ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: Subgroup.map_normalClosure
File: Mathlib/Algebra/Group/Subgroup/Basic.lean

Proof steps:
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁴ : Group G
inst✝³ : Group G'
inst✝² : Group G''
A : Type u_4
inst✝¹ : AddGroup A
N : Type u_5
inst✝ : Group N
s : Set G
f : G →* N
hf : Surjective ⇑f
⊢ map f (normalClosure s) = normalClosure (⇑f '' s)
state_after: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁴ : Group G
inst✝³ : Group G'
inst✝² : Group G''
A : Type u_4
inst✝¹ : AddGroup A
N : Type u_5
inst✝ : Group N
s : Set G
f : G →* N
hf : Surjective ⇑f
this : (map f (normalClosure s)).Normal
⊢ map f (normalClosure s) = normalClosure (⇑f '' s)
tactic: have : Normal (map f (normalClosure s)) := Normal.map inferInstance f hf
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁴ : Group G
inst✝³ : Group G'
inst✝² : Group G''
A : Type u_4
inst✝¹ : AddGroup A
N : Type u_5
inst✝ : Group N
s : Set G
f : G →* N
hf : Surjective ⇑f
this : (map f (normalClosure s)).Normal
⊢ map f (normalClosure s) = normalClosure (⇑f '' s)
state_after: case a
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁴ : Group G
inst✝³ : Group G'
inst✝² : Group G''
A : Type u_4
inst✝¹ : AddGroup A
N : Type u_5
inst✝ : Group N
s : Set G
f : G →* N
hf : Surjective ⇑f
this : (map f (normalClosure s)).Normal
⊢ map f (normalClosure s) ≤ normalClosure (⇑f '' s)

case a
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁴ : Group G
inst✝³ : Group G'
inst✝² : Group G''
A : Type u_4
inst✝¹ : AddGroup A
N : Type u_5
inst✝ : Group N
s : Set G
f : G →* N
hf : Surjective ⇑f
this : (map f (normalClosure s)).Normal
⊢ normalClosure (⇑f '' s) ≤ map f (normalClosure s)
tactic: apply le_antisymm
Theorem: hasConstantSpeedOnWith_zero_iff
File: Mathlib/Analysis/ConstantSpeed.lean

Proof steps:
state_before: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ HasConstantSpeedOnWith f s 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
state_after: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (0 * (y - x))) ↔
    ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
tactic: dsimp [HasConstantSpeedOnWith]
state_before: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (0 * (y - x))) ↔
    ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
state_after: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) ↔ eVariationOn f s = 0
tactic: simp only [zero_mul, ENNReal.ofReal_zero, ← eVariationOn.eq_zero_iff]
state_before: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) ↔ eVariationOn f s = 0
state_after: case mp
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) → eVariationOn f s = 0

case mpr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ eVariationOn f s = 0 → ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0
tactic: constructor
Theorem: Matrix.invOf_submatrix_equiv_eq
File: Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean

Proof steps:
state_before: l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq n
inst✝⁴ : CommRing α
A✝ B : Matrix n n α
inst✝³ : Fintype m
inst✝² : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
inst✝¹ : Invertible A
inst✝ : Invertible (A.submatrix ⇑e₁ ⇑e₂)
⊢ ⅟(A.submatrix ⇑e₁ ⇑e₂) = (⅟A).submatrix ⇑e₂ ⇑e₁
state_after: l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq n
inst✝⁴ : CommRing α
A✝ B : Matrix n n α
inst✝³ : Fintype m
inst✝² : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
inst✝¹ : Invertible A
inst✝ : Invertible (A.submatrix ⇑e₁ ⇑e₂)
this : Invertible (A.submatrix ⇑e₁ ⇑e₂) := A.submatrixEquivInvertible e₁ e₂
⊢ ⅟(A.submatrix ⇑e₁ ⇑e₂) = (⅟A).submatrix ⇑e₂ ⇑e₁
tactic: letI := submatrixEquivInvertible A e₁ e₂
state_before: l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq n
inst✝⁴ : CommRing α
A✝ B : Matrix n n α
inst✝³ : Fintype m
inst✝² : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
inst✝¹ : Invertible A
inst✝ : Invertible (A.submatrix ⇑e₁ ⇑e₂)
this : Invertible (A.submatrix ⇑e₁ ⇑e₂) := A.submatrixEquivInvertible e₁ e₂
⊢ ⅟(A.submatrix ⇑e₁ ⇑e₂) = (⅟A).submatrix ⇑e₂ ⇑e₁
state_after: no goals
tactic: convert (rfl : ⅟ (A.submatrix e₁ e₂) = _)
Theorem: CategoryTheory.GradedObject.ιMapObjOrZero_mapMap
File: Mathlib/CategoryTheory/GradedObject.lean

