ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: FirstOrder.Language.DirectLimit.funMap_unify_equiv
File: Mathlib/ModelTheory/DirectLimit.lean

Proof steps:
state_before: case intro.intro
L : Language
ι : Type v
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
n : ℕ
F : L.Functions n
x : Fin n → Σˣ f
i j : ι
hi : i ∈ upperBounds (range (Sigma.fst ∘ x))
hj : j ∈ upperBounds (range (Sigma.fst ∘ x))
k : ι
ik : i ≤ k
jk : j ≤ k
⊢ Structure.Sigma.mk f i (funMap F (unify f x i hi)) ≈ Structure.Sigma.mk f j (funMap F (unify f x j hj))
state_after: case intro.intro
L : Language
ι : Type v
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
n : ℕ
F : L.Functions n
x : Fin n → Σˣ f
i j : ι
hi : i ∈ upperBounds (range (Sigma.fst ∘ x))
hj : j ∈ upperBounds (range (Sigma.fst ∘ x))
k : ι
ik : i ≤ k
jk : j ≤ k
⊢ (f i k ik) (funMap F (unify f x i hi)) = (f j k jk) (funMap F (unify f x j hj))
tactic: refine ⟨k, ik, jk, ?_⟩
state_before: case intro.intro
L : Language
ι : Type v
inst✝³ : Preorder ι
G : ι → Type w
inst✝² : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
n : ℕ
F : L.Functions n
x : Fin n → Σˣ f
i j : ι
hi : i ∈ upperBounds (range (Sigma.fst ∘ x))
hj : j ∈ upperBounds (range (Sigma.fst ∘ x))
k : ι
ik : i ≤ k
jk : j ≤ k
⊢ (f i k ik) (funMap F (unify f x i hi)) = (f j k jk) (funMap F (unify f x j hj))
state_after: no goals
tactic: rw [(f i k ik).map_fun, (f j k jk).map_fun, comp_unify, comp_unify]
Theorem: PartialHomeomorph.extend_target_mem_nhdsWithin
File: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : TopologicalSpace H'
inst✝ : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
y : M
hy : y ∈ f.source
⊢ (f.extend I).target ∈ 𝓝[range ↑I] ↑(f.extend I) y
state_after: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : TopologicalSpace H'
inst✝ : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
y : M
hy : y ∈ f.source
⊢ ↑(f.extend I) '' (f.extend I).source ∈ map (↑(f.extend I)) (𝓝 y)
tactic: rw [← PartialEquiv.image_source_eq_target, ← map_extend_nhds f I hy]
state_before: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace M
f f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : TopologicalSpace H'
inst✝ : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
y : M
hy : y ∈ f.source
⊢ ↑(f.extend I) '' (f.extend I).source ∈ map (↑(f.extend I)) (𝓝 y)
state_after: no goals
tactic: exact image_mem_map (extend_source_mem_nhds _ _ hy)
Theorem: IsAdjoinRootMonic.minpoly_eq
File: Mathlib/RingTheory/IsAdjoinRoot.lean

Proof steps:
state_before: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
f : R[X]
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : NoZeroSMulDivisors R S
inst✝ : IsIntegrallyClosed R
h : IsAdjoinRootMonic S f
hirr : Irreducible f
q : R[X]
hq : f = minpoly R h.root * q
⊢ Associated f (minpoly R h.root)
state_after: case h.e'_4
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
f : R[X]
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : NoZeroSMulDivisors R S
inst✝ : IsIntegrallyClosed R
h : IsAdjoinRootMonic S f
hirr : Irreducible f
q : R[X]
hq : f = minpoly R h.root * q
⊢ minpoly R h.root = minpoly R h.root * 1
tactic: convert
        Associated.mul_left (minpoly R h.root) <|
          associated_one_iff_isUnit.2 <|
            (hirr.isUnit_or_isUnit hq).resolve_left <| minpoly.not_isUnit R h.root
state_before: case h.e'_4
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
f : R[X]
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : NoZeroSMulDivisors R S
inst✝ : IsIntegrallyClosed R
h : IsAdjoinRootMonic S f
hirr : Irreducible f
q : R[X]
hq : f = minpoly R h.root * q
⊢ minpoly R h.root = minpoly R h.root * 1
state_after: no goals
tactic: rw [mul_one]
Theorem: NormedSpace.isBounded_iff_subset_smul_closedBall
File: Mathlib/Analysis/LocallyConvex/Bounded.lean

