ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: HomologicalComplex₂.flip_d_f
File: Mathlib/Algebra/Homology/HomologicalBicomplex.lean

Proof steps:
state_before: C : Type u_1
inst✝¹ : Category.{?u.12556, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
i i' : I₂
w : ¬c₂.Rel i i'
⊢ (fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i' = 0
state_after: case h
C : Type u_1
inst✝¹ : Category.{?u.12556, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
i i' : I₂
w : ¬c₂.Rel i i'
j : I₁
⊢ ((fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i').f j = Hom.f 0 j
tactic: ext j
state_before: case h
C : Type u_1
inst✝¹ : Category.{?u.12556, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
i i' : I₂
w : ¬c₂.Rel i i'
j : I₁
⊢ ((fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i').f j = Hom.f 0 j
state_after: no goals
tactic: exact (K.X j).shape i i' w
Theorem: Algebra.FormallySmooth.of_equiv
File: Mathlib/RingTheory/Smooth/Basic.lean

Proof steps:
state_before: R : Type u
inst✝⁵ : CommSemiring R
A B : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : FormallySmooth R A
e : A ≃ₐ[R] B
⊢ FormallySmooth R B
state_after: case comp_surjective
R : Type u
inst✝⁵ : CommSemiring R
A B : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : FormallySmooth R A
e : A ≃ₐ[R] B
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),
    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp
tactic: constructor
state_before: case comp_surjective
R : Type u
inst✝⁵ : CommSemiring R
A B : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : FormallySmooth R A
e : A ≃ₐ[R] B
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),
    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp
state_after: case comp_surjective
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f : B →ₐ[R] C ⧸ I
⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f
tactic: intro C _ _ I hI f
state_before: case comp_surjective
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f : B →ₐ[R] C ⧸ I
⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f
state_after: case h
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f : B →ₐ[R] C ⧸ I
⊢ (Ideal.Quotient.mkₐ R I).comp ((lift I ⋯ (f.comp ↑e)).comp ↑e.symm) = f
tactic: use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
state_before: case h
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f : B →ₐ[R] C ⧸ I
⊢ (Ideal.Quotient.mkₐ R I).comp ((lift I ⋯ (f.comp ↑e)).comp ↑e.symm) = f
state_after: no goals
tactic: rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
    AlgHom.comp_id]
Theorem: CategoryTheory.CostructuredArrow.mkPrecomp_id
File: Mathlib/CategoryTheory/Comma/StructuredArrow.lean

Proof steps:
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
T T' T'' : D
Y Y' Y'' : C
S S' : C ⥤ D
f : S.obj Y ⟶ T
⊢ mk (S.map (𝟙 Y) ≫ f) = mk f
state_after: no goals
tactic: aesop_cat
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
T T' T'' : D
Y Y' Y'' : C
S S' : C ⥤ D
f : S.obj Y ⟶ T
⊢ mkPrecomp f (𝟙 Y) = eqToHom ⋯
state_after: no goals
tactic: aesop_cat
Theorem: root_X_pow_sub_C_eq_zero_iff
File: Mathlib/FieldTheory/KummerExtension.lean

