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
hind :
  ∀ {e : V ≃ₗ[K] V},
    e.fixedReduce = 1 → finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
f : Dual K V
v : V
hf : (↑e).fixedSubmodule ≤ ker f
hv : f v = 1
hv_notMem : v ∉ (↑e).fixedSubmodule
⊢ (↑e).fixedSubmodule.mkQ v = (↑e).fixedSubmodule.mkQ (e v)

Tactic:
simp [← fixedReduce_mk, he]

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
μ : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)
hμ : IsProjectiveLimit μ P
⊢ IsFiniteMeasure μ

Tactic:
constructor

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
μ : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)
hμ : IsProjectiveLimit μ P
⊢ μ univ < ⊤
Module: Mathlib.MeasureTheory.Constructions.Projective

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
μ : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)
hμ : IsProjectiveLimit μ P
⊢ IsFiniteMeasure μ

Tactic:
by
  constructor
  rw [hμ.measure_univ_eq (∅ : Finset ι)]
  exact measure_lt_top _ _

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
μ : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsProbabilityMeasure (P i)
hμ : IsProjectiveLimit μ P
⊢ IsProbabilityMeasure μ

Tactic:
constructor

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
μ : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsProbabilityMeasure (P i)
hμ : IsProjectiveLimit μ P
⊢ μ univ = 1
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)
⊢ MeasurableSpace.pi ≤ generateFrom (squareCylinders fun i ↦ {s | MeasurableSet s})

Tactic:
refine iSup_le fun i ↦ ?_

Goal state after:
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})
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)
⊢ MeasurableSpace.pi ≤ generateFrom (measurableCylinders α)

Tactic:
refine iSup_le fun i ↦ ?_

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
X : ι → Type u_3
mα : MeasurableSpace α
m : (i : ι) → MeasurableSpace (X i)
Δ : Set ι
g : α → (i : ι) → X i
⊢ MeasurableSpace.comap g (⨆ i ∈ Δ, MeasurableSpace.comap (fun σ ↦ σ i) (m i)) ≤ mα ↔
    ∀ ⦃i : ι⦄, i ∈ Δ → MeasurableSpace.comap (fun a ↦ g a i) (m i) ≤ mα

Tactic:
MeasurableSpace.comap_iSup,

Goal state after:
α : Type u_1
ι : Type u_2
X : ι → Type u_3
mα : MeasurableSpace α
m : (i : ι) → MeasurableSpace (X i)
Δ : Set ι
g : α → (i : ι) → X i
⊢ ⨆ i ∈ Δ, MeasurableSpace.comap g (MeasurableSpace.comap (fun σ ↦ σ i) (m i)) ≤ mα ↔
    ∀ ⦃i : ι⦄, i ∈ Δ → MeasurableSpace.comap (fun a ↦ g a i) (m i) ≤ mα
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
V : Type u_2
inst✝¹ : AddCommGroup V
inst✝ : Module K V
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
h : 1 < finrank K (V ⧸ (↑e).fixedSubmodule)
u : V
hu : LinearIndependent K ![(↑e).fixedSubmodule.mkQ u, e.fixedReduce ((↑e).fixedSubmodule.mkQ u)]
f : Dual K V
hfu : f u ≠ 0
v : V := (f u)⁻¹ • u
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
v_def : v = (f u)⁻¹ • u
hspan : K ∙ (e v - v) = K ∙ (e u - u)
⊢ LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]

Tactic:
rw [← LinearIndependent.pair_smul_smul_iff (Ne.isUnit hfu).inv (Ne.isUnit hfu).inv] at hu

Goal state after:
K : Type u_1
inst✝² : DivisionRing K
V : Type u_2
inst✝¹ : AddCommGroup V
inst✝ : Module K V
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
h : 1 < finrank K (V ⧸ (↑e).fixedSubmodule)
u : V
f : Dual K V
hu : LinearIndependent K ![(f u)⁻¹ • (↑e).fixedSubmodule.mkQ u, (f u)⁻¹ • e.fixedReduce ((↑e).fixedSubmodule.mkQ u)]
hfu : f u ≠ 0
v : V := (f u)⁻¹ • u
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
v_def : v = (f u)⁻¹ • u
hspan : K ∙ (e v - v) = K ∙ (e u - u)
⊢ LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Set ((i : ι) → α i)
hs : MeasurableSet s
I : Finset ι
t : (i : ι) → Set (α i)
⊢ (↑I).pi t = (↑I).domRestrict ⁻¹' univ.pi fun i ↦ t ↑i

Tactic:
ext

Goal state after:
ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Set ((i : ι) → α i)
hs : MeasurableSet s
I : Finset ι
t : (i : ι) → Set (α i)
x✝ : (i : ι) → α i
⊢ x✝ ∈ (↑I).pi t ↔ x✝ ∈ (↑I).domRestrict ⁻¹' univ.pi fun i ↦ t ↑i
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
⊢ ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v},
    ∀ t ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}, s ∩ t ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}

Tactic:
rintro s ⟨u, v, huv, rfl⟩ t ⟨u', v', hu'v', rfl⟩

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
⊢ Set.Ioc u v ∩ Set.Ioc u' v' ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
⊢ ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v},
    ∀ t ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v},
      ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ s \ t = ⋃₀ ↑I

Tactic:
rintro s ⟨u, v, huv, rfl⟩ t ⟨u', v', hu'v', rfl⟩

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
⊢ ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ Set.Ioc u v \ Set.Ioc u' v' = ⋃₀ ↑I
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl.refine_2
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
⊢ (∀ 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

Tactic:
constructor

Goal state after:
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
⊢ (∀ 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

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
⊢ (∃ 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: 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
⊢ s ∪ t = ⋃ i, ![s, t] i

Tactic:
ext

Goal state after:
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
x✝ : α
⊢ x✝ ∈ s ∪ t ↔ x✝ ∈ ⋃ i, ![s, t] i
Module: Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
⊢ projectiveFamilyFun P (s ∪ t) = projectiveFamilyFun P s + projectiveFamilyFun P t

Tactic:
have hS' : MeasurableSet S' := measurable_restrict₂ _ hS

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
⊢ projectiveFamilyFun P (s ∪ t) = projectiveFamilyFun P s + projectiveFamilyFun P t
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : OrderTop ι
⊢ {⊤} = Ioi ↑⊤

Tactic:
ext

Goal state after:
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : OrderTop ι
x✝ : WithTop ι
⊢ x✝ ∈ {⊤} ↔ x✝ ∈ Ioi ↑⊤
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifIntegrable f p μ
E : Set α
ε : ℝ
hε✝ : 0 < ε
δ : ℝ
hδ_pos : 0 < δ
hε :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ
⊢ eLpNorm (s.indicator ((fun i ↦ E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε

Tactic:
calc
    eLpNorm (s.indicator (E.indicator (f i))) p μ
      = eLpNorm (E.indicator (s.indicator (f i))) p μ := by
      simp only [indicator_indicator, inter_comm]
    _ ≤ eLpNorm (s.indicator (f i)) p μ := eLpNorm_indicator_le _
    _ ≤ ENNReal.ofReal ε := hε _ _ hs hμs

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
⊢ v = w ↔ ∀ (i : Set α), MeasurableSet i → v i = w i

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
⊢ v = w → ∀ (i : Set α), MeasurableSet i → v i = w i

case mpr
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
⊢ (∀ (i : Set α), MeasurableSet i → v i = w i) → v = w
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), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)

Tactic:
rintro i ⟨-, hi⟩

Goal state after:
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)