ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: NonUnitalStarAlgebra.mem_sup_right
File: Mathlib/Algebra/Star/NonUnitalSubalgebra.lean

Proof steps:
state_before: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : NonUnitalSemiring A
inst✝¹³ : StarRing A
inst✝¹² : Module R A
inst✝¹¹ : IsScalarTower R A A
inst✝¹⁰ : SMulCommClass R A A
inst✝⁹ : StarModule R A
inst✝⁸ : NonUnitalSemiring B
inst✝⁷ : StarRing B
inst✝⁶ : Module R B
inst✝⁵ : IsScalarTower R B B
inst✝⁴ : SMulCommClass R B B
inst✝³ : StarModule R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T
state_after: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : NonUnitalSemiring A
inst✝¹³ : StarRing A
inst✝¹² : Module R A
inst✝¹¹ : IsScalarTower R A A
inst✝¹⁰ : SMulCommClass R A A
inst✝⁹ : StarModule R A
inst✝⁸ : NonUnitalSemiring B
inst✝⁷ : StarRing B
inst✝⁶ : Module R B
inst✝⁵ : IsScalarTower R B B
inst✝⁴ : SMulCommClass R B B
inst✝³ : StarModule R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ T ≤ S ⊔ T
tactic: rw [← SetLike.le_def]
state_before: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : NonUnitalSemiring A
inst✝¹³ : StarRing A
inst✝¹² : Module R A
inst✝¹¹ : IsScalarTower R A A
inst✝¹⁰ : SMulCommClass R A A
inst✝⁹ : StarModule R A
inst✝⁸ : NonUnitalSemiring B
inst✝⁷ : StarRing B
inst✝⁶ : Module R B
inst✝⁵ : IsScalarTower R B B
inst✝⁴ : SMulCommClass R B B
inst✝³ : StarModule R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : NonUnitalStarAlgHomClass F R A B
S T : NonUnitalStarSubalgebra R A
⊢ T ≤ S ⊔ T
state_after: no goals
tactic: exact le_sup_right
Theorem: AddGroup.ext
File: Mathlib/Algebra/Group/Ext.lean

Proof steps:
state_before: G : Type u_1
g₁ g₂ : Group G
h_mul : Mul.mul = Mul.mul
h₁ : One.one = One.one
⊢ g₁ = g₂
state_after: G : Type u_1
g₁ g₂ : Group G
h_mul : Mul.mul = Mul.mul
h₁ : One.one = One.one
f : G →* G := { toFun := id, map_one' := h₁, map_mul' := ⋯ }
⊢ g₁ = g₂
tactic: let f : @MonoidHom G G g₁.toMulOneClass g₂.toMulOneClass :=
    @MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h₁)
      (fun x y => congr_fun (congr_fun h_mul x) y)
state_before: G : Type u_1
g₁ g₂ : Group G
h_mul : Mul.mul = Mul.mul
h₁ : One.one = One.one
f : G →* G := { toFun := id, map_one' := h₁, map_mul' := ⋯ }
⊢ g₁ = g₂
state_after: no goals
tactic: exact
    Group.toDivInvMonoid_injective
      (DivInvMonoid.ext h_mul
        (funext <| @MonoidHom.map_inv G G g₁ g₂.toDivisionMonoid f))
Theorem: Matrix.frobeniusBoundedSMul
File: Mathlib/Analysis/Matrix.lean

Proof steps:
state_before: R : Type u_1
l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : Fintype l
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Unique ι
inst✝³ : SeminormedRing R
inst✝² : SeminormedAddCommGroup α
inst✝¹ : Module R α
inst✝ : BoundedSMul R α
⊢ BoundedSMul R (PiLp 2 fun i => PiLp 2 fun j => α)
state_after: no goals
tactic: infer_instance
Theorem: CategoryTheory.ShortComplex.SnakeInput.w₀₂_τ₃
File: Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean

Proof steps:
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : SnakeInput C
⊢ S.v₀₁.τ₃ ≫ S.v₁₂.τ₃ = 0
state_after: no goals
tactic: rw [← comp_τ₃, S.w₀₂, zero_τ₃]
Theorem: cauchy_iInf_uniformSpace'
File: Mathlib/Topology/UniformSpace/Cauchy.lean

Proof steps:
state_before: α : Type u
β : Type v
uniformSpace : UniformSpace α
ι : Sort u_1
u : ι → UniformSpace β
l : Filter β
inst✝ : l.NeBot
⊢ Cauchy l ↔ ∀ (i : ι), Cauchy l
state_after: no goals
tactic: simp_rw [cauchy_iff_le (uniformSpace := _), iInf_uniformity, le_iInf_iff]
Theorem: LipschitzWith.div
File: Mathlib/Analysis/Normed/Group/Uniform.lean

Proof steps:
state_before: 𝓕 : Type u_1
α✝ : Type u_2
E✝ : Type u_3
F : Type u_4
inst✝³ : SeminormedCommGroup E✝
inst✝² : SeminormedCommGroup F
a a₁ a₂ b b₁ b₂ : E✝
r r₁ r₂ : ℝ
α : Type u_5
E : Type u_6
inst✝¹ : SeminormedCommGroup E
inst✝ : PseudoEMetricSpace α
K Kf Kg : ℝ≥0
f g : α → E
s : Set α
x : α
hf : LipschitzWith Kf f
hg : LipschitzWith Kg g
⊢ LipschitzWith (Kf + Kg) fun x => f x / g x
state_after: no goals
tactic: simpa only [div_eq_mul_inv] using hf.mul' hg.inv
Theorem: exists_dist_lt_lt
File: Mathlib/Analysis/NormedSpace/Pointwise.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
⊢ ∃ y, dist x y < δ ∧ dist y z < ε
state_after: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y✝ z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x y < δ ∧ dist y z < ε
tactic: refine (exists_dist_eq x z (div_nonneg hε.le <| add_nonneg hε.le hδ.le)
    (div_nonneg hδ.le <| add_nonneg hε.le hδ.le) <| by
      rw [← add_div, div_self (add_pos hε hδ).ne']).imp
    fun y hy => ?_
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y✝ z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x y < δ ∧ dist y z < ε
state_after: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y✝ z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x z / (ε + δ) * δ < δ ∧ dist x z / (ε + δ) * ε < ε
tactic: rw [hy.1, hy.2, div_mul_comm, div_mul_comm ε]
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y✝ z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x z / (ε + δ) * δ < δ ∧ dist x z / (ε + δ) * ε < ε
state_after: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y✝ z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z / (ε + δ) < 1
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x z / (ε + δ) * δ < δ ∧ dist x z / (ε + δ) * ε < ε
tactic: rw [← div_lt_one (add_pos hε hδ)] at h
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y✝ z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z / (ε + δ) < 1
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x z / (ε + δ) * δ < δ ∧ dist x z / (ε + δ) * ε < ε
state_after: no goals
tactic: exact ⟨mul_lt_of_lt_one_left hδ h, mul_lt_of_lt_one_left hε h⟩
Theorem: Ordinal.opow_principal_add_of_principal_add
File: Mathlib/SetTheory/Ordinal/Principal.lean

Proof steps:
state_before: a : Ordinal.{u_1}
ha : Principal (fun x x_1 => x + x_1) a
b : Ordinal.{u_1}
⊢ Principal (fun x x_1 => x + x_1) (a ^ b)
state_after: case inl
b : Ordinal.{u_1}
ha : Principal (fun x x_1 => x + x_1) 0
⊢ Principal (fun x x_1 => x + x_1) (0 ^ b)

case inr.intro
b c : Ordinal.{u_1}
ha : Principal (fun x x_1 => x + x_1) (ω ^ c)
⊢ Principal (fun x x_1 => x + x_1) ((ω ^ c) ^ b)
tactic: rcases principal_add_iff_zero_or_omega_opow.1 ha with (rfl | ⟨c, rfl⟩)
Theorem: Real.Angle.sign_toReal
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean

Proof steps:
state_before: θ : Angle
h : θ ≠ ↑π
⊢ SignType.sign θ.toReal = θ.sign
state_after: case inl
θ : Angle
h : θ ≠ ↑π
ht : θ.toReal < 0
⊢ SignType.sign θ.toReal = θ.sign

case inr.inl
θ : Angle
h : θ ≠ ↑π
ht : θ.toReal = 0
⊢ SignType.sign θ.toReal = θ.sign

case inr.inr
θ : Angle
h : θ ≠ ↑π
ht : 0 < θ.toReal
⊢ SignType.sign θ.toReal = θ.sign
tactic: rcases lt_trichotomy θ.toReal 0 with (ht | ht | ht)
Theorem: DirichletCharacter.fourierTransform_eq_inv_mul_gaussSum
File: Mathlib/Analysis/Fourier/ZMod.lean

Proof steps:
state_before: N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
k : ZMod N
hχ : χ.IsPrimitive
⊢ 𝓕 (⇑χ) k = χ⁻¹ (-k) * gaussSum χ stdAddChar
state_after: no goals
tactic: rw [fourierTransform_eq_gaussSum_mulShift, gaussSum_mulShift_of_isPrimitive _ hχ]
Theorem: SetTheory.PGame.moveRight_neg'
File: Mathlib/SetTheory/Game/PGame.lean

Proof steps:
state_before: xl xr : Type u
x : PGame
i : (-x).RightMoves
⊢ (-x).moveRight i = -x.moveLeft (toRightMovesNeg.symm i)
state_after: case mk
xl xr : Type u
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
i : (-mk α✝ β✝ a✝¹ a✝).RightMoves
⊢ (-mk α✝ β✝ a✝¹ a✝).moveRight i = -(mk α✝ β✝ a✝¹ a✝).moveLeft (toRightMovesNeg.symm i)
tactic: cases x
state_before: case mk
xl xr : Type u
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
i : (-mk α✝ β✝ a✝¹ a✝).RightMoves
⊢ (-mk α✝ β✝ a✝¹ a✝).moveRight i = -(mk α✝ β✝ a✝¹ a✝).moveLeft (toRightMovesNeg.symm i)
state_after: no goals
tactic: rfl
Theorem: Real.cos_pi_over_two_pow
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

Proof steps:
state_before: x : ℝ
⊢ cos (π / 2 ^ (0 + 1)) = sqrtTwoAddSeries 0 0 / 2
state_after: no goals
tactic: simp
state_before: x : ℝ
n : ℕ
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2
state_after: x : ℝ
n : ℕ
A : 1 < 2 ^ (n + 1)
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2
tactic: have A : (1 : ℝ) < 2 ^ (n + 1) := one_lt_pow one_lt_two n.succ_ne_zero
state_before: x : ℝ
n : ℕ
A : 1 < 2 ^ (n + 1)
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2
state_after: x : ℝ
n : ℕ
A : 1 < 2 ^ (n + 1)
B : π / 2 ^ (n + 1) < π
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2
tactic: have B : π / 2 ^ (n + 1) < π := div_lt_self pi_pos A
state_before: x : ℝ
n : ℕ
A : 1 < 2 ^ (n + 1)
B : π / 2 ^ (n + 1) < π
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2
state_after: x : ℝ
n : ℕ
A : 1 < 2 ^ (n + 1)
B : π / 2 ^ (n + 1) < π
C : 0 < π / 2 ^ (n + 1)
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2
tactic: have C : 0 < π / 2 ^ (n + 1) := by positivity
state_before: x : ℝ
n : ℕ
A : 1 < 2 ^ (n + 1)
B : π / 2 ^ (n + 1) < π
C : 0 < π / 2 ^ (n + 1)
⊢ cos (π / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2
state_after: no goals
tactic: rw [pow_succ, div_mul_eq_div_div, cos_half, cos_pi_over_two_pow n, sqrtTwoAddSeries,
      add_div_eq_mul_add_div, one_mul, ← div_mul_eq_div_div, sqrt_div, sqrt_mul_self] <;>
      linarith [sqrtTwoAddSeries_nonneg le_rfl n]
Theorem: AddCommMonCat.uliftFunctor_map
File: Mathlib/Algebra/Category/MonCat/Basic.lean

