ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: Finsupp.sum_ite_self_eq
File: Mathlib/Algebra/BigOperators/Finsupp.lean

Proof steps:
state_before: α : Type u_1
ι : Type u_2
γ : Type u_3
A : Type u_4
B : Type u_5
C : Type u_6
inst✝⁷ : AddCommMonoid A
inst✝⁶ : AddCommMonoid B
inst✝⁵ : AddCommMonoid C
t : ι → A → C
h0 : ∀ (i : ι), t i 0 = 0
h1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y
s : Finset α
f✝ : α → ι →₀ A
i : ι
g : ι →₀ A
k : ι → A → γ → B
x : γ
β : Type u_7
M : Type u_8
M' : Type u_9
N✝ : Type u_10
P : Type u_11
G : Type u_12
H : Type u_13
R : Type u_14
S : Type u_15
inst✝⁴ : Zero M
inst✝³ : Zero M'
inst✝² : CommMonoid N✝
inst✝¹ : DecidableEq α
N : Type u_16
inst✝ : AddCommMonoid N
f : α →₀ N
a : α
⊢ (f.sum fun x v => if a = x then v else 0) = f a
state_after: no goals
tactic: classical
    convert f.sum_ite_eq a fun _ => id
    simp [ite_eq_right_iff.2 Eq.symm]
Theorem: tprod_eq_finprod
File: Mathlib/Topology/Algebra/InfiniteSum/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f g : β → α
a a₁ a₂ : α
hf : (mulSupport f).Finite
⊢ ∏' (b : β), f b = ∏ᶠ (b : β), f b
state_after: no goals
tactic: simp [tprod_def, multipliable_of_finite_mulSupport hf, hf]
Theorem: Finset.supIndep_univ_bool
File: Mathlib/Order/SupIndep.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
ι' : Type u_4
inst✝¹ : Lattice α
inst✝ : OrderBot α
s t : Finset ι
f✝ : ι → α
i : ι
f : Bool → α
⊢ true ≠ false
state_after: no goals
tactic: simp only [Ne, not_false_iff]
Theorem: support_comp_inv_smul₀
File: Mathlib/Data/Set/Pointwise/Support.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : Zero γ
c : α
hc : c ≠ 0
f : β → γ
⊢ (support fun x => f (c⁻¹ • x)) = c • support f
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : Zero γ
c : α
hc : c ≠ 0
f : β → γ
x : β
⊢ (x ∈ support fun x => f (c⁻¹ • x)) ↔ x ∈ c • support f
tactic: ext x
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : Zero γ
c : α
hc : c ≠ 0
f : β → γ
x : β
⊢ (x ∈ support fun x => f (c⁻¹ • x)) ↔ x ∈ c • support f
state_after: no goals
tactic: simp only [mem_smul_set_iff_inv_smul_mem₀ hc, mem_support]
Theorem: Cycle.induction_on
File: Mathlib/Data/List/Cycle.lean

Proof steps:
state_before: α : Type u_1
C : Cycle α → Prop
s : Cycle α
H0 : C nil
HI : ∀ (a : α) (l : List α), C ↑l → C ↑(a :: l)
l : List α
⊢ C (Quotient.mk'' l)
state_after: case refine_1
α : Type u_1
C : Cycle α → Prop
s : Cycle α
H0 : C nil
HI : ∀ (a : α) (l : List α), C ↑l → C ↑(a :: l)
l : List α
⊢ C nil

case refine_2
α : Type u_1
C : Cycle α → Prop
s : Cycle α
H0 : C nil
HI : ∀ (a : α) (l : List α), C ↑l → C ↑(a :: l)
l : List α
⊢ ∀ (head : α) (tail : List α), C ↑tail → C ↑(head :: tail)
tactic: refine List.recOn l ?_ ?_ <;> simp
state_before: case refine_1
α : Type u_1
C : Cycle α → Prop
s : Cycle α
H0 : C nil
HI : ∀ (a : α) (l : List α), C ↑l → C ↑(a :: l)
l : List α
⊢ C nil

case refine_2
α : Type u_1
C : Cycle α → Prop
s : Cycle α
H0 : C nil
HI : ∀ (a : α) (l : List α), C ↑l → C ↑(a :: l)
l : List α
⊢ ∀ (head : α) (tail : List α), C ↑tail → C ↑(head :: tail)
state_after: no goals
tactic: assumption'
Theorem: Surreal.Multiplication.mulOption_lt
File: Mathlib/SetTheory/Surreal/Multiplication.lean

