1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: OmegaCompletePartialOrder.ωSup_le_iff
File: Mathlib/Order/OmegaCompletePartialOrder.lean

tactic: constructor <;> intros
state_before: α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
⊢ ωSup c ≤ x ↔ ∀ (i : ℕ), c i ≤ x
state_after: case mp
α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ωSup c ≤ x
i✝ : ℕ
⊢ c i✝ ≤ x

case mpr
α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ∀ (i : ℕ), c i ≤ x
⊢ ωSup c ≤ x

tactic: exact ωSup_le _ _ ‹_›
state_before: case mpr
α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ∀ (i : ℕ), c i ≤ x
⊢ ωSup c ≤ x
state_after: no goals

tactic: trans ωSup c
state_before: case mp
α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ωSup c ≤ x
i✝ : ℕ
⊢ c i✝ ≤ x
state_after: α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ωSup c ≤ x
i✝ : ℕ
⊢ c i✝ ≤ ωSup c

α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ωSup c ≤ x
i✝ : ℕ
⊢ ωSup c ≤ x

tactic: exact le_ωSup _ _
state_before: α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ωSup c ≤ x
i✝ : ℕ
⊢ c i✝ ≤ ωSup c
state_after: no goals

tactic: assumption
state_before: α : Type u
β : Type v
γ : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
x : α
a✝ : ωSup c ≤ x
i✝ : ℕ
⊢ ωSup c ≤ x
state_after: no goals
Theorem: Real.sin_nonneg_of_mem_Icc
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

tactic: rw [← closure_Ioo pi_ne_zero.symm] at hx
state_before: x : ℝ
hx : x ∈ Icc 0 π
⊢ 0 ≤ sin x
state_after: x : ℝ
hx : x ∈ closure (Ioo 0 π)
⊢ 0 ≤ sin x

tactic: exact
  closure_lt_subset_le continuous_const continuous_sin
    (closure_mono (fun y => sin_pos_of_mem_Ioo) hx)
state_before: x : ℝ
hx : x ∈ closure (Ioo 0 π)
⊢ 0 ≤ sin x
state_after: no goals
Theorem: NormedField.exists_norm_lt
File: Mathlib/Analysis/Normed/Field/Basic.lean

tactic: rwa [← Set.mem_Ioo, norm_inv, ← Set.mem_inv, Set.inv_Ioo_0_left hr]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝ : NontriviallyNormedField α
r : ℝ
hr : 0 < r
w : α
hw : r⁻¹ < ‖w‖
⊢ 0 < ‖w⁻¹‖ ∧ ‖w⁻¹‖ < r
state_after: no goals
Theorem: IsCompl.symmDiff_eq_top
File: Mathlib/Order/SymmDiff.lean

tactic: rw [h.eq_hnot, hnot_symmDiff_self]
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝ : CoheytingAlgebra α
a✝ a b : α
h : IsCompl a b
⊢ a ∆ b = ⊤
state_after: no goals
Theorem: CategoryTheory.Limits.biprod.lift_desc
File: Mathlib/CategoryTheory/Preadditive/Biproducts.lean

tactic: simp [biprod.lift_eq, biprod.desc_eq]
state_before: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
X Y : C
inst✝ : HasBinaryBiproduct X Y
T U : C
f : T ⟶ X
g : T ⟶ Y
h : X ⟶ U
i : Y ⟶ U
⊢ lift f g ≫ desc h i = f ≫ h + g ≫ i
state_after: no goals
Theorem: Set.ncard_le_ncard
File: Mathlib/Data/Set/Card.lean

tactic: rw [← Nat.cast_le (α := ℕ∞), ht.cast_ncard_eq, (ht.subset hst).cast_ncard_eq]
state_before: α : Type u_1
β : Type u_2
s t : Set α
hst : s ⊆ t
ht : autoParam t.Finite _auto✝
⊢ s.ncard ≤ t.ncard
state_after: α : Type u_1
β : Type u_2
s t : Set α
hst : s ⊆ t
ht : autoParam t.Finite _auto✝
⊢ s.encard ≤ t.encard