Proof steps:
state_before: K : Type u
inst✝ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
⊢ root (X ^ n - C a) = 0 ↔ a = 0
state_after: K : Type u
inst✝ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
hn : 0 < n
⊢ root (X ^ n - C a) = 0 ↔ a = 0
tactic: have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
state_before: K : Type u
inst✝ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
hn : 0 < n
⊢ root (X ^ n - C a) = 0 ↔ a = 0
state_after: K : Type u
inst✝ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
hn : 0 < n
⊢ a = 0 → root (X ^ n - C a) = 0
tactic: refine ⟨not_imp_not.mp (root_X_pow_sub_C_ne_zero' hn), ?_⟩
state_before: K : Type u
inst✝ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
hn : 0 < n
⊢ a = 0 → root (X ^ n - C a) = 0
state_after: K : Type u
inst✝ : Field K
n : ℕ
hn : 0 < n
H : Irreducible (X ^ n - C 0)
⊢ root (X ^ n - C 0) = 0
tactic: rintro rfl
state_before: K : Type u
inst✝ : Field K
n : ℕ
hn : 0 < n
H : Irreducible (X ^ n - C 0)
⊢ root (X ^ n - C 0) = 0
state_after: K : Type u
inst✝ : Field K
n : ℕ
hn : 0 < n
H : Irreducible (X ^ n - C 0)
this : n = 1
⊢ root (X ^ n - C 0) = 0
tactic: have := not_imp_not.mp (fun hn ↦ ne_zero_of_irreducible_X_pow_sub_C' hn H) rfl
state_before: K : Type u
inst✝ : Field K
n : ℕ
hn : 0 < n
H : Irreducible (X ^ n - C 0)
this : n = 1
⊢ root (X ^ n - C 0) = 0
state_after: no goals
tactic: rw [this, pow_one, map_zero, sub_zero, ← mk_X, mk_self]
Theorem: NormedSpace.expSeries_apply_eq
File: Mathlib/Analysis/NormedSpace/Exponential.lean

Proof steps:
state_before: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : TopologicalRing 𝔸
x : 𝔸
n : ℕ
⊢ ((expSeries 𝕂 𝔸 n) fun x_1 => x) = (↑n !)⁻¹ • x ^ n
state_after: no goals
tactic: simp [expSeries]
Theorem: unitary.map_mem
File: Mathlib/Algebra/Star/Unitary.lean

Proof steps:
state_before: R✝ : Type u_1
F : Type u_2
R : Type u_3
S : Type u_4
inst✝⁶ : Monoid R
inst✝⁵ : StarMul R
inst✝⁴ : Monoid S
inst✝³ : StarMul S
inst✝² : FunLike F R S
inst✝¹ : StarHomClass F R S
inst✝ : MonoidHomClass F R S
f : F
r : R
hr : r ∈ unitary R
⊢ f r ∈ unitary S
state_after: R✝ : Type u_1
F : Type u_2
R : Type u_3
S : Type u_4
inst✝⁶ : Monoid R
inst✝⁵ : StarMul R
inst✝⁴ : Monoid S
inst✝³ : StarMul S
inst✝² : FunLike F R S
inst✝¹ : StarHomClass F R S
inst✝ : MonoidHomClass F R S
f : F
r : R
hr : star r * r = 1 ∧ r * star r = 1
⊢ f r ∈ unitary S
tactic: rw [unitary.mem_iff] at hr
state_before: R✝ : Type u_1
F : Type u_2
R : Type u_3
S : Type u_4
inst✝⁶ : Monoid R
inst✝⁵ : StarMul R
inst✝⁴ : Monoid S
inst✝³ : StarMul S
inst✝² : FunLike F R S
inst✝¹ : StarHomClass F R S
inst✝ : MonoidHomClass F R S
f : F
r : R
hr : star r * r = 1 ∧ r * star r = 1
⊢ f r ∈ unitary S
state_after: no goals
tactic: simpa [map_star, map_mul] using And.intro congr(f $(hr.1)) congr(f $(hr.2))
Theorem: InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero
File: Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean

Proof steps:
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖y‖ / Real.tan (angle x (x + y)) = ‖x‖
state_after: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖
tactic: rw [tan_angle_add_of_inner_eq_zero h]
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖
state_after: case inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0
⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖

case inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : y ≠ 0
⊢ ‖y‖ / (‖y‖ / ‖x‖) = ‖x‖
tactic: rcases h0 with (h0 | h0)
Theorem: Fin.removeNth_zero
File: Mathlib/Data/Fin/Tuple/Basic.lean

