ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: ENNReal.fun_eq_funMulInvSnorm_mul_eLpNorm
File: Mathlib/MeasureTheory/Integral/MeanInequalities.lean

Proof steps:
state_before: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
a : α
⊢ f a = funMulInvSnorm f p μ a * (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)
state_after: no goals
tactic: simp [funMulInvSnorm, mul_assoc, ENNReal.inv_mul_cancel, hf_nonzero, hf_top]
Theorem: Profinite.NobelingProof.proj_prop_eq_self
File: Mathlib/Topology/Category/Profinite/Nobeling.lean

Proof steps:
state_before: I : Type u
inst✝⁴ : LinearOrder I
inst✝³ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
J K L : I → Prop
inst✝² : (i : I) → Decidable (J i)
inst✝¹ : (i : I) → Decidable (K i)
inst✝ : (i : I) → Decidable (L i)
hh : ∀ (i : I), ∀ x ∈ C, x i ≠ false → J i
⊢ π C J = C
state_after: case h
I : Type u
inst✝⁴ : LinearOrder I
inst✝³ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
J K L : I → Prop
inst✝² : (i : I) → Decidable (J i)
inst✝¹ : (i : I) → Decidable (K i)
inst✝ : (i : I) → Decidable (L i)
hh : ∀ (i : I), ∀ x ∈ C, x i ≠ false → J i
x : I → Bool
⊢ x ∈ π C J ↔ x ∈ C
tactic: ext x
state_before: case h
I : Type u
inst✝⁴ : LinearOrder I
inst✝³ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
J K L : I → Prop
inst✝² : (i : I) → Decidable (J i)
inst✝¹ : (i : I) → Decidable (K i)
inst✝ : (i : I) → Decidable (L i)
hh : ∀ (i : I), ∀ x ∈ C, x i ≠ false → J i
x : I → Bool
⊢ x ∈ π C J ↔ x ∈ C
state_after: case h.refine_1
I : Type u
inst✝⁴ : LinearOrder I
inst✝³ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
J K L : I → Prop
inst✝² : (i : I) → Decidable (J i)
inst✝¹ : (i : I) → Decidable (K i)
inst✝ : (i : I) → Decidable (L i)
hh : ∀ (i : I), ∀ x ∈ C, x i ≠ false → J i
x : I → Bool
x✝ : x ∈ π C J
y : I → Bool
hy : y ∈ C
h : Proj J y = x
⊢ x ∈ C

case h.refine_2
I : Type u
inst✝⁴ : LinearOrder I
inst✝³ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
J K L : I → Prop
inst✝² : (i : I) → Decidable (J i)
inst✝¹ : (i : I) → Decidable (K i)
inst✝ : (i : I) → Decidable (L i)
hh : ∀ (i : I), ∀ x ∈ C, x i ≠ false → J i
x : I → Bool
h : x ∈ C
⊢ Proj J x = x
tactic: refine ⟨fun ⟨y, hy, h⟩ ↦ ?_, fun h ↦ ⟨x, h, ?_⟩⟩
Theorem: Matrix.isUnit_det_zpow_iff
File: Mathlib/LinearAlgebra/Matrix/ZPow.lean

Proof steps:
state_before: n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A : M
z : ℤ
⊢ IsUnit (A ^ z).det ↔ IsUnit A.det ∨ z = 0
state_after: case hz
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A : M
⊢ IsUnit (A ^ 0).det ↔ IsUnit A.det ∨ 0 = 0

case hp
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A : M
z : ℕ
a✝ : IsUnit (A ^ ↑z).det ↔ IsUnit A.det ∨ ↑z = 0
⊢ IsUnit (A ^ (↑z + 1)).det ↔ IsUnit A.det ∨ ↑z + 1 = 0

case hn
n' : Type u_1
inst✝² : DecidableEq n'
inst✝¹ : Fintype n'
R : Type u_2
inst✝ : CommRing R
A : M
z : ℕ
a✝ : IsUnit (A ^ (-↑z)).det ↔ IsUnit A.det ∨ -↑z = 0
⊢ IsUnit (A ^ (-↑z - 1)).det ↔ IsUnit A.det ∨ -↑z - 1 = 0
tactic: induction' z using Int.induction_on with z _ z _
Theorem: AddSubmonoid.list_sum_mem
File: Mathlib/Algebra/Group/Submonoid/Membership.lean

Proof steps:
state_before: M : Type u_1
A : Type u_2
B : Type u_3
inst✝² : Monoid M
inst✝¹ : SetLike B M
inst✝ : SubmonoidClass B M
S : B
s : Submonoid M
l : List M
hl : ∀ x ∈ l, x ∈ s
⊢ l.prod ∈ s
state_after: case intro
M : Type u_1
A : Type u_2
B : Type u_3
inst✝² : Monoid M
inst✝¹ : SetLike B M
inst✝ : SubmonoidClass B M
S : B
s : Submonoid M
l : List ↥s
⊢ (List.map Subtype.val l).prod ∈ s
tactic: lift l to List s using hl
state_before: case intro
M : Type u_1
A : Type u_2
B : Type u_3
inst✝² : Monoid M
inst✝¹ : SetLike B M
inst✝ : SubmonoidClass B M
S : B
s : Submonoid M
l : List ↥s
⊢ (List.map Subtype.val l).prod ∈ s
state_after: case intro
M : Type u_1
A : Type u_2
B : Type u_3
inst✝² : Monoid M
inst✝¹ : SetLike B M
inst✝ : SubmonoidClass B M
S : B
s : Submonoid M
l : List ↥s
⊢ ↑l.prod ∈ s
tactic: rw [← coe_list_prod]
state_before: case intro
M : Type u_1
A : Type u_2
B : Type u_3
inst✝² : Monoid M
inst✝¹ : SetLike B M
inst✝ : SubmonoidClass B M
S : B
s : Submonoid M
l : List ↥s
⊢ ↑l.prod ∈ s
state_after: no goals
tactic: exact l.prod.coe_prop
Theorem: Basis.equiv_refl
File: Mathlib/LinearAlgebra/Basis.lean

Proof steps:
state_before: ι : Type u_1
ι' : Type u_2
R : Type u_3
R₂ : Type u_4
K : Type u_5
M : Type u_6
M' : Type u_7
M'' : Type u_8
V : Type u
V' : Type u_9
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M'
inst✝² : Module R M'
b b₁ : Basis ι R M
i✝ : ι
c : R
x : M
b' : Basis ι' R M'
e : ι ≃ ι'
inst✝¹ : AddCommMonoid M''
inst✝ : Module R M''
i : ι
⊢ (b.equiv b (Equiv.refl ι)) (b i) = (LinearEquiv.refl R M) (b i)
state_after: no goals
tactic: simp
Theorem: tendsto_comp_coe_Ioi_atBot
File: Mathlib/Topology/Order/DenselyOrdered.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto f (𝓝[>] a) l
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto (f ∘ Subtype.val) atBot l
tactic: rw [← map_coe_Ioi_atBot, tendsto_map'_iff]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto (f ∘ Subtype.val) atBot l
state_after: no goals
tactic: rfl
Theorem: MeasureTheory.lintegral_prod_mul
File: Mathlib/MeasureTheory/Constructions/Prod/Basic.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
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
⊢ ∫⁻ (z : α × β), f z.1 * g z.2 ∂μ.prod ν = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν
state_after: no goals
tactic: simp [lintegral_prod _ (hf.fst.mul hg.snd), lintegral_lintegral_mul hf hg]
Theorem: ZFSet.not_nonempty_empty
File: Mathlib/SetTheory/ZFC/Basic.lean

Proof steps:
state_before: ⊢ ¬∅.Nonempty
state_after: no goals
tactic: simp [ZFSet.Nonempty]
Theorem: mul_ball
File: Mathlib/Analysis/Normed/Group/Pointwise.lean

Proof steps:
state_before: E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x y : E
⊢ s * ball x δ = x • thickening δ s
state_after: no goals
tactic: rw [← smul_ball_one, mul_smul_comm, mul_ball_one]
Theorem: LieSubmodule.baseChange_top
File: Mathlib/Algebra/Lie/BaseChange.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
N : LieSubmodule R L M
⊢ baseChange A ⊤ = ⊤
state_after: R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
N : LieSubmodule R L M
⊢ { toSubmodule := ⊤, lie_mem := ⋯ } = ⊤
tactic: simp only [baseChange, top_coeSubmodule, Submodule.baseChange_top,
    Submodule.bot_toAddSubmonoid]
state_before: R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : CommRing A
inst✝ : Algebra R A
N : LieSubmodule R L M
⊢ { toSubmodule := ⊤, lie_mem := ⋯ } = ⊤
state_after: no goals
tactic: rfl
Theorem: UniqueFactorizationMonoid.normalizedFactors_pos
File: Mathlib/RingTheory/UniqueFactorizationDomain.lean

Proof steps:
state_before: α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
⊢ 0 < normalizedFactors x ↔ ¬IsUnit x
state_after: case mp
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
⊢ 0 < normalizedFactors x → ¬IsUnit x

case mpr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
⊢ ¬IsUnit x → 0 < normalizedFactors x
tactic: constructor
Theorem: Matroid.restrict_restrict_eq
File: Mathlib/Data/Matroid/Restrict.lean

Proof steps:
state_before: α : Type u_1
M✝ : Matroid α
R I J X Y R₁ R₂ : Set α
M : Matroid α
hR : R₂ ⊆ R₁
⊢ M ↾ R₁ ↾ R₂ = M ↾ R₂
state_after: α : Type u_1
M✝ : Matroid α
R I J X Y R₁ R₂ : Set α
M : Matroid α
hR : R₂ ⊆ R₁
⊢ ∀ I ⊆ (M ↾ R₁ ↾ R₂).E, (M ↾ R₁ ↾ R₂).Indep I ↔ (M ↾ R₂).Indep I
tactic: refine eq_of_indep_iff_indep_forall rfl ?_
state_before: α : Type u_1
M✝ : Matroid α
R I J X Y R₁ R₂ : Set α
M : Matroid α
hR : R₂ ⊆ R₁
⊢ ∀ I ⊆ (M ↾ R₁ ↾ R₂).E, (M ↾ R₁ ↾ R₂).Indep I ↔ (M ↾ R₂).Indep I
state_after: α : Type u_1
M✝ : Matroid α
R I J X Y R₁ R₂ : Set α
M : Matroid α
hR : R₂ ⊆ R₁
⊢ ∀ I ⊆ R₂, I ⊆ R₂ → M.Indep I → I ⊆ R₁
tactic: simp only [restrict_ground_eq, restrict_indep_iff, and_congr_left_iff, and_iff_left_iff_imp]
state_before: α : Type u_1
M✝ : Matroid α
R I J X Y R₁ R₂ : Set α
M : Matroid α
hR : R₂ ⊆ R₁
⊢ ∀ I ⊆ R₂, I ⊆ R₂ → M.Indep I → I ⊆ R₁
state_after: no goals
tactic: exact fun _ h _ _ ↦ h.trans hR
Theorem: Finset.centerMass_eq_of_sum_1
File: Mathlib/Analysis/Convex/Combination.lean

Proof steps:
state_before: R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
hw : ∑ i ∈ t, w i = 1
⊢ t.centerMass w z = ∑ i ∈ t, w i • z i
state_after: no goals
tactic: simp only [Finset.centerMass, hw, inv_one, one_smul]
Theorem: CategoryTheory.Limits.inl_pushoutZeroZeroIso_hom
File: Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean

Proof steps:
state_before: C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroObject C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryCoproduct X Y
⊢ pushout.inl 0 0 ≫ (pushoutZeroZeroIso X Y).hom = coprod.inl
state_after: C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroObject C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryCoproduct X Y
⊢ pushout.inl 0 0 ≫
      (colimit.isoColimitCocone
          { cocone := PushoutCocone.mk coprod.inl coprod.inr ⋯,
            isColimit :=
              isPushoutOfIsInitialIsCoproduct coprod.inl coprod.inr 0 0 HasZeroObject.zeroIsInitial
                (coprodIsCoprod X Y) }).hom =
    coprod.inl
tactic: dsimp [pushoutZeroZeroIso]
state_before: C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroObject C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryCoproduct X Y
⊢ pushout.inl 0 0 ≫
      (colimit.isoColimitCocone
          { cocone := PushoutCocone.mk coprod.inl coprod.inr ⋯,
            isColimit :=
              isPushoutOfIsInitialIsCoproduct coprod.inl coprod.inr 0 0 HasZeroObject.zeroIsInitial
                (coprodIsCoprod X Y) }).hom =
    coprod.inl
state_after: no goals
tactic: simp
Theorem: MvPowerSeries.coeff_C_mul
File: Mathlib/RingTheory/MvPowerSeries/Basic.lean

Proof steps:
state_before: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
n : σ →₀ ℕ
φ : MvPowerSeries σ R
a : R
⊢ (coeff R n) ((C σ R) a * φ) = a * (coeff R n) φ
state_after: no goals
tactic: simpa using coeff_add_monomial_mul 0 n φ a
Theorem: ChainComplex.of_d
File: Mathlib/Algebra/Homology/HomologicalComplex.lean

Proof steps:
state_before: ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : HasZeroMorphisms V
α : Type u_2
inst✝² : AddRightCancelSemigroup α
inst✝¹ : One α
inst✝ : DecidableEq α
X : α → V
d : (n : α) → X (n + 1) ⟶ X n
sq : ∀ (n : α), d (n + 1) ≫ d n = 0
j : α
⊢ (of X d sq).d (j + 1) j = d j
state_after: ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : HasZeroMorphisms V
α : Type u_2
inst✝² : AddRightCancelSemigroup α
inst✝¹ : One α
inst✝ : DecidableEq α
X : α → V
d : (n : α) → X (n + 1) ⟶ X n
sq : ∀ (n : α), d (n + 1) ≫ d n = 0
j : α
⊢ (if j + 1 = j + 1 then 𝟙 (X (j + 1)) ≫ d j else 0) = d j
tactic: dsimp [of]
state_before: ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : HasZeroMorphisms V
α : Type u_2
inst✝² : AddRightCancelSemigroup α
inst✝¹ : One α
inst✝ : DecidableEq α
X : α → V
d : (n : α) → X (n + 1) ⟶ X n
sq : ∀ (n : α), d (n + 1) ≫ d n = 0
j : α
⊢ (if j + 1 = j + 1 then 𝟙 (X (j + 1)) ≫ d j else 0) = d j
state_after: no goals
tactic: rw [if_pos rfl, Category.id_comp]
Theorem: AlgebraicGeometry.disjointGlueData'_U
File: Mathlib/AlgebraicGeometry/Limits.lean

Proof steps:
state_before: ι : Type u
f : ι → Scheme
x✝¹ x✝ : ι
⊢ ∀ (h : x✝¹ ≠ x✝), Mono ((fun x x_1 x_2 => (f x).emptyTo) x✝¹ x✝ h)
state_after: ι : Type u
f : ι → Scheme
x✝¹ x✝ : ι
⊢ x✝¹ ≠ x✝ → Mono (f x✝¹).emptyTo
tactic: dsimp only
state_before: ι : Type u
f : ι → Scheme
x✝¹ x✝ : ι
⊢ x✝¹ ≠ x✝ → Mono (f x✝¹).emptyTo
state_after: no goals
tactic: infer_instance
Theorem: Finsupp.prod_eq_single
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✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : 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✝² : Zero M
inst✝¹ : Zero M'
inst✝ : CommMonoid N
f : α →₀ M
a : α
g : α → M → N
h₀ : ∀ (b : α), f b ≠ 0 → b ≠ a → g b (f b) = 1
h₁ : f a = 0 → g a 0 = 1
⊢ f.prod g = g a (f a)
state_after: case refine_1
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✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : 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✝² : Zero M
inst✝¹ : Zero M'
inst✝ : CommMonoid N
f : α →₀ M
a : α
g : α → M → N
h₀ : ∀ (b : α), f b ≠ 0 → b ≠ a → g b (f b) = 1
h₁ : f a = 0 → g a 0 = 1
b : α
hb₁ : b ∈ f.support
hb₂ : b ≠ a
⊢ g b (f b) = 1

case refine_2
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✝⁵ : AddCommMonoid A
inst✝⁴ : AddCommMonoid B
inst✝³ : 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✝² : Zero M
inst✝¹ : Zero M'
inst✝ : CommMonoid N
f : α →₀ M
a : α
g : α → M → N
h₀ : ∀ (b : α), f b ≠ 0 → b ≠ a → g b (f b) = 1
h₁ : f a = 0 → g a 0 = 1
h : a ∉ f.support
⊢ g a (f a) = 1
tactic: refine Finset.prod_eq_single a (fun b hb₁ hb₂ => ?_) (fun h => ?_)
Theorem: IsGalois.fixedField_fixingSubgroup
File: Mathlib/FieldTheory/Galois.lean

Proof steps:
state_before: F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
H : Subgroup (E ≃ₐ[F] E)
K : IntermediateField F E
inst✝ : FiniteDimensional F E
h : IsGalois F E
⊢ IntermediateField.fixedField K.fixingSubgroup = K
state_after: F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
H : Subgroup (E ≃ₐ[F] E)
K : IntermediateField F E
inst✝ : FiniteDimensional F E
h : IsGalois F E
K_le : K ≤ IntermediateField.fixedField K.fixingSubgroup
⊢ IntermediateField.fixedField K.fixingSubgroup = K
tactic: have K_le : K ≤ IntermediateField.fixedField (IntermediateField.fixingSubgroup K) :=
    (IntermediateField.le_iff_le _ _).mpr le_rfl
state_before: F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
H : Subgroup (E ≃ₐ[F] E)
K : IntermediateField F E
inst✝ : FiniteDimensional F E
h : IsGalois F E
K_le : K ≤ IntermediateField.fixedField K.fixingSubgroup
⊢ IntermediateField.fixedField K.fixingSubgroup = K
state_after: F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
H : Subgroup (E ≃ₐ[F] E)
K : IntermediateField F E
inst✝ : FiniteDimensional F E
h : IsGalois F E
K_le : K ≤ IntermediateField.fixedField K.fixingSubgroup
⊢ finrank (↥K) E = finrank (↥(IntermediateField.fixedField K.fixingSubgroup)) E
tactic: suffices
    finrank K E = finrank (IntermediateField.fixedField (IntermediateField.fixingSubgroup K)) E by
    exact (IntermediateField.eq_of_le_of_finrank_eq' K_le this).symm
state_before: F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
H : Subgroup (E ≃ₐ[F] E)
K : IntermediateField F E
inst✝ : FiniteDimensional F E
h : IsGalois F E
K_le : K ≤ IntermediateField.fixedField K.fixingSubgroup
⊢ finrank (↥K) E = finrank (↥(IntermediateField.fixedField K.fixingSubgroup)) E
state_after: no goals
tactic: classical
  rw [IntermediateField.finrank_fixedField_eq_card,
    Fintype.card_congr (IntermediateField.fixingSubgroupEquiv K).toEquiv]
  exact (card_aut_eq_finrank K E).symm
Theorem: MeasureTheory.setLaverage_congr
File: Mathlib/MeasureTheory/Integral/Average.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
μ ν : Measure α
s t : Set α
f g : α → ℝ≥0∞
h : s =ᶠ[ae μ] t
⊢ ⨍⁻ (x : α) in s, f x ∂μ = ⨍⁻ (x : α) in t, f x ∂μ
state_after: no goals
tactic: simp only [setLaverage_eq, setLIntegral_congr h, measure_congr h]
Theorem: BoundedContinuousFunction.nndist_eq_iSup
File: Mathlib/Topology/ContinuousFunction/Bounded.lean

Proof steps:
state_before: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x : α
C : ℝ
⊢ ⨆ x, dist (f x) (g x) = ↑(⨆ x, nndist (f x) (g x))
state_after: no goals
tactic: simp_rw [val_eq_coe, coe_iSup, coe_nndist]
Theorem: Finset.addEnergy_empty_right
File: Mathlib/Combinatorics/Additive/Energy.lean