tactic: exact encard_mono hst
state_before: α : Type u_1
β : Type u_2
s t : Set α
hst : s ⊆ t
ht : autoParam t.Finite _auto✝
⊢ s.encard ≤ t.encard
state_after: no goals
Theorem: MeasureTheory.integral_indicator_const
File: Mathlib/MeasureTheory/Integral/SetIntegral.lean

tactic: rw [integral_indicator s_meas, ← setIntegral_const]
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : X → E
s✝ t : Set X
μ ν : Measure X
l l' : Filter X
inst✝ : CompleteSpace E
e : E
s : Set X
s_meas : MeasurableSet s
⊢ ∫ (x : X), s.indicator (fun x => e) x ∂μ = (μ s).toReal • e
state_after: no goals
Theorem: HasStrictFDerivAt.hasFDerivAt
File: Mathlib/Analysis/Calculus/FDeriv/Basic.lean

tactic: rw [HasFDerivAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff]
state_before: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
⊢ HasFDerivAt f f' x
state_after: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝 x, ‖f x_1 - f x - f' (x_1 - x)‖ ≤ c * ‖x_1 - x‖

tactic: exact fun c hc => tendsto_id.prod_mk_nhds tendsto_const_nhds (isLittleO_iff.1 hf hc)
state_before: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝 x, ‖f x_1 - f x - f' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
state_after: no goals
Theorem: intervalIntegral.integral_deriv_mul_eq_sub_of_hasDeriv_right
File: Mathlib/MeasureTheory/Integral/FundThmCalculus.lean

tactic: apply integral_eq_sub_of_hasDeriv_right (hu.mul hv) fun x hx ↦ (huu' x hx).mul (hvv' x hx)
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝³ : CompleteSpace E
f f' : ℝ → E
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
u v u' v' : ℝ → A
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a
state_after: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝³ : CompleteSpace E
f f' : ℝ → E
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
u v u' v' : ℝ → A
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ IntervalIntegrable (fun x => u' x * v x + u x * v' x) volume a b

tactic: exact (hu'.mul_continuousOn hv).add (hv'.continuousOn_mul hu)
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝³ : CompleteSpace E
f f' : ℝ → E
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
u v u' v' : ℝ → A
hu : ContinuousOn u [[a, b]]
hv : ContinuousOn v [[a, b]]
huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x
hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x
hu' : IntervalIntegrable u' volume a b
hv' : IntervalIntegrable v' volume a b
⊢ IntervalIntegrable (fun x => u' x * v x + u x * v' x) volume a b
state_after: no goals
Theorem: InnerProductGeometry.angle_zero_right
File: Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean

tactic: unfold angle
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y x : V
⊢ angle x 0 = π / 2
state_after: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y x : V
⊢ arccos (⟪x, 0⟫_ℝ / (‖x‖ * ‖0‖)) = π / 2

tactic: rw [inner_zero_right, zero_div, Real.arccos_zero]
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y x : V
⊢ arccos (⟪x, 0⟫_ℝ / (‖x‖ * ‖0‖)) = π / 2
state_after: no goals
Theorem: QuadraticForm.toMatrix'_smul
File: Mathlib/LinearAlgebra/QuadraticForm/Basic.lean

tactic: simp only [toMatrix', LinearEquiv.map_smul, LinearMap.map_smul]
state_before: S : Type u_1
T : Type u_2
R : Type u_3
M : Type u_4
N : Type u_5
n : Type w
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Invertible 2
a : R
Q : QuadraticForm R (n → R)
⊢ (a • Q).toMatrix' = a • Q.toMatrix'
state_after: no goals
Theorem: Set.mulIndicator_union_mul_inter_apply
File: Mathlib/Algebra/Group/Indicator.lean

tactic: by_cases hs : a ∈ s <;> by_cases ht : a ∈ t <;> simp [*]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
M : Type u_4
N : Type u_5
inst✝ : MulOneClass M
s✝ t✝ : Set α
f✝ g : α → M
a✝ : α
f : α → M
s t : Set α
a : α
⊢ (s ∪ t).mulIndicator f a * (s ∩ t).mulIndicator f a = s.mulIndicator f a * t.mulIndicator f a
state_after: no goals
Theorem: DFinsupp.support_subtypeDomain
File: Mathlib/Data/DFinsupp/Basic.lean