Proof steps:
state_before: m n : ℕ
α : Fin (n + 1) → Type u
β : Type v
f : (i : Fin (n + 1)) → α i
⊢ removeNth 0 f = tail f
state_after: case h
m n : ℕ
α : Fin (n + 1) → Type u
β : Type v
f : (i : Fin (n + 1)) → α i
x✝ : Fin n
⊢ removeNth 0 f x✝ = tail f x✝
tactic: ext
state_before: case h
m n : ℕ
α : Fin (n + 1) → Type u
β : Type v
f : (i : Fin (n + 1)) → α i
x✝ : Fin n
⊢ removeNth 0 f x✝ = tail f x✝
state_after: no goals
tactic: simp [tail, removeNth]
Theorem: MeasureTheory.prob_compl_le_one_sub_of_le_prob
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✝¹ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝ : IsProbabilityMeasure μ
p✝ : α → Prop
f : β → α
p : ℝ≥0∞
hμs : p ≤ μ s
s_mble : MeasurableSet s
⊢ μ sᶜ ≤ 1 - p
state_after: no goals
tactic: simpa [prob_compl_eq_one_sub s_mble] using tsub_le_tsub_left hμs 1
Theorem: InnerProductGeometry.angle_add_pos_of_inner_eq_zero
File: Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean

Proof steps:
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ 0 < angle x (x + y)
state_after: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1
tactic: rw [angle_add_eq_arccos_of_inner_eq_zero h, Real.arccos_pos,
    norm_add_eq_sqrt_iff_real_inner_eq_zero.2 h]
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1
state_after: case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
hx : x = 0
⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1

case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
hx : ¬x = 0
⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1
tactic: by_cases hx : x = 0
state_before: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
hx : ¬x = 0
⊢ ‖x‖ / √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) < 1
state_after: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
hx : ¬x = 0
⊢ ‖x‖ * ‖x‖ < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖
tactic: rw [div_lt_one (Real.sqrt_pos.2 (Left.add_pos_of_pos_of_nonneg (mul_self_pos.2
    (norm_ne_zero_iff.2 hx)) (mul_self_nonneg _))), Real.lt_sqrt (norm_nonneg _), pow_two]
state_before: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0
hx : ¬x = 0
⊢ ‖x‖ * ‖x‖ < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖
state_after: no goals
tactic: simpa [hx] using h0
Theorem: Ideal.Polynomial.isJacobson_polynomial_iff_isJacobson
File: Mathlib/RingTheory/Jacobson.lean

Proof steps:
state_before: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹ : CommRing Rₘ
inst✝ : CommRing Sₘ
⊢ IsJacobson R[X] ↔ IsJacobson R
state_after: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹ : CommRing Rₘ
inst✝ : CommRing Sₘ
⊢ IsJacobson R[X] → IsJacobson R
tactic: refine ⟨?_, isJacobson_polynomial_of_isJacobson⟩
state_before: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹ : CommRing Rₘ
inst✝ : CommRing Sₘ
⊢ IsJacobson R[X] → IsJacobson R
state_after: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹ : CommRing Rₘ
inst✝ : CommRing Sₘ
H : IsJacobson R[X]
⊢ IsJacobson R
tactic: intro H
state_before: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹ : CommRing Rₘ
inst✝ : CommRing Sₘ
H : IsJacobson R[X]
⊢ IsJacobson R
state_after: no goals
tactic: exact isJacobson_of_surjective ⟨eval₂RingHom (RingHom.id _) 1, fun x =>
    ⟨C x, by simp only [coe_eval₂RingHom, RingHom.id_apply, eval₂_C]⟩⟩
Theorem: CategoryTheory.Limits.coprod.map_swap
File: Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean

Proof steps:
state_before: C : Type u
inst✝¹ : Category.{v, u} C
X✝ Y✝ A B X Y : C
f : A ⟶ B
g : X ⟶ Y
inst✝ : HasColimitsOfShape (Discrete WalkingPair) C
⊢ map (𝟙 X) f ≫ map g (𝟙 B) = map g (𝟙 A) ≫ map (𝟙 Y) f
state_after: no goals
tactic: simp
Theorem: QuadraticMap.anisotropic_of_prod
File: Mathlib/LinearAlgebra/QuadraticForm/Prod.lean