Proof steps:
state_before: x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
⊢ ⟦x.mulOption y i k⟧ < -⟦x.mulOption (-y) j l⟧
state_after: case inl
x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i < x.moveLeft j
⊢ ⟦x.mulOption y i k⟧ < -⟦x.mulOption (-y) j l⟧

case inr.inl
x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i ≈ x.moveLeft j
⊢ ⟦x.mulOption y i k⟧ < -⟦x.mulOption (-y) j l⟧

case inr.inr
x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
⊢ ⟦x.mulOption y i k⟧ < -⟦x.mulOption (-y) j l⟧
tactic: obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)
Theorem: Units.mk0_mul
File: Mathlib/Algebra/GroupWithZero/Units/Basic.lean

Proof steps:
state_before: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n : ℕ
x y : G₀
hxy : x * y ≠ 0
⊢ mk0 (x * y) hxy = mk0 x ⋯ * mk0 y ⋯
state_after: case a
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n : ℕ
x y : G₀
hxy : x * y ≠ 0
⊢ ↑(mk0 (x * y) hxy) = ↑(mk0 x ⋯ * mk0 y ⋯)
tactic: ext
state_before: case a
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n : ℕ
x y : G₀
hxy : x * y ≠ 0
⊢ ↑(mk0 (x * y) hxy) = ↑(mk0 x ⋯ * mk0 y ⋯)
state_after: no goals
tactic: rfl
Theorem: DFinsupp.toFinsupp_single
File: Mathlib/Data/Finsupp/ToDFinsupp.lean

Proof steps:
state_before: ι : Type u_1
R : Type u_2
M : Type u_3
inst✝² : DecidableEq ι
inst✝¹ : Zero M
inst✝ : (m : M) → Decidable (m ≠ 0)
i : ι
m : M
⊢ (single i m).toFinsupp = Finsupp.single i m
state_after: case h
ι : Type u_1
R : Type u_2
M : Type u_3
inst✝² : DecidableEq ι
inst✝¹ : Zero M
inst✝ : (m : M) → Decidable (m ≠ 0)
i : ι
m : M
a✝ : ι
⊢ (single i m).toFinsupp a✝ = (Finsupp.single i m) a✝
tactic: ext
state_before: case h
ι : Type u_1
R : Type u_2
M : Type u_3
inst✝² : DecidableEq ι
inst✝¹ : Zero M
inst✝ : (m : M) → Decidable (m ≠ 0)
i : ι
m : M
a✝ : ι
⊢ (single i m).toFinsupp a✝ = (Finsupp.single i m) a✝
state_after: no goals
tactic: simp [Finsupp.single_apply, DFinsupp.single_apply]
Theorem: CategoryTheory.Limits.isCokernelEpiComp_desc
File: Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean

Proof steps:
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
c : CokernelCofork f
i : IsColimit c
W : C
g : W ⟶ X
hg : Epi g
h : W ⟶ Y
hh : h = g ≫ f
s : CokernelCofork h
⊢ f ≫ Cofork.π s = 0 ≫ Cofork.π s
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
c : CokernelCofork f
i : IsColimit c
W : C
g : W ⟶ X
hg : Epi g
h : W ⟶ Y
hh : h = g ≫ f
s : CokernelCofork h
⊢ h ≫ Cofork.π s = g ≫ 0 ≫ Cofork.π s
tactic: rw [← cancel_epi g, ← Category.assoc, ← hh]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
X Y : C
f : X ⟶ Y
c : CokernelCofork f
i : IsColimit c
W : C
g : W ⟶ X
hg : Epi g
h : W ⟶ Y
hh : h = g ≫ f
s : CokernelCofork h
⊢ h ≫ Cofork.π s = g ≫ 0 ≫ Cofork.π s
state_after: no goals
tactic: simp
Theorem: CategoryTheory.Square.toArrowArrowFunctor_obj_right_left
File: Mathlib/CategoryTheory/Square.lean

