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
this✝¹ : 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
this : chainBotCoeff i m = chainBotCoeff j (-l)
⊢ (chainBotCoeff i m + 1) * (chainBotCoeff j (-k) + 1) = (chainBotCoeff j (-l) + 1) * (0 + 1)

Tactic:
hjk,

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
this✝¹ : 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
this : chainBotCoeff i m = chainBotCoeff j (-l)
⊢ (chainBotCoeff i m + 1) * (0 + 1) = (chainBotCoeff j (-l) + 1) * (0 + 1)
Module: Mathlib.LinearAlgebra.TensorAlgebra.Grading

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
i : ℕ
x : TensorAlgebra R M
r : R
⊢ (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) 0) (DirectSum.GAlgebra.toFun r) =
    (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) 0) ⟨(algebraMap R (TensorAlgebra R M)) r, ⋯⟩

Tactic:
rfl

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

Goal state before:
pp: case add
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i✝ : ℕ
x✝ x y : TensorAlgebra R M
i : ℕ
hx : x ∈ (ι R).range ^ i
hy : y ∈ (ι R).range ^ i
ihx : ((lift R) (GradedAlgebra.ι R M)) x = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨x, hx⟩
ihy : ((lift R) (GradedAlgebra.ι R M)) y = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨y, hy⟩
⊢ (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) (⟨x, hx⟩ + ⟨y, hy⟩) =
    (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨x + y, ⋯⟩

Tactic:
rfl

Goal state after:
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
h : IsSymmetricAlgebra f
motive : A → Prop
algebraMap : ∀ (r : R), motive ((Algebra.algebraMap R A) r)
ι : ∀ (x : M), motive (f x)
mul : ∀ (a b : A), motive a → motive b → motive (a * b)
add : ∀ (a b : A), motive a → motive b → motive (a + b)
a : A
y : SymmetricAlgebra R M
⊢ motive ((SymmetricAlgebra.lift f) y)

Tactic:
induction y using SymmetricAlgebra.induction with
  | algebraMap r => simpa using algebraMap r
  | ι y => simpa using ι y
  | mul _ _ hx hy => simpa using mul _ _ hx hy
  | add _ _ hx hy => simpa using add _ _ hx hy

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

Goal state before:
pp: case mem_mul
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i✝ : ℕ
x✝ : TensorAlgebra R M
i : ℕ
x : TensorAlgebra R M
hx : x ∈ (ι R).range ^ i
ih : ((lift R) (GradedAlgebra.ι R M)) x = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨x, hx⟩
w✝ : M
⊢ ((lift R) (GradedAlgebra.ι R M)) ((ι R) w✝ * x) = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i.succ) ⟨(ι R) w✝ * x, ⋯⟩

Tactic:
rw [map_mul, ih, lift_ι_apply, GradedAlgebra.ι_apply R M, DirectSum.of_mul_of]

Goal state after:
case mem_mul
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i✝ : ℕ
x✝ : TensorAlgebra R M
i : ℕ
x : TensorAlgebra R M
hx : x ∈ (ι R).range ^ i
ih : ((lift R) (GradedAlgebra.ι R M)) x = (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i) ⟨x, hx⟩
w✝ : M
⊢ (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) (1 + i)) (GradedMonoid.GMul.mul ⟨(ι R) w✝, ⋯⟩ ⟨x, hx⟩) =
    (DirectSum.of (fun i ↦ ↥((ι R).range ^ i)) i.succ) ⟨(ι R) w✝ * x, ⋯⟩
Module: Mathlib.LinearAlgebra.TensorPower.Symmetric

Goal state before:
pp: case e'_2
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
⊢ (r • (PiTensorProduct.tprod R) fun i ↦ f (e i)) = r • (PiTensorProduct.tprod R) (f ∘ ⇑e)

Tactic:
rfl

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
⊢ s ∩ t ∈ C

Tactic:
by
  rw [← sdiff_sdiff_right_self]; exact hC.sdiff_mem hs (hC.sdiff_mem hs ht)

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
h_dis : (↑I).PairwiseDisjoint id
u : Set α
hu : u ∈ ↑I
v : Set α
hv : v ∈ ↑(hC.disjointOfDiffUnion hs hI)
x✝ : u ≠ v
⊢ Disjoint u v

Tactic:
exact disjoint_of_subset (subset_sUnion_of_mem hu) (subset_sUnion_of_mem hv)
    (hC.disjoint_sUnion_disjointOfDiffUnion hs hI)

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
u v : α
⊢ u ≤ max u v

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
u v : α
⊢ Set.Ioc u v = Set.Ioc u (max u v)

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
hu : u < u'
hv : v' < v
⊢ Set.Ioc u v \ Set.Ioc u' v' = Set.Ioc u u' ∪ Set.Ioc v' v

