1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: Computation.orElse_pure
File: Mathlib/Data/Seq/Computation.lean

tactic: unfold_projs
state_before: α : Type u
β : Type v
γ : Type w
c₁ : Computation α
a : α
⊢ (HOrElse.hOrElse c₁.think fun x => pure a).destruct = Sum.inl a
state_after: α : Type u
β : Type v
γ : Type w
c₁ : Computation α
a : α
⊢ (c₁.think.orElse fun x => pure a).destruct = Sum.inl a

tactic: simp [orElse]
state_before: α : Type u
β : Type v
γ : Type w
c₁ : Computation α
a : α
⊢ (c₁.think.orElse fun x => pure a).destruct = Sum.inl a
state_after: no goals
Theorem: Cardinal.aleph0_lt_lift
File: Mathlib/SetTheory/Cardinal/Basic.lean

tactic: rw [← lift_aleph0.{v, u}, lift_lt]
state_before: α β : Type u
c : Cardinal.{u}
⊢ ℵ₀ < lift.{v, u} c ↔ ℵ₀ < c
state_after: no goals
Theorem: Real.sqrt_div_self
File: Mathlib/Data/Real/Sqrt.lean

tactic: rw [sqrt_div_self', one_div]
state_before: x✝ y✝ x y : ℝ
⊢ √x / x = (√x)⁻¹
state_after: no goals
Theorem: Set.biUnion_preimage_singleton
File: Mathlib/Data/Set/Lattice.lean

tactic: rw [← preimage_iUnion₂, biUnion_of_singleton]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
f : α → β
s : Set β
⊢ ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s
state_after: no goals
Theorem: LieIdeal.map_comap_incl
File: Mathlib/Algebra/Lie/IdealOperations.lean

tactic: conv_rhs => rw [← I₁.incl_idealRange]
state_before: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J : LieIdeal R L'
I₁ I₂ : LieIdeal R L
⊢ map I₁.incl (comap I₁.incl I₂) = I₁ ⊓ I₂
state_after: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J : LieIdeal R L'
I₁ I₂ : LieIdeal R L
⊢ map I₁.incl (comap I₁.incl I₂) = I₁.incl.idealRange ⊓ I₂

tactic: rw [← map_comap_eq]
state_before: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J : LieIdeal R L'
I₁ I₂ : LieIdeal R L
⊢ map I₁.incl (comap I₁.incl I₂) = I₁.incl.idealRange ⊓ I₂
state_after: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J : LieIdeal R L'
I₁ I₂ : LieIdeal R L
⊢ I₁.incl.IsIdealMorphism

tactic: exact I₁.incl_isIdealMorphism
state_before: R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
I : LieIdeal R L
J : LieIdeal R L'
I₁ I₂ : LieIdeal R L
⊢ I₁.incl.IsIdealMorphism
state_after: no goals
Theorem: WeierstrassCurve.map_ψ₂
File: Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean

tactic: simp only [ψ₂, Affine.map_polynomialY]
state_before: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W : WeierstrassCurve R
f : R →+* S
⊢ (W.map f).ψ₂ = Polynomial.map (mapRingHom f) W.ψ₂
state_after: no goals
Theorem: neg_one_lt_goldConj
File: Mathlib/Data/Real/GoldenRatio.lean

tactic: rw [neg_lt, ← inv_gold]
state_before: ⊢ -1 < ψ
state_after: ⊢ φ⁻¹ < 1

tactic: exact inv_lt_one one_lt_gold
state_before: ⊢ φ⁻¹ < 1
state_after: no goals
Theorem: HurwitzZeta.isBigO_atTop_evenKernel_sub
File: Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean

tactic: induction' a using QuotientAddGroup.induction_on with b
state_before: a : UnitAddCircle
⊢ ∃ p, 0 < p ∧ (fun x => evenKernel a x - if a = 0 then 1 else 0) =O[atTop] fun x => rexp (-p * x)
state_after: case H
b : ℝ
⊢ ∃ p, 0 < p ∧ (fun x => evenKernel (↑b) x - if ↑b = 0 then 1 else 0) =O[atTop] fun x => rexp (-p * x)

