ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse_obj_mul_app
File: Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean

Proof steps:
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : MonoidalCategory D
inst✝ : BraidedCategory D
F : C ⥤ CommMon_ D
⊢ (β_ __src✝.X __src✝.X).hom ≫ __src✝.mul = __src✝.mul
state_after: case w.h
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : MonoidalCategory D
inst✝ : BraidedCategory D
F : C ⥤ CommMon_ D
X : C
⊢ ((β_ __src✝.X __src✝.X).hom ≫ __src✝.mul).app X = __src✝.mul.app X
tactic: ext X
state_before: case w.h
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : MonoidalCategory D
inst✝ : BraidedCategory D
F : C ⥤ CommMon_ D
X : C
⊢ ((β_ __src✝.X __src✝.X).hom ≫ __src✝.mul).app X = __src✝.mul.app X
state_after: no goals
tactic: exact (F.obj X).mul_comm
Theorem: ProbabilityTheory.Kernel.setLIntegral_rnDerivAux
File: Mathlib/Probability/Kernel/RadonNikodym.lean

Proof steps:
state_before: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (x : γ) in s, ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) ∂(κ + η) a = (κ a) s
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
h_le : κ ≤ κ + η
⊢ ∫⁻ (x : γ) in s, ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) ∂(κ + η) a = (κ a) s
tactic: have h_le : κ ≤ κ + η := le_add_of_nonneg_right bot_le
state_before: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
h_le : κ ≤ κ + η
⊢ ∫⁻ (x : γ) in s, ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) ∂(κ + η) a = (κ a) s
state_after: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
h_le : κ ≤ κ + η
⊢ ∫⁻ (x : γ) in s,
      ENNReal.ofReal
        (if hα : Countable α then ((∂κ a/∂(κ + η) a) x).toReal
        else (κ.map (fun a => (a, ())) ⋯).density (κ + η) a x univ) ∂(κ + η) a =
    (κ a) s
tactic: simp_rw [rnDerivAux]
state_before: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
h_le : κ ≤ κ + η
⊢ ∫⁻ (x : γ) in s,
      ENNReal.ofReal
        (if hα : Countable α then ((∂κ a/∂(κ + η) a) x).toReal
        else (κ.map (fun a => (a, ())) ⋯).density (κ + η) a x univ) ∂(κ + η) a =
    (κ a) s
state_after: case pos
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
h_le : κ ≤ κ + η
hα : Countable α
⊢ ∫⁻ (x : γ) in s, ENNReal.ofReal ((∂κ a/∂(κ + η) a) x).toReal ∂(κ + η) a = (κ a) s

case neg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ✝ η✝ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
h_le : κ ≤ κ + η
hα : ¬Countable α
⊢ ∫⁻ (x : γ) in s, ENNReal.ofReal ((κ.map (fun a => (a, ())) ⋯).density (κ + η) a x univ) ∂(κ + η) a = (κ a) s
tactic: split_ifs with hα
Theorem: LinearEquiv.isUnit_det
File: Mathlib/LinearAlgebra/Determinant.lean

Proof steps:
state_before: R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
f : M ≃ₗ[R] M'
v : Basis ι R M
v' : Basis ι R M'
⊢ IsUnit ((toMatrix v v') ↑f).det
state_after: case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
f : M ≃ₗ[R] M'
v : Basis ι R M
v' : Basis ι R M'
⊢ ?B * (toMatrix v v') ↑f = 1

case B
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
f : M ≃ₗ[R] M'
v : Basis ι R M
v' : Basis ι R M'
⊢ Matrix ι ι R
tactic: apply isUnit_det_of_left_inverse
state_before: case h
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
f : M ≃ₗ[R] M'
v : Basis ι R M
v' : Basis ι R M'
⊢ ?B * (toMatrix v v') ↑f = 1