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
hu : u < u'
hv : v' < v
⊢ ↑{Set.Ioc u u', Set.Ioc v' v} ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
hu : u < u'
hv : v' < v
a : Set α
ha : a ∈ ↑{Set.Ioc u u', Set.Ioc v' v}
b : Set α
hb : b ∈ ↑{Set.Ioc u u', Set.Ioc v' v}
hab : a ≠ b
⊢ Disjoint a b

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl
α : Type u_1
𝒜 ℬ : Set (Set α)
h : 𝒜 ⊆ ℬ
s t : Set α
hs✝ : generateSetAlgebra 𝒜 t
t_mem : t ∈ generateSetAlgebra ℬ
⊢ tᶜ ∈ generateSetAlgebra ℬ

Tactic:
| compl t _ t_mem =>

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl
α : Type u_1
𝒜 ℬ : Set (Set α)
h : 𝒜 ⊆ ℬ
hℬ : IsSetAlgebra ℬ
s t : Set α
hs✝ : generateSetAlgebra 𝒜 t
t_mem : t ∈ ℬ
⊢ tᶜ ∈ ℬ

Tactic:
| compl t _ t_mem =>

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

Goal state before:
pp: case h_ss
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ↑(Finset.image f a) ⊆ C

Tactic:
grind

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

Goal state before:
pp: case convert_2.«0».«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) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ (Disjoint on ![s, t]) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
grind

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

Goal state before:
pp: case convert_2.«1».«1»
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) ⟨1, ⋯⟩
⊢ (Disjoint on ![s, t]) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
grind

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

Goal state before:
pp: case a.inl.coe.coe.refine_2.hs.h
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 := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
i : ι
hi : i ∈ f
⊢ Ioi ↑i ∈ {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}

Tactic:
grind

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

Goal state before:
pp: case a.inl.coe.coe.refine_2.ht.h
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 := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
i : ι
hi : i ∈ g
⊢ Iio ↑i ∈ {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}

Tactic:
grind

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

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
u v : α
h : u ≤ v
h' : ∃ p, p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2
u' : α := h'.choose.1
v' : α := h'.choose.2
hu'v' : u' ≤ v' ∧ u = u' ∧ v = v'
huv : u < v
⊢ f v' - f u' = f v - f u

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
h''I : ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
hn : #I = 0
⊢ I = ∅

Tactic:
grind

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

Goal state before:
pp: case a.inr.coe.coe.refine_2.hs.hs.h
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 := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
i : ι
hi : i ∈ f
⊢ Ioi ↑i ∈ {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' v' : α
hu'v' : u' ≤ v'
tI : Set.Ioc u' v' ∈ I
ht : v ∈ Set.Ioc u' v'
⊢ u' < v'

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' v' : α
hu'v' : u' ≤ v'
tI : Set.Ioc u' v' ∈ I
ht : v ∈ Set.Ioc u' v'
⊢ Set.Ioc u' v' ⊆ Set.Ioc u v

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' v' : α
hu'v' : u' ≤ v'
tI : Set.Ioc u' v' ∈ I
ht : v ∈ Set.Ioc u' v'
left✝ : v' ≤ v
uu' : u ≤ u'
⊢ v = v'

Tactic:
grind

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

Goal state before:
pp: case a.inr.coe.coe.refine_2.hs.ht.h
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 := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
i : ι
hi : i ∈ g
⊢ Iio ↑i ∈ {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
huv : u < v
this✝ : v ∈ ⋃₀ ↑I
u' : α
uu' : u ≤ u'
hu'v' : u' ≤ v
tI : Set.Ioc u' v ∈ I
ht : v ∈ Set.Ioc u' v
left✝ : v ≤ v
I' : Finset (Set α) := I.erase (Set.Ioc u' v)
I'I : I' ⊆ I
I_eq_insert : I = insert (Set.Ioc u' v) I'
this : (Set.Ioc u' v ∪ ⋃₀ ↑I') \ Set.Ioc u' v = ⋃₀ ↑I'
h'uv : Set.Ioc u' v ∪ ⋃₀ ↑I' = Set.Ioc u v
⊢ ⋃₀ ↑I' = Set.Ioc u u'

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' : α
uu' : u ≤ u'
hu'v' : u' ≤ v
tI : Set.Ioc u' v ∈ I
ht : v ∈ Set.Ioc u' v
left✝ : v ≤ v
I' : Finset (Set α) := I.erase (Set.Ioc u' v)
I'I : I' ⊆ I
I_eq_insert : I = insert (Set.Ioc u' v) I'
UI' : ⋃₀ ↑I' = Set.Ioc u u'
⊢ ⋃₀ ↑I' ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' : α
uu' : u ≤ u'
hu'v' : u' ≤ v
tI : Set.Ioc u' v ∈ I
ht : v ∈ Set.Ioc u' v
left✝ : v ≤ v
I' : Finset (Set α) := I.erase (Set.Ioc u' v)
I'I : I' ⊆ I
I_eq_insert : I = insert (Set.Ioc u' v) I'
UI' : ⋃₀ ↑I' = Set.Ioc u u'
⊢ #I' = n