tactic: ext i
state_before: ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
p : ι → Prop
inst✝ : DecidablePred p
f : Π₀ (i : ι), β i
⊢ (subtypeDomain p f).support = Finset.subtype p f.support
state_after: case a
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
p : ι → Prop
inst✝ : DecidablePred p
f : Π₀ (i : ι), β i
i : Subtype p
⊢ i ∈ (subtypeDomain p f).support ↔ i ∈ Finset.subtype p f.support

tactic: simp
state_before: case a
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → Zero (β i)
inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
p : ι → Prop
inst✝ : DecidablePred p
f : Π₀ (i : ι), β i
i : Subtype p
⊢ i ∈ (subtypeDomain p f).support ↔ i ∈ Finset.subtype p f.support
state_after: no goals
Theorem: PolyEquivTensor.toFunLinear_mul_tmul_mul_aux_1
File: Mathlib/RingTheory/PolynomialAlgebra.lean

tactic: classical split_ifs <;> simp [*]
state_before: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
p : R[X]
k : ℕ
h : Decidable ¬p.coeff k = 0
a : A
⊢ (if ¬p.coeff k = 0 then a * (algebraMap R A) (p.coeff k) else 0) = a * (algebraMap R A) (p.coeff k)
state_after: no goals

tactic: split_ifs <;> simp [*]
state_before: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
p : R[X]
k : ℕ
h : Decidable ¬p.coeff k = 0
a : A
⊢ (if ¬p.coeff k = 0 then a * (algebraMap R A) (p.coeff k) else 0) = a * (algebraMap R A) (p.coeff k)
state_after: no goals
Theorem: Int.csInf_eq_least_of_bdd
File: Mathlib/Data/Int/ConditionallyCompleteOrder.lean

tactic: have : s.Nonempty ∧ BddBelow s := ⟨Hinh, b, Hb⟩
state_before: s : Set ℤ
inst✝ : DecidablePred fun x => x ∈ s
b : ℤ
Hb : ∀ z ∈ s, b ≤ z
Hinh : ∃ z, z ∈ s
⊢ sInf s = ↑(b.leastOfBdd Hb Hinh)
state_after: s : Set ℤ
inst✝ : DecidablePred fun x => x ∈ s
b : ℤ
Hb : ∀ z ∈ s, b ≤ z
Hinh : ∃ z, z ∈ s
this : s.Nonempty ∧ BddBelow s
⊢ sInf s = ↑(b.leastOfBdd Hb Hinh)

tactic: simp only [sInf, this, and_self, dite_true]
state_before: s : Set ℤ
inst✝ : DecidablePred fun x => x ∈ s
b : ℤ
Hb : ∀ z ∈ s, b ≤ z
Hinh : ∃ z, z ∈ s
this : s.Nonempty ∧ BddBelow s
⊢ sInf s = ↑(b.leastOfBdd Hb Hinh)
state_after: s : Set ℤ
inst✝ : DecidablePred fun x => x ∈ s
b : ℤ
Hb : ∀ z ∈ s, b ≤ z
Hinh : ∃ z, z ∈ s
this : s.Nonempty ∧ BddBelow s
⊢ ↑((Classical.choose ⋯).leastOfBdd ⋯ ⋯) = ↑(b.leastOfBdd Hb Hinh)

tactic: convert (coe_leastOfBdd_eq Hb (Classical.choose_spec (⟨b, Hb⟩ : BddBelow s)) Hinh).symm
state_before: s : Set ℤ
inst✝ : DecidablePred fun x => x ∈ s
b : ℤ
Hb : ∀ z ∈ s, b ≤ z
Hinh : ∃ z, z ∈ s
this : s.Nonempty ∧ BddBelow s
⊢ ↑((Classical.choose ⋯).leastOfBdd ⋯ ⋯) = ↑(b.leastOfBdd Hb Hinh)
state_after: no goals
Theorem: upperClosure_image
File: Mathlib/Order/UpperLower/Basic.lean

