ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: Composition.single_embedding
File: Mathlib/Combinatorics/Enumerative/Composition.lean

Proof steps:
state_before: n✝ : ℕ
c : Composition n✝
n : ℕ
h : 0 < n
i : Fin n
⊢ ((single n h).embedding 0) i = i
state_after: case h
n✝ : ℕ
c : Composition n✝
n : ℕ
h : 0 < n
i : Fin n
⊢ ↑(((single n h).embedding 0) i) = ↑i
tactic: ext
state_before: case h
n✝ : ℕ
c : Composition n✝
n : ℕ
h : 0 < n
i : Fin n
⊢ ↑(((single n h).embedding 0) i) = ↑i
state_after: no goals
tactic: simp
Theorem: Orientation.two_zsmul_oangle_neg_right
File: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean

Proof steps:
state_before: V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ 2 • o.oangle x (-y) = 2 • o.oangle x y
state_after: case pos
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x = 0
⊢ 2 • o.oangle x (-y) = 2 • o.oangle x y

case neg
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
⊢ 2 • o.oangle x (-y) = 2 • o.oangle x y
tactic: by_cases hx : x = 0
Theorem: WeierstrassCurve.Jacobian.dblU_smul
File: Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean

Proof steps:
state_before: R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → R
u : R
⊢ W'.dblU (u • P) = u ^ 4 * W'.dblU P
state_after: R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → R
u : R
⊢ W'.a₁ * (u ^ 3 * P y) * (u * P z) -
      (3 * (u ^ 2 * P x) ^ 2 + 2 * W'.a₂ * (u ^ 2 * P x) * (u * P z) ^ 2 + W'.a₄ * (u * P z) ^ 4) =
    u ^ 4 * (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4))
tactic: simp only [dblU_eq, smul_fin3_ext]
state_before: R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → R
u : R
⊢ W'.a₁ * (u ^ 3 * P y) * (u * P z) -
      (3 * (u ^ 2 * P x) ^ 2 + 2 * W'.a₂ * (u ^ 2 * P x) * (u * P z) ^ 2 + W'.a₄ * (u * P z) ^ 4) =
    u ^ 4 * (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4))
state_after: no goals
tactic: ring1
Theorem: irrational_sqrt_natCast_iff
File: Mathlib/Data/Real/Irrational.lean

Proof steps:
state_before: n : ℕ
⊢ Irrational √↑n ↔ ¬IsSquare n
state_after: no goals
tactic: rw [← Rat.isSquare_natCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg n.cast_nonneg,
    Rat.cast_natCast]
Theorem: Matroid.Indep.indep_restriction
File: Mathlib/Data/Matroid/Restrict.lean

Proof steps:
state_before: α : Type u_1
M : Matroid α
R I J X Y : Set α
N : Matroid α
hI : M.Indep I
hNM : N ≤r M
hIN : I ⊆ N.E
⊢ N.Indep I
state_after: case intro.intro
α : Type u_1
M : Matroid α
R✝ I J X Y : Set α
hI : M.Indep I
R : Set α
hIN : I ⊆ (M ↾ R).E
⊢ (M ↾ R).Indep I
tactic: obtain ⟨R, -, rfl⟩ := hNM
state_before: case intro.intro
α : Type u_1
M : Matroid α
R✝ I J X Y : Set α
hI : M.Indep I
R : Set α
hIN : I ⊆ (M ↾ R).E
⊢ (M ↾ R).Indep I
state_after: no goals
tactic: simpa [hI]
Theorem: Polynomial.eval₂_X
File: Mathlib/Algebra/Polynomial/Eval.lean

Proof steps:
state_before: R : Type u
S : Type v
T : Type w
ι : Type y
a b : R
m n : ℕ
inst✝¹ : Semiring R
p q r : R[X]
inst✝ : Semiring S
f : R →+* S
x : S
⊢ eval₂ f x X = x
state_after: no goals
tactic: simp [eval₂_eq_sum]
Theorem: IsCoprime.mul_add_right_right
File: Mathlib/RingTheory/Coprime/Basic.lean

