ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: toIcoDiv_zsmul_sub_self
File: Mathlib/Algebra/Order/ToIntervalMod.lean

Proof steps:
state_before: α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a✝ b✝ c : α
n : ℤ
a b : α
⊢ toIcoDiv hp a b • p - b = -toIcoMod hp a b
state_after: no goals
tactic: rw [toIcoMod, neg_sub]
Theorem: Filter.EventuallyLE.cardinal_bInter
File: Mathlib/Order/Filter/CardinalInter.lean

Proof steps:
state_before: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
s t : (i : ι) → i ∈ S → Set α
h : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi
⊢ ⋂ i, ⋂ (h : i ∈ S), s i h ≤ᶠ[l] ⋂ i, ⋂ (h : i ∈ S), t i h
state_after: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
s t : (i : ι) → i ∈ S → Set α
h : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi
⊢ ⋂ x, s ↑x ⋯ ≤ᶠ[l] ⋂ x, t ↑x ⋯
tactic: simp only [biInter_eq_iInter]
state_before: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
s t : (i : ι) → i ∈ S → Set α
h : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi
⊢ ⋂ x, s ↑x ⋯ ≤ᶠ[l] ⋂ x, t ↑x ⋯
state_after: no goals
tactic: exact EventuallyLE.cardinal_iInter hS fun i => h i i.2
Theorem: Int.floor_add_int
File: Mathlib/Algebra/Order/Floor.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z✝ : ℤ
a✝¹ a✝ : α
z a : ℤ
⊢ a ≤ ⌊a✝ + ↑z⌋ ↔ a ≤ ⌊a✝⌋ + z
state_after: no goals
tactic: rw [le_floor, ← sub_le_iff_le_add, ← sub_le_iff_le_add, le_floor, Int.cast_sub]
Theorem: MonoidHom.compAddChar_injective_left
File: Mathlib/Algebra/Group/AddChar.lean

Proof steps:
state_before: A : Type u_1
B : Type u_2
M : Type u_3
N : Type u_4
inst✝³ : AddMonoid A
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
ψ✝ ψ : AddChar A M
hψ : Surjective ⇑ψ
⊢ Injective fun f => f.compAddChar ψ
state_after: A : Type u_1
B : Type u_2
M : Type u_3
N : Type u_4
inst✝³ : AddMonoid A
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
ψ✝ ψ : AddChar A M
hψ : Surjective ⇑ψ
f g : M →* N
h : (fun f => f.compAddChar ψ) f = (fun f => f.compAddChar ψ) g
⊢ f = g
tactic: rintro f g h
state_before: A : Type u_1
B : Type u_2
M : Type u_3
N : Type u_4
inst✝³ : AddMonoid A
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
ψ✝ ψ : AddChar A M
hψ : Surjective ⇑ψ
f g : M →* N
h : (fun f => f.compAddChar ψ) f = (fun f => f.compAddChar ψ) g
⊢ f = g
state_after: A : Type u_1
B : Type u_2
M : Type u_3
N : Type u_4
inst✝³ : AddMonoid A
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
ψ✝ ψ : AddChar A M
hψ : Surjective ⇑ψ
f g : M →* N
h : ⇑((fun f => f.compAddChar ψ) f) = ⇑((fun f => f.compAddChar ψ) g)
⊢ ⇑f = ⇑g
tactic: rw [DFunLike.ext'_iff] at h ⊢
state_before: A : Type u_1
B : Type u_2
M : Type u_3
N : Type u_4
inst✝³ : AddMonoid A
inst✝² : AddMonoid B
inst✝¹ : Monoid M
inst✝ : Monoid N
ψ✝ ψ : AddChar A M
hψ : Surjective ⇑ψ
f g : M →* N
h : ⇑((fun f => f.compAddChar ψ) f) = ⇑((fun f => f.compAddChar ψ) g)
⊢ ⇑f = ⇑g
state_after: no goals
tactic: exact hψ.injective_comp_right h
Theorem: Set.uIoc_injective_left
File: Mathlib/Order/Interval/Set/UnorderedInterval.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
s : Set α
a✝ a₁ a₂ b b₁ b₂ c d x a : α
⊢ Injective (Ι a)
state_after: no goals
tactic: simpa only [uIoc_comm] using uIoc_injective_right a
Theorem: LinearMap.BilinForm.linearIndependent_of_iIsOrtho
File: Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean

Proof steps:
state_before: R : Type u_1
M : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝⁹ : CommRing R₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : Module R₁ M₁
V : Type u_5
K : Type u_6
inst✝⁶ : Field K
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
R₄ : Type u_7
M₄ : Type u_8
inst✝³ : CommRing R₄
inst✝² : IsDomain R₄
inst✝¹ : AddCommGroup M₄
inst✝ : Module R₄ M₄
G : BilinForm R₄ M₄
n : Type w
B : BilinForm K V
v : n → V
hv₁ : B.iIsOrtho v
hv₂ : ∀ (i : n), ¬B.IsOrtho (v i) (v i)
⊢ LinearIndependent K v
state_after: no goals
tactic: classical
    rw [linearIndependent_iff']
    intro s w hs i hi
    have : B (s.sum fun i : n => w i • v i) (v i) = 0 := by rw [hs, zero_left]
    have hsum : (s.sum fun j : n => w j * B (v j) (v i)) = w i * B (v i) (v i) := by
      apply Finset.sum_eq_single_of_mem i hi
      intro j _ hij
      rw [iIsOrtho_def.1 hv₁ _ _ hij, mul_zero]
    simp_rw [sum_left, smul_left, hsum] at this
    exact eq_zero_of_ne_zero_of_mul_right_eq_zero (hv₂ i) this
Theorem: le_neg_add_iff_le
File: Mathlib/Algebra/Order/Group/Unbundled/Basic.lean

Proof steps:
state_before: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c d : α
⊢ 1 ≤ b⁻¹ * a ↔ b ≤ a
state_after: no goals
tactic: rw [← mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left]
Theorem: ContinuousLinearEquiv.comp_hasFDerivAt_iff
File: Mathlib/Analysis/Calculus/FDeriv/Equiv.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f✝ f₀ f₁ g : E → F
f'✝ f₀' f₁' g' e : E →L[𝕜] F
x✝ : E
s t : Set E
L L₁ L₂ : Filter E
iso : E ≃L[𝕜] F
f : G → E
x : G
f' : G →L[𝕜] E
⊢ HasFDerivAt (⇑iso ∘ f) ((↑iso).comp f') x ↔ HasFDerivAt f f' x
state_after: no goals
tactic: simp_rw [← hasFDerivWithinAt_univ, iso.comp_hasFDerivWithinAt_iff]
Theorem: Set.mem_of_mem_coe
File: Mathlib/Data/Set/Functor.lean

Proof steps:
state_before: α β✝ : Type u
s : Set α
f : α → Set β✝
g : Set (α → β✝)
β : Set α
γ : Set ↑β
a : α
ha :
  a ∈ do
    let a ← γ
    pure ↑a
⊢ ⟨a, ⋯⟩ ∈ γ
state_after: case intro.intro.intro.intro.intro.intro
α β✝ : Type u
s : Set α
f : α → Set β✝
g : Set (α → β✝)
β : Set α
γ : Set ↑β
a : α
w✝ : ↑β
ha : w✝ ∈ γ
right✝ : a ∈ (fun h => (fun a => pure ↑a) w✝) ha
⊢ ⟨a, ⋯⟩ ∈ γ
tactic: rcases ha with ⟨_, ⟨_, rfl⟩, _, ⟨ha, rfl⟩, _⟩
state_before: case intro.intro.intro.intro.intro.intro
α β✝ : Type u
s : Set α
f : α → Set β✝
g : Set (α → β✝)
β : Set α
γ : Set ↑β
a : α
w✝ : ↑β
ha : w✝ ∈ γ
right✝ : a ∈ (fun h => (fun a => pure ↑a) w✝) ha
⊢ ⟨a, ⋯⟩ ∈ γ
state_after: no goals
tactic: convert ha
Theorem: Set.insert_diff_singleton
File: Mathlib/Data/Set/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
γ : Type w
ι : Sort x
a✝ b : α
s✝ s₁ s₂ t t₁ t₂ u : Set α
a : α
s : Set α
⊢ insert a (s \ {a}) = insert a s
state_after: no goals
tactic: simp [insert_eq, union_diff_self, -union_singleton, -singleton_union]
Theorem: LinearMap.nilRankAux_le
File: Mathlib/Algebra/Module/LinearMap/Polynomial.lean