case B
R : Type u_1
inst✝⁶ : CommRing R
M : Type u_2
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
M' : Type u_3
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
f : M ≃ₗ[R] M'
v : Basis ι R M
v' : Basis ι R M'
⊢ Matrix ι ι R
state_after: no goals
tactic: simpa using (LinearMap.toMatrix_comp v v' v f.symm f).symm
Theorem: Finset.filter_mem_eq_inter
File: Mathlib/Data/Finset/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
p q : α → Prop
inst✝³ : DecidablePred p
inst✝² : DecidablePred q
s✝ t✝ : Finset α
inst✝¹ : DecidableEq α
s t : Finset α
inst✝ : (i : α) → Decidable (i ∈ t)
i : α
⊢ i ∈ filter (fun i => i ∈ t) s ↔ i ∈ s ∩ t
state_after: no goals
tactic: simp [mem_filter, mem_inter]
Theorem: LowerSet.coe_iSup₂
File: Mathlib/Order/UpperLower/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝ : LE α
S : Set (LowerSet α)
s t : LowerSet α
a : α
f : (i : ι) → κ i → LowerSet α
⊢ ↑(⨆ i, ⨆ j, f i j) = ⋃ i, ⋃ j, ↑(f i j)
state_after: no goals
tactic: simp_rw [coe_iSup]
Theorem: zero_eq_dist
File: Mathlib/Topology/MetricSpace/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
γ : Type w
inst✝ : MetricSpace γ
x y : γ
⊢ 0 = dist x y ↔ x = y
state_after: no goals
tactic: rw [eq_comm, dist_eq_zero]
Theorem: Hyperreal.infinitePos_mul_of_infinitePos_not_infinitesimal_pos
File: Mathlib/Data/Real/Hyperreal.lean

Proof steps:
state_before: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
⊢ ↑r < x * y
state_after: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
⊢ ↑r < x * y
tactic: have hy₁' := not_forall.mp (mt infinitesimal_def.2 hy₁)
state_before: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
⊢ ↑r < x * y
state_after: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
r₁ : ℝ
hy₁'' : ¬(0 < r₁ → -↑r₁ < y ∧ y < ↑r₁)
⊢ ↑r < x * y
tactic: let ⟨r₁, hy₁''⟩ := hy₁'
state_before: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
r₁ : ℝ
hy₁'' : ¬(0 < r₁ → -↑r₁ < y ∧ y < ↑r₁)
⊢ ↑r < x * y
state_after: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
r₁ : ℝ
hy₁'' : ¬(0 < r₁ → -↑r₁ < y ∧ y < ↑r₁)
hyr : 0 < r₁ ∧ ↑r₁ ≤ y
⊢ ↑r < x * y
tactic: have hyr : 0 < r₁ ∧ ↑r₁ ≤ y := by
    rwa [Classical.not_imp, ← abs_lt, not_lt, abs_of_pos hy₂] at hy₁''
state_before: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
r₁ : ℝ
hy₁'' : ¬(0 < r₁ → -↑r₁ < y ∧ y < ↑r₁)
hyr : 0 < r₁ ∧ ↑r₁ ≤ y
⊢ ↑r < x * y
state_after: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
r₁ : ℝ
hy₁'' : ¬(0 < r₁ → -↑r₁ < y ∧ y < ↑r₁)
hyr : 0 < r₁ ∧ ↑r₁ ≤ y
⊢ ↑(r / r₁) * ↑r₁ < x * y
tactic: rw [← div_mul_cancel₀ r (ne_of_gt hyr.1), coe_mul]
state_before: x y : ℝ*
hx : x.InfinitePos
hy₁ : ¬y.Infinitesimal
hy₂ : 0 < y
r : ℝ
hy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)
r₁ : ℝ
hy₁'' : ¬(0 < r₁ → -↑r₁ < y ∧ y < ↑r₁)
hyr : 0 < r₁ ∧ ↑r₁ ≤ y
⊢ ↑(r / r₁) * ↑r₁ < x * y
state_after: no goals
tactic: exact mul_lt_mul (hx (r / r₁)) hyr.2 (coe_lt_coe.2 hyr.1) (le_of_lt (hx 0))
Theorem: Subsemiring.coe_sSup_of_directedOn
File: Mathlib/Algebra/Ring/Subsemiring/Basic.lean

Proof steps:
state_before: R : Type u
S✝ : Type v
T : Type w
inst✝² : NonAssocSemiring R
M : Submonoid R
inst✝¹ : NonAssocSemiring S✝
inst✝ : NonAssocSemiring T
S : Set (Subsemiring R)
Sne : S.Nonempty
hS : DirectedOn (fun x x_1 => x ≤ x_1) S
x : R
⊢ x ∈ ↑(sSup S) ↔ x ∈ ⋃ s ∈ S, ↑s
state_after: no goals
tactic: simp [mem_sSup_of_directedOn Sne hS]
Theorem: UniformFun.hasBasis_nhds_one_of_basis
File: Mathlib/Topology/Algebra/UniformConvergence.lean