Proof steps:
state_before: ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M₁
inst✝⁸ : AddCommMonoid M₂
inst✝⁷ : AddCommMonoid N₁
inst✝⁶ : AddCommMonoid N₂
inst✝⁵ : AddCommMonoid P
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Module R N₁
inst✝¹ : Module R N₂
inst✝ : Module R P
Q₁ : QuadraticMap R M₁ P
Q₂ : QuadraticMap R M₂ P
h : (Q₁.prod Q₂).Anisotropic
⊢ Q₁.Anisotropic ∧ Q₂.Anisotropic
state_after: ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M₁
inst✝⁸ : AddCommMonoid M₂
inst✝⁷ : AddCommMonoid N₁
inst✝⁶ : AddCommMonoid N₂
inst✝⁵ : AddCommMonoid P
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Module R N₁
inst✝¹ : Module R N₂
inst✝ : Module R P
Q₁ : QuadraticMap R M₁ P
Q₂ : QuadraticMap R M₂ P
h : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0
⊢ Q₁.Anisotropic ∧ Q₂.Anisotropic
tactic: simp_rw [Anisotropic, prod_apply, Prod.forall, Prod.mk_eq_zero] at h
state_before: ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M₁
inst✝⁸ : AddCommMonoid M₂
inst✝⁷ : AddCommMonoid N₁
inst✝⁶ : AddCommMonoid N₂
inst✝⁵ : AddCommMonoid P
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Module R N₁
inst✝¹ : Module R N₂
inst✝ : Module R P
Q₁ : QuadraticMap R M₁ P
Q₂ : QuadraticMap R M₂ P
h : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0
⊢ Q₁.Anisotropic ∧ Q₂.Anisotropic
state_after: case left
ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M₁
inst✝⁸ : AddCommMonoid M₂
inst✝⁷ : AddCommMonoid N₁
inst✝⁶ : AddCommMonoid N₂
inst✝⁵ : AddCommMonoid P
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Module R N₁
inst✝¹ : Module R N₂
inst✝ : Module R P
Q₁ : QuadraticMap R M₁ P
Q₂ : QuadraticMap R M₂ P
h : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0
⊢ Q₁.Anisotropic

case right
ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M₁
inst✝⁸ : AddCommMonoid M₂
inst✝⁷ : AddCommMonoid N₁
inst✝⁶ : AddCommMonoid N₂
inst✝⁵ : AddCommMonoid P
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Module R N₁
inst✝¹ : Module R N₂
inst✝ : Module R P
Q₁ : QuadraticMap R M₁ P
Q₂ : QuadraticMap R M₂ P
h : ∀ (a : M₁) (b : M₂), Q₁ a + Q₂ b = 0 → a = 0 ∧ b = 0
⊢ Q₂.Anisotropic
tactic: constructor
Theorem: Sym2.other_spec
File: Mathlib/Data/Sym/Sym2.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a : α
z : Sym2 α
h : a ∈ z
⊢ s(a, Mem.other h) = z
state_after: no goals
tactic: erw [← Classical.choose_spec h]
Theorem: Finsupp.support_sup
File: Mathlib/Data/Finsupp/Order.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : CanonicallyLinearOrderedAddCommMonoid α
inst✝ : DecidableEq ι
f g : ι →₀ α
⊢ (f ⊔ g).support = f.support ∪ g.support
state_after: case a
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : CanonicallyLinearOrderedAddCommMonoid α
inst✝ : DecidableEq ι
f g : ι →₀ α
a✝ : ι
⊢ a✝ ∈ (f ⊔ g).support ↔ a✝ ∈ f.support ∪ g.support
tactic: ext
state_before: case a
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : CanonicallyLinearOrderedAddCommMonoid α
inst✝ : DecidableEq ι
f g : ι →₀ α
a✝ : ι
⊢ a✝ ∈ (f ⊔ g).support ↔ a✝ ∈ f.support ∪ g.support
state_after: case a
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : CanonicallyLinearOrderedAddCommMonoid α
inst✝ : DecidableEq ι
f g : ι →₀ α
a✝ : ι
⊢ ¬f a✝ ⊔ g a✝ = ⊥ ↔ ¬f a✝ = ⊥ ∨ ¬g a✝ = ⊥
tactic: simp only [Finset.mem_union, mem_support_iff, sup_apply, Ne, ← bot_eq_zero]
state_before: case a
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : CanonicallyLinearOrderedAddCommMonoid α
inst✝ : DecidableEq ι
f g : ι →₀ α
a✝ : ι
⊢ ¬f a✝ ⊔ g a✝ = ⊥ ↔ ¬f a✝ = ⊥ ∨ ¬g a✝ = ⊥
state_after: no goals
tactic: rw [_root_.sup_eq_bot_iff, not_and_or]
Theorem: MulChar.IsQuadratic.comp
File: Mathlib/NumberTheory/MulChar/Basic.lean