Proof steps:
state_before: R : Type u_1
L : Type u_2
M : Type u_3
n : Type u_4
ι : Type u_5
ι' : Type u_6
ιM : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup L
inst✝¹⁰ : Module R L
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁷ : Fintype ι
inst✝⁶ : Fintype ι'
inst✝⁵ : Fintype ιM
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
b✝ : Basis ι R L
inst✝ : Nontrivial R
b : Basis ι R L
b' : Basis ι' R L
⊢ φ.nilRankAux b ≤ φ.nilRankAux b'
state_after: case h
R : Type u_1
L : Type u_2
M : Type u_3
n : Type u_4
ι : Type u_5
ι' : Type u_6
ιM : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup L
inst✝¹⁰ : Module R L
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁷ : Fintype ι
inst✝⁶ : Fintype ι'
inst✝⁵ : Fintype ιM
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
b✝ : Basis ι R L
inst✝ : Nontrivial R
b : Basis ι R L
b' : Basis ι' R L
⊢ (φ.polyCharpoly b).coeff (φ.nilRankAux b') ≠ 0
tactic: apply Polynomial.natTrailingDegree_le_of_ne_zero
state_before: case h
R : Type u_1
L : Type u_2
M : Type u_3
n : Type u_4
ι : Type u_5
ι' : Type u_6
ιM : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup L
inst✝¹⁰ : Module R L
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁷ : Fintype ι
inst✝⁶ : Fintype ι'
inst✝⁵ : Fintype ιM
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
b✝ : Basis ι R L
inst✝ : Nontrivial R
b : Basis ι R L
b' : Basis ι' R L
⊢ (φ.polyCharpoly b).coeff (φ.nilRankAux b') ≠ 0
state_after: case h
R : Type u_1
L : Type u_2
M : Type u_3
n : Type u_4
ι : Type u_5
ι' : Type u_6
ιM : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup L
inst✝¹⁰ : Module R L
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁷ : Fintype ι
inst✝⁶ : Fintype ι'
inst✝⁵ : Fintype ιM
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
b✝ : Basis ι R L
inst✝ : Nontrivial R
b : Basis ι R L
b' : Basis ι' R L
⊢ ¬(φ.polyCharpoly b').coeff (φ.nilRankAux b') = 0
tactic: rw [Ne, (polyCharpoly_coeff_eq_zero_iff_of_basis φ b b' _).not]
state_before: case h
R : Type u_1
L : Type u_2
M : Type u_3
n : Type u_4
ι : Type u_5
ι' : Type u_6
ιM : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup L
inst✝¹⁰ : Module R L
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁷ : Fintype ι
inst✝⁶ : Fintype ι'
inst✝⁵ : Fintype ιM
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
b✝ : Basis ι R L
inst✝ : Nontrivial R
b : Basis ι R L
b' : Basis ι' R L
⊢ ¬(φ.polyCharpoly b').coeff (φ.nilRankAux b') = 0
state_after: no goals
tactic: apply polyCharpoly_coeff_nilRankAux_ne_zero
Theorem: StarAlgHom.snd_prod
File: Mathlib/Algebra/Star/StarAlgHom.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁹ : CommSemiring R
inst✝⁸ : Semiring A
inst✝⁷ : Algebra R A
inst✝⁶ : Star A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Star B
inst✝² : Semiring C
inst✝¹ : Algebra R C
inst✝ : Star C
f : A →⋆ₐ[R] B
g : A →⋆ₐ[R] C
⊢ (snd R B C).comp (f.prod g) = g
state_after: case h
R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁹ : CommSemiring R
inst✝⁸ : Semiring A
inst✝⁷ : Algebra R A
inst✝⁶ : Star A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Star B
inst✝² : Semiring C
inst✝¹ : Algebra R C
inst✝ : Star C
f : A →⋆ₐ[R] B
g : A →⋆ₐ[R] C
x✝ : A
⊢ ((snd R B C).comp (f.prod g)) x✝ = g x✝
tactic: ext
state_before: case h
R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁹ : CommSemiring R
inst✝⁸ : Semiring A
inst✝⁷ : Algebra R A
inst✝⁶ : Star A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Star B
inst✝² : Semiring C
inst✝¹ : Algebra R C
inst✝ : Star C
f : A →⋆ₐ[R] B
g : A →⋆ₐ[R] C
x✝ : A
⊢ ((snd R B C).comp (f.prod g)) x✝ = g x✝
state_after: no goals
tactic: rfl
Theorem: PartialHomeomorph.eventually_ne_nhdsWithin
File: Mathlib/Topology/PartialHomeomorph.lean