Proof steps:
state_before: R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (z * x + y)
state_after: R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + z * x)
tactic: rw [add_comm]
state_before: R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + z * x)
state_after: no goals
tactic: exact h.add_mul_right_right z
Theorem: MeasureTheory.tendsto_setLIntegral_zero
File: Mathlib/MeasureTheory/Integral/Lebesgue.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
ι : Type u_5
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
l : Filter ι
s : ι → Set α
hl : Tendsto (⇑μ ∘ s) l (𝓝 0)
⊢ Tendsto (fun i => ∫⁻ (x : α) in s i, f x ∂μ) l (𝓝 0)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
ι : Type u_5
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
l : Filter ι
s : ι → Set α
hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i
⊢ ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < i
tactic: simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio,
    ← pos_iff_ne_zero] at hl ⊢
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
ι : Type u_5
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
l : Filter ι
s : ι → Set α
hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i
⊢ ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < i
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
ι : Type u_5
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
l : Filter ι
s : ι → Set α
hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i
ε : ℝ≥0∞
ε0 : 0 < ε
⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε
tactic: intro ε ε0
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
ι : Type u_5
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
l : Filter ι
s : ι → Set α
hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i
ε : ℝ≥0∞
ε0 : 0 < ε
⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
ι : Type u_5
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
l : Filter ι
s : ι → Set α
hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i
ε : ℝ≥0∞
ε0 : 0 < ε
δ : ℝ≥0∞
δ0 : δ > 0
hδ : ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε
⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε
tactic: rcases exists_pos_setLIntegral_lt_of_measure_lt h ε0.ne' with ⟨δ, δ0, hδ⟩
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
ι : Type u_5
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ⊤
l : Filter ι
s : ι → Set α
hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i
ε : ℝ≥0∞
ε0 : 0 < ε
δ : ℝ≥0∞
δ0 : δ > 0
hδ : ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε
⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε
state_after: no goals
tactic: exact (hl δ δ0).mono fun i => hδ _
Theorem: one_div_lt_one_div_of_lt
File: Mathlib/Algebra/Order/Field/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
ha : 0 < a
h : a < b
⊢ 1 / b < 1 / a
state_after: no goals
tactic: rwa [lt_div_iff' ha, ← div_eq_mul_one_div, div_lt_one (ha.trans h)]
Theorem: MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff
File: Mathlib/Algebra/Order/Rearrangement.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s
tactic: have hσinv : { x | σ⁻¹ x ≠ x } ⊆ s := (set_support_inv_eq _).subset.trans hσ
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ ⇑σ) g ↑s
state_after: case refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
⊢ ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ ∑ i ∈ s, f i • g (σ⁻¹ i) = ∑ i ∈ s, f i • g i

case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
h : MonovaryOn f (g ∘ ⇑σ⁻¹) ↑s
⊢ MonovaryOn (f ∘ ⇑σ) g ↑s

case refine_3
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : LinearOrderedRing α
inst✝² : LinearOrderedAddCommGroup β
inst✝¹ : Module α β
inst✝ : OrderedSMul α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
h : MonovaryOn (f ∘ ⇑σ) g ↑s
⊢ MonovaryOn f (g ∘ ⇑σ⁻¹) ↑s
tactic: refine (Iff.trans ?_ <| hfg.sum_smul_comp_perm_eq_sum_smul_iff hσinv).trans
    ⟨fun h ↦ ?_, fun h ↦ ?_⟩
Theorem: dist_birkhoffAverage_birkhoffAverage
File: Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : BoundedSMul 𝕜 E
f : α → α
g : α → E
n : ℕ
x y : α
⊢ dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y) = dist (birkhoffSum f g n x) (birkhoffSum f g n y) / ↑n
state_after: no goals
tactic: simp [birkhoffAverage, dist_smul₀, div_eq_inv_mul]
Theorem: DFA.accepts_comap
File: Mathlib/Computability/DFA.lean

Proof steps:
state_before: α : Type u
σ : Type v
M : DFA α σ
α' : Type u_1
σ' : Type u_2
f : α' → α
⊢ (comap f M).accepts = List.map f ⁻¹' M.accepts
state_after: case h
α : Type u
σ : Type v
M : DFA α σ
α' : Type u_1
σ' : Type u_2
f : α' → α
x : List α'
⊢ x ∈ (comap f M).accepts ↔ x ∈ List.map f ⁻¹' M.accepts
tactic: ext x
state_before: case h
α : Type u
σ : Type v
M : DFA α σ
α' : Type u_1
σ' : Type u_2
f : α' → α
x : List α'
⊢ x ∈ (comap f M).accepts ↔ x ∈ List.map f ⁻¹' M.accepts
state_after: case h
α : Type u
σ : Type v
M : DFA α σ
α' : Type u_1
σ' : Type u_2
f : α' → α
x : List α'
⊢ x ∈ (comap f M).accepts ↔ M.eval (List.map f x) ∈ M.accept
tactic: conv =>
    rhs
    rw [Set.mem_preimage, mem_accepts]
state_before: case h
α : Type u
σ : Type v
M : DFA α σ
α' : Type u_1
σ' : Type u_2
f : α' → α
x : List α'
⊢ x ∈ (comap f M).accepts ↔ M.eval (List.map f x) ∈ M.accept
state_after: no goals
tactic: simp [mem_accepts]
Theorem: Multiset.mem_ndinter
File: Mathlib/Data/Multiset/FinsetOps.lean