Proof steps:
state_before: I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝⁵ : Category.{u_5, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j✝ : J
inst✝⁴ : X.HasMap p
inst✝³ : Y.HasMap p
inst✝² : Z.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
inst✝¹ : HasZeroMorphisms C
inst✝ : DecidableEq J
i : I
j : J
⊢ X.ιMapObjOrZero p i j ≫ mapMap φ p j = φ i ≫ Y.ιMapObjOrZero p i j
state_after: case pos
I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝⁵ : Category.{u_5, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j✝ : J
inst✝⁴ : X.HasMap p
inst✝³ : Y.HasMap p
inst✝² : Z.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
inst✝¹ : HasZeroMorphisms C
inst✝ : DecidableEq J
i : I
j : J
h : p i = j
⊢ X.ιMapObjOrZero p i j ≫ mapMap φ p j = φ i ≫ Y.ιMapObjOrZero p i j

case neg
I : Type u_1
J : Type u_2
K : Type u_3
C : Type u_4
inst✝⁵ : Category.{u_5, u_4} C
X Y Z : GradedObject I C
φ : X ⟶ Y
e : X ≅ Y
ψ : Y ⟶ Z
p : I → J
j✝ : J
inst✝⁴ : X.HasMap p
inst✝³ : Y.HasMap p
inst✝² : Z.HasMap p
q : J → K
r : I → K
hpqr : ∀ (i : I), q (p i) = r i
inst✝¹ : HasZeroMorphisms C
inst✝ : DecidableEq J
i : I
j : J
h : ¬p i = j
⊢ X.ιMapObjOrZero p i j ≫ mapMap φ p j = φ i ≫ Y.ιMapObjOrZero p i j
tactic: by_cases h : p i = j
Theorem: Cardinal.aleph0_le
File: Mathlib/SetTheory/Cardinal/Basic.lean

Proof steps:
state_before: α β : Type u
c : Cardinal.{u_1}
h : ∀ (n : ℕ), ↑n ≤ c
hn : c < ℵ₀
⊢ False
state_after: case intro
α β : Type u
n : ℕ
h : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n
hn : ↑n < ℵ₀
⊢ False
tactic: rcases lt_aleph0.1 hn with ⟨n, rfl⟩
state_before: case intro
α β : Type u
n : ℕ
h : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n
hn : ↑n < ℵ₀
⊢ False
state_after: no goals
tactic: exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1)))
Theorem: Matroid.uniqueBaseOn_base_iff
File: Mathlib/Data/Matroid/Constructions.lean

Proof steps:
state_before: α : Type u_1
M : Matroid α
E B I X R J : Set α
hIE : I ⊆ E
⊢ (uniqueBaseOn I E).Base B ↔ B = I
state_after: no goals
tactic: rw [uniqueBaseOn, base_restrict_iff', freeOn_basis'_iff, inter_eq_self_of_subset_right hIE]
Theorem: Submonoid.LocalizationMap.lift_comp
File: Mathlib/GroupTheory/MonoidLocalization/Basic.lean

Proof steps:
state_before: M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoid N
P : Type u_3
inst✝ : CommMonoid P
f : S.LocalizationMap N
g : M →* P
hg : ∀ (y : ↥S), IsUnit (g ↑y)
⊢ (f.lift hg).comp f.toMap = g
state_after: case h
M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoid N
P : Type u_3
inst✝ : CommMonoid P
f : S.LocalizationMap N
g : M →* P
hg : ∀ (y : ↥S), IsUnit (g ↑y)
x✝ : M
⊢ ((f.lift hg).comp f.toMap) x✝ = g x✝
tactic: ext
state_before: case h
M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoid N
P : Type u_3
inst✝ : CommMonoid P
f : S.LocalizationMap N
g : M →* P
hg : ∀ (y : ↥S), IsUnit (g ↑y)
x✝ : M
⊢ ((f.lift hg).comp f.toMap) x✝ = g x✝
state_after: no goals
tactic: exact f.lift_eq hg _
Theorem: Real.pi_gt_3141592
File: Mathlib/Data/Real/Pi/Bounds.lean