Tactic:
grind

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

Goal state before:
pp: case a.inr.coe
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 : ι
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio ↑a)

Tactic:
rw [@isOpen_iff_forall_mem_open]
      intro b hb
      induction b with
      | top => simp at hb
      | coe b =>
        -- `b` can not be equal to `⊤`, so it comes from `ι`. In `ι` we can find a finite
        -- intersection of our sets containing `b` and contained in `Iio a`. We lift it
        -- to `WithTop ι`, and intersect with `Iio ⊤` (which is also in our set) to exclude `⊤`.
        simp only [mem_Iio, WithTop.coe_lt_coe] at hb
        obtain ⟨t, ⟨f, g, hfc, hgc, f_fin, g_fin, rfl⟩, hb, hfg⟩ :
          ∃ t ∈ basis, b ∈ t ∧ t ⊆ Iio a := h_basis.isOpen_iff.1 (isOpen_Iio' a) b hb
        refine ⟨(⋂ z ∈ f, Ioi z) ∩ (⋂ z ∈ g, Iio z) ∩ Iio ⊤, ?_, ?_, by simpa using hb⟩
        · intro w hw
          induction w with
          | top => simp at hw
          | coe w =>
            simp only [mem_Iio, WithTop.coe_lt_coe]
            apply hfg
            simpa using hw
        · apply @IsOpen.inter _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}); swap
          · apply isOpen_generateFrom_of_mem
            simp [d]
          apply @IsOpen.inter _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a})
          · apply @Finite.isOpen_biInter _ _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a})
            · exact f_fin
            · intro i hi
              apply isOpen_generateFrom_of_mem
              grind
          · apply @Finite.isOpen_biInter _ _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a})
            · exact g_fin
            · intro i hi
              apply isOpen_generateFrom_of_mem
              grind

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
n✝ : ℕ
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
n : ℕ
hind :
  ∀ {e : V ≃ₗ[K] V},
    (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →
      finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
f : Dual K V
v : V
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
hv_notMem : v ∉ (↑e).fixedSubmodule
hrank : finrank K (V ⧸ (↑e).fixedSubmodule ⊔ K ∙ v) = n + 1
⊢ ∃ x ∈ transvections K V, ∃ y ∈ transvections K V ^ n * dilatransvections K V, x * y = e

Tactic:
rcases Nat.eq_zero_or_pos n with hn0 | hn_pos

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
n✝ : ℕ
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
n : ℕ
hind :
  ∀ {e : V ≃ₗ[K] V},
    (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →
      finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
f : Dual K V
v : V
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
hv_notMem : v ∉ (↑e).fixedSubmodule
hrank : finrank K (V ⧸ (↑e).fixedSubmodule ⊔ K ∙ v) = n + 1
hn0 : n = 0
⊢ ∃ x ∈ transvections K V, ∃ y ∈ transvections K V ^ n * dilatransvections K V, x * y = e

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
n✝ : ℕ
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
n : ℕ
hind :
  ∀ {e : V ≃ₗ[K] V},
    (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →
      finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
f : Dual K V
v : V
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
hv_notMem : v ∉ (↑e).fixedSubmodule
hrank : finrank K (V ⧸ (↑e).fixedSubmodule ⊔ K ∙ v) = n + 1
hn_pos : n > 0
⊢ ∃ x ∈ transvections K V, ∃ y ∈ transvections K V ^ n * dilatransvections K V, x * y = e
Module: Mathlib.MeasureTheory.Function.AEEqFun.DomAct

Goal state before:
pp: case mk
M : Type u_3
N : Type u_1
α : Type u_4
β : Type u_2
inst✝⁷ : MeasurableSpace N
inst✝⁶ : MeasurableSpace α
μ : Measure α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : SMul M α
inst✝³ : MeasurableConstSMul M α
inst✝² : SMulInvariantMeasure M α μ
inst✝¹ : SMul N β
inst✝ : ContinuousConstSMul N β
m✝ : Mᵈᵐᵃ
n✝ : N
a✝¹ : α →ₘ[μ] β
a✝ : { f // AEStronglyMeasurable f μ }
⊢ m✝ • n✝ • Quot.mk (⇑(Measure.aeEqSetoid β μ)) a✝ = n✝ • m✝ • Quot.mk (⇑(Measure.aeEqSetoid β μ)) a✝

Tactic:
rfl

Goal state after: