ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: CategoryTheory.Triangulated.TStructure.predicateShift_GE
File: Mathlib/CategoryTheory/Triangulated/TStructure/Basic.lean

Proof steps:
state_before: C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
⊢ PredicateShift (t.GE n) a = t.GE n'
state_after: case h.a
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
⊢ PredicateShift (t.GE n) a X ↔ t.GE n' X
tactic: ext X
state_before: case h.a
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
⊢ PredicateShift (t.GE n) a X ↔ t.GE n' X
state_after: case h.a.mp
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
⊢ PredicateShift (t.GE n) a X → t.GE n' X

case h.a.mpr
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
⊢ t.GE n' X → PredicateShift (t.GE n) a X
tactic: constructor
Theorem: Int.lor_bit
File: Mathlib/Data/Int/Bitwise.lean

Proof steps:
state_before: a : Bool
m : ℤ
b : Bool
n : ℤ
⊢ (bit a m).lor (bit b n) = bit (a || b) (m.lor n)
state_after: no goals
tactic: rw [← bitwise_or, bitwise_bit]
Theorem: MeasureTheory.integral_sub_right_eq_self
File: Mathlib/MeasureTheory/Group/Integral.lean

Proof steps:
state_before: 𝕜 : Type u_1
M : Type u_2
α : Type u_3
G : Type u_4
E : Type u_5
F : Type u_6
inst✝⁷ : MeasurableSpace G
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
μ : Measure G
f✝ : G → E
g✝ : G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulRightInvariant
f : G → E
g : G
⊢ ∫ (x : G), f (x / g) ∂μ = ∫ (x : G), f x ∂μ
state_after: 𝕜 : Type u_1
M : Type u_2
α : Type u_3
G : Type u_4
E : Type u_5
F : Type u_6
inst✝⁷ : MeasurableSpace G
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
μ : Measure G
f✝ : G → E
g✝ : G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulRightInvariant
f : G → E
g : G
⊢ ∫ (x : G), f (x * g⁻¹) ∂μ = ∫ (x : G), f x ∂μ
tactic: simp_rw [div_eq_mul_inv]
state_before: 𝕜 : Type u_1
M : Type u_2
α : Type u_3
G : Type u_4
E : Type u_5
F : Type u_6
inst✝⁷ : MeasurableSpace G
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
μ : Measure G
f✝ : G → E
g✝ : G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulRightInvariant
f : G → E
g : G
⊢ ∫ (x : G), f (x * g⁻¹) ∂μ = ∫ (x : G), f x ∂μ
state_after: no goals
tactic: rw [integral_mul_right_eq_self f g⁻¹]
Theorem: rank_span
File: Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean

Proof steps:
state_before: R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type w
ι' : Type w'
inst✝ : StrongRankCondition R
v : ι → M
hv : LinearIndependent R v
⊢ Module.rank R ↥(span R (range v)) = #↑(range v)
state_after: R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type w
ι' : Type w'
inst✝ : StrongRankCondition R
v : ι → M
hv : LinearIndependent R v
this : Nontrivial R
⊢ Module.rank R ↥(span R (range v)) = #↑(range v)
tactic: haveI := nontrivial_of_invariantBasisNumber R
state_before: R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type w
ι' : Type w'
inst✝ : StrongRankCondition R
v : ι → M
hv : LinearIndependent R v
this : Nontrivial R
⊢ Module.rank R ↥(span R (range v)) = #↑(range v)
state_after: no goals
tactic: rw [← Cardinal.lift_inj, ← (Basis.span hv).mk_eq_rank,
    Cardinal.mk_range_eq_of_injective (@LinearIndependent.injective ι R M v _ _ _ _ hv)]
Theorem: Finset.image₂_distrib_subset_left
File: Mathlib/Data/Finset/NAry.lean

Proof steps:
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ✝
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f✝ f' : α → β → γ✝
g✝ g'✝ : α → β → γ✝ → δ
s s' : Finset α
t t' : Finset β
u✝ u' : Finset γ✝
a a' : α
b b' : β
c : γ✝
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f₁ : α → β → β'
f₂ : α → γ → γ'
g' : β' → γ' → ε
h_distrib : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' (f₁ a b) (f₂ a c)
⊢ ↑(image₂ f s (image₂ g t u)) ⊆ ↑(image₂ g' (image₂ f₁ s t) (image₂ f₂ s u))
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ✝
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f✝ f' : α → β → γ✝
g✝ g'✝ : α → β → γ✝ → δ
s s' : Finset α
t t' : Finset β
u✝ u' : Finset γ✝
a a' : α
b b' : β
c : γ✝
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f₁ : α → β → β'
f₂ : α → γ → γ'
g' : β' → γ' → ε
h_distrib : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' (f₁ a b) (f₂ a c)
⊢ image2 f (↑s) (image2 g ↑t ↑u) ⊆ image2 g' (image2 f₁ ↑s ↑t) (image2 f₂ ↑s ↑u)
tactic: push_cast
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ✝
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f✝ f' : α → β → γ✝
g✝ g'✝ : α → β → γ✝ → δ
s s' : Finset α
t t' : Finset β
u✝ u' : Finset γ✝
a a' : α
b b' : β
c : γ✝
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f₁ : α → β → β'
f₂ : α → γ → γ'
g' : β' → γ' → ε
h_distrib : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' (f₁ a b) (f₂ a c)
⊢ image2 f (↑s) (image2 g ↑t ↑u) ⊆ image2 g' (image2 f₁ ↑s ↑t) (image2 f₂ ↑s ↑u)
state_after: no goals
tactic: exact Set.image2_distrib_subset_left h_distrib
Theorem: MvPolynomial.finSuccEquiv_X_zero
File: Mathlib/Algebra/MvPolynomial/Equiv.lean

Proof steps:
state_before: R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
⊢ (finSuccEquiv R n) (X 0) = Polynomial.X
state_after: no goals
tactic: simp
Theorem: SatisfiesM.pure
File: .lake/packages/batteries/Batteries/Classes/SatisfiesM.lean

Proof steps:
state_before: m : Type u_1 → Type u_2
α✝ : Type u_1
p : α✝ → Prop
a : α✝
inst✝¹ : Applicative m
inst✝ : LawfulApplicative m
h : p a
⊢ Subtype.val <$> pure ⟨a, h⟩ = pure a
state_after: no goals
tactic: simp
Theorem: Set.Ioi_add_Ici_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
⊢ Ioi a * Ici b ⊆ Ioi (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.11614) (N : Type ?u.11613) (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
⊢ Ioi a * Ici b ⊆ Ioi (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.11614) (N : Type ?u.11613) (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
⊢ Ioi a * Ici b ⊆ Ioi (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.11614) (N : Type ?u.11613) (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 ∈ Ioi a
z : α
hzb : z ∈ Ici b
⊢ (fun x x_1 => x * x_1) y z ∈ Ioi (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.11614) (N : Type ?u.11613) (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 ∈ Ioi a
z : α
hzb : z ∈ Ici b
⊢ (fun x x_1 => x * x_1) y z ∈ Ioi (a * b)
state_after: no goals
tactic: exact mul_lt_mul_of_lt_of_le hya hzb
Theorem: Polynomial.toFinsupp_erase
File: Mathlib/Algebra/Polynomial/Basic.lean