Proof steps:
state_before: ⊢ 3.141592 < π
state_after: no goals
tactic: pi_lower_bound
        [11482 / 8119, 7792 / 4217, 54055 / 27557, 949247 / 476920, 3310126 / 1657059,
          2635492 / 1318143, 1580265 / 790192, 1221775 / 610899, 3612247 / 1806132, 849943 / 424972]
Theorem: LinearMap.det_zero
File: Mathlib/LinearAlgebra/Determinant.lean

Proof steps:
state_before: R : Type u_1
inst✝¹² : CommRing R
M✝ : Type u_2
inst✝¹¹ : AddCommGroup M✝
inst✝¹⁰ : Module R M✝
M' : Type u_3
inst✝⁹ : AddCommGroup M'
inst✝⁸ : Module R M'
ι : Type u_4
inst✝⁷ : DecidableEq ι
inst✝⁶ : Fintype ι
e : Basis ι R M✝
A : Type u_5
inst✝⁵ : CommRing A
inst✝⁴ : Module A M✝
κ : Type u_6
inst✝³ : Fintype κ
𝕜 : Type u_7
inst✝² : Field 𝕜
M : Type u_8
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
⊢ LinearMap.det 0 = 0 ^ FiniteDimensional.finrank 𝕜 M
state_after: no goals
tactic: simp only [← zero_smul 𝕜 (1 : M →ₗ[𝕜] M), det_smul, mul_one, MonoidHom.map_one]
Theorem: Submodule.orthogonal_eq_inter
File: Mathlib/Analysis/InnerProductSpace/Orthogonal.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ Kᗮ = ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)
state_after: case a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ Kᗮ ≤ ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)

case a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v) ≤ Kᗮ
tactic: apply le_antisymm
Theorem: Function.Antiperiodic.const_add
File: Mathlib/Algebra/Periodic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → β
c c₁ c₂ x✝ : α
inst✝¹ : AddSemigroup α
inst✝ : Neg β
h : Antiperiodic f c
a x : α
⊢ (fun x => f (a + x)) (x + c) = -(fun x => f (a + x)) x
state_after: no goals
tactic: simpa [add_assoc] using h (a + x)
Theorem: Part.toOption_eq_some_iff
File: Mathlib/Data/Part.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
o : Part α
inst✝ : Decidable o.Dom
a : α
⊢ o.toOption = Option.some a ↔ a ∈ o
state_after: no goals
tactic: rw [← Option.mem_def, mem_toOption]
Theorem: Nat.Primrec'.head
File: Mathlib/Computability/Primrec.lean

Proof steps:
state_before: n : ℕ
v : Vector ℕ n.succ
⊢ v.get 0 = v.head
state_after: no goals
tactic: simp [get_zero]
Theorem: ProbabilityTheory.lintegral_stieltjesOfMeasurableRat
File: Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
⊢ ∫⁻ (b : β), ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (univ ×ˢ Iic x)
state_after: no goals
tactic: rw [← setLIntegral_univ, setLIntegral_stieltjesOfMeasurableRat hf _ _ MeasurableSet.univ]
Theorem: DualNumber.algHom_ext
File: Mathlib/Algebra/DualNumber.lean