Proof steps:
state_before: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
s s₁ s₂ t t₁ t₂ : Finset α
⊢ Eₘ[s, ∅] = 0
state_after: no goals
tactic: simp [mulEnergy]
Theorem: Set.OrdConnected.image_hasDerivWithinAt
File: Mathlib/Analysis/Calculus/Darboux.lean

Proof steps:
state_before: a b : ℝ
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
⊢ (f' '' s).OrdConnected
state_after: case hs
a b : ℝ
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
⊢ ∀ x ∈ f' '' s, ∀ y ∈ f' '' s, x < y → Ioo x y ⊆ f' '' s
tactic: apply ordConnected_of_Ioo
state_before: case hs
a b : ℝ
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
⊢ ∀ x ∈ f' '' s, ∀ y ∈ f' '' s, x < y → Ioo x y ⊆ f' '' s
state_after: case hs.intro.intro.intro.intro.intro
a✝ b✝ : ℝ
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : ℝ
ha : a ∈ s
b : ℝ
hb : b ∈ s
m : ℝ
hma : f' a < m
hmb : m < f' b
⊢ m ∈ f' '' s
tactic: rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ - m ⟨hma, hmb⟩
state_before: case hs.intro.intro.intro.intro.intro
a✝ b✝ : ℝ
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : ℝ
ha : a ∈ s
b : ℝ
hb : b ∈ s
m : ℝ
hma : f' a < m
hmb : m < f' b
⊢ m ∈ f' '' s
state_after: case hs.intro.intro.intro.intro.intro.inl
a✝ b✝ : ℝ
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : ℝ
ha : a ∈ s
b : ℝ
hb : b ∈ s
m : ℝ
hma : f' a < m
hmb : m < f' b
hab : a ≤ b
⊢ m ∈ f' '' s

case hs.intro.intro.intro.intro.intro.inr
a✝ b✝ : ℝ
f f' : ℝ → ℝ
s : Set ℝ
hs : s.OrdConnected
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
a : ℝ
ha : a ∈ s
b : ℝ
hb : b ∈ s
m : ℝ
hma : f' a < m
hmb : m < f' b
hab : b ≤ a
⊢ m ∈ f' '' s
tactic: rcases le_total a b with hab | hab
Theorem: Ideal.isCompactElement_top
File: Mathlib/RingTheory/Ideal/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
inst✝ : Semiring α
I : Ideal α
a b : α
⊢ CompleteLattice.IsCompactElement ⊤
state_after: no goals
tactic: simpa only [← span_singleton_one] using Submodule.singleton_span_isCompactElement 1
Theorem: Finset.image₂_swap
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
f : α → β → γ
s : Finset α
t : Finset β
⊢ ↑(image₂ f s t) = ↑(image₂ (fun a b => f b a) t s)
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
f : α → β → γ
s : Finset α
t : Finset β
⊢ image2 f ↑s ↑t = image2 (fun a b => f b a) ↑t ↑s
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
f : α → β → γ
s : Finset α
t : Finset β
⊢ image2 f ↑s ↑t = image2 (fun a b => f b a) ↑t ↑s
state_after: no goals
tactic: exact image2_swap _ _ _
Theorem: List.rotate_append_length_eq
File: Mathlib/Data/List/Rotate.lean

Proof steps:
state_before: α : Type u
l l' : List α
⊢ (l ++ l').rotate l.length = l' ++ l
state_after: α : Type u
l l' : List α
⊢ (l ++ l').rotate' l.length = l' ++ l
tactic: rw [rotate_eq_rotate']
state_before: α : Type u
l l' : List α
⊢ (l ++ l').rotate' l.length = l' ++ l
state_after: case nil
α : Type u
l' : List α
⊢ ([] ++ l').rotate' [].length = l' ++ []

case cons
α : Type u
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ (l' : List α), (tail✝ ++ l').rotate' tail✝.length = l' ++ tail✝
l' : List α
⊢ (head✝ :: tail✝ ++ l').rotate' (head✝ :: tail✝).length = l' ++ head✝ :: tail✝
tactic: induction l generalizing l'