Proof steps:
state_before: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.closedBall 0 1
state_after: case mp
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ Bornology.IsBounded s → ∃ a, s ⊆ a • Metric.closedBall 0 1

case mpr
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_5
ι : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsBounded s
tactic: constructor
Theorem: _private.Mathlib.Tactic.CongrExclamation.0.eq_imp_of_iff_imp
File: Mathlib/Tactic/CongrExclamation.lean

Proof steps:
state_before: x y : Prop
p : x = y → Prop
h : ∀ (he : x ↔ y), p ⋯
he : x = y
⊢ p he
state_after: case refl
x : Prop
p : x = x → Prop
h : ∀ (he : x ↔ x), p ⋯
⊢ p ⋯
tactic: cases he
state_before: case refl
x : Prop
p : x = x → Prop
h : ∀ (he : x ↔ x), p ⋯
⊢ p ⋯
state_after: no goals
tactic: exact h Iff.rfl
Theorem: Set.pow_subset_pow_of_one_mem
File: Mathlib/Data/Set/Pointwise/Basic.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
hn : m ≤ n
⊢ s ^ m ⊆ s ^ n
state_after: case base
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
⊢ s ^ m ⊆ s ^ m

case succ
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
n✝ : ℕ
hmn✝ : m ≤ n✝
ih : s ^ m ⊆ s ^ n✝
⊢ s ^ m ⊆ s ^ (n✝ + 1)
tactic: induction' n, hn using Nat.le_induction with _ _ ih
Theorem: addOrderOf_dvd_natCard
File: Mathlib/GroupTheory/OrderOfElement.lean

Proof steps:
state_before: G✝ : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝² : Group G✝
inst✝¹ : Fintype G✝
x✝ : G✝
n : ℕ
G : Type u_6
inst✝ : Group G
x : G
⊢ orderOf x ∣ Nat.card G
state_after: case inl
G✝ : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝² : Group G✝
inst✝¹ : Fintype G✝
x✝ : G✝
n : ℕ
G : Type u_6
inst✝ : Group G
x : G
h : Fintype G
⊢ orderOf x ∣ Nat.card G

case inr
G✝ : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝² : Group G✝
inst✝¹ : Fintype G✝
x✝ : G✝
n : ℕ
G : Type u_6
inst✝ : Group G
x : G
h : Infinite G
⊢ orderOf x ∣ Nat.card G
tactic: cases' fintypeOrInfinite G with h h
Theorem: CategoryTheory.MonoidalCategory.pentagon_inv
File: Mathlib/CategoryTheory/Monoidal/Category.lean

Proof steps:
state_before: C✝ : Type u
𝒞 : Category.{v, u} C✝
inst✝² : MonoidalCategory C✝
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
U V W X Y Z : C
⊢ inv (W ◁ (α_ X Y Z).inv ≫ (α_ W (X ⊗ Y) Z).inv ≫ (α_ W X Y).inv ▷ Z) =
    inv ((α_ W X (Y ⊗ Z)).inv ≫ (α_ (W ⊗ X) Y Z).inv)
state_after: no goals
tactic: simp
Theorem: Filter.map_comap_of_mem
File: Mathlib/Order/Filter/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f✝ f₁ f₂ : Filter α
g g₁ g₂ : Filter β
m✝ : α → β
m' : β → γ
s : Set α
t : Set β
f : Filter β
m : α → β
hf : range m ∈ f
⊢ map m (comap m f) = f
state_after: no goals
tactic: rw [map_comap, inf_eq_left.2 (le_principal_iff.2 hf)]
Theorem: InfPrime.infIrred
File: Mathlib/Order/Irreducible.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
inst✝ : SemilatticeInf α
a b✝ c✝ : α
h : ∀ ⦃b c : α⦄, b ⊓ c ≤ a → b ≤ a ∨ c ≤ a
b c : α
ha : b ⊓ c = a
⊢ b = a ∨ c = a
state_after: no goals
tactic: simpa [← ha] using h ha.le
Theorem: CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono_g'
File: Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean

Proof steps:
state_before: C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ (ofEpiOfIsIsoOfMono φ h).g' = h.g' ≫ φ.τ₃
state_after: no goals
tactic: simp [ofEpiOfIsIsoOfMono, opMap]
Theorem: Complex.normSq_eq_norm_sq
File: Mathlib/Analysis/Complex/Basic.lean

Proof steps:
state_before: z✝ : ℂ
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
⊢ normSq z = ‖z‖ ^ 2
state_after: no goals
tactic: rw [normSq_eq_abs, norm_eq_abs]
Theorem: Subalgebra.mvPolynomial_aeval_coe
File: Mathlib/RingTheory/MvPolynomial/Tower.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
B : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S : Subalgebra R A
x : σ → ↥S
p : MvPolynomial σ R
⊢ (aeval fun i => ↑(x i)) p = ↑((aeval x) p)
state_after: no goals
tactic: convert aeval_algebraMap_apply A x p
Theorem: Ordnode.all_node4R
File: Mathlib/Data/Ordmap/Ordset.lean

Proof steps:
state_before: α : Type u_1
P : α → Prop
l : Ordnode α
x : α
m : Ordnode α
y : α
r : Ordnode α
⊢ All P (l.node4R x m y r) ↔ All P l ∧ P x ∧ All P m ∧ P y ∧ All P r
state_after: no goals
tactic: cases m <;> simp [node4R, all_node', All, all_node3R, and_assoc]
Theorem: MeasureTheory.limsup_lintegral_le
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
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ⨅ n, ∫⁻ (a : α), ⨆ i, ⨆ (_ : i ≥ n), f i a ∂μ = ∫⁻ (a : α), ⨅ n, ⨆ i, ⨆ (_ : i ≥ n), f i a ∂μ
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∀ (n : ℕ), Measurable fun a => ⨆ i, ⨆ (_ : i ≥ n), f i a

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ Antitone fun n a => ⨆ i, ⨆ (_ : i ≥ n), f i a

case refine_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), ⨆ i, ⨆ (_ : i ≥ 0), f i a ∂μ ≠ ⊤
tactic: refine (lintegral_iInf ?_ ?_ ?_).symm
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
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), ⨅ n, ⨆ i, ⨆ (_ : i ≥ n), f i a ∂μ = ∫⁻ (a : α), limsup (fun n => f n a) atTop ∂μ
state_after: no goals
tactic: simp only [limsup_eq_iInf_iSup_of_nat]
Theorem: OrthogonalFamily.range_linearIsometry
File: Mathlib/Analysis/InnerProductSpace/l2Space.lean

Proof steps:
state_before: ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
cplt : CompleteSpace E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
⊢ LinearMap.range hV.linearIsometry.toLinearMap = (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure
state_after: case refine_1
ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
cplt : CompleteSpace E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
⊢ LinearMap.range hV.linearIsometry.toLinearMap ≤ (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure

case refine_2
ι : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
cplt : CompleteSpace E
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : ∀ (i : ι), CompleteSpace (G i)
⊢ (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure ≤ LinearMap.range hV.linearIsometry.toLinearMap
tactic: refine le_antisymm ?_ ?_
Theorem: CategoryTheory.Limits.epi_image_of_epi
File: Mathlib/CategoryTheory/Limits/Shapes/Images.lean

Proof steps:
state_before: C : Type u
inst✝¹ : Category.{v, u} C
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X Y : C
f : X ⟶ Y
inst✝ : HasImage f
E : Epi f
⊢ Epi (image.ι f)
state_after: C : Type u
inst✝¹ : Category.{v, u} C
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X Y : C
f : X ⟶ Y
inst✝ : HasImage f
E : Epi (factorThruImage f ≫ image.ι f)
⊢ Epi (image.ι f)
tactic: rw [← image.fac f] at E
state_before: C : Type u
inst✝¹ : Category.{v, u} C
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X Y : C
f : X ⟶ Y
inst✝ : HasImage f
E : Epi (factorThruImage f ≫ image.ι f)
⊢ Epi (image.ι f)
state_after: no goals
tactic: exact epi_of_epi (factorThruImage f) (image.ι f)
Theorem: VectorFourier.norm_fourierPowSMulRight_le
File: Mathlib/Analysis/Fourier/FourierTransformDeriv.lean