tactic: obtain ⟨p, hp, hp'⟩ := HurwitzKernelBounds.isBigO_atTop_F_int_zero_sub b
state_before: case H
b : ℝ
⊢ ∃ p, 0 < p ∧ (fun x => evenKernel (↑b) x - if ↑b = 0 then 1 else 0) =O[atTop] fun x => rexp (-p * x)
state_after: case H.intro.intro
b p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
⊢ ∃ p, 0 < p ∧ (fun x => evenKernel (↑b) x - if ↑b = 0 then 1 else 0) =O[atTop] fun x => rexp (-p * x)

tactic: refine ⟨p, hp, (EventuallyEq.isBigO ?_).trans hp'⟩
state_before: case H.intro.intro
b p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
⊢ ∃ p, 0 < p ∧ (fun x => evenKernel (↑b) x - if ↑b = 0 then 1 else 0) =O[atTop] fun x => rexp (-p * x)
state_after: case H.intro.intro
b p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
⊢ (fun x => evenKernel (↑b) x - if ↑b = 0 then 1 else 0) =ᶠ[atTop] fun t =>
    HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0

tactic: filter_upwards [eventually_gt_atTop 0] with t ht
state_before: case H.intro.intro
b p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
⊢ (fun x => evenKernel (↑b) x - if ↑b = 0 then 1 else 0) =ᶠ[atTop] fun t =>
    HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0
state_after: case h
b p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
t : ℝ
ht : 0 < t
⊢ (evenKernel (↑b) t - if ↑b = 0 then 1 else 0) = HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0

tactic: simp only [← (hasSum_int_evenKernel b ht).tsum_eq, HurwitzKernelBounds.F_int,
  HurwitzKernelBounds.f_int, pow_zero, one_mul, Function.Periodic.lift_coe]
state_before: case h
b p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
t : ℝ
ht : 0 < t
⊢ (evenKernel (↑b) t - if ↑b = 0 then 1 else 0) = HurwitzKernelBounds.F_int 0 (↑b) t - if ↑b = 0 then 1 else 0
state_after: no goals
Theorem: IsCoprime.ne_zero
File: Mathlib/RingTheory/Coprime/Basic.lean

tactic: rintro rfl
state_before: R : Type u
inst✝¹ : CommSemiring R
x y z : R
inst✝ : Nontrivial R
p : Fin 2 → R
h : IsCoprime (p 0) (p 1)
⊢ p ≠ 0
state_after: R : Type u
inst✝¹ : CommSemiring R
x y z : R
inst✝ : Nontrivial R
h : IsCoprime (0 0) (0 1)
⊢ False

tactic: exact not_isCoprime_zero_zero h
state_before: R : Type u
inst✝¹ : CommSemiring R
x y z : R
inst✝ : Nontrivial R
h : IsCoprime (0 0) (0 1)
⊢ False
state_after: no goals
Theorem: List.mem_pure
File: Mathlib/Data/List/Basic.lean

tactic: simp
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
x y : α
⊢ x ∈ pure y ↔ x = y
state_after: no goals
Theorem: BoundedContinuousFunction.integrable
File: Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean

tactic: refine ⟨f.continuous.measurable.aestronglyMeasurable, (hasFiniteIntegral_def _ _).mp ?_⟩
state_before: X : Type u_1
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : OpensMeasurableSpace X
μ : Measure X
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : IsFiniteMeasure μ
f : X →ᵇ E
⊢ Integrable (⇑f) μ
state_after: X : Type u_1
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : OpensMeasurableSpace X
μ : Measure X
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : IsFiniteMeasure μ
f : X →ᵇ E
⊢ HasFiniteIntegral (⇑f) μ