Proof steps:
state_before: R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
p✝ q p : R[X]
n : ℕ
⊢ (erase n p).toFinsupp = Finsupp.erase n p.toFinsupp
state_after: case ofFinsupp
R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
toFinsupp✝ : R[ℕ]
⊢ (erase n { toFinsupp := toFinsupp✝ }).toFinsupp = Finsupp.erase n { toFinsupp := toFinsupp✝ }.toFinsupp
tactic: rcases p with ⟨⟩
state_before: case ofFinsupp
R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
toFinsupp✝ : R[ℕ]
⊢ (erase n { toFinsupp := toFinsupp✝ }).toFinsupp = Finsupp.erase n { toFinsupp := toFinsupp✝ }.toFinsupp
state_after: no goals
tactic: simp only [erase_def]
Theorem: Odd.pos
File: Mathlib/Algebra/Order/Ring/Canonical.lean

Proof steps:
state_before: α : Type u
β : Type u_1
inst✝¹ : CanonicallyOrderedCommSemiring α
a b c d : α
inst✝ : Nontrivial α
⊢ Odd a → 0 < a
state_after: case intro
α : Type u
β : Type u_1
inst✝¹ : CanonicallyOrderedCommSemiring α
b c d : α
inst✝ : Nontrivial α
k : α
⊢ 0 < 2 * k + 1
tactic: rintro ⟨k, rfl⟩
state_before: case intro
α : Type u
β : Type u_1
inst✝¹ : CanonicallyOrderedCommSemiring α
b c d : α
inst✝ : Nontrivial α
k : α
⊢ 0 < 2 * k + 1
state_after: no goals
tactic: simp [pos_iff_ne_zero]
Theorem: polynomialFunctions.comap_compRightAlgHom_iccHomeoI
File: Mathlib/Topology/ContinuousFunction/Polynomial.lean

Proof steps:
state_before: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
⊢ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) =
    polynomialFunctions (Set.Icc a b)
state_after: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) ↔
    f ∈ polynomialFunctions (Set.Icc a b)
tactic: ext f
state_before: case h
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) ↔
    f ∈ polynomialFunctions (Set.Icc a b)
state_after: case h.mp
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I) →
    f ∈ polynomialFunctions (Set.Icc a b)

case h.mpr
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSemiring R
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
⊢ f ∈ polynomialFunctions (Set.Icc a b) →
    f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) (polynomialFunctions I)
tactic: fconstructor
Theorem: SetTheory.PGame.toRightMovesNim_one_symm
File: Mathlib/SetTheory/Game/Nim.lean

Proof steps:
state_before: i : (nim 1).RightMoves
⊢ toRightMovesNim.symm i = ⟨0, ⋯⟩
state_after: no goals
tactic: simp [eq_iff_true_of_subsingleton]
Theorem: CategoryTheory.ShortComplex.exact_iff_homology_iso_zero
File: Mathlib/Algebra/Homology/ShortComplex/Exact.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.4862, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
inst✝¹ : S.HasHomology
inst✝ : HasZeroObject C
⊢ S.Exact ↔ Nonempty (S.homology ≅ 0)
state_after: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.4862, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
inst✝¹ : S.HasHomology
inst✝ : HasZeroObject C
⊢ IsZero S.homology ↔ Nonempty (S.homology ≅ 0)
tactic: rw [exact_iff_isZero_homology]
state_before: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.4862, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
inst✝¹ : S.HasHomology
inst✝ : HasZeroObject C
⊢ IsZero S.homology ↔ Nonempty (S.homology ≅ 0)
state_after: case mp
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.4862, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
inst✝¹ : S.HasHomology
inst✝ : HasZeroObject C
⊢ IsZero S.homology → Nonempty (S.homology ≅ 0)

case mpr
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_3, u_1} C
inst✝⁴ : Category.{?u.4862, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
inst✝¹ : S.HasHomology
inst✝ : HasZeroObject C
⊢ Nonempty (S.homology ≅ 0) → IsZero S.homology
tactic: constructor
Theorem: AddMonoidHom.closure_preimage_le
File: Mathlib/Algebra/Group/Subgroup/Basic.lean