Proof steps:
state_before: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
v : V
n : ℕ
⊢ ‖fourierPowSMulRight L f v n‖ ≤ (2 * π * ‖L‖) ^ n * ‖v‖ ^ n * ‖f v‖
state_after: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
v : V
n : ℕ
m : Fin n → W
⊢ ‖(fourierPowSMulRight L f v n) m‖ ≤ (2 * π * ‖L‖) ^ n * ‖v‖ ^ n * ‖f v‖ * ∏ i : Fin n, ‖m i‖
tactic: apply ContinuousMultilinearMap.opNorm_le_bound _ (by positivity) (fun m ↦ ?_)
Theorem: CategoryTheory.Limits.InitialMonoClass.of_isInitial
File: Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean

Proof steps:
state_before: C : Type u₁
inst✝ : Category.{v₁, u₁} C
I : C
hI : IsInitial I
h : ∀ (X : C), Mono (hI.to X)
I' X : C
hI' : IsInitial I'
⊢ Mono (hI'.to X)
state_after: C : Type u₁
inst✝ : Category.{v₁, u₁} C
I : C
hI : IsInitial I
h : ∀ (X : C), Mono (hI.to X)
I' X : C
hI' : IsInitial I'
⊢ Mono ((hI'.uniqueUpToIso hI).hom ≫ hI.to X)
tactic: rw [hI'.hom_ext (hI'.to X) ((hI'.uniqueUpToIso hI).hom ≫ hI.to X)]
state_before: C : Type u₁
inst✝ : Category.{v₁, u₁} C
I : C
hI : IsInitial I
h : ∀ (X : C), Mono (hI.to X)
I' X : C
hI' : IsInitial I'
⊢ Mono ((hI'.uniqueUpToIso hI).hom ≫ hI.to X)
state_after: no goals
tactic: apply mono_comp
Theorem: exists_abs_lt
File: Mathlib/Algebra/Order/Ring/Abs.lean

Proof steps:
state_before: α✝ : Type u_1
inst✝¹ : LinearOrderedRing α✝
n : ℕ
a✝ b c : α✝
α : Type u_2
inst✝ : LinearOrderedRing α
a : α
⊢ 1 ≤ |a| + 1
state_after: no goals
tactic: simp
Theorem: ZMod.ringEquivCongr_val
File: Mathlib/Data/ZMod/Basic.lean

Proof steps:
state_before: n : ℕ
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R n
a b : ℕ
h : a = b
x : ZMod a
⊢ ((ringEquivCongr h) x).val = x.val
state_after: n : ℕ
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R n
a : ℕ
x : ZMod a
⊢ ((ringEquivCongr ⋯) x).val = x.val
tactic: subst h
state_before: n : ℕ
R : Type u_1
inst✝¹ : Ring R
inst✝ : CharP R n
a : ℕ
x : ZMod a
⊢ ((ringEquivCongr ⋯) x).val = x.val
state_after: no goals
tactic: cases a <;> rfl
Theorem: ProbabilityTheory.measure_condCDF_univ
File: Mathlib/Probability/Kernel/Disintegration/CondCdf.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
⊢ (condCDF ρ a).measure univ = 1
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
⊢ (condCDF ρ a).measure univ = ENNReal.ofReal (1 - 0)
tactic: rw [← ENNReal.ofReal_one, ← sub_zero (1 : ℝ)]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
⊢ (condCDF ρ a).measure univ = ENNReal.ofReal (1 - 0)
state_after: no goals
tactic: exact StieltjesFunction.measure_univ _ (tendsto_condCDF_atBot ρ a) (tendsto_condCDF_atTop ρ a)
Theorem: StarAlgEquiv.nnnorm_map
File: Mathlib/Analysis/CstarAlgebra/Spectrum.lean

Proof steps:
state_before: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : EquivLike F A B
inst✝¹ : NonUnitalAlgEquivClass F ℂ A B
inst✝ : StarAlgEquivClass F ℂ A B
φ : F
a : A
⊢ ‖a‖₊ ≤ ‖φ a‖₊
state_after: no goals
tactic: simpa using StarAlgHom.nnnorm_apply_le (symm (φ : A ≃⋆ₐ[ℂ] B)) ((φ : A ≃⋆ₐ[ℂ] B) a)
Theorem: integral_Ioi_rpow_of_lt
File: Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean

Proof steps:
state_before: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)
state_after: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)
tactic: have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
    intro x hx
    convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
    field_simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]
state_before: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)
state_after: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1)))
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)
tactic: have ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1))) := by
    apply Tendsto.div_const
    simpa only [neg_neg] using tendsto_rpow_neg_atTop (by linarith : 0 < -(a + 1))
state_before: a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1)))
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)
state_after: case h.e'_3
a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1)))
⊢ -c ^ (a + 1) / (a + 1) = 0 / (a + 1) - c ^ (a + 1) / (a + 1)
tactic: convert integral_Ioi_of_hasDerivAt_of_tendsto' hd (integrableOn_Ioi_rpow_of_lt ha hc) ht using 1
state_before: case h.e'_3
a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1)))
⊢ -c ^ (a + 1) / (a + 1) = 0 / (a + 1) - c ^ (a + 1) / (a + 1)
state_after: no goals
tactic: simp only [neg_div, zero_div, zero_sub]
Theorem: CategoryTheory.Limits.HasBiproduct.of_hasCoproduct
File: Mathlib/CategoryTheory/Preadditive/Biproducts.lean

Proof steps:
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Finite J
f : J → C
inst✝ : HasCoproduct f
⊢ HasBiproduct f
state_after: case intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Finite J
f : J → C
inst✝ : HasCoproduct f
val✝ : Fintype J
⊢ HasBiproduct f
tactic: cases nonempty_fintype J
state_before: case intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
J : Type
inst✝¹ : Finite J
f : J → C
inst✝ : HasCoproduct f
val✝ : Fintype J
⊢ HasBiproduct f
state_after: no goals
tactic: exact HasBiproduct.mk
    { bicone := _
      isBilimit := biconeIsBilimitOfColimitCoconeOfIsColimit (colimit.isColimit _) }
Theorem: HasProd.nat_mul_neg_add_one
File: Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean

Proof steps:
state_before: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
⊢ HasProd (fun n => f ↑n * f (-(↑n + 1))) m
state_after: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
⊢ HasProd (fun n => f ↑n * f (Int.negSucc n)) m
tactic: change HasProd (fun n : ℕ ↦ f n * f (Int.negSucc n)) m
state_before: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
⊢ HasProd (fun n => f ↑n * f (Int.negSucc n)) m
state_after: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
this : Injective Int.negSucc
⊢ HasProd (fun n => f ↑n * f (Int.negSucc n)) m
tactic: have : Injective Int.negSucc := @Int.negSucc.inj
state_before: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
this : Injective Int.negSucc
⊢ HasProd (fun n => f ↑n * f (Int.negSucc n)) m
state_after: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
this : Injective Int.negSucc
u : Finset ℤ
⊢ ∃ v, ∀ (v' : Finset ℕ), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', f x = ∏ b ∈ v', f ↑b * f (Int.negSucc b)
tactic: refine hf.hasProd_of_prod_eq fun u ↦ ?_
state_before: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
this : Injective Int.negSucc
u : Finset ℤ
⊢ ∃ v, ∀ (v' : Finset ℕ), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', f x = ∏ b ∈ v', f ↑b * f (Int.negSucc b)
state_after: case refine_1
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
this : Injective Int.negSucc
u : Finset ℤ
v' : Finset ℕ
hv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'
x : ℤ
hx : x ∈ u
⊢ x ∈ image Nat.cast v' ∪ image Int.negSucc v'

case refine_2
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m m' : M
G : Type u_2
inst✝ : CommGroup G
g g' : G
f : ℤ → M
hf : HasProd f m
this : Injective Int.negSucc
u : Finset ℤ
v' : Finset ℕ
hv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'
⊢ ∏ x ∈ image Nat.cast v' ∪ image Int.negSucc v', f x = ∏ b ∈ v', f ↑b * f (Int.negSucc b)
tactic: refine ⟨u.preimage _ Nat.cast_injective.injOn ∪ u.preimage _ this.injOn,
      fun v' hv' ↦ ⟨v'.image Nat.cast ∪ v'.image Int.negSucc, fun x hx ↦ ?_, ?_⟩⟩