Proof steps:
state_before: R : Type u_1
inst✝² : CommMonoid R
R' : Type u_2
inst✝¹ : CommRing R'
R'' : Type u_3
inst✝ : CommRing R''
χ : MulChar R R'
hχ : χ.IsQuadratic
f : R' →+* R''
⊢ (χ.ringHomComp f).IsQuadratic
state_after: R : Type u_1
inst✝² : CommMonoid R
R' : Type u_2
inst✝¹ : CommRing R'
R'' : Type u_3
inst✝ : CommRing R''
χ : MulChar R R'
hχ : χ.IsQuadratic
f : R' →+* R''
a : R
⊢ (χ.ringHomComp f) a = 0 ∨ (χ.ringHomComp f) a = 1 ∨ (χ.ringHomComp f) a = -1
tactic: intro a
state_before: R : Type u_1
inst✝² : CommMonoid R
R' : Type u_2
inst✝¹ : CommRing R'
R'' : Type u_3
inst✝ : CommRing R''
χ : MulChar R R'
hχ : χ.IsQuadratic
f : R' →+* R''
a : R
⊢ (χ.ringHomComp f) a = 0 ∨ (χ.ringHomComp f) a = 1 ∨ (χ.ringHomComp f) a = -1
state_after: no goals
tactic: rcases hχ a with (ha | ha | ha) <;> simp [ha]
Theorem: Polynomial.card_support_eraseLead'
File: Mathlib/Algebra/Polynomial/EraseLead.lean

Proof steps:
state_before: R : Type u_1
inst✝ : Semiring R
f : R[X]
c : ℕ
fc : f.support.card = c + 1
⊢ f.eraseLead.support.card = c
state_after: no goals
tactic: rw [card_support_eraseLead, fc, add_tsub_cancel_right]
Theorem: AlgebraicGeometry.Scheme.OpenCover.add_map
File: Mathlib/AlgebraicGeometry/Cover/Open.lean

Proof steps:
state_before: X✝ Y✝ Z : Scheme
𝒰✝ : X✝.OpenCover
f✝ : X✝ ⟶ Z
g : Y✝ ⟶ Z
inst✝¹ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g
X Y : Scheme
𝒰 : X.OpenCover
f : Y ⟶ X
inst✝ : IsOpenImmersion f
⊢ ∀ (x : Option 𝒰.J), IsOpenImmersion ((fun i => Option.rec f 𝒰.map i) x)
state_after: no goals
tactic: rintro (_ | _) <;> dsimp <;> infer_instance
Theorem: Complex.cos_eq_one_iff
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean

Proof steps:
state_before: x : ℂ
⊢ cos x = 1 ↔ ∃ k, ↑k * (2 * ↑π) = x
state_after: x : ℂ
⊢ (∃ k, x = 2 * ↑k * ↑π + 0 ∨ x = 2 * ↑k * ↑π - 0) ↔ ∃ k, ↑k * (2 * ↑π) = x
tactic: rw [← cos_zero, eq_comm, cos_eq_cos_iff]
state_before: x : ℂ
⊢ (∃ k, x = 2 * ↑k * ↑π + 0 ∨ x = 2 * ↑k * ↑π - 0) ↔ ∃ k, ↑k * (2 * ↑π) = x
state_after: no goals
tactic: simp [mul_assoc, mul_left_comm, eq_comm]
Theorem: not_summable_indicator_mod_of_antitone_of_neg
File: Mathlib/Analysis/SumOverResidueClass.lean