Proof steps:
state_before: R : Type u_1
A✝ : Type u_2
B : Type u_3
A : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f g : R[ε] →ₐ[R] A
hε : f ε = g ε
⊢ f = g
state_after: case hinr.h
R : Type u_1
A✝ : Type u_2
B : Type u_3
A : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f g : R[ε] →ₐ[R] A
hε : f ε = g ε
⊢ (f.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1 =
    (g.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1
tactic: ext
state_before: case hinr.h
R : Type u_1
A✝ : Type u_2
B : Type u_3
A : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f g : R[ε] →ₐ[R] A
hε : f ε = g ε
⊢ (f.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1 =
    (g.toLinearMap ∘ₗ ↑R (LinearMap.toSpanSingleton R R[ε] ε)) 1
state_after: case hinr.h
R : Type u_1
A✝ : Type u_2
B : Type u_3
A : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f g : R[ε] →ₐ[R] A
hε : f ε = g ε
⊢ f (1 • ε) = g (1 • ε)
tactic: dsimp
state_before: case hinr.h
R : Type u_1
A✝ : Type u_2
B : Type u_3
A : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f g : R[ε] →ₐ[R] A
hε : f ε = g ε
⊢ f (1 • ε) = g (1 • ε)
state_after: no goals
tactic: simp only [one_smul, hε]
Theorem: SimpleGraph.Walk.mem_support_iff
File: Mathlib/Combinatorics/SimpleGraph/Walk.lean

Proof steps:
state_before: V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
u v w : V
p : G.Walk u v
⊢ w ∈ p.support ↔ w = u ∨ w ∈ p.support.tail
state_after: no goals
tactic: cases p <;> simp
Theorem: sdiff_bot
File: Mathlib/Order/Heyting/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedCoheytingAlgebra α
a b✝ c d b : α
⊢ a \ ⊥ ≤ b ↔ a ≤ b
state_after: no goals
tactic: rw [sdiff_le_iff, bot_sup_eq]
Theorem: Finset.Icc_diff_both
File: Mathlib/Order/Interval/Finset/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
a✝ b✝ c : α
inst✝ : DecidableEq α
a b : α
⊢ Icc a b \ {a, b} = Ioo a b
state_after: no goals
tactic: simp [← coe_inj]
Theorem: Finset.prod_piecewise
File: Mathlib/Algebra/BigOperators/Group/Finset.lean

Proof steps:
state_before: ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s✝ s₁ s₂ : Finset α
a : α
f✝ g✝ : α → β
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s t : Finset α
f g : α → β
⊢ ∏ x ∈ s, t.piecewise f g x = (∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \ t, g x
state_after: no goals
tactic: erw [prod_ite, filter_mem_eq_inter, ← sdiff_eq_filter]
Theorem: CategoryTheory.tensorLeftHomEquiv_symm_coevaluation_comp_whiskerRight
File: Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean

Proof steps:
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
X Y : C
inst✝¹ : HasRightDual X
inst✝ : HasRightDual Y
f : X ⟶ Y
⊢ (tensorLeftHomEquiv (𝟙_ C) Y Yᘁ Xᘁ).symm (η_ X Xᘁ ≫ f ▷ Xᘁ) = (ρ_ Yᘁ).hom ≫ fᘁ
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
X Y : C
inst✝¹ : HasRightDual X
inst✝ : HasRightDual Y
f : X ⟶ Y
⊢ Yᘁ ◁ (η_ X Xᘁ ≫ f ▷ Xᘁ) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom =
    (ρ_ Yᘁ).hom ≫ (ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom
tactic: dsimp [tensorLeftHomEquiv, rightAdjointMate]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
X Y : C
inst✝¹ : HasRightDual X
inst✝ : HasRightDual Y
f : X ⟶ Y
⊢ Yᘁ ◁ (η_ X Xᘁ ≫ f ▷ Xᘁ) ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom =
    (ρ_ Yᘁ).hom ≫ (ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom
state_after: no goals
tactic: simp
Theorem: CategoryTheory.OverPresheafAux.YonedaCollection.map₁_yonedaEquivFst
File: Mathlib/CategoryTheory/Comma/Presheaf.lean