Proof steps:
state_before: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
x : X
hx : x ∈ e.source
x' : X
hx' : ↑e.symm (↑e x') = x'
h : ↑e x' = ↑e x
⊢ x' ∈ {x}
state_after: no goals
tactic: rw [mem_singleton_iff, ← e.left_inv hx, ← h, hx']
Theorem: CategoryTheory.Equivalence.cancel_counit_right
File: Mathlib/CategoryTheory/Equivalence.lean

Proof steps:
state_before: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
e : C ≌ D
X Y : D
f f' : X ⟶ e.functor.obj (e.inverse.obj Y)
⊢ f ≫ e.counit.app Y = f' ≫ e.counit.app Y ↔ f = f'
state_after: no goals
tactic: simp only [cancel_mono]
Theorem: Finset.disjoint_erase_insert
File: Mathlib/Data/Finset/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
ha : a ∉ s
⊢ _root_.Disjoint (insert a s) (t.erase a) ↔ _root_.Disjoint s t
state_after: no goals
tactic: rw [← disjoint_erase_comm, erase_insert ha]
Theorem: List.eq_cons_of_length_one
File: Mathlib/Data/List/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
⊢ 0 < l.length
state_after: no goals
tactic: omega
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
⊢ l = [l.nthLe 0 ⋯]
state_after: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
n : ℕ
h₁ : n < l.length
h₂ : n < [l.nthLe 0 ⋯].length
⊢ l.get ⟨n, h₁⟩ = [l.nthLe 0 ⋯].get ⟨n, h₂⟩
tactic: refine ext_get (by convert h) fun n h₁ h₂ => ?_
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
n : ℕ
h₁ : n < l.length
h₂ : n < [l.nthLe 0 ⋯].length
⊢ l.get ⟨n, h₁⟩ = [l.nthLe 0 ⋯].get ⟨n, h₂⟩
state_after: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
n : ℕ
h₁ : n < l.length
h₂ : n < [l.nthLe 0 ⋯].length
⊢ l.get ⟨n, h₁⟩ = l.nthLe 0 ⋯
tactic: simp only [get_singleton]
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
n : ℕ
h₁ : n < l.length
h₂ : n < [l.nthLe 0 ⋯].length
⊢ l.get ⟨n, h₁⟩ = l.nthLe 0 ⋯
state_after: case e_a.e_val
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
n : ℕ
h₁ : n < l.length
h₂ : n < [l.nthLe 0 ⋯].length
⊢ n = 0
tactic: congr
state_before: case e_a.e_val
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ l : List α
h : l.length = 1
n : ℕ
h₁ : n < l.length
h₂ : n < [l.nthLe 0 ⋯].length
⊢ n = 0
state_after: no goals
tactic: omega
Theorem: FormalMultilinearSeries.partialSum_continuous
File: Mathlib/Analysis/Analytic/Basic.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁰ : Ring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup E
inst✝² : TopologicalAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
⊢ Continuous (p.partialSum n)
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁰ : Ring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup E
inst✝² : TopologicalAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
⊢ Continuous fun x => ∑ k ∈ Finset.range n, (p k) fun x_1 => x
tactic: unfold partialSum
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁰ : Ring 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalAddGroup E
inst✝² : TopologicalAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
⊢ Continuous fun x => ∑ k ∈ Finset.range n, (p k) fun x_1 => x
state_after: no goals
tactic: continuity
Theorem: Filter.tendsto_div_const_atBot_of_pos
File: Mathlib/Order/Filter/AtTopBot.lean