Proof steps:
state_before: α : Type u_1
inst✝ : DecidableEq α
s✝ s t : Multiset α
a : α
⊢ a ∈ s.ndinter t ↔ a ∈ s ∧ a ∈ t
state_after: no goals
tactic: simp [ndinter, mem_filter]
Theorem: SimpleGraph.card_edgeFinset_sup_edge
File: Mathlib/Combinatorics/SimpleGraph/Operations.lean

Proof steps:
state_before: V : Type u_1
inst✝³ : DecidableEq V
G : SimpleGraph V
s t : V
inst✝² : Fintype V
inst✝¹ : DecidableRel G.Adj
inst✝ : Fintype ↑(G ⊔ edge s t).edgeSet
hn : ¬G.Adj s t
h : s ≠ t
⊢ (G ⊔ edge s t).edgeFinset.card = G.edgeFinset.card + 1
state_after: no goals
tactic: rw [G.edgeFinset_sup_edge hn h, card_cons]
Theorem: prevD_comp_right
File: Mathlib/Algebra/Homology/Homotopy.lean

Proof steps:
state_before: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f✝ g✝ : C ⟶ D
h k : D ⟶ E
i : ι
f : (i j : ι) → C.X i ⟶ D.X j
g : D ⟶ E
j : ι
⊢ ((prevD j) fun i j => f i j ≫ g.f j) = (prevD j) f ≫ g.f j
state_after: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f✝ g✝ : C ⟶ D
h k : D ⟶ E
i : ι
f : (i j : ι) → C.X i ⟶ D.X j
g : D ⟶ E
j : ι
⊢ (f j (c.prev j) ≫ g.f (c.prev j)) ≫ E.d (c.prev j) j = (f j (c.prev j) ≫ D.d (c.prev j) j) ≫ g.f j
tactic: dsimp [prevD]
state_before: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f✝ g✝ : C ⟶ D
h k : D ⟶ E
i : ι
f : (i j : ι) → C.X i ⟶ D.X j
g : D ⟶ E
j : ι
⊢ (f j (c.prev j) ≫ g.f (c.prev j)) ≫ E.d (c.prev j) j = (f j (c.prev j) ≫ D.d (c.prev j) j) ≫ g.f j
state_after: no goals
tactic: simp only [assoc, g.comm]
Theorem: EuclideanDomain.xgcdAux_val
File: Mathlib/Algebra/EuclideanDomain/Basic.lean

Proof steps:
state_before: R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
⊢ xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y)
state_after: no goals
tactic: rw [xgcd, ← xgcdAux_fst x y 1 0 0 1]
Theorem: eq_one_of_mul_left
File: Mathlib/Algebra/Group/Units.lean

Proof steps:
state_before: α : Type u
inst✝¹ : CommMonoid α
inst✝ : Subsingleton αˣ
a b : α
h : a * b = 1
⊢ b * a = 1
state_after: no goals
tactic: rwa [mul_comm]
Theorem: Multiset.Subset.dedup_add_right
File: Mathlib/Data/Multiset/Dedup.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s t : Multiset α
h : s ⊆ t
⊢ (s + t).dedup = t.dedup
state_after: case h
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a✝ b✝ : List α
h : Quot.mk Setoid.r a✝ ⊆ Quot.mk Setoid.r b✝
⊢ (Quot.mk Setoid.r a✝ + Quot.mk Setoid.r b✝).dedup = dedup (Quot.mk Setoid.r b✝)
tactic: induction s, t using Quot.induction_on₂
state_before: case h
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a✝ b✝ : List α
h : Quot.mk Setoid.r a✝ ⊆ Quot.mk Setoid.r b✝
⊢ (Quot.mk Setoid.r a✝ + Quot.mk Setoid.r b✝).dedup = dedup (Quot.mk Setoid.r b✝)
state_after: no goals
tactic: exact congr_arg ((↑) : List α → Multiset α) <| List.Subset.dedup_append_right h
Theorem: SimpleGraph.Walk.isEulerian_iff
File: Mathlib/Combinatorics/SimpleGraph/Trails.lean