Proof steps:
state_before: C : Type u
inst✝ : Category.{v, u} C
A : Cᵒᵖ ⥤ Type v
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
G : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
p : YonedaCollection F X
⊢ (map₁ η p).yonedaEquivFst = p.yonedaEquivFst
state_after: no goals
tactic: simp only [YonedaCollection.yonedaEquivFst_eq, map₁_fst]
Theorem: Nat.even_or_odd'
File: Mathlib/Algebra/Ring/Parity.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
m n✝ n : ℕ
⊢ ∃ k, n = 2 * k ∨ n = 2 * k + 1
state_after: no goals
tactic: simpa only [← two_mul, exists_or, Odd, Even] using even_or_odd n
Theorem: MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict'
File: Mathlib/MeasureTheory/Measure/Typeclasses.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν✝ ν₁ ν₂ : Measure α
s t : Set α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
P : α → Prop
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x
⊢ ∀ᵐ (x : α) ∂μ, P x
state_after: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν✝ ν₁ ν₂ : Measure α
s t : Set α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
P : α → Prop
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x
this : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x
⊢ ∀ᵐ (x : α) ∂μ, P x
tactic: have : ∀ n, ∀ᵐ x ∂μ, x ∈ spanningSets (μ + ν) n → P x := by
    intro n
    have := h
      (spanningSets (μ + ν) n) (measurable_spanningSets _ _)
      ((self_le_add_right _ _).trans_lt (measure_spanningSets_lt_top (μ + ν) _))
      ((self_le_add_left _ _).trans_lt (measure_spanningSets_lt_top (μ + ν) _))
    exact (ae_restrict_iff' (measurable_spanningSets _ _)).mp this
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ✝ ν✝ ν₁ ν₂ : Measure α
s t : Set α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
P : α → Prop
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x
this : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x
⊢ ∀ᵐ (x : α) ∂μ, P x
state_after: no goals
tactic: filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)
Theorem: ENNReal.toReal_sub_of_le
File: Mathlib/Data/ENNReal/Real.lean

Proof steps:
state_before: a✝ b✝ c d : ℝ≥0∞
r p q : ℝ≥0
a b : ℝ≥0∞
h : b ≤ a
ha : a ≠ ⊤
⊢ (a - b).toReal = a.toReal - b.toReal
state_after: case intro
a✝ b✝ c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ≥0∞
ha : a ≠ ⊤
b : ℝ≥0
h : ↑b ≤ a
⊢ (a - ↑b).toReal = a.toReal - (↑b).toReal
tactic: lift b to ℝ≥0 using ne_top_of_le_ne_top ha h
state_before: case intro
a✝ b✝ c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ≥0∞
ha : a ≠ ⊤
b : ℝ≥0
h : ↑b ≤ a
⊢ (a - ↑b).toReal = a.toReal - (↑b).toReal
state_after: case intro.intro
a✝ b✝ c d : ℝ≥0∞
r p q b a : ℝ≥0
h : ↑b ≤ ↑a
⊢ (↑a - ↑b).toReal = (↑a).toReal - (↑b).toReal
tactic: lift a to ℝ≥0 using ha
state_before: case intro.intro
a✝ b✝ c d : ℝ≥0∞
r p q b a : ℝ≥0
h : ↑b ≤ ↑a
⊢ (↑a - ↑b).toReal = (↑a).toReal - (↑b).toReal
state_after: no goals
tactic: simp only [← ENNReal.coe_sub, ENNReal.coe_toReal, NNReal.coe_sub (ENNReal.coe_le_coe.mp h)]
Theorem: List.Perm.filterMap
File: .lake/packages/batteries/Batteries/Data/List/Perm.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
f : α → Option β
l₁ l₂ : List α
p : l₁ ~ l₂
⊢ List.filterMap f l₁ ~ List.filterMap f l₂
state_after: no goals
tactic: induction p with
  | nil => simp
  | cons x _p IH => cases h : f x <;> simp [h, filterMap_cons, IH, Perm.cons]
  | swap x y l₂ => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap_cons, swap]
  | trans _p₁ _p₂ IH₁ IH₂ => exact IH₁.trans IH₂
Theorem: Function.Periodic.sub_antiperiod
File: Mathlib/Algebra/Periodic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → β
c c₁ c₂ x : α
inst✝¹ : AddGroup α
inst✝ : InvolutiveNeg β
h1 : Periodic f c₁
h2 : Antiperiodic f c₂
⊢ Antiperiodic f (c₁ - c₂)
state_after: no goals
tactic: simpa only [sub_eq_add_neg] using h1.add_antiperiod h2.neg
Theorem: FractionalIdeal.mapEquiv_refl
File: Mathlib/RingTheory/FractionalIdeal/Operations.lean