tactic: calc  ∫⁻ x, ‖f x‖₊ ∂μ
  _ ≤ ‖f‖₊ * (μ Set.univ)   := f.lintegral_nnnorm_le μ
  _ < ∞                     := ENNReal.mul_lt_top ENNReal.coe_ne_top (measure_ne_top μ Set.univ)
state_before: X : Type u_1
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : OpensMeasurableSpace X
μ : Measure X
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : IsFiniteMeasure μ
f : X →ᵇ E
⊢ HasFiniteIntegral (⇑f) μ
state_after: no goals
Theorem: UniformOnFun.gen_mem_uniformity
File: Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean

tactic: rw [UniformOnFun.uniformity_eq]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s s' : Set α
x : α
p : Filter ι
g : ι → α
inst✝ : UniformSpace β
𝔖 : Set (Set α)
hs : s ∈ 𝔖
V : Set (β × β)
hV : V ∈ 𝓤 β
⊢ UniformOnFun.gen 𝔖 s V ∈ 𝓤 (α →ᵤ[𝔖] β)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s s' : Set α
x : α
p : Filter ι
g : ι → α
inst✝ : UniformSpace β
𝔖 : Set (Set α)
hs : s ∈ 𝔖
V : Set (β × β)
hV : V ∈ 𝓤 β
⊢ UniformOnFun.gen 𝔖 s V ∈ ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 (UniformOnFun.gen 𝔖 s V)

tactic: apply_rules [mem_iInf_of_mem, mem_principal_self]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s s' : Set α
x : α
p : Filter ι
g : ι → α
inst✝ : UniformSpace β
𝔖 : Set (Set α)
hs : s ∈ 𝔖
V : Set (β × β)
hV : V ∈ 𝓤 β
⊢ UniformOnFun.gen 𝔖 s V ∈ ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 (UniformOnFun.gen 𝔖 s V)
state_after: no goals
Theorem: Ordinal.le_cof_iff_lsub
File: Mathlib/SetTheory/Cardinal/Cofinality.lean

tactic: rw [cof_eq_sInf_lsub]
state_before: α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
a : Cardinal.{u}
⊢ a ≤ o.cof ↔ ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι
state_after: α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
a : Cardinal.{u}
⊢ a ≤ sInf {a | ∃ ι f, lsub f = o ∧ #ι = a} ↔ ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι

tactic: exact
  (le_csInf_iff'' (cof_lsub_def_nonempty o)).trans
    ⟨fun H ι f hf => H _ ⟨ι, f, hf, rfl⟩, fun H b ⟨ι, f, hf, hb⟩ => by
      rw [← hb]
      exact H _ hf⟩
state_before: α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
a : Cardinal.{u}
⊢ a ≤ sInf {a | ∃ ι f, lsub f = o ∧ #ι = a} ↔ ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι
state_after: no goals

tactic: rw [← hb]
state_before: α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
a : Cardinal.{u}
H : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι
b : Cardinal.{u}
x✝ : b ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a}
ι : Type u
f : ι → Ordinal.{u}
hf : lsub f = o
hb : #ι = b
⊢ a ≤ b
state_after: α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
a : Cardinal.{u}
H : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι
b : Cardinal.{u}
x✝ : b ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a}
ι : Type u
f : ι → Ordinal.{u}
hf : lsub f = o
hb : #ι = b
⊢ a ≤ #ι

tactic: exact H _ hf
state_before: α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
a : Cardinal.{u}
H : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι
b : Cardinal.{u}
x✝ : b ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a}
ι : Type u
f : ι → Ordinal.{u}
hf : lsub f = o
hb : #ι = b
⊢ a ≤ #ι
state_after: no goals
Theorem: Set.image_sub_const_uIcc
File: Mathlib/Data/Set/Pointwise/Interval.lean