Proof steps:
state_before: m : ℕ
hm : NeZero m
f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
k : ZMod m
⊢ ¬Summable ({n | ↑n = k}.indicator f)
state_after: m : ℕ
hm : NeZero m
f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
k : ZMod m
⊢ ¬Summable fun n => f (m * n + k.val)
tactic: rw [← ZMod.natCast_zmod_val k, summable_indicator_mod_iff_summable]
state_before: m : ℕ
hm : NeZero m
f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
k : ZMod m
⊢ ¬Summable fun n => f (m * n + k.val)
state_after: no goals
tactic: exact not_summable_of_antitone_of_neg
    (hf.comp_monotone <| (Covariant.monotone_of_const m).add_const k.val) <|
    (hf <| (Nat.le_mul_of_pos_left n Fin.size_pos').trans <| Nat.le_add_right ..).trans_lt hn
Theorem: continuousWithinAt_iff_continuousAt
File: Mathlib/Topology/ContinuousOn.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
x : α
h : s ∈ 𝓝 x
⊢ ContinuousWithinAt f s x ↔ ContinuousAt f x
state_after: no goals
tactic: rw [← univ_inter s, continuousWithinAt_inter h, continuousWithinAt_univ]
Theorem: div_mul_eq_mul_div
File: Mathlib/Algebra/Group/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : DivisionCommMonoid α
a b c d : α
⊢ a / b * c = a * c / b
state_after: no goals
tactic: simp
Theorem: Subspace.dualAnnihilator_dualAnnihilator_eq
File: Mathlib/LinearAlgebra/Dual.lean

Proof steps:
state_before: K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K V
⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W
state_after: K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K V
this : (dualAnnihilator W).dualCoannihilator = W
⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W
tactic: have : _ = W := Subspace.dualAnnihilator_dualCoannihilator_eq
state_before: K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K V
this : (dualAnnihilator W).dualCoannihilator = W
⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W
state_after: K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K V
this : (Module.mapEvalEquiv K V).symm (dualAnnihilator W).dualAnnihilator = W
⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W
tactic: rw [dualCoannihilator, ← Module.mapEvalEquiv_symm_apply] at this
state_before: K : Type u
V : Type v
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W✝ : Subspace K V
inst✝ : FiniteDimensional K V
W : Subspace K V
this : (Module.mapEvalEquiv K V).symm (dualAnnihilator W).dualAnnihilator = W
⊢ (dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W
state_after: no goals
tactic: rwa [← OrderIso.symm_apply_eq]
Theorem: Polynomial.eq_zero_of_dvd_of_degree_lt
File: Mathlib/Algebra/Polynomial/RingDivision.lean

Proof steps:
state_before: R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h₁ : p ∣ q
h₂ : q.degree < p.degree
⊢ q = 0
state_after: R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h₁ : p ∣ q
h₂ : q.degree < p.degree
hc : ¬q = 0
⊢ False
tactic: by_contra hc
state_before: R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p✝ q✝ p q : R[X]
h₁ : p ∣ q
h₂ : q.degree < p.degree
hc : ¬q = 0
⊢ False
state_after: no goals
tactic: exact (lt_iff_not_ge _ _).mp h₂ (degree_le_of_dvd h₁ hc)
Theorem: ContinuousLinearMap.hasStrictDerivAt_of_bilinear
File: Mathlib/Analysis/Calculus/Deriv/Mul.lean

Proof steps:
state_before: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 → F
u' : E
v' : F
hu : HasStrictDerivAt u u' x
hv : HasStrictDerivAt v v' x
⊢ HasStrictDerivAt (fun x => (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) x
state_after: no goals
tactic: simpa using
    (B.hasStrictFDerivAt_of_bilinear hu.hasStrictFDerivAt hv.hasStrictFDerivAt).hasStrictDerivAt