1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: Equicontinuous.uniformInducing_uniformFun_iff_pi
File: Mathlib/Topology/UniformSpace/Ascoli.lean

tactic: rw [uniformInducing_iff_uniformSpace, uniformInducing_iff_uniformSpace,
    ← F_eqcont.comap_uniformFun_eq]
state_before: ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
F : ι → X → α
G : ι → β → α
inst✝¹ : UniformSpace ι
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
⊢ UniformInducing (⇑UniformFun.ofFun ∘ F) ↔ UniformInducing F
state_after: ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
F : ι → X → α
G : ι → β → α
inst✝¹ : UniformSpace ι
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
⊢ UniformSpace.comap (⇑UniformFun.ofFun ∘ F) (UniformFun.uniformSpace X α) = inst✝¹ ↔
    UniformSpace.comap F (UniformFun.uniformSpace X α) = inst✝¹

tactic: rfl
state_before: ι : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
β : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
F : ι → X → α
G : ι → β → α
inst✝¹ : UniformSpace ι
inst✝ : CompactSpace X
F_eqcont : Equicontinuous F
⊢ UniformSpace.comap (⇑UniformFun.ofFun ∘ F) (UniformFun.uniformSpace X α) = inst✝¹ ↔
    UniformSpace.comap F (UniformFun.uniformSpace X α) = inst✝¹
state_after: no goals
Theorem: LocallyFinite.finite_nonempty_of_compact
File: Mathlib/Topology/Compactness/Compact.lean

tactic: simpa only [inter_univ] using hf.finite_nonempty_inter_compact isCompact_univ
state_before: X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
f : ι → Set X
hf : LocallyFinite f
⊢ {i | (f i).Nonempty}.Finite
state_after: no goals
Theorem: EuclideanGeometry.image_inversion_sphere_dist_center
File: Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean

tactic: rw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR),
  preimage_inversion_sphere_dist_center hR hy]
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c x y : P
R : ℝ
hR : R ≠ 0
hy : y ≠ c
⊢ inversion c R '' sphere y (dist y c) = insert c ↑(perpBisector c (inversion c R y))
state_after: no goals
Theorem: Rat.numberField_discr
File: Mathlib/NumberTheory/NumberField/Discriminant.lean

tactic: let b : Basis (Fin 1) ℤ (𝓞 ℚ) :=
  Basis.map (Basis.singleton (Fin 1) ℤ) ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
state_before: ⊢ discr ℚ = 1
state_after: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ discr ℚ = 1

tactic: calc NumberField.discr ℚ
  _ = Algebra.discr ℤ b := by convert (discr_eq_discr ℚ b).symm
  _ = Algebra.trace ℤ (𝓞 ℚ) (b default * b default) := by
    rw [Algebra.discr_def, Matrix.det_unique, Algebra.traceMatrix_apply, Algebra.traceForm_apply]
  _ = Algebra.trace ℤ (𝓞 ℚ) 1 := by
    rw [Basis.map_apply, RingEquiv.toAddEquiv_eq_coe, AddEquiv.toIntLinearEquiv_symm,
      AddEquiv.coe_toIntLinearEquiv, Basis.singleton_apply,
      show (AddEquiv.symm ↑ringOfIntegersEquiv) (1 : ℤ) = ringOfIntegersEquiv.symm 1 by rfl,
      map_one, mul_one]
  _ = 1 := by rw [Algebra.trace_eq_matrix_trace b]; norm_num
state_before: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ discr ℚ = 1
state_after: no goals

tactic: convert (discr_eq_discr ℚ b).symm
state_before: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ discr ℚ = Algebra.discr ℤ ⇑b
state_after: no goals

tactic: rw [Algebra.discr_def, Matrix.det_unique, Algebra.traceMatrix_apply, Algebra.traceForm_apply]
state_before: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ Algebra.discr ℤ ⇑b = (Algebra.trace ℤ (𝓞 ℚ)) (b default * b default)
state_after: no goals

tactic: rw [Basis.map_apply, RingEquiv.toAddEquiv_eq_coe, AddEquiv.toIntLinearEquiv_symm,
  AddEquiv.coe_toIntLinearEquiv, Basis.singleton_apply,
  show (AddEquiv.symm ↑ringOfIntegersEquiv) (1 : ℤ) = ringOfIntegersEquiv.symm 1 by rfl,
  map_one, mul_one]
state_before: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ (Algebra.trace ℤ (𝓞 ℚ)) (b default * b default) = (Algebra.trace ℤ (𝓞 ℚ)) 1
state_after: no goals