tactic: simp [sub_eq_add_neg, add_comm]
state_before: α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
a b c d : α
⊢ (fun x => x - a) '' [[b, c]] = [[b - a, c - a]]
state_after: no goals
Theorem: nndist_pi_lt_iff
File: Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean

tactic: rw [← bot_eq_zero'] at hr
state_before: α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
f g : (b : β) → π b
r : ℝ≥0
hr : 0 < r
⊢ nndist f g < r ↔ ∀ (b : β), nndist (f b) (g b) < r
state_after: α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
f g : (b : β) → π b
r : ℝ≥0
hr : ⊥ < r
⊢ nndist f g < r ↔ ∀ (b : β), nndist (f b) (g b) < r

tactic: simp [nndist_pi_def, Finset.sup_lt_iff hr]
state_before: α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
π : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (π b)
f g : (b : β) → π b
r : ℝ≥0
hr : ⊥ < r
⊢ nndist f g < r ↔ ∀ (b : β), nndist (f b) (g b) < r
state_after: no goals
Theorem: WeierstrassCurve.Jacobian.eval_polynomialX_of_Z_ne_zero
File: Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean

tactic: linear_combination (norm := (rw [eval_polynomialX, Affine.evalEval_polynomialX]; ring1))
  W.a₁ * P y / P z ^ 3 * div_self hPz - 2 * W.a₂ * P x / P z ^ 2 * div_self (pow_ne_zero 2 hPz)
    - W.a₄ * div_self (pow_ne_zero 4 hPz)
state_before: R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (eval P) W.polynomialX / P z ^ 4 = Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) W.toAffine.polynomialX
state_after: no goals

tactic: rw [eval_polynomialX, Affine.evalEval_polynomialX]
state_before: case a
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (eval P) W.polynomialX / P z ^ 4 - Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) W.toAffine.polynomialX -
      (W.a₁ * P y / P z ^ 3 * (P z / P z) - 2 * W.a₂ * P x / P z ^ 2 * (P z ^ 2 / P z ^ 2) -
          W.a₄ * (P z ^ 4 / P z ^ 4) -
        (W.a₁ * P y / P z ^ 3 * 1 - 2 * W.a₂ * P x / P z ^ 2 * 1 - W.a₄ * 1)) =
    0
state_after: case a
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (W.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W.a₂ * P x * P z ^ 2 + W.a₄ * P z ^ 4)) / P z ^ 4 -
        (W.toAffine.a₁ * (P y / P z ^ 3) -
          (3 * (P x / P z ^ 2) ^ 2 + 2 * W.toAffine.a₂ * (P x / P z ^ 2) + W.toAffine.a₄)) -
      (W.a₁ * P y / P z ^ 3 * (P z / P z) - 2 * W.a₂ * P x / P z ^ 2 * (P z ^ 2 / P z ^ 2) -
          W.a₄ * (P z ^ 4 / P z ^ 4) -
        (W.a₁ * P y / P z ^ 3 * 1 - 2 * W.a₂ * P x / P z ^ 2 * 1 - W.a₄ * 1)) =
    0

tactic: ring1
state_before: case a
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (W.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W.a₂ * P x * P z ^ 2 + W.a₄ * P z ^ 4)) / P z ^ 4 -
        (W.toAffine.a₁ * (P y / P z ^ 3) -
          (3 * (P x / P z ^ 2) ^ 2 + 2 * W.toAffine.a₂ * (P x / P z ^ 2) + W.toAffine.a₄)) -
      (W.a₁ * P y / P z ^ 3 * (P z / P z) - 2 * W.a₂ * P x / P z ^ 2 * (P z ^ 2 / P z ^ 2) -
          W.a₄ * (P z ^ 4 / P z ^ 4) -
        (W.a₁ * P y / P z ^ 3 * 1 - 2 * W.a₂ * P x / P z ^ 2 * 1 - W.a₄ * 1)) =
    0
state_after: no goals
Theorem: MeasureTheory.snorm'_measure_zero_of_pos
File: Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean

tactic: simp [snorm', hq_pos]
state_before: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : MeasurableSpace α
f : α → F
hq_pos : 0 < q
⊢ snorm' f q 0 = 0
state_after: no goals
Theorem: ExistsContDiffBumpBase.u_int_pos
File: Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean

tactic: refine (integral_pos_iff_support_of_nonneg u_nonneg ?_).mpr ?_
state_before: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < ∫ (x : E), u x ∂μ
state_after: case refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ Integrable u μ

case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < μ (support u)

tactic: exact (u_continuous E).integrable_of_hasCompactSupport (u_compact_support E)
state_before: case refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ Integrable u μ
state_after: no goals

tactic: rw [u_support]
state_before: case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < μ (support u)
state_after: case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < μ (ball 0 1)

tactic: exact measure_ball_pos _ _ zero_lt_one
state_before: case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < μ (ball 0 1)
state_after: no goals
Theorem: root_X_pow_sub_C_pow
File: Mathlib/FieldTheory/KummerExtension.lean

tactic: rw [← sub_eq_zero, ← AdjoinRoot.eval₂_root, eval₂_sub, eval₂_C, eval₂_pow, eval₂_X]
state_before: K : Type u
inst✝ : Field K
n : ℕ
a : K
⊢ root (X ^ n - C a) ^ n = (of (X ^ n - C a)) a
state_after: no goals
Theorem: ProbabilityTheory.kernel.snorm_densityProcess_le
File: Mathlib/Probability/Kernel/Disintegration/Density.lean

tactic: refine (snorm_le_of_ae_bound (C := 1) (ae_of_all _ (fun x ↦ ?_))).trans ?_
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
s : Set β
⊢ snorm (fun x => densityProcess κ ν n a x s) 1 (ν a) ≤ (ν a) univ
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
s : Set β
x : γ
⊢ ‖densityProcess κ ν n a x s‖ ≤ 1

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
s : Set β
⊢ (ν a) univ ^ (ENNReal.toReal 1)⁻¹ * ENNReal.ofReal 1 ≤ (ν a) univ

tactic: simp only [Real.norm_eq_abs, abs_of_nonneg (densityProcess_nonneg κ ν n a x s),
  densityProcess_le_one hκν n a x s]
state_before: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
s : Set β
x : γ
⊢ ‖densityProcess κ ν n a x s‖ ≤ 1
state_after: no goals

tactic: simp
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : ↥(kernel α (γ × β))
ν : ↥(kernel α γ)
hκν : fst κ ≤ ν
n : ℕ
a : α
s : Set β
⊢ (ν a) univ ^ (ENNReal.toReal 1)⁻¹ * ENNReal.ofReal 1 ≤ (ν a) univ
state_after: no goals
Theorem: Real.tendsto_sum_range_one_div_nat_succ_atTop
File: Mathlib/Analysis/PSeries.lean

tactic: rw [← not_summable_iff_tendsto_nat_atTop_of_nonneg]
state_before: ⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, 1 / (↑i + 1)) atTop atTop
state_after: ⊢ ¬Summable fun i => 1 / (↑i + 1)


⊢ ∀ (n : ℕ), 0 ≤ 1 / (↑n + 1)

tactic: exact_mod_cast mt (_root_.summable_nat_add_iff 1).1 not_summable_one_div_natCast
state_before: ⊢ ¬Summable fun i => 1 / (↑i + 1)
state_after: no goals

tactic: exact fun i => by positivity
state_before: ⊢ ∀ (n : ℕ), 0 ≤ 1 / (↑n + 1)
state_after: no goals

tactic: positivity
state_before: i : ℕ
⊢ 0 ≤ 1 / (↑i + 1)
state_after: no goals
Theorem: LinearMap.conjugate_i
File: Mathlib/RepresentationTheory/Maschke.lean