Proof steps:
state_before: X : CommMonCat
⊢ { obj := fun X => of (ULift.{v, u} ↑X),
          map := fun {X Y} f =>
            ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
      (𝟙 X) =
    𝟙
      ({ obj := fun X => of (ULift.{v, u} ↑X),
            map := fun {X Y} f =>
              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.obj
        X)
state_after: no goals
tactic: rfl
state_before: X Y Z : CommMonCat
f : X ⟶ Y
g : Y ⟶ Z
⊢ { obj := fun X => of (ULift.{v, u} ↑X),
          map := fun {X Y} f =>
            ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
      (f ≫ g) =
    { obj := fun X => of (ULift.{v, u} ↑X),
            map := fun {X Y} f =>
              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
        f ≫
      { obj := fun X => of (ULift.{v, u} ↑X),
            map := fun {X Y} f =>
              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
        g
state_after: no goals
tactic: rfl
Theorem: MeasureTheory.memℒp_one_iff_integrable
File: Mathlib/MeasureTheory/Function/L1Space.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
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → β
⊢ Memℒp f 1 μ ↔ Integrable f μ
state_after: no goals
tactic: simp_rw [Integrable, HasFiniteIntegral, Memℒp, eLpNorm_one_eq_lintegral_nnnorm]
Theorem: FintypeCat.toLightProfinite_obj_toTop_α
File: Mathlib/Topology/Category/LightProfinite/Basic.lean

Proof steps:
state_before: X✝ Y✝ : FintypeCat
f : X✝ ⟶ Y✝
⊢ Continuous f
state_after: no goals
tactic: continuity
Theorem: Bool.coe_sort_true
File: Mathlib/Data/Bool/Basic.lean

Proof steps:
state_before: ⊢ (true = true) = True
state_after: no goals
tactic: simp
Theorem: mdifferentiable_iff_differentiable
File: Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
f : E → E'
s : Set E
x : E
⊢ MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') f ↔ Differentiable 𝕜 f
state_after: no goals
tactic: simp only [MDifferentiable, Differentiable, mdifferentiableAt_iff_differentiableAt]
Theorem: CategoryTheory.shift_shiftFunctorCompIsoId_add_neg_self_hom_app
File: Mathlib/CategoryTheory/Shift/Basic.lean

Proof steps:
state_before: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
X✝ Y : C
f : X✝ ⟶ Y
n : A
X : C
⊢ (shiftFunctor C n).map ((shiftFunctorCompIsoId C n (-n) ⋯).hom.app X) =
    (shiftFunctorCompIsoId C (-n) n ⋯).hom.app ((shiftFunctor C n).obj X)
state_after: no goals
tactic: apply shift_shiftFunctorCompIsoId_hom_app
Theorem: Matrix.Pivot.isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow
File: Mathlib/LinearAlgebra/Matrix/Transvection.lean

Proof steps:
state_before: n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝³ : Field 𝕜
inst✝² : DecidableEq n
inst✝¹ : DecidableEq p
inst✝ : CommRing R
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
⊢ ((listTransvecCol M).prod * M * (listTransvecRow M).prod).IsTwoBlockDiagonal
state_after: case left
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝³ : Field 𝕜
inst✝² : DecidableEq n
inst✝¹ : DecidableEq p
inst✝ : CommRing R
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
⊢ ((listTransvecCol M).prod * M * (listTransvecRow M).prod).toBlocks₁₂ = 0

case right
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝³ : Field 𝕜
inst✝² : DecidableEq n
inst✝¹ : DecidableEq p
inst✝ : CommRing R
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
⊢ ((listTransvecCol M).prod * M * (listTransvecRow M).prod).toBlocks₂₁ = 0
tactic: constructor
Theorem: CategoryTheory.ShortComplex.Homotopy.add_h₁
File: Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean

Proof steps:
state_before: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ (φ₁ + φ₃).τ₁ = S₁.f ≫ (h.h₁ + h'.h₁) + (h.h₀ + h'.h₀) + (φ₂ + φ₄).τ₁
state_after: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁ + (S₁.f ≫ h'.h₁ + h'.h₀ + φ₄.τ₁) =
    S₁.f ≫ h.h₁ + S₁.f ≫ h'.h₁ + (h.h₀ + h'.h₀) + (φ₂.τ₁ + φ₄.τ₁)
tactic: rw [add_τ₁, add_τ₁, h.comm₁, h'.comm₁, comp_add]
state_before: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁ + (S₁.f ≫ h'.h₁ + h'.h₀ + φ₄.τ₁) =
    S₁.f ≫ h.h₁ + S₁.f ≫ h'.h₁ + (h.h₀ + h'.h₀) + (φ₂.τ₁ + φ₄.τ₁)
state_after: no goals
tactic: abel
state_before: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ (φ₁ + φ₃).τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂ + φ₄).τ₂
state_after: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂ + (S₁.g ≫ h'.h₂ + h'.h₁ ≫ S₂.f + φ₄.τ₂) =
    S₁.g ≫ h.h₂ + S₁.g ≫ h'.h₂ + (h.h₁ ≫ S₂.f + h'.h₁ ≫ S₂.f) + (φ₂.τ₂ + φ₄.τ₂)
tactic: rw [add_τ₂, add_τ₂, h.comm₂, h'.comm₂, comp_add, add_comp]
state_before: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂ + (S₁.g ≫ h'.h₂ + h'.h₁ ≫ S₂.f + φ₄.τ₂) =
    S₁.g ≫ h.h₂ + S₁.g ≫ h'.h₂ + (h.h₁ ≫ S₂.f + h'.h₁ ≫ S₂.f) + (φ₂.τ₂ + φ₄.τ₂)
state_after: no goals
tactic: abel
state_before: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ (φ₁ + φ₃).τ₃ = h.h₃ + h'.h₃ + (h.h₂ + h'.h₂) ≫ S₂.g + (φ₂ + φ₄).τ₃
state_after: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃ + (h'.h₃ + h'.h₂ ≫ S₂.g + φ₄.τ₃) =
    h.h₃ + h'.h₃ + (h.h₂ ≫ S₂.g + h'.h₂ ≫ S₂.g) + (φ₂.τ₃ + φ₄.τ₃)
tactic: rw [add_τ₃, add_τ₃, h.comm₃, h'.comm₃, add_comp]
state_before: C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃ + (h'.h₃ + h'.h₂ ≫ S₂.g + φ₄.τ₃) =
    h.h₃ + h'.h₃ + (h.h₂ ≫ S₂.g + h'.h₂ ≫ S₂.g) + (φ₂.τ₃ + φ₄.τ₃)
state_after: no goals
tactic: abel
Theorem: ENNReal.Tendsto.mul_const
File: Mathlib/Topology/Instances/ENNReal.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a✝ b✝ c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f : Filter α
m : α → ℝ≥0∞
a b : ℝ≥0∞
hm : Tendsto m f (𝓝 a)
ha : a ≠ 0 ∨ b ≠ ⊤
⊢ Tendsto (fun x => m x * b) f (𝓝 (a * b))
state_after: no goals
tactic: simpa only [mul_comm] using ENNReal.Tendsto.const_mul hm ha
Theorem: CategoryTheory.ShortComplex.Exact.map_of_preservesLeftHomologyOf
File: Mathlib/Algebra/Homology/ShortComplex/Exact.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{u_4, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.Exact
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
⊢ (S.map F).Exact
state_after: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{u_4, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.Exact
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
⊢ (S.map F).Exact
tactic: have := h.hasHomology
state_before: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{u_4, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.Exact
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
⊢ (S.map F).Exact
state_after: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{u_4, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
h : 𝟙 S.leftHomologyData.H = 0
⊢ (S.map F).Exact
tactic: rw [S.leftHomologyData.exact_iff, IsZero.iff_id_eq_zero] at h
state_before: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Category.{u_4, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
h : 𝟙 S.leftHomologyData.H = 0
⊢ (S.map F).Exact
state_after: no goals
tactic: rw [S.leftHomologyData.exact_map_iff F, IsZero.iff_id_eq_zero,
    ← F.map_id, h, F.map_zero]
Theorem: Rat.inv_divInt
File: .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean

Proof steps:
state_before: n d : Int
⊢ (n /. d).inv = d /. n
state_after: no goals
tactic: if z : d = 0 then simp [z] else
  cases e : n /. d; rcases divInt_num_den z e with ⟨g, zg, rfl, rfl⟩
  simp [inv_def, divInt_mul_right zg]
Theorem: omegaLimit_subset_closure_fw_image
File: Mathlib/Dynamics/OmegaLimit.lean

Proof steps:
state_before: τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
u : Set τ
hu : u ∈ f
⊢ ω f ϕ s ⊆ closure (image2 ϕ u s)
state_after: τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
u : Set τ
hu : u ∈ f
⊢ ⋂ u, closure (image2 ϕ (↑u) s) ⊆ closure (image2 ϕ u s)
tactic: rw [omegaLimit_eq_iInter]
state_before: τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
u : Set τ
hu : u ∈ f
⊢ ⋂ u, closure (image2 ϕ (↑u) s) ⊆ closure (image2 ϕ u s)
state_after: τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
u : Set τ
hu : u ∈ f
a✝ : β
hx : a✝ ∈ ⋂ u, closure (image2 ϕ (↑u) s)
⊢ a✝ ∈ closure (image2 ϕ u s)
tactic: intro _ hx
state_before: τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
u : Set τ
hu : u ∈ f
a✝ : β
hx : a✝ ∈ ⋂ u, closure (image2 ϕ (↑u) s)
⊢ a✝ ∈ closure (image2 ϕ u s)
state_after: τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
u : Set τ
hu : u ∈ f
a✝ : β
hx : ∀ (i : ↑f.sets), a✝ ∈ closure (image2 ϕ (↑i) s)
⊢ a✝ ∈ closure (image2 ϕ u s)
tactic: rw [mem_iInter] at hx
state_before: τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
u : Set τ
hu : u ∈ f
a✝ : β
hx : ∀ (i : ↑f.sets), a✝ ∈ closure (image2 ϕ (↑i) s)
⊢ a✝ ∈ closure (image2 ϕ u s)
state_after: no goals
tactic: exact hx ⟨u, hu⟩
Theorem: AdjoinRoot.mk_leftInverse
File: Mathlib/RingTheory/AdjoinRoot.lean

Proof steps:
state_before: R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
g : R[X]
hg : g.Monic
⊢ Function.LeftInverse ⇑(mk g) ⇑(modByMonicHom hg)
state_after: R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
g : R[X]
hg : g.Monic
f : AdjoinRoot g
⊢ (mk g) ((modByMonicHom hg) f) = f
tactic: intro f
state_before: R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
g : R[X]
hg : g.Monic
f : AdjoinRoot g
⊢ (mk g) ((modByMonicHom hg) f) = f
state_after: case ih
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
g : R[X]
hg : g.Monic
p✝ : R[X]
⊢ (mk g) ((modByMonicHom hg) ((mk g) p✝)) = (mk g) p✝
tactic: induction f using AdjoinRoot.induction_on
state_before: case ih
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
g : R[X]
hg : g.Monic
p✝ : R[X]
⊢ (mk g) ((modByMonicHom hg) ((mk g) p✝)) = (mk g) p✝
state_after: case ih
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
g : R[X]
hg : g.Monic
p✝ : R[X]
⊢ g ∣ g * (p✝ /ₘ g)
tactic: rw [modByMonicHom_mk hg, mk_eq_mk, modByMonic_eq_sub_mul_div _ hg, sub_sub_cancel_left,
    dvd_neg]
state_before: case ih
R : Type u
S : Type v
K : Type w
inst✝ : CommRing R
g : R[X]
hg : g.Monic
p✝ : R[X]
⊢ g ∣ g * (p✝ /ₘ g)
state_after: no goals
tactic: apply dvd_mul_right