Proof steps:
state_before: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : Square C
φ : X✝ ⟶ Y✝
⊢ Arrow.homMk ⋯ ≫ ((fun sq => Arrow.mk (Arrow.homMk ⋯)) Y✝).hom =
    ((fun sq => Arrow.mk (Arrow.homMk ⋯)) X✝).hom ≫ Arrow.homMk ⋯
state_after: no goals
tactic: aesop_cat
Theorem: uniqueDiffOn_extChartAt_target
File: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
x : M
⊢ UniqueDiffOn 𝕜 (extChartAt I x).target
state_after: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
x : M
⊢ UniqueDiffOn 𝕜 (↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I)
tactic: rw [extChartAt_target]
state_before: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
x : M
⊢ UniqueDiffOn 𝕜 (↑I.symm ⁻¹' (chartAt H x).target ∩ range ↑I)
state_after: no goals
tactic: exact I.unique_diff_preimage (chartAt H x).open_target
Theorem: MulEquiv.op_symm_apply_symm_apply
File: Mathlib/Algebra/Group/Opposite.lean

Proof steps:
state_before: α✝ : Type u_1
α : Type ?u.39365
β : Type ?u.39368
inst✝¹ : Mul α
inst✝ : Mul β
f : αᵐᵒᵖ ≃* βᵐᵒᵖ
x : α
⊢ (unop ∘ ⇑f.symm ∘ MulOpposite.op) ((unop ∘ ⇑f ∘ MulOpposite.op) x) = x
state_after: no goals
tactic: simp
state_before: α✝ : Type u_1
α : Type ?u.39365
β : Type ?u.39368
inst✝¹ : Mul α
inst✝ : Mul β
f : αᵐᵒᵖ ≃* βᵐᵒᵖ
x : β
⊢ (unop ∘ ⇑f ∘ MulOpposite.op) ((unop ∘ ⇑f.symm ∘ MulOpposite.op) x) = x
state_after: no goals
tactic: simp
Theorem: Multiset.mem_Iio
File: Mathlib/Order/Interval/Multiset.lean

Proof steps:
state_before: α : Type u_1
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrderBot α
b x : α
⊢ x ∈ Iio b ↔ x < b
state_after: no goals
tactic: rw [Iio, ← Finset.mem_def, Finset.mem_Iio]
Theorem: MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre
File: Mathlib/MeasureTheory/Measure/Hausdorff.lean

Proof steps:
state_before: ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m✝ : Set X → ℝ≥0∞
r : ℝ≥0∞
μ : OuterMeasure X
s✝ : Set X
m : Set X → ℝ≥0∞
s : Set X
⊢ Tendsto (fun r => (pre m r) s) (𝓝[>] 0) (𝓝 ((mkMetric' m) s))
state_after: ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m✝ : Set X → ℝ≥0∞
r : ℝ≥0∞
μ : OuterMeasure X
s✝ : Set X
m : Set X → ℝ≥0∞
s : Set X
⊢ Tendsto ((fun r => (pre m r) s) ∘ Subtype.val) atBot (𝓝 ((mkMetric' m) s))
tactic: rw [← map_coe_Ioi_atBot, tendsto_map'_iff]
state_before: ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m✝ : Set X → ℝ≥0∞
r : ℝ≥0∞
μ : OuterMeasure X
s✝ : Set X
m : Set X → ℝ≥0∞
s : Set X
⊢ Tendsto ((fun r => (pre m r) s) ∘ Subtype.val) atBot (𝓝 ((mkMetric' m) s))
state_after: ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m✝ : Set X → ℝ≥0∞
r : ℝ≥0∞
μ : OuterMeasure X
s✝ : Set X
m : Set X → ℝ≥0∞
s : Set X
⊢ Tendsto ((fun r => (pre m r) s) ∘ Subtype.val) atBot (𝓝 (⨆ i, (pre m ↑i) s))
tactic: simp only [mkMetric', OuterMeasure.iSup_apply, iSup_subtype']
state_before: ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m✝ : Set X → ℝ≥0∞
r : ℝ≥0∞
μ : OuterMeasure X
s✝ : Set X
m : Set X → ℝ≥0∞
s : Set X
⊢ Tendsto ((fun r => (pre m r) s) ∘ Subtype.val) atBot (𝓝 (⨆ i, (pre m ↑i) s))
state_after: no goals
tactic: exact tendsto_atBot_iSup fun r r' hr => mono_pre _ hr _
Theorem: MulAction.orbitRel.Quotient.subgroup_quotient_eq_iff
File: Mathlib/GroupTheory/GroupAction/Basic.lean