tactic: rw [conjugate_apply, ← i.map_smul, h, ← mul_smul, single_mul_single, mul_one, mul_left_inv,
  ← one_def, one_smul]
state_before: k : Type u
inst✝⁹ : CommRing k
G : Type u
inst✝⁸ : Group G
V : Type v
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : Module (MonoidAlgebra k G) V
inst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V
W : Type w
inst✝³ : AddCommGroup W
inst✝² : Module k W
inst✝¹ : Module (MonoidAlgebra k G) W
inst✝ : IsScalarTower k (MonoidAlgebra k G) W
π : W →ₗ[k] V
i : V →ₗ[MonoidAlgebra k G] W
h : ∀ (v : V), π (i v) = v
g : G
v : V
⊢ (π.conjugate g) (i v) = v
state_after: no goals
Theorem: Right.pow_le_one_of_le
File: Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean

tactic: rw [pow_succ]
state_before: β : Type u_1
G : Type u_2
M : Type u_3
inst✝² : Monoid M
inst✝¹ : Preorder M
inst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : M
hx : x ≤ 1
n : ℕ
⊢ x ^ (n + 1) ≤ 1
state_after: β : Type u_1
G : Type u_2
M : Type u_3
inst✝² : Monoid M
inst✝¹ : Preorder M
inst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : M
hx : x ≤ 1
n : ℕ
⊢ x ^ n * x ≤ 1

tactic: exact Right.mul_le_one (Right.pow_le_one_of_le hx) hx
state_before: β : Type u_1
G : Type u_2
M : Type u_3
inst✝² : Monoid M
inst✝¹ : Preorder M
inst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : M
hx : x ≤ 1
n : ℕ
⊢ x ^ n * x ≤ 1
state_after: no goals
Theorem: ContDiffBump.support_normed_eq
File: Mathlib/Analysis/Calculus/BumpFunction/Normed.lean

tactic: unfold ContDiffBump.normed
state_before: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
x : E
n : ℕ∞
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
⊢ support (f.normed μ) = ball c f.rOut
state_after: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
x : E
n : ℕ∞
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
⊢ (support fun x => ↑f x / ∫ (x : E), ↑f x ∂μ) = ball c f.rOut

tactic: rw [support_div, f.support_eq, support_const f.integral_pos.ne', inter_univ]
state_before: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
x : E
n : ℕ∞
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
⊢ (support fun x => ↑f x / ∫ (x : E), ↑f x ∂μ) = ball c f.rOut
state_after: no goals
Theorem: Finset.prod_hom_rel
File: Mathlib/Algebra/BigOperators/Group/Finset.lean

tactic: delta Finset.prod
state_before: ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s✝ s₁ s₂ : Finset α
a : α
f✝ g✝ : α → β
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
r : β → γ → Prop
f : α → β
g : α → γ
s : Finset α
h₁ : r 1 1
h₂ : ∀ (a : α) (b : β) (c : γ), r b c → r (f a * b) (g a * c)
⊢ r (∏ x ∈ s, f x) (∏ x ∈ s, g x)
state_after: ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s✝ s₁ s₂ : Finset α
a : α
f✝ g✝ : α → β
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
r : β → γ → Prop
f : α → β
g : α → γ
s : Finset α
h₁ : r 1 1
h₂ : ∀ (a : α) (b : β) (c : γ), r b c → r (f a * b) (g a * c)
⊢ r (Multiset.map (fun x => f x) s.val).prod (Multiset.map (fun x => g x) s.val).prod

tactic: apply Multiset.prod_hom_rel <;> assumption
state_before: ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s✝ s₁ s₂ : Finset α
a : α
f✝ g✝ : α → β
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
r : β → γ → Prop
f : α → β
g : α → γ
s : Finset α
h₁ : r 1 1
h₂ : ∀ (a : α) (b : β) (c : γ), r b c → r (f a * b) (g a * c)
⊢ r (Multiset.map (fun x => f x) s.val).prod (Multiset.map (fun x => g x) s.val).prod
state_after: no goals