Proof steps:
state_before: ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝ : LinearOrderedField α
l : Filter β
f : β → α
r : α
hr : 0 < r
⊢ Tendsto (fun x => f x / r) l atBot ↔ Tendsto f l atBot
state_after: no goals
tactic: simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_pos, hr]
Theorem: IsFractionRing.div_surjective
File: Mathlib/RingTheory/Localization/FractionRing.lean

Proof steps:
state_before: R : Type u_1
inst✝¹¹ : CommRing R
M : Submonoid R
S : Type u_2
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
P : Type u_3
inst✝⁸ : CommRing P
A : Type u_4
inst✝⁷ : CommRing A
inst✝⁶ : IsDomain A
K : Type u_5
B : Type u_6
inst✝⁵ : CommRing B
inst✝⁴ : IsDomain B
inst✝³ : Field K
L : Type u_7
inst✝² : Field L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
g : A →+* L
z : K
x y : A
hy : y ∈ nonZeroDivisors A
h : mk' K x ⟨y, hy⟩ = z
⊢ (algebraMap A K) x / (algebraMap A K) y = z
state_after: no goals
tactic: rwa [mk'_eq_div] at h
Theorem: MeasureTheory.convolution_eq_swap
File: Mathlib/Analysis/Convolution.lean

Proof steps:
state_before: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : NormedAddCommGroup E''
inst✝¹² : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ ν : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddCommGroup G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : μ.IsNegInvariant
inst✝¹ : MeasurableNeg G
inst✝ : MeasurableAdd G
⊢ (f ⋆[L, μ] g) x = ∫ (t : G), (L (f (x - t))) (g t) ∂μ
state_after: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : NormedAddCommGroup E''
inst✝¹² : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ ν : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddCommGroup G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : μ.IsNegInvariant
inst✝¹ : MeasurableNeg G
inst✝ : MeasurableAdd G
⊢ (g ⋆[L.flip, μ] f) x = ∫ (t : G), (L (f (x - t))) (g t) ∂μ
tactic: rw [← convolution_flip]
state_before: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : NormedAddCommGroup E''
inst✝¹² : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ ν : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddCommGroup G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : μ.IsNegInvariant
inst✝¹ : MeasurableNeg G
inst✝ : MeasurableAdd G
⊢ (g ⋆[L.flip, μ] f) x = ∫ (t : G), (L (f (x - t))) (g t) ∂μ
state_after: no goals
tactic: rfl
Theorem: Cardinal.nfp_lt_ord_of_isRegular
File: Mathlib/SetTheory/Cardinal/Cofinality.lean