Proof steps:
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
A : Type u_4
inst✝² : AddGroup A
N : Type u_5
P : Type u_6
inst✝¹ : Group N
inst✝ : Group P
K : Subgroup G
f : G →* N
s : Set N
x : G
hx : x ∈ ⇑f ⁻¹' s
⊢ x ∈ ↑(comap f (closure s))
state_after: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
A : Type u_4
inst✝² : AddGroup A
N : Type u_5
P : Type u_6
inst✝¹ : Group N
inst✝ : Group P
K : Subgroup G
f : G →* N
s : Set N
x : G
hx : x ∈ ⇑f ⁻¹' s
⊢ f x ∈ closure s
tactic: rw [SetLike.mem_coe, mem_comap]
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
A : Type u_4
inst✝² : AddGroup A
N : Type u_5
P : Type u_6
inst✝¹ : Group N
inst✝ : Group P
K : Subgroup G
f : G →* N
s : Set N
x : G
hx : x ∈ ⇑f ⁻¹' s
⊢ f x ∈ closure s
state_after: no goals
tactic: exact subset_closure hx
Theorem: contMDiff_finsum_cond
File: Mathlib/Geometry/Manifold/Algebra/Monoid.lean

Proof steps:
state_before: ι : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_3
inst✝¹¹ : TopologicalSpace H
E : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
inst✝⁸ : CommMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : SmoothMul I G
E' : Type u_6
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
s : Set M
x x₀ : M
t : Finset ι
f : ι → M → G
n : ℕ∞
p : ι → Prop
hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)
hf : LocallyFinite fun i => mulSupport (f i)
⊢ ContMDiff I' I n fun x => ∏ᶠ (i : ι) (_ : p i), f i x
state_after: ι : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_3
inst✝¹¹ : TopologicalSpace H
E : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
inst✝⁸ : CommMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : SmoothMul I G
E' : Type u_6
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
s : Set M
x x₀ : M
t : Finset ι
f : ι → M → G
n : ℕ∞
p : ι → Prop
hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)
hf : LocallyFinite fun i => mulSupport (f i)
⊢ ContMDiff I' I n fun x => ∏ᶠ (j : Subtype p), f (↑j) x
tactic: simp only [← finprod_subtype_eq_finprod_cond]
state_before: ι : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
H : Type u_3
inst✝¹¹ : TopologicalSpace H
E : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
inst✝⁸ : CommMonoid G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : ChartedSpace H G
inst✝⁵ : SmoothMul I G
E' : Type u_6
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
s : Set M
x x₀ : M
t : Finset ι
f : ι → M → G
n : ℕ∞
p : ι → Prop
hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)
hf : LocallyFinite fun i => mulSupport (f i)
⊢ ContMDiff I' I n fun x => ∏ᶠ (j : Subtype p), f (↑j) x
state_after: no goals
tactic: exact contMDiff_finprod (fun i => hc i i.2) (hf.comp_injective Subtype.coe_injective)
Theorem: CategoryTheory.Sigma.desc_obj
File: Mathlib/CategoryTheory/Sigma/Basic.lean