tactic: rfl
state_before: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ (↑ringOfIntegersEquiv).symm 1 = ringOfIntegersEquiv.symm 1
state_after: no goals

tactic: rw [Algebra.trace_eq_matrix_trace b]
state_before: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ (Algebra.trace ℤ (𝓞 ℚ)) 1 = 1
state_after: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ ((Algebra.leftMulMatrix b) 1).trace = 1

tactic: norm_num
state_before: b : Basis (Fin 1) ℤ (𝓞 ℚ) := (Basis.singleton (Fin 1) ℤ).map ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
⊢ ((Algebra.leftMulMatrix b) 1).trace = 1
state_after: no goals
Theorem: CategoryTheory.PreGaloisCategory.has_decomp_connected_components
File: Mathlib/CategoryTheory/Galois/Decomposition.lean

tactic: let F := GaloisCategory.getFiberFunctor C
state_before: C : Type u₁
inst✝¹ : Category.{u₂, u₁} C
inst✝ : GaloisCategory C
X : C
⊢ ∃ ι f g x, (∀ (i : ι), IsConnected (f i)) ∧ Finite ι
state_after: C : Type u₁
inst✝¹ : Category.{u₂, u₁} C
inst✝ : GaloisCategory C
X : C
F : C ⥤ FintypeCat := GaloisCategory.getFiberFunctor C
⊢ ∃ ι f g x, (∀ (i : ι), IsConnected (f i)) ∧ Finite ι

tactic: exact has_decomp_connected_components_aux F (Nat.card <| F.obj X) X rfl
state_before: C : Type u₁
inst✝¹ : Category.{u₂, u₁} C
inst✝ : GaloisCategory C
X : C
F : C ⥤ FintypeCat := GaloisCategory.getFiberFunctor C
⊢ ∃ ι f g x, (∀ (i : ι), IsConnected (f i)) ∧ Finite ι
state_after: no goals
Theorem: Finset.exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum
File: Mathlib/Combinatorics/Pigeonhole.lean

tactic: simpa
state_before: α : Type u
β : Type v
M : Type w
inst✝¹ : DecidableEq β
s : Finset α
t : Finset β
f : α → β
w : α → M
b : M
n : ℕ
inst✝ : LinearOrderedCancelAddCommMonoid M
ht : ∀ y ∉ t, ∑ x ∈ filter (fun x => f x = y) s, w x ≤ 0
hb : t.card • b < ∑ x ∈ s, w x
⊢ ∑ _y ∈ t, b < ∑ x ∈ s, w x
state_after: no goals
Theorem: Matrix.kroneckerMap_diagonal_diagonal
File: Mathlib/Data/Matrix/Kronecker.lean

tactic: ext ⟨i₁, i₂⟩ ⟨j₁, j₂⟩
state_before: R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝⁴ : Zero α
inst✝³ : Zero β
inst✝² : Zero γ
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
f : α → β → γ
hf₁ : ∀ (b : β), f 0 b = 0
hf₂ : ∀ (a : α), f a 0 = 0
a : m → α
b : n → β
⊢ kroneckerMap f (diagonal a) (diagonal b) = diagonal fun mn => f (a mn.1) (b mn.2)
state_after: case a.mk.mk
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝⁴ : Zero α
inst✝³ : Zero β
inst✝² : Zero γ
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
f : α → β → γ
hf₁ : ∀ (b : β), f 0 b = 0
hf₂ : ∀ (a : α), f a 0 = 0
a : m → α
b : n → β
i₁ : m
i₂ : n
j₁ : m
j₂ : n
⊢ kroneckerMap f (diagonal a) (diagonal b) (i₁, i₂) (j₁, j₂) =
    diagonal (fun mn => f (a mn.1) (b mn.2)) (i₁, i₂) (j₁, j₂)

tactic: simp [diagonal, apply_ite f, ite_and, ite_apply, apply_ite (f (a i₁)), hf₁, hf₂]
state_before: case a.mk.mk
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝⁴ : Zero α
inst✝³ : Zero β
inst✝² : Zero γ
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
f : α → β → γ
hf₁ : ∀ (b : β), f 0 b = 0
hf₂ : ∀ (a : α), f a 0 = 0
a : m → α
b : n → β
i₁ : m
i₂ : n
j₁ : m
j₂ : n
⊢ kroneckerMap f (diagonal a) (diagonal b) (i₁, i₂) (j₁, j₂) =
    diagonal (fun mn => f (a mn.1) (b mn.2)) (i₁, i₂) (j₁, j₂)
state_after: no goals
Theorem: SemiNormedGrp.explicitCokernelDesc_unique
File: Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean

tactic: apply (isColimitCokernelCocone f).uniq (Cofork.ofπ g (by simp [w]))
state_before: X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ e = explicitCokernelDesc w
state_after: case x
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ ∀ (j : WalkingParallelPair), (cokernelCocone f).ι.app j ≫ e = (Cofork.ofπ g ⋯).ι.app j

tactic: rintro (_ | _)
state_before: case x
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ ∀ (j : WalkingParallelPair), (cokernelCocone f).ι.app j ≫ e = (Cofork.ofπ g ⋯).ι.app j
state_after: case x.zero
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ (cokernelCocone f).ι.app WalkingParallelPair.zero ≫ e = (Cofork.ofπ g ⋯).ι.app WalkingParallelPair.zero

case x.one
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ (cokernelCocone f).ι.app WalkingParallelPair.one ≫ e = (Cofork.ofπ g ⋯).ι.app WalkingParallelPair.one

tactic: simp [w]
state_before: X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ f ≫ g = 0 ≫ g
state_after: no goals

tactic: convert w.symm
state_before: case x.zero
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ (cokernelCocone f).ι.app WalkingParallelPair.zero ≫ e = (Cofork.ofπ g ⋯).ι.app WalkingParallelPair.zero
state_after: case h.e'_2.h
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
e_1✝ : ((parallelPair f 0).obj WalkingParallelPair.zero ⟶ (Cofork.ofπ g ⋯).pt) = (X ⟶ Z)
⊢ (cokernelCocone f).ι.app WalkingParallelPair.zero ≫ e = 0

tactic: simp
state_before: case h.e'_2.h
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
e_1✝ : ((parallelPair f 0).obj WalkingParallelPair.zero ⟶ (Cofork.ofπ g ⋯).pt) = (X ⟶ Z)
⊢ (cokernelCocone f).ι.app WalkingParallelPair.zero ≫ e = 0
state_after: no goals

tactic: exact he
state_before: case x.one
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ (cokernelCocone f).ι.app WalkingParallelPair.one ≫ e = (Cofork.ofπ g ⋯).ι.app WalkingParallelPair.one
state_after: no goals
Theorem: CategoryTheory.SmallObject.functorMap_comm
File: Mathlib/CategoryTheory/SmallObject/Construction.lean

tactic: ext ⟨i, t, b, w⟩
state_before: C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X Y Z : C
πX : X ⟶ S
πY : Y ⟶ S
φ : X ⟶ Y
hφ : φ ≫ πY = πX
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
⊢ functorObjLeft f πX ≫ functorMapTgt f πX πY φ hφ = functorMapSrc f πX πY φ hφ ≫ functorObjLeft f πY
state_after: case h.mk
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X Y Z : C
πX : X ⟶ S
πY : Y ⟶ S
φ : X ⟶ Y
hφ : φ ≫ πY = πX
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
i : I
t : A i ⟶ X
b : B i ⟶ S
w : t ≫ πX = f i ≫ b
⊢ Sigma.ι (functorObjSrcFamily f πX) { i := i, t := t, b := b, w := w } ≫
      functorObjLeft f πX ≫ functorMapTgt f πX πY φ hφ =
    Sigma.ι (functorObjSrcFamily f πX) { i := i, t := t, b := b, w := w } ≫
      functorMapSrc f πX πY φ hφ ≫ functorObjLeft f πY

tactic: simp only [ι_colimMap_assoc, Discrete.natTrans_app, ι_colimMap,
  ι_functorMapTgt f πX πY φ hφ i t b w _ rfl,
  ι_functorMapSrc_assoc f πX πY φ hφ i t b w _ rfl]
state_before: case h.mk
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X Y Z : C
πX : X ⟶ S
πY : Y ⟶ S
φ : X ⟶ Y
hφ : φ ≫ πY = πX
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
i : I
t : A i ⟶ X
b : B i ⟶ S
w : t ≫ πX = f i ≫ b
⊢ Sigma.ι (functorObjSrcFamily f πX) { i := i, t := t, b := b, w := w } ≫
      functorObjLeft f πX ≫ functorMapTgt f πX πY φ hφ =
    Sigma.ι (functorObjSrcFamily f πX) { i := i, t := t, b := b, w := w } ≫
      functorMapSrc f πX πY φ hφ ≫ functorObjLeft f πY
state_after: no goals
Theorem: BoxIntegral.Prepartition.tag_biUnionTagged
File: Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean

tactic: rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
state_before: ι : Type u_1
I J : Box ι
π : Prepartition I
πi : (J : Box ι) → TaggedPrepartition J
hJ : J ∈ π
J' : Box ι
hJ' : J' ∈ πi J
⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'
state_after: ι : Type u_1
I J : Box ι
π : Prepartition I
πi : (J : Box ι) → TaggedPrepartition J
hJ : J ∈ π
J' : Box ι
hJ' : J' ∈ πi J
⊢ (π.biUnionTagged πi).tag J' = (πi (π.biUnionIndex (fun J => (πi J).toPrepartition) J')).tag J'

tactic: rfl
state_before: ι : Type u_1
I J : Box ι
π : Prepartition I
πi : (J : Box ι) → TaggedPrepartition J
hJ : J ∈ π
J' : Box ι
hJ' : J' ∈ πi J
⊢ (π.biUnionTagged πi).tag J' = (πi (π.biUnionIndex (fun J => (πi J).toPrepartition) J')).tag J'
state_after: no goals
Theorem: AlgEquiv.symm_comp
File: Mathlib/Algebra/Algebra/Equiv.lean

tactic: ext
state_before: R : Type uR
A₁ : Type uA₁
A₂ : Type uA₂
A₃ : Type uA₃
A₁' : Type uA₁'
A₂' : Type uA₂'
A₃' : Type uA₃'
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A₁
inst✝¹⁰ : Semiring A₂
inst✝⁹ : Semiring A₃
inst✝⁸ : Semiring A₁'
inst✝⁷ : Semiring A₂'
inst✝⁶ : Semiring A₃'
inst✝⁵ : Algebra R A₁
inst✝⁴ : Algebra R A₂
inst✝³ : Algebra R A₃
inst✝² : Algebra R A₁'
inst✝¹ : Algebra R A₂'
inst✝ : Algebra R A₃'
e✝ e : A₁ ≃ₐ[R] A₂
⊢ (↑e.symm).comp ↑e = AlgHom.id R A₁
state_after: case H
R : Type uR
A₁ : Type uA₁
A₂ : Type uA₂
A₃ : Type uA₃
A₁' : Type uA₁'
A₂' : Type uA₂'
A₃' : Type uA₃'
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A₁
inst✝¹⁰ : Semiring A₂
inst✝⁹ : Semiring A₃
inst✝⁸ : Semiring A₁'
inst✝⁷ : Semiring A₂'
inst✝⁶ : Semiring A₃'
inst✝⁵ : Algebra R A₁
inst✝⁴ : Algebra R A₂
inst✝³ : Algebra R A₃
inst✝² : Algebra R A₁'
inst✝¹ : Algebra R A₂'
inst✝ : Algebra R A₃'
e✝ e : A₁ ≃ₐ[R] A₂
x✝ : A₁
⊢ ((↑e.symm).comp ↑e) x✝ = (AlgHom.id R A₁) x✝

tactic: simp
state_before: case H
R : Type uR
A₁ : Type uA₁
A₂ : Type uA₂
A₃ : Type uA₃
A₁' : Type uA₁'
A₂' : Type uA₂'
A₃' : Type uA₃'
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A₁
inst✝¹⁰ : Semiring A₂
inst✝⁹ : Semiring A₃
inst✝⁸ : Semiring A₁'
inst✝⁷ : Semiring A₂'
inst✝⁶ : Semiring A₃'
inst✝⁵ : Algebra R A₁
inst✝⁴ : Algebra R A₂
inst✝³ : Algebra R A₃
inst✝² : Algebra R A₁'
inst✝¹ : Algebra R A₂'
inst✝ : Algebra R A₃'
e✝ e : A₁ ≃ₐ[R] A₂
x✝ : A₁
⊢ ((↑e.symm).comp ↑e) x✝ = (AlgHom.id R A₁) x✝
state_after: no goals
Theorem: MeasureTheory.q
File: Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean

tactic: rw [memℒp_one_iff_integrable]
state_before: α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
⊢ Memℒp (↑↑(condexpIndSMul hm hs hμs x)) 1 μ
state_after: α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
⊢ Integrable (↑↑(condexpIndSMul hm hs hμs x)) μ

tactic: apply integrable_condexpIndSMul
state_before: α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
⊢ Integrable (↑↑(condexpIndSMul hm hs hμs x)) μ
state_after: no goals
Theorem: Polynomial.leadingCoeff_linear
File: Mathlib/Algebra/Polynomial/Degree/Definitions.lean

tactic: rw [add_comm, leadingCoeff_add_of_degree_lt (degree_C_lt_degree_C_mul_X ha),
  leadingCoeff_C_mul_X]
state_before: R : Type u
S : Type v
a b c d : R
n m : ℕ
inst✝ : Semiring R
p✝ p q : R[X]
ι : Type u_1
ha : a ≠ 0
⊢ (C a * X + C b).leadingCoeff = a
state_after: no goals
Theorem: Filter.Eventually.intCast_atBot
File: Mathlib/Order/Filter/Archimedean.lean

tactic: rw [← Int.comap_cast_atBot (R := R)]
state_before: α : Type u_1
R : Type u_2
inst✝¹ : StrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
⊢ ∀ᶠ (n : ℤ) in atBot, p ↑n
state_after: α : Type u_1
R : Type u_2
inst✝¹ : StrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
⊢ ∀ᶠ (n : ℤ) in Filter.comap Int.cast atBot, p ↑n

tactic: exact h.comap _
state_before: α : Type u_1
R : Type u_2
inst✝¹ : StrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
⊢ ∀ᶠ (n : ℤ) in Filter.comap Int.cast atBot, p ↑n
state_after: no goals
Theorem: AffineIndependent.mem_affineSpan_iff
File: Mathlib/LinearAlgebra/AffineSpace/Independent.lean

tactic: constructor
state_before: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
⊢ p i ∈ affineSpan k (p '' s) ↔ i ∈ s
state_after: case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
⊢ p i ∈ affineSpan k (p '' s) → i ∈ s

case mpr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
⊢ i ∈ s → p i ∈ affineSpan k (p '' s)

tactic: intro hs
state_before: case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
⊢ p i ∈ affineSpan k (p '' s) → i ∈ s
state_after: case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
hs : p i ∈ affineSpan k (p '' s)
⊢ i ∈ s

tactic: have h :=
  AffineIndependent.exists_mem_inter_of_exists_mem_inter_affineSpan ha hs
    (mem_affineSpan k (Set.mem_image_of_mem _ (Set.mem_singleton _)))
state_before: case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
hs : p i ∈ affineSpan k (p '' s)
⊢ i ∈ s
state_after: case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
hs : p i ∈ affineSpan k (p '' s)
h : ∃ i_1, i_1 ∈ s ∩ {i}
⊢ i ∈ s

tactic: rwa [← Set.nonempty_def, Set.inter_singleton_nonempty] at h
state_before: case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
hs : p i ∈ affineSpan k (p '' s)
h : ∃ i_1, i_1 ∈ s ∩ {i}
⊢ i ∈ s
state_after: no goals

tactic: exact fun h => mem_affineSpan k (Set.mem_image_of_mem p h)
state_before: case mpr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
i : ι
s : Set ι
⊢ i ∈ s → p i ∈ affineSpan k (p '' s)
state_after: no goals
Theorem: LinearMap.toMatrixAlgEquiv_toLinAlgEquiv
File: Mathlib/LinearAlgebra/Matrix/ToLin.lean

tactic: rw [← Matrix.toLinAlgEquiv_symm, AlgEquiv.symm_apply_apply]
state_before: R : Type u_1
inst✝⁹ : CommSemiring R
l : Type u_2
m : Type u_3
n : Type u_4
inst✝⁸ : Fintype n
inst✝⁷ : Fintype m
inst✝⁶ : DecidableEq n
M₁ : Type u_5
M₂ : Type u_6
inst✝⁵ : AddCommMonoid M₁
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₁
inst✝² : Module R M₂
v₁ : Basis n R M₁
v₂ : Basis m R M₂
M₃ : Type u_7
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
v₃ : Basis l R M₃
M : Matrix n n R
⊢ (toMatrixAlgEquiv v₁) ((toLinAlgEquiv v₁) M) = M
state_after: no goals