Proof steps:
state_before: α : Type u_1
G : Type u_2
ι : Type u_3
inst✝² : Group G
𝔖 : Set (Set α)
inst✝¹ : UniformSpace G
inst✝ : UniformGroup G
p : ι → Prop
b : ι → Set G
h : (𝓝 1).HasBasis p b
⊢ (𝓝 1).HasBasis p fun i => {f | ∀ (x : α), toFun f x ∈ b i}
state_after: case h.e'_5.h.h.e'_2.h.a
α : Type u_1
G : Type u_2
ι : Type u_3
inst✝² : Group G
𝔖 : Set (Set α)
inst✝¹ : UniformSpace G
inst✝ : UniformGroup G
p : ι → Prop
b : ι → Set G
h : (𝓝 1).HasBasis p b
x✝¹ : ι
x✝ : α →ᵤ G
⊢ (∀ (x : α), toFun x✝ x ∈ b x✝¹) ↔ (1, x✝) ∈ UniformFun.gen α G {x | x.2 / x.1 ∈ b x✝¹}
tactic: convert UniformFun.hasBasis_nhds_of_basis α _ (1 : α →ᵤ G) h.uniformity_of_nhds_one
state_before: case h.e'_5.h.h.e'_2.h.a
α : Type u_1
G : Type u_2
ι : Type u_3
inst✝² : Group G
𝔖 : Set (Set α)
inst✝¹ : UniformSpace G
inst✝ : UniformGroup G
p : ι → Prop
b : ι → Set G
h : (𝓝 1).HasBasis p b
x✝¹ : ι
x✝ : α →ᵤ G
⊢ (∀ (x : α), toFun x✝ x ∈ b x✝¹) ↔ (1, x✝) ∈ UniformFun.gen α G {x | x.2 / x.1 ∈ b x✝¹}
state_after: no goals
tactic: simp
Theorem: Cardinal.mk_finsupp_of_infinite
File: Mathlib/SetTheory/Cardinal/Ordinal.lean

Proof steps:
state_before: α β : Type u
inst✝² : Infinite α
inst✝¹ : Zero β
inst✝ : Nontrivial β
⊢ #(α →₀ β) = max #α #β
state_after: no goals
tactic: simp
Theorem: MeasureTheory.setLintegral_compl
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
s : Set α
hsm : MeasurableSet s
hfs : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤
⊢ ∫⁻ (x : α) in sᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ - ∫⁻ (x : α) in s, f x ∂μ
state_after: no goals
tactic: rw [← lintegral_add_compl (μ := μ) f hsm, ENNReal.add_sub_cancel_left hfs]
Theorem: Behrend.map_apply
File: Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
n d✝ k N : ℕ
x : Fin n → ℕ
d : ℕ
⊢ (fun a => ∑ i : Fin n, a i * d ^ ↑i) 0 = 0
state_after: no goals
tactic: simp_rw [Pi.zero_apply, zero_mul, sum_const_zero]
state_before: α : Type u_1
β : Type u_2
n d✝ k N : ℕ
x : Fin n → ℕ
d : ℕ
a b : Fin n → ℕ
⊢ { toFun := fun a => ∑ i : Fin n, a i * d ^ ↑i, map_zero' := ⋯ }.toFun (a + b) =
    { toFun := fun a => ∑ i : Fin n, a i * d ^ ↑i, map_zero' := ⋯ }.toFun a +
      { toFun := fun a => ∑ i : Fin n, a i * d ^ ↑i, map_zero' := ⋯ }.toFun b
state_after: no goals
tactic: simp_rw [Pi.add_apply, add_mul, sum_add_distrib]
Theorem: Orientation.oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two
File: Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean

Proof steps:
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
⊢ o.oangle y (y - x) = ↑(Real.arctan (‖x‖ / ‖y‖))
state_after: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle y (y - x)).sign = 1
⊢ o.oangle y (y - x) = ↑(Real.arctan (‖x‖ / ‖y‖))
tactic: have hs : (o.oangle y (y - x)).sign = 1 := by
    rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle y (y - x)).sign = 1
⊢ o.oangle y (y - x) = ↑(Real.arctan (‖x‖ / ‖y‖))
state_after: no goals
tactic: rw [o.oangle_eq_angle_of_sign_eq_one hs,
    InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero
      (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h) (o.right_ne_zero_of_oangle_eq_pi_div_two h)]
Theorem: Set.Iic_add_Iio_subset
File: Mathlib/Data/Set/Pointwise/Interval.lean