Proof steps:
state_before: G : Type u_1
α : Type u_2
β : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : MulAction G β
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
⊢ ⟦a⟧ = ⟦b⟧ ↔ ⟦↑a⟧ = ⟦↑b⟧
state_after: G : Type u_1
α : Type u_2
β : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : MulAction G β
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
⊢ Setoid.r a b ↔ Setoid.r ↑a ↑b
tactic: simp_rw [← @Quotient.mk''_eq_mk, Quotient.eq'']
state_before: G : Type u_1
α : Type u_2
β : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : MulAction G β
H : Subgroup G
x : Quotient G α
a b : ↑x.orbit
⊢ Setoid.r a b ↔ Setoid.r ↑a ↑b
state_after: no goals
tactic: exact orbitRel.Quotient.mem_subgroup_orbit_iff.symm
Theorem: ExistsContDiffBumpBase.u_int_pos
File: Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean

Proof steps:
state_before: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < ∫ (x : E), u x ∂μ
state_after: case refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ Integrable u μ

case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < μ (support u)
tactic: refine (integral_pos_iff_support_of_nonneg u_nonneg ?_).mpr ?_
Theorem: PiTensorProduct.lift_comp_reindex_symm
File: Mathlib/LinearAlgebra/PiTensorProduct.lean

Proof steps:
state_before: ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝⁷ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁶ : (i : ι) → AddCommMonoid (s i)
inst✝⁵ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
E : Type u_9
inst✝² : AddCommMonoid E
inst✝¹ : Module R E
F : Type u_10
inst✝ : AddCommMonoid F
e : ι ≃ ι₂
φ : MultilinearMap R s E
⊢ lift φ ∘ₗ ↑(reindex R s e).symm = lift ((domDomCongrLinearEquiv' R R s E e) φ)
state_after: case H.H
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝⁷ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁶ : (i : ι) → AddCommMonoid (s i)
inst✝⁵ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
E : Type u_9
inst✝² : AddCommMonoid E
inst✝¹ : Module R E
F : Type u_10
inst✝ : AddCommMonoid F
e : ι ≃ ι₂
φ : MultilinearMap R s E
x✝ : (i : ι₂) → s (e.symm i)
⊢ ((lift φ ∘ₗ ↑(reindex R s e).symm).compMultilinearMap (tprod R)) x✝ =
    ((lift ((domDomCongrLinearEquiv' R R s E e) φ)).compMultilinearMap (tprod R)) x✝
tactic: ext
state_before: case H.H
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝⁷ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁶ : (i : ι) → AddCommMonoid (s i)
inst✝⁵ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
E : Type u_9
inst✝² : AddCommMonoid E
inst✝¹ : Module R E
F : Type u_10
inst✝ : AddCommMonoid F
e : ι ≃ ι₂
φ : MultilinearMap R s E
x✝ : (i : ι₂) → s (e.symm i)
⊢ ((lift φ ∘ₗ ↑(reindex R s e).symm).compMultilinearMap (tprod R)) x✝ =
    ((lift ((domDomCongrLinearEquiv' R R s E e) φ)).compMultilinearMap (tprod R)) x✝
state_after: no goals
tactic: simp [reindex]
Theorem: Finset.iSup_insert
File: Mathlib/Data/Finset/Lattice.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝¹ : CompleteLattice β
inst✝ : DecidableEq α
a : α
s : Finset α
t : α → β
⊢ ⨆ x ∈ insert a s, t x = t a ⊔ ⨆ x ∈ s, t x
state_after: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝¹ : CompleteLattice β
inst✝ : DecidableEq α
a : α
s : Finset α
t : α → β
⊢ ⨆ x ∈ {a} ∪ s, t x = t a ⊔ ⨆ x ∈ s, t x
tactic: rw [insert_eq]
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝¹ : CompleteLattice β
inst✝ : DecidableEq α
a : α
s : Finset α
t : α → β
⊢ ⨆ x ∈ {a} ∪ s, t x = t a ⊔ ⨆ x ∈ s, t x
state_after: no goals
tactic: simp only [iSup_union, Finset.iSup_singleton]
Theorem: mem_own_rightCoset
File: Mathlib/GroupTheory/Coset.lean