tactic: rw [← f.symm_symm, ← UpperSet.symm_map, f.symm_symm]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
s t : Set α
x : α
f : α ≃o β
⊢ upperClosure (⇑f '' s) = (UpperSet.map f) (upperClosure s)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
s t : Set α
x : α
f : α ≃o β
⊢ upperClosure (⇑f '' s) = (UpperSet.map f.symm).symm (upperClosure s)

tactic: ext
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
s t : Set α
x : α
f : α ≃o β
⊢ upperClosure (⇑f '' s) = (UpperSet.map f.symm).symm (upperClosure s)
state_after: case a.h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
s t : Set α
x : α
f : α ≃o β
x✝ : β
⊢ x✝ ∈ ↑(upperClosure (⇑f '' s)) ↔ x✝ ∈ ↑((UpperSet.map f.symm).symm (upperClosure s))

tactic: simp [-UpperSet.symm_map, UpperSet.map, OrderIso.symm, ← f.le_symm_apply]
state_before: case a.h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
s t : Set α
x : α
f : α ≃o β
x✝ : β
⊢ x✝ ∈ ↑(upperClosure (⇑f '' s)) ↔ x✝ ∈ ↑((UpperSet.map f.symm).symm (upperClosure s))
state_after: no goals
Theorem: GeneralizedContinuedFraction.IntFractPair.coe_of_rat_eq
File: Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean

tactic: simp [IntFractPair.of, v_eq_q]
state_before: K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
⊢ mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v
state_after: no goals
Theorem: IsFractional.nsmul
File: Mathlib/RingTheory/FractionalIdeal/Basic.lean

tactic: rw [zero_smul]
state_before: R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ IsFractional S (0 • I)
state_after: R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ IsFractional S 0

tactic: convert ((0 : Ideal R) : FractionalIdeal S P).isFractional
state_before: R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ IsFractional S 0
state_after: case h.e'_7
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ 0 = ↑↑0

tactic: simp
state_before: case h.e'_7
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
x✝ : IsFractional S I
⊢ 0 = ↑↑0
state_after: no goals

tactic: rw [succ_nsmul]
state_before: R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
n : ℕ
h : IsFractional S I
⊢ IsFractional S ((n + 1) • I)
state_after: R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
n : ℕ
h : IsFractional S I
⊢ IsFractional S (n • I + I)

tactic: exact (IsFractional.nsmul n h).sup h
state_before: R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
loc : IsLocalization S P
I : Submodule R P
n : ℕ
h : IsFractional S I
⊢ IsFractional S (n • I + I)
state_after: no goals
Theorem: CategoryTheory.regularTopology.eq_induced
File: Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean

tactic: ext X S
state_before: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
⊢ regularTopology C = F.inducedTopologyOfIsCoverDense (regularTopology D)
state_after: case h.h.h
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
X : C
S : Sieve X
⊢ S ∈ (regularTopology C).sieves X ↔ S ∈ (F.inducedTopologyOfIsCoverDense (regularTopology D)).sieves X

tactic: have := F.reflects_preregular
state_before: case h.h.h
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
X : C
S : Sieve X
⊢ S ∈ (regularTopology C).sieves X ↔ S ∈ (F.inducedTopologyOfIsCoverDense (regularTopology D)).sieves X
state_after: case h.h.h
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
X : C
S : Sieve X
this : Preregular C
⊢ S ∈ (regularTopology C).sieves X ↔ S ∈ (F.inducedTopologyOfIsCoverDense (regularTopology D)).sieves X

tactic: rw [← exists_effectiveEpi_iff_mem_induced F X]
state_before: case h.h.h
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
X : C
S : Sieve X
this : Preregular C
⊢ S ∈ (regularTopology C).sieves X ↔ S ∈ (F.inducedTopologyOfIsCoverDense (regularTopology D)).sieves X
state_after: case h.h.h
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
X : C
S : Sieve X
this : Preregular C
⊢ S ∈ (regularTopology C).sieves X ↔ ∃ Y π, EffectiveEpi π ∧ S.arrows π