Proof steps:
state_before: V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : G.Walk u v
⊢ p.IsEulerian ↔ p.IsTrail ∧ ∀ e ∈ G.edgeSet, e ∈ p.edges
state_after: case mp
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : G.Walk u v
⊢ p.IsEulerian → p.IsTrail ∧ ∀ e ∈ G.edgeSet, e ∈ p.edges

case mpr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : G.Walk u v
⊢ (p.IsTrail ∧ ∀ e ∈ G.edgeSet, e ∈ p.edges) → p.IsEulerian
tactic: constructor
Theorem: ENNReal.toNNReal_eq_one_iff
File: Mathlib/Data/ENNReal/Basic.lean

Proof steps:
state_before: α : Type u_1
a b c d : ℝ≥0∞
r p q : ℝ≥0
x : ℝ≥0∞
⊢ x = ↑1 ∨ x = ⊤ ∧ 1 = 0 ↔ x = 1
state_after: no goals
tactic: simp
Theorem: RatFunc.denom_one
File: Mathlib/FieldTheory/RatFunc/Basic.lean

Proof steps:
state_before: K : Type u
inst✝ : Field K
⊢ denom 1 = 1
state_after: no goals
tactic: convert denom_div (1 : K[X]) one_ne_zero <;> simp
Theorem: SigmaCompactSpace_iff_exists_compact_covering
File: Mathlib/Topology/Compactness/SigmaCompact.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
⊢ SigmaCompactSpace X ↔ ∃ K, (∀ (n : ℕ), IsCompact (K n)) ∧ ⋃ n, K n = univ
state_after: no goals
tactic: rw [← isSigmaCompact_univ_iff, IsSigmaCompact]
Theorem: MeasureTheory.Measure.IicSnd_mono
File: Mathlib/Probability/Kernel/Disintegration/CondCdf.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
⊢ ρ.IicSnd r ≤ ρ.IicSnd r'
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
s : Set α
hs : MeasurableSet s
⊢ (ρ.IicSnd r) s ≤ (ρ.IicSnd r') s
tactic: refine Measure.le_iff.2 fun s hs ↦ ?_
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
s : Set α
hs : MeasurableSet s
⊢ (ρ.IicSnd r) s ≤ (ρ.IicSnd r') s
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
s : Set α
hs : MeasurableSet s
⊢ ρ (s ×ˢ Iic r) ≤ ρ (s ×ˢ Iic r')
tactic: simp_rw [IicSnd_apply ρ _ hs]
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
s : Set α
hs : MeasurableSet s
⊢ ρ (s ×ˢ Iic r) ≤ ρ (s ×ˢ Iic r')
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
s : Set α
hs : MeasurableSet s
⊢ r ≤ r'
tactic: refine measure_mono (prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, Iic_subset_Iic.mpr ?_⟩))
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r r' : ℝ
h_le : r ≤ r'
s : Set α
hs : MeasurableSet s
⊢ r ≤ r'
state_after: no goals
tactic: exact mod_cast h_le
Theorem: LieSubalgebra.submodule_span_le_lieSpan
File: Mathlib/Algebra/Lie/Subalgebra.lean

Proof steps:
state_before: R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
s : Set L
⊢ Submodule.span R s ≤ (lieSpan R L s).toSubmodule
state_after: R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
s : Set L
⊢ s ⊆ ↑(lieSpan R L s).toSubmodule
tactic: rw [Submodule.span_le]
state_before: R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
K K' : LieSubalgebra R L
K₂ : LieSubalgebra R L₂
s : Set L
⊢ s ⊆ ↑(lieSpan R L s).toSubmodule
state_after: no goals
tactic: apply subset_lieSpan
Theorem: isLUB_iff_sSup_eq
File: Mathlib/Order/CompleteLattice.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : CompleteSemilatticeSup α
s t : Set α
a b : α
⊢ sSup s = a → IsLUB s a
state_after: α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : CompleteSemilatticeSup α
s t : Set α
b : α
⊢ IsLUB s (sSup s)
tactic: rintro rfl
state_before: α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : CompleteSemilatticeSup α
s t : Set α
b : α
⊢ IsLUB s (sSup s)
state_after: no goals
tactic: exact isLUB_sSup _
Theorem: MvPolynomial.coeff_X
File: Mathlib/Algebra/MvPolynomial/Basic.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
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
p q : MvPolynomial σ R
i : σ
⊢ coeff (Finsupp.single i 1) (X i) = 1
state_after: no goals
tactic: classical rw [coeff_X', if_pos rfl]
Theorem: Subgroup.commutator_commutator_eq_bot_of_rotate
File: Mathlib/GroupTheory/Commutator.lean