Proof steps:
state_before: α : Type u_1
inst✝ : Monoid α
s : Submonoid α
a : α
this : 1 * a ∈ op a • ↑s
⊢ a ∈ op a • ↑s
state_after: no goals
tactic: simpa
Theorem: List.sublist_or_mem_of_sublist
File: .lake/packages/batteries/Batteries/Data/List/Lemmas.lean

Proof steps:
state_before: α✝ : Type u_1
l l₁ : List α✝
a : α✝
l₂ : List α✝
h : l <+ l₁ ++ a :: l₂
⊢ l <+ l₁ ++ l₂ ∨ a ∈ l
state_after: no goals
tactic: induction l₁ generalizing l with
  | nil => match h with
    | .cons _ h => exact .inl h
    | .cons₂ _ h => exact .inr (.head ..)
  | cons b l₁ IH =>
    match h with
    | .cons _ h => exact (IH h).imp_left (Sublist.cons _)
    | .cons₂ _ h => exact (IH h).imp (Sublist.cons₂ _) (.tail _)
Theorem: NNReal.tsum_eq_add_tsum_ite
File: Mathlib/Topology/Instances/ENNReal.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
f : α → ℝ≥0
hf : Summable f
i : α
⊢ ∑' (x : α), f x = f i + ∑' (x : α), if x = i then 0 else f x
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
f : α → ℝ≥0
hf : Summable f
i i' : α
⊢ update f i 0 i' ≤ f i'
tactic: refine tsum_eq_add_tsum_ite' i (NNReal.summable_of_le (fun i' => ?_) hf)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
f : α → ℝ≥0
hf : Summable f
i i' : α
⊢ update f i 0 i' ≤ f i'
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
f : α → ℝ≥0
hf : Summable f
i i' : α
⊢ (if i' = i then 0 else f i') ≤ f i'
tactic: rw [Function.update_apply]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
f : α → ℝ≥0
hf : Summable f
i i' : α
⊢ (if i' = i then 0 else f i') ≤ f i'
state_after: no goals
tactic: split_ifs <;> simp only [zero_le', le_rfl]
Theorem: CategoryTheory.Limits.Bicone.π_of_isColimit
File: Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean

Proof steps:
state_before: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
F f : J → C
t : Bicone f
ht : IsColimit t.toCocone
j : J
j' : Discrete J
⊢ t.toCocone.ι.app j' ≫ t.π j =
    t.toCocone.ι.app j' ≫ ht.desc (Cofan.mk (f j) fun j' => if h : j' = j then eqToHom ⋯ else 0)
state_after: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
F f : J → C
t : Bicone f
ht : IsColimit t.toCocone
j : J
j' : Discrete J
⊢ t.toCocone.ι.app j' ≫ t.π j = (Cofan.mk (f j) fun j' => if h : j' = j then eqToHom ⋯ else 0).ι.app j'
tactic: rw [ht.fac]
state_before: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
F f : J → C
t : Bicone f
ht : IsColimit t.toCocone
j : J
j' : Discrete J
⊢ t.toCocone.ι.app j' ≫ t.π j = (Cofan.mk (f j) fun j' => if h : j' = j then eqToHom ⋯ else 0).ι.app j'
state_after: no goals
tactic: simp [t.ι_π]
Theorem: isPathConnected_range
File: Mathlib/Topology/Connected/PathConnected.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : PathConnectedSpace X
f : X → Y
hf : Continuous f
⊢ IsPathConnected (range f)
state_after: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : PathConnectedSpace X
f : X → Y
hf : Continuous f
⊢ IsPathConnected (f '' univ)
tactic: rw [← image_univ]
state_before: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : PathConnectedSpace X
f : X → Y
hf : Continuous f
⊢ IsPathConnected (f '' univ)
state_after: no goals
tactic: exact isPathConnected_univ.image hf
Theorem: Set.einfsep_pos
File: Mathlib/Topology/MetricSpace/Infsep.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝ : EDist α
x y : α
s t : Set α
⊢ 0 < s.einfsep ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y
state_after: α : Type u_1
β : Type u_2
inst✝ : EDist α
x y : α
s t : Set α
⊢ (¬∀ C > 0, ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < C) ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y
tactic: rw [pos_iff_ne_zero, Ne, einfsep_zero]
state_before: α : Type u_1
β : Type u_2
inst✝ : EDist α
x y : α
s t : Set α
⊢ (¬∀ C > 0, ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < C) ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y
state_after: no goals
tactic: simp only [not_forall, not_exists, not_lt, exists_prop, not_and]
Theorem: AlgebraicGeometry.Scheme.inv_app_top
File: Mathlib/AlgebraicGeometry/Scheme.lean

Proof steps:
state_before: X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
⊢ Hom.app (inv f) ⊤ = inv (Hom.app f ⊤)
state_after: no goals
tactic: simp
Theorem: FirstOrder.Language.DirectLimit.unify_sigma_mk_self
File: Mathlib/ModelTheory/DirectLimit.lean

Proof steps:
state_before: L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
⊢ unify f (fun a => Structure.Sigma.mk f i (x a)) i ⋯ = x
state_after: case h
L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
a : α
⊢ unify f (fun a => Structure.Sigma.mk f i (x a)) i ⋯ a = x a
tactic: ext a
state_before: case h
L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
a : α
⊢ unify f (fun a => Structure.Sigma.mk f i (x a)) i ⋯ a = x a
state_after: case h
L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
a : α
⊢ (f (Structure.Sigma.mk f i (x a)).fst i ⋯) (Structure.Sigma.mk f i (x a)).snd = x a
tactic: rw [unify]
state_before: case h
L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
a : α
⊢ (f (Structure.Sigma.mk f i (x a)).fst i ⋯) (Structure.Sigma.mk f i (x a)).snd = x a
state_after: no goals
tactic: apply DirectedSystem.map_self
Theorem: _private.Mathlib.Data.Complex.Exponential.0.Complex.sinh_add_aux
File: Mathlib/Data/Complex/Exponential.lean

Proof steps:
state_before: x y a b c d : ℂ
⊢ (a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d)
state_after: no goals
tactic: ring
Theorem: Finset.disjoint_sup_left
File: Mathlib/Data/Finset/Lattice.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝¹ : DistribLattice α
inst✝ : OrderBot α
s : Finset ι
t : Finset κ
f : ι → α
g : κ → α
a : α
⊢ _root_.Disjoint (s.sup f) a ↔ ∀ ⦃i : ι⦄, i ∈ s → _root_.Disjoint (f i) a
state_after: no goals
tactic: simp only [disjoint_iff, sup_inf_distrib_right, Finset.sup_eq_bot_iff]