Proof steps:
state_before: R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I J : FractionalIdeal S P
g : P →ₐ[R] P'
x : FractionalIdeal S P
⊢ (mapEquiv AlgEquiv.refl) x = (RingEquiv.refl (FractionalIdeal S P)) x
state_after: no goals
tactic: simp
Theorem: MeasurableSet.measurableSet_bliminf
File: Mathlib/MeasureTheory/MeasurableSpace/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
inst✝ : MeasurableSpace α
s : ℕ → Set α
p : ℕ → Prop
h : ∀ (n : ℕ), p n → MeasurableSet (s n)
⊢ MeasurableSet (bliminf s atTop p)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
inst✝ : MeasurableSpace α
s : ℕ → Set α
p : ℕ → Prop
h : ∀ (n : ℕ), p n → MeasurableSet (s n)
⊢ MeasurableSet (⋃ i, ⋂ j, ⋂ (_ : p j ∧ i ≤ j), s j)
tactic: simp only [Filter.bliminf_eq_iSup_biInf_of_nat, iInf_eq_iInter, iSup_eq_iUnion]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t u : Set α
inst✝ : MeasurableSpace α
s : ℕ → Set α
p : ℕ → Prop
h : ∀ (n : ℕ), p n → MeasurableSet (s n)
⊢ MeasurableSet (⋃ i, ⋂ j, ⋂ (_ : p j ∧ i ≤ j), s j)
state_after: no goals
tactic: exact .iUnion fun n => .iInter fun m => .iInter fun hm => h m hm.1
Theorem: PiLp.nnnorm_eq_ciSup
File: Mathlib/Analysis/Normed/Lp/PiLp.lean

Proof steps:
state_before: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
f : PiLp ⊤ β
⊢ ‖f‖₊ = ⨆ i, ‖f i‖₊
state_after: case a
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
f : PiLp ⊤ β
⊢ ↑‖f‖₊ = ↑(⨆ i, ‖f i‖₊)
tactic: ext
state_before: case a
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
f : PiLp ⊤ β
⊢ ↑‖f‖₊ = ↑(⨆ i, ‖f i‖₊)
state_after: no goals
tactic: simp [NNReal.coe_iSup, norm_eq_ciSup]
Theorem: LinearMap.stdBasis_apply'
File: Mathlib/LinearAlgebra/StdBasis.lean

Proof steps:
state_before: R : Type u_1
ι : Type u_2
inst✝³ : Semiring R
φ : ι → Type u_3
inst✝² : (i : ι) → AddCommMonoid (φ i)
inst✝¹ : (i : ι) → Module R (φ i)
inst✝ : DecidableEq ι
i i' : ι
⊢ (stdBasis R (fun _x => R) i) 1 i' = if i = i' then 1 else 0
state_after: R : Type u_1
ι : Type u_2
inst✝³ : Semiring R
φ : ι → Type u_3
inst✝² : (i : ι) → AddCommMonoid (φ i)
inst✝¹ : (i : ι) → Module R (φ i)
inst✝ : DecidableEq ι
i i' : ι
⊢ (if i' = i then 1 else 0) = if i = i' then 1 else 0
tactic: rw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]
state_before: R : Type u_1
ι : Type u_2
inst✝³ : Semiring R
φ : ι → Type u_3
inst✝² : (i : ι) → AddCommMonoid (φ i)
inst✝¹ : (i : ι) → Module R (φ i)
inst✝ : DecidableEq ι
i i' : ι
⊢ (if i' = i then 1 else 0) = if i = i' then 1 else 0
state_after: case e_c
R : Type u_1
ι : Type u_2
inst✝³ : Semiring R
φ : ι → Type u_3
inst✝² : (i : ι) → AddCommMonoid (φ i)
inst✝¹ : (i : ι) → Module R (φ i)
inst✝ : DecidableEq ι
i i' : ι
⊢ (i' = i) = (i = i')
tactic: congr 1
state_before: case e_c
R : Type u_1
ι : Type u_2
inst✝³ : Semiring R
φ : ι → Type u_3
inst✝² : (i : ι) → AddCommMonoid (φ i)
inst✝¹ : (i : ι) → Module R (φ i)
inst✝ : DecidableEq ι
i i' : ι
⊢ (i' = i) = (i = i')
state_after: no goals
tactic: rw [eq_iff_iff, eq_comm]