Proof steps:
state_before: α : Type u_1
r : α → α → Prop
f : Ordinal.{u_2} → Ordinal.{u_2}
c : Cardinal.{u_2}
hc : c.IsRegular
hc' : c ≠ ℵ₀
hf : ∀ i < c.ord, f i < c.ord
a : Ordinal.{u_2}
⊢ ℵ₀ < c.ord.cof
state_after: α : Type u_1
r : α → α → Prop
f : Ordinal.{u_2} → Ordinal.{u_2}
c : Cardinal.{u_2}
hc : c.IsRegular
hc' : c ≠ ℵ₀
hf : ∀ i < c.ord, f i < c.ord
a : Ordinal.{u_2}
⊢ ℵ₀ < c
tactic: rw [hc.cof_eq]
state_before: α : Type u_1
r : α → α → Prop
f : Ordinal.{u_2} → Ordinal.{u_2}
c : Cardinal.{u_2}
hc : c.IsRegular
hc' : c ≠ ℵ₀
hf : ∀ i < c.ord, f i < c.ord
a : Ordinal.{u_2}
⊢ ℵ₀ < c
state_after: no goals
tactic: exact lt_of_le_of_ne hc.1 hc'.symm
Theorem: continuous_algebraMap_iff_smul
File: Mathlib/Topology/Algebra/Algebra.lean

Proof steps:
state_before: R : Type u_1
A : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : TopologicalSpace R
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSemiring A
⊢ Continuous ⇑(algebraMap R A) ↔ Continuous fun p => p.1 • p.2
state_after: R : Type u_1
A : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : TopologicalSpace R
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSemiring A
h : Continuous ⇑(algebraMap R A)
⊢ Continuous fun p => p.1 • p.2
tactic: refine ⟨fun h => ?_, fun h => have : ContinuousSMul R A := ⟨h⟩; continuous_algebraMap _ _⟩
state_before: R : Type u_1
A : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : TopologicalSpace R
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSemiring A
h : Continuous ⇑(algebraMap R A)
⊢ Continuous fun p => p.1 • p.2
state_after: R : Type u_1
A : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : TopologicalSpace R
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSemiring A
h : Continuous ⇑(algebraMap R A)
⊢ Continuous fun p => (algebraMap R A) p.1 * p.2
tactic: simp only [Algebra.smul_def]
state_before: R : Type u_1
A : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : TopologicalSpace R
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSemiring A
h : Continuous ⇑(algebraMap R A)
⊢ Continuous fun p => (algebraMap R A) p.1 * p.2
state_after: no goals
tactic: exact (h.comp continuous_fst).mul continuous_snd
Theorem: div_le_iff
File: Mathlib/Algebra/Order/Field/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
hb : 0 < b
h : a / b ≤ c
⊢ a = a / b * b
state_after: no goals
tactic: rw [div_mul_cancel₀ _ (ne_of_lt hb).symm]
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
hb : 0 < b
h : a ≤ c * b
⊢ c * b / b = c
state_after: no goals
tactic: refine (div_eq_iff (ne_of_gt hb)).mpr rfl
Theorem: MeasureTheory.Measure.rnDeriv_le_one_of_le
File: Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ≤ ν
inst✝ : SigmaFinite ν
⊢ μ.rnDeriv ν ≤ᶠ[ae ν] 1
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ≤ ν
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ⊤
⊢ ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν ≤ ∫⁻ (x : α) in s, 1 x ∂ν
tactic: refine ae_le_of_forall_setLIntegral_le_of_sigmaFinite (μ.measurable_rnDeriv ν) fun s _ _ ↦ ?_
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ≤ ν
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ⊤
⊢ ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν ≤ ∫⁻ (x : α) in s, 1 x ∂ν
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ≤ ν
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ⊤
⊢ ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν ≤ ν s
tactic: simp only [Pi.one_apply, MeasureTheory.setLIntegral_one]
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
hμν : μ ≤ ν
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ⊤
⊢ ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν ≤ ν s
state_after: no goals
tactic: exact (Measure.setLIntegral_rnDeriv_le s).trans (hμν s)