tactic: rw [← mem_sieves_iff_hasEffectiveEpi S]
state_before: case h.h.h
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
X : C
S : Sieve X
this : Preregular C
⊢ S ∈ (regularTopology C).sieves X ↔ ∃ Y π, EffectiveEpi π ∧ S.arrows π
state_after: no goals
Theorem: Real.volume_preimage_mul_right
File: Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean

tactic: rw [map_volume_mul_right h]
state_before: ι : Type u_1
inst✝ : Fintype ι
a : ℝ
h : a ≠ 0
s : Set ℝ
⊢ (Measure.map (fun x => x * a) volume) s = ofReal |a⁻¹| * volume s
state_after: ι : Type u_1
inst✝ : Fintype ι
a : ℝ
h : a ≠ 0
s : Set ℝ
⊢ (ofReal |a⁻¹| • volume) s = ofReal |a⁻¹| * volume s

tactic: rfl
state_before: ι : Type u_1
inst✝ : Fintype ι
a : ℝ
h : a ≠ 0
s : Set ℝ
⊢ (ofReal |a⁻¹| • volume) s = ofReal |a⁻¹| * volume s
state_after: no goals
Theorem: one_lt_pow
File: Mathlib/Algebra/Order/Ring/Basic.lean

tactic: rw [pow_succ']
state_before: α : Type u_1
M : Type u_2
R : Type u_3
inst✝ : OrderedSemiring R
a b x y : R
n✝ m : ℕ
ha : 1 < a
n : ℕ
x✝ : n + 1 ≠ 0
⊢ 1 < a ^ (n + 1)
state_after: α : Type u_1
M : Type u_2
R : Type u_3
inst✝ : OrderedSemiring R
a b x y : R
n✝ m : ℕ
ha : 1 < a
n : ℕ
x✝ : n + 1 ≠ 0
⊢ 1 < a * a ^ n

tactic: exact one_lt_mul_of_lt_of_le ha (one_le_pow_of_one_le ha.le _)
state_before: α : Type u_1
M : Type u_2
R : Type u_3
inst✝ : OrderedSemiring R
a b x y : R
n✝ m : ℕ
ha : 1 < a
n : ℕ
x✝ : n + 1 ≠ 0
⊢ 1 < a * a ^ n
state_after: no goals
Theorem: IsLocalMin.hasFDerivAt_eq_zero
File: Mathlib/Analysis/Calculus/LocalExtr/Basic.lean

tactic: ext y
state_before: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
a : E
f' : E →L[ℝ] ℝ
h : IsLocalMin f a
hf : HasFDerivAt f f' a
⊢ f' = 0
state_after: case h
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
a : E
f' : E →L[ℝ] ℝ
h : IsLocalMin f a
hf : HasFDerivAt f f' a
y : E
⊢ f' y = 0 y

tactic: apply (h.on univ).hasFDerivWithinAt_eq_zero hf.hasFDerivWithinAt <;>
    rw [posTangentConeAt_univ] <;>
  apply mem_univ
state_before: case h
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
a : E
f' : E →L[ℝ] ℝ
h : IsLocalMin f a
hf : HasFDerivAt f f' a
y : E
⊢ f' y = 0 y
state_after: no goals
Theorem: IsOpen.tendstoLocallyUniformlyOn_iff_forall_tendsto
File: Mathlib/Topology/UniformSpace/UniformConvergence.lean

tactic: rw [hs.nhdsWithin_eq hx]
state_before: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x✝ : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
hs : IsOpen s
x : α
hx : x ∈ s
⊢ Tendsto (fun y => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β) ↔ Tendsto (fun y => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β)
state_after: no goals
Theorem: Complex.re_ofReal_mul
File: Mathlib/Data/Complex/Basic.lean

tactic: simp [ofReal']
state_before: r : ℝ
z : ℂ
⊢ (↑r * z).re = r * z.re
state_after: no goals
Theorem: MeasureTheory.OuterMeasure.top_apply
File: Mathlib/MeasureTheory/OuterMeasure/Operations.lean

tactic: simp [smul_dirac_apply, as]
state_before: α : Type u_1
β : Type u_2
m : OuterMeasure α
s : Set α
h : s.Nonempty
a : α
as : a ∈ s
⊢ ⊤ ≤ (⊤ • dirac a) s
state_after: no goals