Proof steps:
state_before: I : Type w₁
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : (i : I) → C i ⥤ D
⊢ ∀ (X : (i : I) × C i),
    { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (𝟙 X) =
      𝟙 ({ obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.obj X)
state_after: case mk
I : Type w₁
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : (i : I) → C i ⥤ D
i : I
X : C i
⊢ { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (𝟙 ⟨i, X⟩) =
    𝟙 ({ obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.obj ⟨i, X⟩)
tactic: rintro ⟨i, X⟩
state_before: case mk
I : Type w₁
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : (i : I) → C i ⥤ D
i : I
X : C i
⊢ { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (𝟙 ⟨i, X⟩) =
    𝟙 ({ obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.obj ⟨i, X⟩)
state_after: no goals
tactic: apply (F i).map_id
state_before: I : Type w₁
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : (i : I) → C i ⥤ D
⊢ ∀ {X Y Z : (i : I) × C i} (f : X ⟶ Y) (g : Y ⟶ Z),
    { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (f ≫ g) =
      { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map f ≫
        { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map g
state_after: case mk.mk.mk.mk.mk
I : Type w₁
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : (i : I) → C i ⥤ D
i : I
X Y✝¹ : C i
f : X ⟶ Y✝¹
Y✝ : C i
g : Y✝¹ ⟶ Y✝
⊢ { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk f ≫ SigmaHom.mk g) =
    { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk f) ≫
      { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk g)
tactic: rintro ⟨i, X⟩ ⟨_, Y⟩ ⟨_, Z⟩ ⟨f⟩ ⟨g⟩
state_before: case mk.mk.mk.mk.mk
I : Type w₁
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : (i : I) → C i ⥤ D
i : I
X Y✝¹ : C i
f : X ⟶ Y✝¹
Y✝ : C i
g : Y✝¹ ⟶ Y✝
⊢ { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk f ≫ SigmaHom.mk g) =
    { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk f) ≫
      { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk g)
state_after: no goals
tactic: apply (F i).map_comp
Theorem: AddMagma.FreeAddSemigroup.lift_symm_apply
File: Mathlib/Algebra/Free.lean

Proof steps:
state_before: α : Type u
inst✝¹ : Mul α
β : Type v
inst✝ : Semigroup β
f✝ f : α →ₙ* β
x : AssocQuotient α
⊢ ∀ (a b : α), AssocRel α a b → f a = f b
state_after: no goals
tactic: rintro a b (⟨c, d, e⟩ | ⟨c, d, e, f⟩) <;> simp only [map_mul, mul_assoc]
Theorem: EMetric.closedBall_prod_same
File: Mathlib/Topology/EMetricSpace/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α
x✝ y✝ z✝ : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
inst✝ : PseudoEMetricSpace β
x : α
y : β
r : ℝ≥0∞
z : α × β
⊢ z ∈ closedBall x r ×ˢ closedBall y r ↔ z ∈ closedBall (x, y) r
state_after: no goals
tactic: simp [Prod.edist_eq]
Theorem: EuclideanGeometry.angle_eq_zero_of_angle_eq_pi_right
File: Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean

Proof steps:
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p p₀ p₁ p₂ p1 p2 p3 : P
h : ∠ p1 p2 p3 = π
⊢ ∠ p2 p3 p1 = 0
state_after: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p p₀ p₁ p₂ p1 p2 p3 : P
h : ∠ p3 p2 p1 = π
⊢ ∠ p2 p3 p1 = 0
tactic: rw [angle_comm] at h
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p p₀ p₁ p₂ p1 p2 p3 : P
h : ∠ p3 p2 p1 = π
⊢ ∠ p2 p3 p1 = 0
state_after: no goals
tactic: exact angle_eq_zero_of_angle_eq_pi_left h
Theorem: NumberField.InfinitePlace.mk_eq_iff
File: Mathlib/NumberTheory/NumberField/Embeddings.lean

Proof steps:
state_before: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
F : Type u_3
inst✝ : Field F
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ
state_after: case mp
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
F : Type u_3
inst✝ : Field F
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ → φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

case mpr
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
F : Type u_3
inst✝ : Field F
φ ψ : K →+* ℂ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ → mk φ = mk ψ
tactic: constructor
Theorem: NormedAddGroupHom.opNorm_zero
File: Mathlib/Analysis/Normed/Group/Hom.lean

Proof steps:
state_before: V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f g : NormedAddGroupHom V₁ V₂
x✝ : V₁
⊢ ‖0 x✝‖ = 0 * ‖x✝‖
state_after: V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f g : NormedAddGroupHom V₁ V₂
x✝ : V₁
⊢ ‖0 x✝‖ = 0
tactic: rw [zero_mul]
state_before: V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f g : NormedAddGroupHom V₁ V₂
x✝ : V₁
⊢ ‖0 x✝‖ = 0
state_after: no goals
tactic: exact norm_zero