Proof steps:
state_before: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt
a b : α
⊢ Iic a * Iio b ⊆ Iio (a * b)
state_after: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt
a b : α
this :
  ∀ (M : Type ?u.9430) (N : Type ?u.9429) (μ : M → N → N) [inst : PartialOrder N]
    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1
⊢ Iic a * Iio b ⊆ Iio (a * b)
tactic: haveI := covariantClass_le_of_lt
state_before: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt
a b : α
this :
  ∀ (M : Type ?u.9430) (N : Type ?u.9429) (μ : M → N → N) [inst : PartialOrder N]
    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1
⊢ Iic a * Iio b ⊆ Iio (a * b)
state_after: case intro.intro.intro.intro
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt
a b : α
this :
  ∀ (M : Type ?u.9430) (N : Type ?u.9429) (μ : M → N → N) [inst : PartialOrder N]
    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1
y : α
hya : y ∈ Iic a
z : α
hzb : z ∈ Iio b
⊢ (fun x x_1 => x * x_1) y z ∈ Iio (a * b)
tactic: rintro x ⟨y, hya, z, hzb, rfl⟩
state_before: case intro.intro.intro.intro
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt
a b : α
this :
  ∀ (M : Type ?u.9430) (N : Type ?u.9429) (μ : M → N → N) [inst : PartialOrder N]
    [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1
y : α
hya : y ∈ Iic a
z : α
hzb : z ∈ Iio b
⊢ (fun x x_1 => x * x_1) y z ∈ Iio (a * b)
state_after: no goals
tactic: exact mul_lt_mul_of_le_of_lt hya hzb
Theorem: continuous_pow
File: Mathlib/Topology/Algebra/Monoid.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
⊢ Continuous fun a => a ^ 0
state_after: no goals
tactic: simpa using continuous_const
state_before: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
k : ℕ
⊢ Continuous fun a => a ^ (k + 1)
state_after: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
k : ℕ
⊢ Continuous fun a => a * a ^ k
tactic: simp only [pow_succ']
state_before: ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
k : ℕ
⊢ Continuous fun a => a * a ^ k
state_after: no goals
tactic: exact continuous_id.mul (continuous_pow _)
Theorem: det_ne_zero_of_sum_col_lt_diag
File: Mathlib/LinearAlgebra/Matrix/Gershgorin.lean

Proof steps:
state_before: K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : ∀ (k : n), ∑ i ∈ Finset.univ.erase k, ‖A i k‖ < ‖A k k‖
⊢ A.det ≠ 0
state_after: K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : ∀ (k : n), ∑ i ∈ Finset.univ.erase k, ‖A i k‖ < ‖A k k‖
⊢ A.transpose.det ≠ 0
tactic: rw [← Matrix.det_transpose]
state_before: K : Type u_1
n : Type u_2
inst✝² : NormedField K
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n K
h : ∀ (k : n), ∑ i ∈ Finset.univ.erase k, ‖A i k‖ < ‖A k k‖
⊢ A.transpose.det ≠ 0
state_after: no goals
tactic: exact det_ne_zero_of_sum_row_lt_diag (by simp_rw [Matrix.transpose_apply]; exact h)
Theorem: Monoid.exponent_pi_eq_zero
File: Mathlib/GroupTheory/Exponent.lean

Proof steps:
state_before: G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : exponent (M j) = 0
⊢ exponent ((i : ι) → M i) = 0
state_after: G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ¬∃ n, 0 < n ∧ ∀ (g : M j), g ^ n = 1
⊢ ¬∃ n, 0 < n ∧ ∀ (g : (i : ι) → M i), g ^ n = 1
tactic: rw [@exponent_eq_zero_iff, ExponentExists] at hj ⊢
state_before: G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ¬∃ n, 0 < n ∧ ∀ (g : M j), g ^ n = 1
⊢ ¬∃ n, 0 < n ∧ ∀ (g : (i : ι) → M i), g ^ n = 1
state_after: G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
⊢ ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
tactic: push_neg at hj ⊢
state_before: G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
⊢ ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
state_after: case h.h
G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
n : ℕ
hn : 0 < n
this : ∃ g, g ^ n ≠ 1
⊢ ∃ g, g ^ n ≠ 1
tactic: peel hj with n hn _
state_before: case h.h
G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
n : ℕ
hn : 0 < n
this : ∃ g, g ^ n ≠ 1
⊢ ∃ g, g ^ n ≠ 1
state_after: case h.h.intro
G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
n : ℕ
hn : 0 < n
m : M j
hm : m ^ n ≠ 1
⊢ ∃ g, g ^ n ≠ 1
tactic: obtain ⟨m, hm⟩ := this
state_before: case h.h.intro
G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
n : ℕ
hn : 0 < n
m : M j
hm : m ^ n ≠ 1
⊢ ∃ g, g ^ n ≠ 1
state_after: case h.h.intro
G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
n : ℕ
hn : 0 < n
m : M j
hm : m ^ n ≠ 1
h : Pi.mulSingle j m ^ n = 1
⊢ m ^ n = 1
tactic: refine ⟨Pi.mulSingle j m, fun h ↦ hm ?_⟩
state_before: case h.h.intro
G : Type u
ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
j : ι
hj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1
n : ℕ
hn : 0 < n
m : M j
hm : m ^ n ≠ 1
h : Pi.mulSingle j m ^ n = 1
⊢ m ^ n = 1
state_after: no goals
tactic: simpa using congr_fun h j
Theorem: AddCircle.coe_image_Ioc_eq
File: Mathlib/Topology/Instances/AddCircle.lean