Proof steps:
state_before: G : Type u_1
G' : Type u_2
F : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : FunLike F G G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥
h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥
⊢ ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥
state_after: G : Type u_1
G' : Type u_2
F : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : FunLike F G G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
⊢ ∀ g₁ ∈ H₁, ∀ g₂ ∈ H₂, ∀ h ∈ ↑H₃, ⁅h, ⁅g₁, g₂⁆⁆ = 1
tactic: simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
    mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
state_before: G : Type u_1
G' : Type u_2
F : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : FunLike F G G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
⊢ ∀ g₁ ∈ H₁, ∀ g₂ ∈ H₂, ∀ h ∈ ↑H₃, ⁅h, ⁅g₁, g₂⁆⁆ = 1
state_after: G : Type u_1
G' : Type u_2
F : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : FunLike F G G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ ⁅z, ⁅x, y⁆⁆ = 1
tactic: intro x hx y hy z hz
state_before: G : Type u_1
G' : Type u_2
F : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : FunLike F G G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ ⁅z, ⁅x, y⁆⁆ = 1
state_after: G : Type u_1
G' : Type u_2
F : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : FunLike F G G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ ⁅z, ⁅x, y⁆⁆ = x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹

G : Type u_1
G' : Type u_2
F : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : FunLike F G G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹ = 1
tactic: trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹
Theorem: CategoryTheory.CategoryOfElements.structuredArrowEquivalence_functor_obj
File: Mathlib/CategoryTheory/Elements.lean

Proof steps:
state_before: C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Type w
X : F.Elements
⊢ (𝟭 F.Elements).obj X = (toStructuredArrow F ⋙ fromStructuredArrow F).obj X
state_after: no goals
tactic: aesop_cat
Theorem: Down.mem_compression
File: Mathlib/Combinatorics/SetFamily/Compression/Down.lean

Proof steps:
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s : Finset α
a : α
⊢ s ∈ 𝓓 a 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∉ 𝒜 ∧ insert a s ∈ 𝒜
state_after: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s : Finset α
a : α
⊢ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ (∃ a_1 ∈ 𝒜, a_1.erase a = s) ∧ s ∉ 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ insert a s ∈ 𝒜 ∧ s ∉ 𝒜
tactic: simp_rw [compression, mem_disjUnion, mem_filter, mem_image, and_comm (a := (¬ s ∈ 𝒜))]
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s : Finset α
a : α
⊢ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ (∃ a_1 ∈ 𝒜, a_1.erase a = s) ∧ s ∉ 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ insert a s ∈ 𝒜 ∧ s ∉ 𝒜
state_after: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s : Finset α
a : α
hs : s ∉ 𝒜
⊢ (∃ a_1 ∈ 𝒜, a_1.erase a = s) → insert a s ∈ 𝒜
tactic: refine
    or_congr_right
      (and_congr_left fun hs =>
        ⟨?_, fun h => ⟨_, h, erase_insert <| insert_ne_self.1 <| ne_of_mem_of_not_mem h hs⟩⟩)
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s : Finset α
a : α
hs : s ∉ 𝒜
⊢ (∃ a_1 ∈ 𝒜, a_1.erase a = s) → insert a s ∈ 𝒜
state_after: case intro.intro
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
a : α
t : Finset α
ht : t ∈ 𝒜
hs : t.erase a ∉ 𝒜
⊢ insert a (t.erase a) ∈ 𝒜
tactic: rintro ⟨t, ht, rfl⟩
state_before: case intro.intro
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
a : α
t : Finset α
ht : t ∈ 𝒜
hs : t.erase a ∉ 𝒜
⊢ insert a (t.erase a) ∈ 𝒜
state_after: no goals
tactic: rwa [insert_erase (erase_ne_self.1 (ne_of_mem_of_not_mem ht hs).symm)]