Proof steps:
state_before: 𝕜 : Type u_1
B : Type u_2
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
p : 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
⊢ QuotientAddGroup.mk '' Ioc a (a + p) = univ
state_after: 𝕜 : Type u_1
B : Type u_2
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
p : 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
⊢ (range fun x => ↑↑x) = univ
tactic: rw [image_eq_range]
state_before: 𝕜 : Type u_1
B : Type u_2
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
p : 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
⊢ (range fun x => ↑↑x) = univ
state_after: no goals
tactic: exact (equivIoc p a).symm.range_eq_univ
Theorem: Cardinal.ord_nat
File: Mathlib/SetTheory/Ordinal/Basic.lean

Proof steps:
state_before: α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
n : ℕ
⊢ ↑n ≤ (↑n).ord
state_after: case zero
α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
⊢ ↑0 ≤ (↑0).ord

case succ
α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
n : ℕ
IH : ↑n ≤ (↑n).ord
⊢ ↑(n + 1) ≤ (↑(n + 1)).ord
tactic: induction' n with n IH
Theorem: Monotone.ne_of_lt_of_lt_int
File: Mathlib/Order/Monotone/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
δ : Type u_2
π : ι → Type u_3
r : α → α → Prop
inst✝ : Preorder α
f : ℤ → α
hf : Monotone f
n : ℤ
x : α
h1 : f n < x
h2 : x < f (n + 1)
a : ℤ
⊢ f a ≠ x
state_after: ι : Type u_1
α : Type u
β : Type v
γ : Type w
δ : Type u_2
π : ι → Type u_3
r : α → α → Prop
inst✝ : Preorder α
f : ℤ → α
hf : Monotone f
n a : ℤ
h1 : f n < f a
h2 : f a < f (n + 1)
⊢ False
tactic: rintro rfl
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
δ : Type u_2
π : ι → Type u_3
r : α → α → Prop
inst✝ : Preorder α
f : ℤ → α
hf : Monotone f
n a : ℤ
h1 : f n < f a
h2 : f a < f (n + 1)
⊢ False
state_after: no goals
tactic: exact (hf.reflect_lt h1).not_le (Int.le_of_lt_add_one <| hf.reflect_lt h2)
Theorem: QuadraticMap.associated_isOrtho
File: Mathlib/LinearAlgebra/QuadraticForm/Basic.lean

Proof steps:
state_before: S : Type u_1
T : Type u_2
R : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
A : Type u_7
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
Q : QuadraticMap R M N
inst✝ : Invertible 2
x y : M
⊢ LinearMap.IsOrtho (associated Q) x y ↔ Q.IsOrtho x y
state_after: no goals
tactic: simp_rw [isOrtho_def, LinearMap.isOrtho_def, associated_apply, invOf_smul_eq_iff,
    smul_zero, sub_sub, sub_eq_zero]
Theorem: CategoryTheory.Abelian.imageStrongEpiMonoFactorisation_I
File: Mathlib/CategoryTheory/Abelian/Basic.lean

Proof steps:
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
P Q : C
f : P ⟶ Q
⊢ Mono (image.ι f)
state_after: no goals
tactic: infer_instance
Theorem: CategoryTheory.Functor.uncurry_obj_injective
File: Mathlib/CategoryTheory/Functor/Currying.lean

Proof steps:
state_before: B : Type u₁
inst✝⁴ : Category.{v₁, u₁} B
C : Type u₂
inst✝³ : Category.{v₂, u₂} C
D : Type u₃
inst✝² : Category.{v₃, u₃} D
E : Type u₄
inst✝¹ : Category.{v₄, u₄} E
H : Type u₅
inst✝ : Category.{v₅, u₅} H
F₁ F₂ : B ⥤ C ⥤ D
h : uncurry.obj F₁ = uncurry.obj F₂
⊢ F₁ = F₂
state_after: no goals
tactic: rw [← curry_obj_uncurry_obj F₁, ← curry_obj_uncurry_obj F₂, h]